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A ROBOTIC MUSCLE SPINDLE: NEUROMECHANICS OF INDIVIDUAL AND ENSEMBLE RESPONSE

by

Kristen Nicole Jaax

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

University of Washington

2001

Program Authorized to Offer Degree: Department of Bioengineering

Copyright 2001

Kristen Nicole Jaax

University of Washington Graduate School

This is to certify that I have examined this copy of a doctoral dissertation by

Kristen Nicole Jaax

and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.

Chair of Supervisory Committee:

Blake Hannaford

Reading Committee:

Blake Hannaford

Martin Kushmerick

Francis Spelman

Date:

In presenting this dissertation in partial fulfillment of the requirements for the Doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that the extensive copying of the dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying of reproduction of this dissertation may be referred to Bell and Howell Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.”

Signature Date

University of Washington

Abstract

A Robotic Muscle Spindle: Neuromechanics of Individual and Ensemble Response

By Kristen Nicole Jaax

Chairperson of Supervisory Committee:

Professor Blake Hannaford Department of Electrical Engineering and Department of Bioengineering (adjunct)

A mechatronic structural model of the mammalian muscle spindle Ia response was developed and used to

investigate neuromechanical mechanisms contributing to individual spindle dynamics and the information

content of spindle ensemble response. Engineering specifications were derived from displacement,

receptor potential and Ia data in the muscle spindle literature, allowing reproduction of core muscle

spindle behavior directly in hardware. A linear actuator controlled by a software muscle model replicated

intrafusal contractile behavior; a cantilever-based transducer reproduced sensory membrane

depolarization; a voltage-controlled oscillator encoded strain into a frequency signal. Results of

engineering tests met all performance specifications. Data from the biological literature was used first to

tune the model against 5 measures of ramp and hold response, then to validate the fully tuned model

against ramp and hold, sinusoidal and fusimotor response experiments. The response with dynamic or

static fusimotor input was excellent across all studies. The passive spindle response matched well in 5 of

9 measures. Dynamic intramuscular strain data from 28 locations on the surface of a contracting rat

medial gastrocnemius was sent sequentially through the model to reconstruct the Ia ensemble response of

a large population of muscle spindles. Results showed that under dynamic fusimotor stimulation, the

ensemble significantly increased Ia correlation to whole muscle kinematic inputs and that homogeneously

distributed dynamic fusimotor stimulation increased Ia ensemble correlation to muscle velocity in a dose-

dependent manner. Proposed mechanisms include decorrelation of spindle noise by intramuscular strain

inhomogeneities and fusimotor-dependent noise and nonlinear gains, as well as fusimotor-dependent

velocity selectivity. Potential applications for the robotic model include basic science motor control

research and applied research in prosthetics and robotics.

2001, K.N. Jaax Ph.D. Dissertation University of Washington

i

TABLE OF CONTENTS

List of Figures................................................................................................................ iii List of Tables ...................................................................................................................v

Chapter 1: Introduction .................................................................................................1 1.1 Problem Statement ........................................................................................1 1.2 Specific Aims ................................................................................................2 1.3 Dissertation Overview...................................................................................3

Chapter 2: Literature Review........................................................................................5 2.1 The Mammalian Muscle Spindle ..................................................................5

2.1.1 Overview...................................................................................................5 2.1.2 Intrafusal Muscle.......................................................................................7 2.1.3 Neural Transduction and Encoding...........................................................8

2.2 Muscle Spindle Modeling .............................................................................8 2.2.1 Intrafusal Muscle Models..........................................................................8 2.2.2 Transducer and Encoder Models.............................................................14 2.2.3 Biorobotic Models...................................................................................15

2.3 Muscle Spindle Ensemble Response...........................................................16 2.3.1 Ensemble Information Content ...............................................................17 2.3.2 Experimental Data...................................................................................17 2.3.3 Modeling .................................................................................................18

Chapter 3: Mechatronic Design of an Actuated Biomimetic Length and Velocity Sensor ..........................................................................................20

3.1 Abstract .......................................................................................................20 3.2 Introduction .................................................................................................21

3.2.1 Background .............................................................................................22 3.3 Methods .......................................................................................................23

3.3.1 Design .....................................................................................................23 3.3.2 Implementation .......................................................................................24 3.3.3 Linear Positioning Device.......................................................................27 3.3.4 Modeling .................................................................................................28

3.4 Results .........................................................................................................30 3.4.1 Actuator Performance .............................................................................30 3.4.2 Transducer and Encoder Calibration.......................................................31 3.4.3 Linear Positioning Device Performance .................................................32 3.4.4 Integrated Performance ...........................................................................32

3.5 Discussion ...................................................................................................34


ii

Chapter 4: A Biorobotic Structural Model of the Mammalian Muscle Spindle Primary Afferent Response ......................................................................37

4.1 Abstract .......................................................................................................37 4.2 Introduction .................................................................................................37

4.2.1 Prior Literature........................................................................................39 4.2.2 Approach.................................................................................................41

4.3 Methods .......................................................................................................42 4.3.1 Design .....................................................................................................42 4.3.2 Experimental Methods ............................................................................49

4.4 Results .........................................................................................................50 4.4.1 Model Tuning Studies.............................................................................50 4.4.2 Model Validation Studies........................................................................55

4.5 Discussion ...................................................................................................58 4.5.1 Model Tuning..........................................................................................59 4.5.2 Model Validation ....................................................................................65 4.5.3 Summary of Contributions......................................................................69

Chapter 5: Fusimotor Effect on Signal Information Content of Ia Ensemble Model Reconstructed from Dynamic Intramuscular Strain Data........72

5.1 Summary .....................................................................................................72 5.2 Introduction .................................................................................................73 5.3 Methods .......................................................................................................76

5.3.1 Collecting Local Muscle Fiber Strain Data.............................................76 5.3.2 Calculating Muscle Spindle Ensemble Response ...................................78 5.3.3 Data Analysis: Nonlinearity of Spindle Ensemble Output. ....................79 5.3.4 Data Analysis: Effect of Fixed Fusimotor Stimulation Rate ..................80

5.4 Results .........................................................................................................81 5.4.1 Local Strain Data ....................................................................................81 5.4.2 Ensemble Reconstruction........................................................................82 5.4.3 Nonlinearity of Spindle Ensemble Output. .............................................84 5.4.4 Effect of Fixed Fusimotor Stimulation Rate ...........................................86

5.5 Discussion ...................................................................................................87 5.5.1 Reconstructing the Ensemble Response .................................................88 5.5.2 Effect of Ensemble on Kinematic Information Content .........................91 5.5.3 Dose-Dependent Effect of Homogeneous Fusimotor Stimulation .........91 5.5.4 Conclusions.............................................................................................93

Chapter 6: Conclusions ................................................................................................94 6.1 Summary .....................................................................................................94 6.2 Future Work ................................................................................................96

Bibliography ................................................................................................................100 Appendix A: Technical Drawings..............................................................................112


iii

LIST OF FIGURES

Figure 2.1: Mammalian muscle spindle...........................................................................6 Figure 2.2: Artificial Muscle Spindle ............................................................................16 Figure 2.3: Robotic Muscle Spindle ..............................................................................16 Figure 3.1: CAD model of biomimetic sensor...............................................................20 Figure 3.2: Mammalian muscle spindle anatomy ..........................................................22 Figure 3.3: Linear actuator and transducer assembly ....................................................24 Figure 3.4: Transducer platform ....................................................................................25 Figure 3.5: CAD drawing of transducer platform..........................................................26 Figure 3.6: Encoder circuit diagram ..............................................................................27 Figure 3.7: Linear Positioning Device and Robotic Muscle Spindle.............................28 Figure 3.8: Time response of linear actuator ................................................................31 Figure 3.9: Calibration plots for transducer and encoder...............................................31 Figure 3.10: Waveform of frequency modulated square wave......................................32 Figure 3.11: Time response of LPD...............................................................................32 Figure 3.12: Test of integrated engineering hardware ...................................................33 Figure 3.13: Effect of ramp speed and γ mn input on robotic Ia Response during 6 mm

amplitude ramp and hold.........................................................................................34 Figure 3.14: Comparison of robotic and biological Ia response to sinusoidal stretch

input. .......................................................................................................................35 Figure 4.1: Mammalian muscle spindle. ........................................................................39 Figure 4.2: CAD drawing of sensory element design....................................................44 Figure 4.3: CAD drawing of linear actuator design.......................................................45 Figure 4.4: Block diagram of linear actuator controller.................................................47 Figure 4.5: Model parameter tuning study. Ia output metrics during ramp and hold

experiment...............................................................................................................52 Figure 4.6: Model parameter tuning study. Comparison of Ia responses during ramp

and hold input..........................................................................................................53 Figure 4.7: Model parameter tuning study. Sensory region stretch during ramp and

hold stretch applied across whole muscle spindle ..................................................54 Figure 4.8: Completed model validation study. Comparison of Ia response to ramp and

hold position input ..................................................................................................55 Figure 4.9: Completed model validation study. Comparison of depth of modulation of

Ia output in response to varying amplitude of sinusoidal stretch input ..................57 Figure 4.10: Completed model validation study. Comparison of effect of varying γmn

stimulation level on Ia response..............................................................................58 Figure 5.1: Location of 28 markers on surface of rat medial gastrocnemius muscle

fibers used to reconstruct mechanical input to 28 hypothetical muscle spindles....81 Figure 5.2: Dynamic strains recorded at the 28 markers shown in Figure 5.1 ..............82


iv

Figure 5.3: Sequence of 28 displacement trajectories laid out in manner in which they were physically applied to muscle spindle model...................................................82

Figure 5.4: Comparison of ensemble response to kinematic inputs ..............................83 Figure 5.5: Correlation coefficients for multiple regression on whole muscle position

and velocity .............................................................................................................85 Figure 5.6: Correlation between ensemble response and whole muscle velocity under

dynamic fusimotor stimulation ...............................................................................86 Figure 5.7: Correlation between ensemble response and whole muscle position under

static fusimotor stimulation.....................................................................................88


v

LIST OF TABLES

Table 2.1: Anatomical elements included in nonlinear structural models .....................10 Table 2.2: Processes modeled in structural models to generate neural output ..............14 Table 4.1: Parameter values changed during tuning of robotic muscle spindle ............51


vi

ACKNOWLEDGEMENTS

My thanks go first to my parents, Jim and Suzanne Jaax, whose support and

encouragement of my science education has spanned twenty-three years. Suzanne has

served as an expert proofreader throughout those years, including this dissertation and

the technical papers it has produced.

My love and gratitude go to Ryan Campbell, who served as my sounding board, my

resident computer expert, and in the final months of this dissertation, my right hand.

His help in finding voice recognition software, formatting this dissertation and

performing the many two-handed tasks of daily living were invaluable in completing

this dissertation.

I wish to thank Prof. Blake Hannaford for providing both the guidance and the freedom

we graduate students needed to develop as independent researchers. I also wish to

thank my committee members, Prof. Francis Spelman, Prof. Martin Kushmerick, Prof.

Deirdre Meldrum and Prof. Peter Detwiler for contributing their time and wisdom to the

completion of this dissertation.

My thanks goes out to the members of the BioRobotics Lab for making this journey fun

every step of the way. In particular I wish to thank Glenn Klute, Dan Ferris, Thavida

Maneewarn and Steven Venema. Their informal mentoring was a cornerstone of my

education.

My collaborators have been instrumental in the development of this dissertation.

Pierre-Henry Marbot laid the foundation for this work and has been a source of

knowledge and experience throughout the building of the robotic muscle spindle. C.C.

van Donkelaar and M.R. Drost allowed my vision of the ensemble study to become a


vii

reality by making available their intramuscular strain data, collected at a remarkable

seventy sites on the surface of the muscle.

Finally, I am grateful to the Whitaker Graduate Fellowship Program and the University

of Washington Medical Scientist Training Program for their financial support.


viii

DEDICATION

To my parents, Jim and Suzanne Jaax, for their unwavering dedication to

& enthusiasm for my education.

1


Chapter 1:

Introduction

1.1 Problem Statement

The biological sensor responsible for measuring muscle length, the muscle spindle, is a

complex neuromuscular organ. The focus of the muscle spindle is a central sensory

region whose strains determine the muscle spindle output, the Ia response. Surrounding

the sensory region, the muscle spindle has an internal muscle, the intrafusal muscle,

whose sole purpose is to mechanically filter the spindle’s position and velocity inputs

thereby shaping the character of the strains that reach the central sensory region. The

muscle spindle also has a dedicated input from the central nervous system (CNS), the

gamma motorneuron, whose sole function is to modulate the intrafusal muscle’s

mechanical properties, creating a way for the organism to actively control the

mechanical filtering of the intrafusal muscle. The complexity of these systems shows

that the organism devotes substantial neurological and muscular resources toward the

goal of controlling the shape of the signal it receives from these sensors. From this

expenditure of resources, one would assume that the spindle’s response is carefully

sculpted to maximize information content. Yet these sensors are known to be noisy and

nonlinear transducers, raising the question of how the central nervous system extracts a

decipherable signal from their response [1].

Recent advances in technology have allowed researchers to implement the mechanisms

of integrated physiological systems, such as the muscle spindle, in robotic hardware.

This field, known as biorobotics, has grown rapidly on the principle that engineers in

the field of robotics are often trying to find solutions to problems that have already been

solved in physiological systems. Biologists, meanwhile, are often working on the

problem of unraveling the mechanisms underlying these same systems. Biorobotics

2


brings together the knowledge and experience of both fields to address these shared

problems by using engineering hardware to recreate mechanisms used in biology. In

developing this technology, bioroboticists address several interrelated goals: increasing

understanding of biological systems, discovering novel solutions for engineering

problems and developing components for prosthetic devices. The muscle spindle, with

its complex neuromechanical systems and unusual transducer behavior, is an ideal

candidate for such an approach.

The problem, then, is two-fold. First, to advance the state-of-the-art in biorobotics by

developing the technology needed to accurately reproduce the behavior of the muscle

spindle in precision engineering hardware. Second, to apply the biorobotic muscle

spindle model to the basic science question of whether the CNS could use the ensemble

response of a population of muscle spindles to extract a more decipherable signal of

muscle length and velocity from its muscle spindles.

1.2 Specific Aims

The Specific Aims of this dissertation include:

1) To identify the core neural and mechanical elements of the muscle spindle

necessary to elicit their characteristic Ia response and develop precision

engineering hardware capable of replicating the performance of these core

elements.

2) To integrate the core engineering components into a structural model of the

individual muscle spindle Ia response.

3) To tune and validate the physiological faithfulness of the individual muscle

spindle model against biological Ia data from the literature, proposing

modifications to the underlying biological mechanisms when supported by

evidence from model behavior as well as data from the biological literature.

3


4) To use the mechatronic model to reconstruct the ensemble response that would

be generated by a population of muscle spindles residing in a single muscle

body.

5) To test whether the ensemble model’s response exhibits increased correlation to

mechanical inputs, length and velocity, as compared to the individual response.

6) To test whether gamma motorneuron stimulation, when homogeneously

distributed across a population of muscle spindles, improves the correlation of

the ensemble model’s response to position or velocity.

1.3 Dissertation Overview

The research content of this dissertation is organized as distinct chapters written in a

manner suitable for independent publication. The chapters collectively describe the

development of a robotic muscle spindle, starting with engineering hardware

development and concluding with a reconstruction of the response of a population of

muscle spindles.

Chapter 2 reviews background and literature pertinent to this dissertation.

Chapter 3, entitled “Mechatronic Design of an Actuated Biomimetic Length and

Velocity Sensor,” describes the engineering aspects of the robotic muscle spindle’s

development. This chapter describes the design and implementation of mechatronic

systems created to capture the behavior of the core elements of the muscle spindle’s

anatomy and physiology. Engineering performance data are presented as well as tests

of the integrated system to demonstrate feasibility.

Chapter 4, entitled “A Biorobotic Structural Model of the Mammalian Muscle Spindle

Primary Afferent Response,” describes the biological modeling aspects of the robotic

muscle spindle’s development. The methods used to integrate the engineering hardware

4


into a biological model are described including a modification to current muscle spindle

theory proposed during the tuning process. Results from the tuning studies are

presented as well as the performance of the fully tuned model in a battery of validation

experiments. Biological data from the literature accompany all results to facilitate

comparison.

Chapter 5, entitled “Fusimotor Effect on Signal Information Content of Ia Ensemble

Model Reconstructed from Dynamic Intramuscular Strain Data,” describes the

reconstruction of the ensemble response of a population of muscle spindles.

Collaborators provided data describing the mechanical strains experienced at 28

locations on the surface of a muscle during the course of a muscular contraction. These

data were used to reconstruct the ensemble response of a hypothetical population of 28

muscle spindles. Using this novel methodology, this chapter investigates the

information content of the ensemble response including the influence of the fusimotor

system. The chapter concludes by proposing neuromechanical mechanisms to explain

the observed behavior.

Chapter 6 summarizes the major findings of Chapters 3-5. It also suggests directions for

future work both in muscle spindle physiology and in the development and application

of biorobotic length and velocity sensors.

5


Chapter 2:

Literature Review

The first section of this chapter describes the anatomy and physiology of the muscle

spindle as well as a review of pertinent literature on the physiology of these

mechanoreceptors. Section two reviews muscle spindle modeling efforts to date, with

emphasis on structural nonlinear models including the prototype artificial muscle

spindle. Section three reviews the topic of muscle spindle ensemble response including

theory, experimental data and modeling efforts.

2.1 The Mammalian Muscle Spindle

2.1.1 Overview

The mammalian muscle spindle, shown in Figure 2.1, resides in the body of its host

muscle. The fusiform-shaped organ consists of long muscle fibers that run the length of

the spindle. Those fibers are called intrafusal muscle fibers, and can be divided both

anatomically and functionally into the sensory region, in the center, and the contractile

region, lying at either end. The sensory region of these fibers is devoid of contractile

tissue, instead behaving in a spring-like manner. The muscle spindle has two types of

sensory nerve endings. Primary endings, or group Ia neurons, wrap around the sensory

region; secondary endings, group II neurons, terminate in endings called flower spray

endings that adhere to the intrafusal fiber. As the cell membranes of these nerve

endings are stretched, strain dependent ion channels in the membrane open. The

resulting flow of ions across the membrane causes local depolarization of the cell,

transducing the strain into an analog receptor potential. This receptor potential is then

encoded at the heminodes of the nerve ending, translating the analog potential into a

train of action potentials, or voltage spikes, whose frequency is proportional to the

6


applied strain. This frequency modulated spike train then travels down the Ia or II

nerve axon to the spinal cord.

The contractile regions of the muscle spindle

are essentially muscle fibers. They are filled

with actin and myosin, the force generating

proteins of muscles. These molecules are

aligned to generate tension along the long

axis of the spindle. The γ motorneuron, a

motor nerve fiber specific to muscle

spindles, transmits control signals to the

contractile region from the central nervous

system[2]. These commands govern the

contraction of the intrafusal fiber. The

kinematics of these contractile regions is

further modulated by the unique viscoelastic

properties of muscle tissue.

The function of the contractile regions is to

filter incoming displacements, thereby conditioning the nature of the signal reported by

the sensory transducer. An example of this filtering is to keep this central sensory

region taut as the host muscle changes lengths. Hence, if the CNS commands the

biceps to contract by 10%, a γ motorneuron can command the intrafusal fibers of the

bicep’s muscle spindles to also contract by 10%. More sophisticated filtering can be

achieved by driving the host muscle and intrafusal muscle fibers independently. A

simple example of this is to contract the intrafusal fiber by 12% instead of 10% in the

scenario described above, making the nerve endings very taut and thereby increasing

the gain. This is observed during uncertain kinematic situations, such as in a cat being

held by a human[3].

Figure 2.1: Mammalian muscle spindle

7


A further nuance of control in muscle spindles is the presence of three different classes

of fibers within a given muscle spindle, each classified as velocity or position sensitive.

The static nuclear bag and nuclear chain fibers are position sensitive, producing

primarily a response proportional to the magnitude of strain. The dynamic nuclear bag

fibers are velocity sensitive, producing a more complex output approximated by a

weighted sum of strain magnitude and its first derivative. The γ motorneuron system

has separate inputs to the static fibers and the dynamic fibers, allowing the central

nervous system to preferentially amplify the response of just one type of fiber.

Further information on the basic biology of muscle spindles can be found in Kandel et

al.[3], Gladden[4] or the exhaustive review by Hunt[5].

2.1.2 Intrafusal Muscle

Researchers have sought to identify which aspects of the muscle spindle’s behavior

arise from the mechanical properties of the intrafusal muscle since Matthews first

proposed that intrafusal muscles might be responsible for the muscle spindle’s position

and velocity sensitivity[6]. Ottoson and Shepherd opened the door to examining this

question directly by visually recording changes in intrafusal muscle length with

stroboscopic photomicroscopy[7], a technique which has been applied extensively in

the ensuing years[8-11]. This experimental work was driven by the investigation of

several hypotheses about intrafusal muscle mechanisms. Short-range stiffness is one of

the most popular, with the theory that many of the actin-myosin cross-bridges remain

bound during displacements <0.2%[12]. This results in a highly sensitive linear region

of Ia output during small displacements and the initial burst seen during ramp and

holds[13, 14]. Another theory tested with photomicroscopy data is stretch activation,

the theory that stretching a passive dynamic nuclear bag fiber will cause it to contract.

This theory was first proposed by Boyd[15] and has been controversial ever since. The

only definitive evidence for it was obtained with an experimental technique that

damaged the muscle spindle and is thus inconclusive[10, 16].

8


Researchers have shown that the distinctive mechanical properties associated with each

of the intrafusal muscle types correlate well to their respective myosin isoforms:

dynamic bag myosin is similar to slow tonic extrafusal fibers, static bag myosin is a

unique isoform similar to extrafusal slow twitch fibers, and nuclear chain fibers are a

fast form similar to developing muscle[4]. This provides support for the assumption

that the differing mechanical properties of the various intrafusal fibers are the result of

mechanisms similar to those of the well studied extrafusal muscle.

2.1.3 Neural Transduction and Encoding

Investigators have sought to understand the mechanisms causing the dependency of Ia

action potential frequency on the receptor potential rate of change. One possible

mechanism, a decrease in AP initiation threshold voltage during dynamic increases in

receptor potential, has been noted in three different studies[17-19]. To date, though, no

investigation has specifically addressed the possibility of unidirectional behavior in the

encoder’s rate dependency.

Researchers have also sought to isolate the source of various spindle behaviors to the

mechanical system vs. the sensory system. Many have concluded that almost all

behaviors, including short-range stiffness, must be mechanical in origin because they

are clearly parallel in the tension record and the receptor potential record[18, 20]. The

exception to this is undershoots which are deemed of chemical origin because they do

not appear in the tension record[14]. Although some work has been done in identifying

the types of ions that are involved in neural transduction, there are no data available on

specific ion channel types or numbers[21].

2.2 Muscle Spindle Modeling

2.2.1 Intrafusal Muscle Models

2.2.1.1 Linear Models

Since Matthew’s observation in the 1930’s that the position and velocity sensitivity of

9


the muscle spindle could be caused by mechanical properties of the intrafusal fiber,

virtually all structural (homeomorphic) muscle spindle models have attempted to

account for the mechanics of the intrafusal fiber. Early models ranged from 1st order to

6th order linear models. The linearity limited their applicability to small subsets of the

spindle’s behavior in which the output was known to be linear. One example is

Poppele’s 1970 model[22] which used system identification techniques to empirically

fit transfer functions to Bode plots of deefferented spindles’ responses to sinusoidal

length inputs in the spindle’s linear range. Through his experimental data he concluded

that primary and secondary afferents shared a common mechanical filtering system, but

differed in their transduction and encoding. Hence, he generated one common

mechanical filtering transfer function and two unique transduction/encoding transfer

functions. While the resulting transfer functions matched small amplitude sinusoidal

behavior quite well, the range was limited, applying only to deefferented muscle

spindles in their linear region.

In 1970, Rudjord introduced a structural linear model[23], one in which model elements

corresponded to selected physiological entities in the muscle spindle. The model

consisted of 2 fibers: a nuclear bag fiber and a nuclear chain fiber. Like the Poppele

model, it also generated both primary and secondary output. Modeling of fusimotor

input was omitted from early versions of the model, then included later under the

condition of constant length. Rudjord established an arrangement of springs and

dampers based on spindle anatomy and physiology. He then used experimental data

from the ubiquitous ramp and hold experiment to tune his model parameters. This

model achieved good performance in the small amplitude linear region, but lacked

general applicability.

A few of these early linear models took changes in fusimotor activation, not length, as

their input. Andersson et al.[24] presented such a model in 1968. This model

successfully reproduced spindle response to sinusoidal fusimotor activation of either the

gamma static or gamma dynamic system, but could not account for the nonlinearities

10


observed when both systems were stimulated simultaneously.

2.2.1.2 Nonlinear Models

Nonlinear models began appearing in 1981, with the introduction of three non-structural

empirical models with small ranges of applicability. The Houk model[25] defined

power laws capable of describing the spindle’s response during the constant velocity

phase of a ramp and hold after the initial burst has dissipated. Poppele and Quick

introduced a model very similar to his 1970 model[26], except that the transfer

functions were fit to experimental data generated by subjecting the spindle to band-

limited white noise inputs rather than slow sinusoids.

2.2.1.2.1 Hasan Model

In 1983, Hasan published the first structural nonlinear model, responding to the lack of

a comprehensive set of rules describing the dependence of firing output on stretch

input[27]. Table 2.1 gives an overview of the major anatomical features included in this

and the other nonlinear structural models.

Table 2.1: Anatomical elements included in nonlinear structural models

Model Gamma Static

Gamma Dynamic

Ia Fibers

II Fibers

Dynamic Bag 1

Static Bag 2

Nuclear Chain

Schaafsma C C X X H H Winters C X X Hasan D D X X X Robotic C C X X H H

Where: C=continuous range of levels, D= discrete levels, X=explicitly included in model, H=included in model as part of hybrid static fiber.

Hasan’s model of the Dynamic Nuclear Bag 1 fiber incorporated a nonlinear

mechanical filter component and a linear transducer/encoder component. Like all

models to date, the Hasan model treated the contractile tissue as an extrafusal muscle.

The transducer/encoder was implemented as a series elastic element. In theory, the

model accounted for both static and dynamic fusimotor input as well as primary and

11


secondary output. In practice, however, the model can only represent one fusimotor

state at a time, since the modelers recorded a set of experimental data in each of the 4

desired fusimotor states and retuned their constants to match each of the four states.

One feature that captures some of the complexity of muscle spindles is “resetting” of

short-range stiffness. Hence, if the spindle is stretched slowly enough, the sensitivity

can stay very high. The mathematical expression of the model cannot be solved

analytically and results are obtained through numerical simulation. The model enjoys a

broad applicability to a range of types of motion, successfully reproducing both

sinusoids and ramp and hold.

2.2.1.2.2 Schaafsma Model

Schaafsma et al. developed the most complete mechanical model introduced to

date[28]. Since its introduction in 1991, the same group has introduced three additional

submodels relating the neural aspects of spindle behavior. All of the models are based

on known micro-physiological or micro-anatomical concepts. The complete set of the

four submodels, the Integrated Model of the Mammalian Muscle Spindle, is described

in Otten et al.[29]. The Schaafsma submodel is discussed in this section while the

remaining submodels are presented below in the transducer/encoder section.

The Schaafsma muscle spindle model was the first model to structurally incorporate

fusimotor stimulation during dynamic length changes and widely varying testing

protocols. The model was founded on the belief that complex spindle behavior arises

from the mechanical interaction between the intrafusal muscle tissue and the sensory

region. The Schaafsma mechanical model models only the primary afferent fiber and

consists of two submodels: (a) Bag1: analogous to the dynamic bag 1 fiber, and

responsible for the spindle’s dynamic fusimotor response, and (b) Bag2, a composite of

the bag2 fiber and the nuclear chain fiber, and responsible for the spindle’s static

fusimotor response. Each submodel then consists of a sensory region in series with a

muscular region. The sensory region is represented mechanically as a simple linear

12


spring, while the muscular region is represented by an extrafusal muscle model

developed by Otten incorporating sensitivity to length, velocity and activation[30]. The

primary afferent output is computed as a simple linear function of the length of the

sensory region and its first derivative. At any given moment the model’s output is

either entirely due to the Bag1 model or the Bag2 model, whichever is larger. This is an

early implementation of the competitive pacemaking concept elaborated further in one

of the Otten group’s submodels[31]. Schaafsma has incorporated a short-range stiffness

model consisting of 100 fused cross-bridges that make the intrafusal muscle

indistensible until a force exceeding a threshold ruptures one or more of the cross-

bridges. Parameter values for the Schaafsma model were obtained via a parametric

search using metrics from experimental ramp and hold data from muscle spindles under

a variety of velocities and fusimotor activation levels.

This model has been also been adapted to mimic fusimotor driving of Ia output in

nuclear chain fibers[32]. The model was moderately successful, though it did not

capture some of the subtleties of the biological system such as robustness to length

change.

2.2.1.2.3 Winters Model

Two further models have been introduced since the Schaafsma model, both as parts of

models describing a larger segment of the neuromuscular control system. The Winters

model[33] was developed in the context of providing closed loop feedback for a large-

scale neuro-musculoskeletal model of the shoulder. Since he was focusing on posture

control studies, Winters chose to model the secondary afferent output of a static nuclear

bag fiber under static gamma motorneuron input. The basic structure of the model

consists of a contractile region in series with a series elastic region. The combined

elements are assumed to span the full length of the host extrafusal muscle. The

contractile element is essentially modeled as a shorter version of the host extrafusal

muscle model with a few basic modifications: (a) no damping, since this is a model of

13


static position sensing, (b) the force-length curve of the parallel elastic element has only

a positive slope, required for the stability of the system, (c) constant strain across the

length of the fiber. The muscle model is quite detailed, even including nerve activation

dynamics and calcium dynamics. The sensory element is modeled as a series elastic

element. Its sensitivity is distributed as a gaussian function with strain, with peak

sensitivity at mid strain. The secondary afferent output is computed as a function of the

sensory element’s length plus a very small function of the sensory element’s rate of

change. This model presents an interesting engineering based treatment of many of the

issues addressed in other spindle models.

2.2.1.2.4 Wallace Model

In 1996, Wallace and Kerr developed a model of the ensemble response of ten muscle

spindles, each from a different muscle[34]. He intentionally chose to use a simple

model of individual spindle response rather than a more detailed model such as

Schaafsma’s or Hasan’s. Wallace’s model predicts primary and secondary afferent

output with no fusimotor input. It is based on Houk’s[27] empirical model which used

a power law to describe the spindle output during the constant velocity region of a ramp

and hold after the initial response has died out. Wallace augmented this model by

introducing a term causing the spindle to fall silent when shortening velocities dropped

below a threshold. He also removed the spindle length dependency of the output,

making it purely a function of velocity. His companion paper to the ensemble model

does a sensitivity analysis and concludes that, in the context of ensemble encoding, the

information transfer is independent of both the fractional power of velocity and absolute

firing levels of the afferents. He did, however, experiment with reintroducing length

sensitivity to the model with three different types of mathematical expressions and

found a small change in observed correlation coefficients resulting from the inclusion of

explicit length-dependent terms.

14


2.2.2 Transducer and Encoder Models

Classically, the transducer and encoder functions have been lumped together, if they are

distinguished from the mechanical filtering function at all. Major models such as the

Hasan model and the original Schaafsma model depict the generator potential as simply

the stretch across the sensory region and the Ia firing rate as the sum of the generator

potential and its first derivative.

The recent models introduced by the Otten group have placed a new focus on the

transduction and encoding process. The contents of these models, as well as the neural

transduction and encoding aspects of the Schaafsma and Winters models, are shown in

Table 2.2.

Table 2.2: Processes modeled in structural models to generate neural output

Model Nerve Strain

Nerve Strain Rate

Membrane Depolar- ization

AP Encoding

Pacemaker Sites

Nerve Modeled

Otten Integrated Model X X X X X Ia Schaafsma X X Ia Winters X X II Otten, K+ Conductance X X Ia Banks X Ia

The Otten transduction model[35] uses modified Frakenhauser-Huxley equations to

model ion channel dynamics, focusing on the impact of slow potassium conductance

channels. This model is able to account for many of the nonlinear phenomenon

attributed to the mechanical system including the slow decay during hold and the

silence upon release of holds. Although there are little data on ion channel composition

to validate such a model[36], its results suggest an interesting hypothesis worthy of

experimental investigation regarding how much the mechanical vs. neural systems

contribute to the overall dynamics of the muscle spindle.

The encoder models originated with an interesting study correlating histological data

15


and neurophysiological recordings. Banks et al.[31] showed that, depending on the

number of nodes of Ranvier separating two sensory endings, the two signals would

either electrotonically couple or exhibit competitive pacemaker interaction. In the first

case, when separated by one node of Ranvier or less, the signals from the separate

nodes would have an averaging effect, modeled as an analog resistive circuit. When

separated by two or more nodes of Ranvier, they instead interacted competitively, with

the faster node sending its action potential antidromically down to the adjacent nodes

thereby inhibiting their output. Otten’s group has implemented a hybrid of the

occlusion (pacemaker) submodel and the electrotonic coupling submodel in their

integrated model, determining the relative contribution of each submodel according to

the number of nodes of Ranvier separating the two sensory endings. Again, there are

few data available regarding pacemaker membrane kinetics and channel composition to

validate the results of such a model. However, it is a useful tool for suggesting new

experiments and as such is a promising step towards a more detailed understanding of

the encoding process.

2.2.3 Biorobotic Models

Biorobotic hardware is a new medium for muscle spindle modeling. In 1993, Marbot

and Hannaford [37, 38] presented the first prototype of a biorobotic muscle spindle

model, the Artificial Muscle Spindle. The device, shown in Figure 2.2, uses a lead

screw actuator for the mechanical filter, a strain gage for the sensory transducer, and an

onboard printed circuit board for encoding the transducer output into a frequency

modulated square wave. This model demonstrated well the feasibility of reproducing

muscle spindle behavior in engineering hardware in tests spanning a wide range of

experimental protocols.

16


The mechanical design of the Artificial Muscle Spindle was precarious, though,

resulting in problems with noise, repeatability and mechanical failure. Most notably,

the noise arising from the mechanical design was too great to allow inclusion of the

derivative term in the computation of muscle spindle Ia output. These design

limitations prevented the Artificial Muscle Spindle from fully implementing a structural

model of Ia response, limiting its applicability as a testbed for asking basic science

questions about motor control.

The robotic muscle spindle presented in this dissertation, Figure 2.3, extends this initial

hardware design by implementing a reengineered design in precision hardware and

validating each engineering subsystem against biological performance specifications.

The resulting design has alleviated the previous limitations and provides a robust

platform for modeling all aspects of spindle behavior.

2.3 Muscle Spindle Ensemble Response

The ensemble response of muscle spindles is a relatively new field that has risen to

prominence during the 1990s. The information that can be extracted from a single

spindle’s Ia response is sharply limited by noise and nonlinearities[1]. As a result,

Figure 2.2: Artificial Muscle Spindle, prototype biorobotic muscle spindle.

Figure 2.3: Robotic Muscle Spindle

17


researchers look to the ensemble response of a population of spindles as a way for the

central nervous system to obtain a decipherable signal of muscle kinematics[39-46].

Unfortunately, the technical difficulty of recording from multiple muscle spindles from

a single muscle has limited the size of simultaneously recorded ensembles to

populations of ten spindles or less[41-44, 47]. This makes ensemble modeling an

attractive alternative, though to date only two models of spindle populations have been

published, both using simple models of individual spindle behavior to examine limb

position encoding by spindle populations spanning multiple muscles [34, 48].

2.3.1 Ensemble Information Content

The question of what parameters might increase ensemble information has received

considerable attention. Ensemble size, simultaneous recording and an intact fusimotor

system all have been shown to increase the ensemble’s ability to discriminate between

sinusoids of varying amplitude[41, 43]. The fusimotor system has been further

implicated as a mechanism by which ketamine application[44] and heteronymous

muscle fatigue[47] degrade ensemble information content. Several investigators have

raised the issue of decorrelating individual muscle spindle responses as a means to

improve spatial filtering of ensemble information content. Proposed mechanisms for

introducing the decorrelation include the fusimotor system behaving as a neural

network [43, 47], random noise introduced by the active fusimotor system [49] and

membrane firing threshold variability [50]. To date, though, such decorrelation

mechanisms have only been tested indirectly [47], theoretically [50] or in small

populations [49].

2.3.2 Experimental Data

Recording from populations of muscle spindles is a relatively new field, with the

majority of the work being done in the 1990s. The amount of experimental data is

limited by the difficulty of recording from a sufficient number of primary muscle

spindle afferents from a single muscle during a reproducible motor task[51]. Within

18


this body of work there are two subfields, intra- and intermuscular populations. These

two fields tend to be split into non-human vs. human data, respectively, due to increased

technical difficulties in recording from humans. There are also two further divisions in

the experimental literature: (a) those that record from one spindle at a time, accruing

“ensemble” data over sequential repetitions of the same behavior[51, 52] and (b) those

that record simultaneously from multiple Ia fibers[41, 43].

In terms of specific data, there are several studies by Bergenheim and Johansson

describing recordings of both simultaneous and sequential data from multiple muscle

spindles in anesthetized cats[41-43, 47]. Prochazka et al. set out to compile an

extensive “look-up chart” of data from muscle spindle ensembles during the cat step

cycle[52]. Since then, they have recorded firing profiles of 47 muscle afferents during

the cat step cycle. These data are a sequential recording under a similar scenario, as

opposed to a simultaneous recording, using 34 cats to collect data on the 47 muscle

spindles during free locomotion[53]. In terms of human data, the studies available are

limited to recordings of single muscle spindles from multiple muscles using the

microneurographic technique[54, 55].

2.3.3 Modeling

Two models have been published describing the response of a population of muscle

spindles. Both articles use populations across multiple muscles to examine limb

position encoding, emphasizing distribution of the spindles within the limb. The Scott

and Loeb[48] study focuses on the reasons underlying muscle spindle distribution

across the muscles of the human body. The study models the individual spindle

secondary response using a simple model which includes a sensory element and a noise

source. The variation between spindle outputs is based on the location of the host

muscle and the variation in the injected noise. Wallace and Kerr[34] present a model of

ensemble muscle spindle output using 10 different muscles with a single muscle spindle

per muscle. Spindle output is calculated as a power law of velocity, adapting the model

19


from the work of Houk et al.[25]. They are able to show that the ensemble metric,

calculated as the average of the individual spindle outputs, is well correlated with joint

angular velocity, but not joint angular position.

20


Chapter 3:

Mechatronic Design of an Actuated

Biomimetic Length and Velocity Sensor

3.1 Abstract

Drawing from the rich source of proven and often novel mechanisms in the

biological realm, biomimetic sensors are being successfully developed for

many different transduction tasks. This paper presents such a sensor for

transducing displacements. Our sensor, Figure 3.1, is a robotic analog of

the biological muscle spindle, an actuated sensor which transduces muscle

displacement for kinesthetic awareness.

The mechanical filter exhibits the desired step response with Tr=26 msec,

Ts=54 msec, P.O. = 9.2%, Ess=6.8x10-3mm. The transducer possesses the

desired linear response with a sensitivity of 34nm/Hz. Finally, the encoder

circuitry successfully maps the millivolt output to a pulse frequency range

of 1150Hz to 12.5kHz. Results from integrated system tests show that

with a traditional engineering-based controller the sensor can successfully

detect errors in trajectory tracking introduced by both phase lag and

perturbations. With a physiologically-based controller, it successfully

replicates the major features of muscle spindle response. By physically

realizing the hypothesized core features of a biological muscle spindle in

engineering hardware, we have evoked the type of actuated sensor output

seen in the biological muscle spindle, a widely utilized tool of biological

motor control.

Figure 3.1: CAD model of biomimetic sensor

21


3.2 Introduction

In biorobotics research, engineers and biologists come together to implement in steel

and silicon the researcher’s vision of the mechanisms driving a biological process.

Such a project is instructive for all parties involved. Biologists are able to test the

viability and coherency of the proposed mechanisms when challenged with the demands

of the physical world. Meanwhile, engineers are able to draw from this process novel

approaches to age-old tasks such as detecting the properties of the physical

environment.

This paper describes the development of such a biorobotic device, an actuated

biomimetic length and velocity sensor. The design is inspired by length and velocity

sensors found in mammalian muscle tissue called muscle spindles. These organs

contain a spring-like transducer region which lies in series with an internal actuator, the

intrafusal muscle. This tiny actuator receives motor commands from the central

nervous system (CNS), allowing the brain to actively modulate the nature of the output

of the transducer’s sensory region.

The development of an engineering implementation of these sensors poses the

following questions: What elements of the muscle spindle represent core functionality?

How are these functional elements best implemented to form a robust robotic sensor?

Finally, can a non-back-driveable electromechanical system yield the active filtering

and transduction behavior of living muscle and nervous tissue?

We thus address the following hypotheses:

(a) The core functions of a robotic length and velocity sensor based around a structural

model of muscle spindles are mechanical filtering, transduction and encoding. A sensor

which captures these methods can exhibit the type of response seen in muscle spindles.

(b) The electromechanical systems presented here are capable of achieving the

performance specifications necessary to match the physiology of mammalian muscle

22


spindles.

3.2.1 Background

The mammalian muscle spindle, shown in

Figure 3.2, consists of long muscle fibers,

called intrafusal fibers, which run the

length of the spindle. Each fiber contains a

sensory region, in the center, and an

actuator region, lying at either end. The

sensory region acts as a passive linear

elastic spring. Ia sensory nerve endings

wrap around these fibers and transduce

stretch of the sensory region into a

depolarization of their membrane.

Heminodes on the Ia axon then encode this

analog depolarization into a frequency

modulated spike train of action potentials

which travel up to the spinal cord.

The actuator region is essentially a normal

muscle fiber, controlled by the input of a

dedicated signal from the spinal cord, the γ

motor neuron. The function of the actuator

region is to filter incoming displacements, thereby conditioning the nature of the signal

reported by the sensory transducer. The γ motor neuron control of the actuator’s force

production allows the CNS to finely control this process. For instance, it can raise the

sensor’s gain during uncertain kinematic situations by increasing contraction, thereby

increasing the stretch of the sensory region[56].

Several mathematical models of the muscle spindle have been developed[23, 27]. The

Figure 3.2: Mammalian muscle spindle anatomy

23


Schaafsma model [28] is one of the most sophisticated models, consisting of a nonlinear

extrafusal muscle in series with the non-contractile sensory element. This model, unlike

the many linear models, is able to approximate muscle spindle behavior for a wide

range of stimuli.

3.3 Methods

3.3.1 Design

Based on an earlier prototype[37, 38], we abstracted the three core elements of muscle

spindle function: mechanical filtering, transduction and encoding. The mechanical

filtering is performed in the biological muscle spindle by the contractile region of the

intrafusal fibers. Our goal is to create an internal actuator with performance

specifications sufficient to mimic intrafusal muscle dynamics. Based on the kinematics

observed in intrafusal muscle response during direct observations[10, 16], this requires

a rise time (Tr) <30 msec, settling time (Ts) <150 msec, percent overshoot (P.O.) =10%,

steady state error (Ess) =0.

The transduction role in the biological muscle spindle is performed by strain-sensitive

ion channels which cause depolarization of the Ia nerve membrane in direct proportion

to the strain applied across the sensory region[3]. The biological transducer exhibits a

resolution of better than 20µm[4] with linear output at small displacements, known as

short-range stiffness. With large displacements, the transducer stiffens, exhibiting

decreased sensitivity[4]. Our design goal is a transducer with similar resolution and a

large linear region, stiffening to lower sensitivity output at the end of its range.

24


The final role, encoding, is performed at the Ia axon

heminodes in the biological spindle. The analog

depolarization of the transduction sites is translated into a

frequency modulated spike train with a range of

approximately 0 to 400 Hz. Our goal is to create an encoder

that also produces a frequency modulated spike train

proportional to the analog voltage of the transducer.

3.3.2 Implementation

3.3.2.1 Mechanical Filter

The mechanical filtering task is implemented with a low

inertia, direct drive lead screw linear actuator system, Figure

3.3, to achieve the rapid response times of the intrafusal

muscle. A miniature ironless core dc motor (1016-N-006,

MicroMo, Clearwater, FL) is coupled to a cold rolled

stainless steel 2-56 lead screw with a flexible helical

coupling. The system is mounted between a pair of

semicylindrical stainless steel guides with a Delrin AF

bushing aligning the tip of the lead screw and the guides.

The transducer element is mounted on a platform machined

from Delrin AF with a 2-56 thread tapped through its center.

This platform has integral linear bushings which ride in the

track formed between the two semicylindrical housings,

allowing the lead screw rotation to be transformed into linear

motion of the platform. Mounted onto the lead screw system,

the transducer platform forms the end point of the intrafusal

muscle implementation, which lies in series with the transducer.

Figure 3.3: Linear actuator and transducer assembly

25


An encoder (HEM-1016-N-10, MicroMo, Clearwater, FL) is mounted directly to the

motor. Its quadrature signal is read by a dSPACE 1102 controller board (dSPACE

GmbH, Paderborn DE). The encoder data are filtered with a 25Hz 4th order digital

elliptic filter. A PID controller for the complete linear actuator was designed in

MATLAB and Simulink then implemented in C (Real Time Workshop, MathWorks).

3.3.2.2 Transducer

The transduction element, shown in

Figure 3.4 and Figure 3.5, lies in series

between the linear actuator and the distal

end of the position sensor. We

implemented high resolution transduction

between strain and analog voltage with a

pair of strain gaged cantilevers mounted

perpendicular to the axis of sensing. The

cantilevers are machined from stainless

steel shim stock 51 microns thick. An

aluminum stop 0.16 mm above the plane

of the cantilever, machined to a 6.6°

angle, is used to keep the cantilever’s

deflection within its linear elastic range.

One uniaxial polyimide and constantan

alloy self-temperature compensated 120

Ohm strain gage (EA 06 031CF 120, Measurements Group, Raleigh, NC) is mounted to

the bottom surface of each cantilever. The dimensions and materials of the cantilever

were selected such that a maximum of +2000/-0 µstrain would be applied to the foil

matrix of the strain gages during deflection, giving a fatigue life of 108 cycles.

Figure 3.4: Transducer platform. Consists of Delrin AF bushing and aluminum stop. Strain gaged transducer is visible in the gap between bushing and stop

26


Input displacements are

applied to the cantilevers by

means of nylon coated 3x7

stainless-steel cables.

These cables run through

guide holes both in the

Delrin AF bushing at the

distal end of the position

sensor as well as in the

aluminum stop immediately

adjacent to the cantilevers

to ensure robust and

repeatable performance,

unmarred by tangling of the cables in the lead screw. The tension of the cables is

transmitted to the cantilever by means of a steel compression sleeve crimped to form a

solid beam. The beam runs the width of the cantilever, thereby minimizing edge effects

on the strain gage film.

3.3.2.3 Encoder

The encoding of the analog voltage into a frequency modulated spike train is

implemented with surface mount integrated circuit (IC) chips on a printed circuit board

mounted directly to the sensor platform. The circuit, Figure 3.6, uses a Wheatstone

bridge configured as a half bridge and is zeroed by a 60 kΩ resistor in parallel with one

of the 120Ω bridge completion resistors. The resulting signal is then immediately

amplified with a gain of 430. The amplified signal is then sent into a 7555 IC, wired in

voltage controlled oscillator mode, resulting in a frequency modulated square wave with

a range of 1150 to 12500 Hz.

Figure 3.5: CAD drawing of transducer platform.

27


3.3.3 Linear Positioning Device

We designed and built a linear positioning device (LPD), shown in Figure 3.7, to

provide position inputs to the robotic spindle. The actuator was based around a 5.25in

hard drive actuator. The rotary displacement of the precision hard drive motor is

converted to linear displacement by wrapping a metal ribbon 3.17mm wide and 0.10mm

thick around a metal drum rigidly mounted to the motor. A slot machined in the outer

circumference of the drum aligns the metal ribbon with the linear axis. The metal

ribbon is then rigidly mounted to the ball slide of a miniature linear guide. The motion

of this ball slide is defined as the linear position output of the actuator.

Strain Gage Y

R4b100k

R1120

Strain Gage X

R4a150k

R2120

R51k

R61k

R7a470k

R7b5100k

C41nF

R8a470k

R8b5100k

C31nF

C522pF

7555

R9390k

VCC7

4

6

2

5

1

R1010k

Out

C16.8nF

LM 308

Figure 3.6: Encoder circuit diagram. Strain sensed by strain gages generates a millivolt potential across a Wheatstone Bridge. That signal is amplified (LM308 chip) then converted (7555 chip) to a frequency modulated square wave.

28


To control the actuator’s position, we use an LVDT (LD100-20, Omega, Stamford, CT)

rigidly mounted to the LPD base such that its axis is parallel to the linear slide rail. The

aluminum core of the LVDT is then rigidly fixed to the ball slide. The data are filtered

with a 3rd order 40 Hz Butterworth filter. A separate PID controller was designed for

the LPD and implemented in the same dSPACE system.

To perform experiments on the robotic spindle, the cable from the robotic muscle

spindle is fixed directly to the ball slide of the LPD.

3.3.4 Modeling

3.3.4.1 Mechanical Filtering

The transfer function for position control of the linear actuator is:

sKKRBsLBRJJLs

PKsVsX

mb

m

IN

A

)()()()(

23 ++++= ( 3.1 )

Where J=inertia, L=motor inductance, R=motor resistance, B=damping, Kb=back EMF

5 cm

Figure 3.7: Linear Positioning Device and Robotic Muscle Spindle

29


constant, Km=torque constant, P=thread pitch, XA=linear actuator position and

VIN=motor voltage.

With the parameter values for our system inserted, the transfer function for position

control of the linear actuator is:

VmmssssV

sX

IN

A /1074.010*4.2

217)()(

236 ++= −

( 3.2 )

Our desired step response was Tr<30 msec, Ts<150 msec, P.O.=10%, Ess=0. A PID

controller was designed iteratively both in simulation and on the physical hardware to

meet these specifications. The resulting controller gains are KP=100, KI=10, KD=0.5.

This design gives the following theoretical step response: Tr=3.5 msec, Ts=18 msec,

PO=25%, Ess=0.

3.3.4.2 Transducer

We derived the linear relationship between displacement of the transducer, xC, and the

strain of the strain gages, ε, to be:

Cth x

LdLC

32)(3 −

=ε ( 3.3 )

Where xC = overall input displacement, Cth = cantilever thickness, L = distance from

cantilever base to load, d=distance from cantilever base to center of strain gage.

We designed the transducer with values for L, d, and Cth such that the displacement

range, xC, yielded the desired 2000 µstrain at full-scale deflection. As this transducer is

analog, it has continuous resolution. Hence, the resolution goals were met. Finally, this

model shows the response is linear throughout the transducer’s primary range.

3.3.4.3 Encoder

Accounting for the interaction between the strain gage and the pulse generation circuit,

30


we derived the relationship between strain, ε, and frequency, F, as:

( )

1

1

2

1

2

5*2

*5

5*2

*25

ln69.

−

−

+

−

+++=

GG

RR

GG

RR

CRRCRF CBB

εε

εε

( 3.4 )

Where: RB=resistor between 7555 VCC and discharge pins, RC=resistor between 7555

discharge and threshold pins, C=capacitor across 7555 trigger and ground pins, R2/R1 =

amplifier gain, G=gage factor of strain gages.

Values for these parameters were selected to give a frequency range of approximately

1kHz to 14kHz.

3.4 Results

3.4.1 Actuator Performance

Figure 3.8a shows the step response of the linear actuator. The performance metrics for

this step function are: Tr=26msec, Ts=54msec, PO=9.2%, Ess=6.8x10-3mm, which meets

our goal of: Tr<30 msec, Ts<150 msec, P.O.=10%, Ess~0.

31


On a 30mm/sec ramp trajectory,

Figure 3.8b, the performance metrics

are P.O.=0.30% of absolute position,

overshoot=0.089mm, maximum

error=0.15mm, mean absolute Ess =

0.041mm.

3.4.2 Transducer and Encoder

Calibration

Figure 3.9 shows the combined

calibration of the transducer and

encoder systems. Calibration is

depicted between Displacement and

Frequency, Figure 3.9a, and Force

and Frequency, Figure 3.9b. In each,

the response is linear at small to

moderate displacements and forces,

followed by a region at the end of the

range exhibiting decreased

sensitivity, reflecting the design

specifications. Figure 3.10

demonstrates the waveform

generated by the encoder circuitry at

both the low and high ends of the

encoder’s working range.

Figure 3.8: Time response of linear actuator implementation of intrafusal muscle (solid line). (a) 1 mm step position input (dotted line), (b) 30 mm/sec ramp position input (dotted line).

Figure 3.9: Calibration plots for transducer and encoder, (a) Frequency vs. Displacement, (b)

Frequency vs. Force

32


The observed range of strain across

the strain gages is 66µstrain-

1700µstrain, within the targeted 0-

2000 µstrain range.

3.4.3 Linear Positioning Device

Performance

Figure 3.11 shows the ramp response

of the linear positioning device

during a 6 mm/sec ramp and hold.

For PID controller values of P=10,

I=140, D=0.1, the 6mm/sec ramp

performance metrics are

P.O.=0.75%, overshoot=0.017mm,

and mean absolute Ess =0.018mm.

3.4.4 Integrated Performance

To initially test the performance of

the three core elements as an

integrated system, we programmed our sensor and the LPD testing machine to move

with the same sinusoidal trajectory, separated only by a phase lead. The resulting

performance is shown in Figure 3.12.

Figure 3.10: Waveform of frequency modulated square wave at small sensor displacement (top graph) and large sensor displacement (bottom graph).

Figure 3.11: Time response of LPD (solid line) to 6mm/sec ramp and hold position input (dotted line).

33


As is shown in the Figure 3.12, the

transducer produces a response

proportional to the “error” in the host

muscle’s displacement, as created by

the phase lead of the robotic sensor’s

movement. Like the biological

spindle, it only detects stretching and

not compression forces. Additionally,

the frequency response reflects any

transient perturbations between the

robotic sensor’s motion and the host

muscle’s motion. An example of this

is the local peaks produced at 1.95

sec when the LPD experiences

stiction and briefly deviates from the

sinusoidal trajectory.

Figure 3.13 and Figure 3.14 show the performance of the hardware elements following

full integration and validation with a physiologically based controller. Integration,

tuning and validation details are in Chapter 4. Figure 3.13 shows that the response of

the robotic system to variation in ramp velocities and γ motorneuron (γ mn) activation

levels is well tuned to match the current theory regarding muscle spindle behavior.

Position gain is independent of speed, but dependent on γ mn activation level, which

alters the properties of the linear actuator’s control algorithm. The velocity gain

produces a velocity-dependent offset during the ramps whose magnitude is dependent

on γ mn input rate. The noise exhibited is normally distributed with a standard

deviation of 10.5 Hz, which is typical of active biological muscle spindles which exhibit

normally distributed noise with a standard deviation of ~8 Hz[1]. The model’s time

domain sinusoidal response, Figure 3.14, shows good qualitative correspondence to the

Figure 3.12: Test of integrated engineering hardware. (a) Trajectory of robotic sensor (solid line) and LPD (dotted line) for phase lead of 20°°°°. (b) Frequency output for phase leads of 8.6°°°°, 14.3°°°°, and 20.0°°°°.

34


biological response, including similarities in phase lead and relative amplitudes under

different γ mn levels. In the passive case, response amplitude varies from the biological

data, revealing a limitation of the device. Noise is absent in the biological cases

because these data are the average response of multiple trials.

3.5 Discussion

This paper presents a physically

realized robotic implementation of a

biological length and velocity sensor,

the mammalian muscle spindle. We

set out two hypotheses in this paper.

First, that a sensor that captured the

three core behaviors of mechanical

filtering, transduction and encoding

could exhibit the type of behavior

seen in muscle spindles. Second,

that the electromechanical devices

we selected to implement each of

these core functions could meet the

performance specifications necessary

to express each of these behaviors.

Figure 3.13: Effect of ramp speed and γγγγ mn input on robotic Ia Response during 6 mm amplitude ramp and hold. Left column: No γγγγ mn input (passive); Middle column: 100 Hz dynamic, 0 Hz static (dynamic); Right column: 0 Hz dynamic, 100 Hz static (static).

35


Each of the three electromechanical

subsystems met the required

performance specifications to

replicate their biological analogs.

The linear actuator met the desired

time response criteria. The

transducer detected displacements

with the desired resolution and

linearity.

For the encoder subsystem, we met

our desired frequency range,

although we had intentionally chosen

a range substantially different from

the biological encoder range. First of

all, we desired a frequency range of

several kilohertz, whereas the

biological encoder range is

approximately 0-400Hz. This increase in range is a consequence of needing to increase

sensitivity beyond that of biological muscle spindles. This was necessary because our

sensor will be used in a 1:1 ratio with the host muscle, while biological muscle spindles

are often found in much higher densities. Secondly, our displacement-frequency

relationship is the inverse of the biological spindle’s relationship: in our system,

increasing displacements lead to decreasing frequencies. This choice was made to

minimize the number of integrated circuit chips in the pulse generation circuitry. This,

in turn, allowed the circuit to be mounted directly to the transducer platform. Based on

the fact that all three subsystems met the performance specifications of their biological

analog, the second hypothesis is confirmed.

The results from our test of the integrated system support the first hypothesis as well.

Figure 3.14: Comparison of robotic and biological (cf. Hulliger et al. 1977[57]) Ia response to sinusoidal stretch input. Robotic response (top row) matches phase lead and shape, but not amplitude, of cat soleus muscle spindle response (middle row) to sinusoidal position input (bottom row) under different γγγγ mn levels: Left column: 0 Hz dynamic, 0 Hz static, Center column: 87Hz dynamic, 0 Hz static, Right column: 0 Hz dynamic, 100 Hz static. Lengths are reported as displacements to cat soleus.

36


When the three elements are integrated, they produce an output proportional to the

positive displacement discrepancy between the actuator and the LPD, as seen in

biological muscle spindles. Further, they detect this equally well during low frequency

sinusoids and transient perturbations. When integrated using a physiologically-based

controller, the system is able to replicate the major features of the performance of the

full mammalian muscle spindle. Hence, we have shown that the first hypothesis is

correct, these three hardware subsystems are capable of exhibiting the type of sensing

behavior seen in biological muscle spindles.

In conclusion, we have implemented in mechatronic hardware a sensor which replicates

the transducer behavior of a biological length and velocity sensor, the muscle spindle.

Such a device has applications in basic science, as a testbed for studying motor control,

and in prosthetics, as a sensor which communicates in the language of the user’s motor

control system. The question remains, though, as to the suitability of such a device for

engineering applications. An actuated sensor for kinematic measurements such as this

is not commonly employed in engineering applications. We propose that such a system

might be advantageous in situations where the range of the actual transducer is limited,

or for real-time tuning of the sensor’s output to a variety of different kinematic

variables, e.g. length, velocity, or perturbations from a desired length.

37


Chapter 4:

A Biorobotic Structural Model of the Mammalian Muscle

Spindle Primary Afferent Response

4.1 Abstract

A biorobotic model of the mammalian muscle spindle Ia response was implemented in

precision hardware. We derived engineering specifications from displacement, receptor

potential and Ia data in the muscle spindle literature, allowing reproduction of muscle

spindle behavior directly in the robot’s hardware; a linear actuator replicated intrafusal

contractile behavior, a cantilever-based transducer reproduced sensory membrane

depolarization, and a voltage-controlled oscillator encoded strain into a frequency

signal. Aspects of muscle spindle behavior not intrinsic to the physical design were

added in control software using an adaptation of Schaafsma’s mathematical model. We

tuned the response to biological ramp and hold metrics including peak, mean, dynamic

index, time domain response and sensory region displacement. The model was

validated against biological Ia response to ramp and holds, sinusoids and fusimotor

input. The response with dynamic or static gamma motorneuron input was excellent

across all studies. The passive spindle response matched well in 5 of the 9 measures.

Potential applications include basic science muscle spindle research and applied

research in prosthetics and robotics.

4.2 Introduction

Investigators have been studying the muscle spindle for many years, developing and

testing theories about the physiological origins of its unique transducer properties. One

means of testing these theories has been synthesizing them into a structural model, a set

of mathematical expressions that have direct analogs in the physiological system, to see

38


if they exhibit muscle spindle like behavior. While these structural models have offered

substantial insight into the physiology of the muscle spindle, they are limited by their

abstraction from the physical world. This barrier limits their ability to rigorously test

under strict adherence to all physical laws and to apply physically realistic experimental

inputs, e.g. limited bandwidth of stretch inputs. It also deprives them of the opportunity

to gain insights into the muscle spindle through physically implementing their theories

in hardware.

A number of researchers have recognized the potential of building models which span

that gap between idealized mathematical theory and the physical world. The models

these investigators have built implement hypotheses regarding biological mechanisms

on robotic hardware. The primary goal for this breed of biorobotics researchers is to

increase their understanding of biological mechanisms by testing the ability of their

proposed mechanisms to drive real systems replete with physical obstacles such as

friction and inertia. Spin-off applications, though, are inherent to the nature of such a

project. Biorobotic devices are attractive candidates for prosthetics as they are designed

to use the language of the body to replicate its behavior. These devices also offer novel

mechanisms for engineering applications.

The robotic muscle spindle project was thus conceived with the following objectives:

(a) implementing a state-of-the-art structural model in precision robotic hardware and

(b) testing the biological theories which drive the model by rigorously validating the

model’s behavior against biological data from a wide range of experimental protocols.

39


4.2.1 Prior Literature

4.2.1.1 Biological Muscle Spindles

The mammalian muscle spindle, Figure

4.1, is a mechanoreceptor that resides in

the body of extrafusal muscle and

transduces muscle length. Intrafusal

muscle fibers span the length of the

spindle and are divided anatomically and

functionally into a sensory region and a

contractile region, which lie in series. The

contractile region is a muscle fiber

aligned to generate tension along the long

axis of the spindle. The sensory region

is a linearly elastic spring devoid of

contractile tissue. Group Ia afferent

neurons wrap around the sensory region,

linearly transducing sensory region strain

into receptor potential. This analog

potential is then encoded into an action

potential train, the Ia response, whose

frequency is thought to be a function of

the receptor potential and its first

derivative[5, 14]. This frequency

modulated spike train then travels down

the Ia axon to the spinal cord. There are

three types of intrafusal fibers: static nuclear bag and nuclear chain fibers transduce

primarily position information while dynamic nuclear bag fibers transduce primarily

velocity information. Commands from the γ motorneuron (γmn) descend from the

Static NuclearBag Fiber

DynamicNuclear Bag

Fiber

Nuclear ChainFibers

Capsule

IaNeuronOutput

Gamma MotorNeuron Input

Figure 4.1: Mammalian muscle spindle. Strain applied across the organ is transduced into primary (Ia) afferent output. Input from the γγγγmn contracts intrafusal fiber tissue at distal ends, modulating the Ia response.

40


spinal cord and control the contraction of the intrafusal muscle. Two types of γ

motorneurons exist, static and dynamic, which innervate position sensitive fibers and

velocity sensitive fibers, respectively.

B. H. C. Matthews first proposed in the 1930’s that the position and velocity sensitivity

of the muscle spindle could arise from the differing mechanical properties between the

intrafusal muscle and the sensory region[6]. Studies using stroboscopic

photomicroscopy [7, 10, 58] and force transducers [18, 20] to study intrafusal fibers

support this hypothesis, which forms the foundation of the structural model presented in

this paper.

4.2.1.2 Modeling

Researchers have been developing models of the muscle spindle for decades. A large

number of linear models have been developed, but exhibit limited ranges due to the

spindle’s nonlinear behavior[22, 23]. Empirical nonlinear models [25] are in common

use in large neuromuscular models[34] due to their computational simplicity and

broader range. Structural nonlinear models, though computationally intensive, offer a

unique opportunity in that specific model behaviors can be correlated to analogous

physiological mechanisms. A small number of these models have been published

describing all [27-29] or part[31, 35] of the muscle spindle. One such model, the

Schaafsma model[28], was built upon the widely held theory that complex spindle

behavior arises from mechanical interaction between the intrafusal muscle tissue and

the sensory region. It models the primary (Ia) response of a dynamic (bag1) fiber and

static (bag2 and nuclear chain) fiber. Each fiber consists of a linear elastic sensory

region in series with a contractile region. The primary afferent output is computed as a

function of sensory region length and its first derivative. Our robotic muscle spindle

model incorporates parts of the Schaafsma model for aspects of spindle behavior not

intrinsic to the mechatronic design.

41


4.2.1.3 Robotics

Biorobotic devices are being developed to replicate a variety of aspects of the peripheral

motor control system. Projects include an analog VLSI based model of motorneuron

pools[59], a robotic replica of the upper arm[60, 61] and pneumatic artificial

muscles[60, 62]. The robotic muscle spindle project, initiated by Marbot and

Hannaford[38], presents the first biorobotic model of a muscle spindle. This device

offers the precision engineering and validation required for using it both as a platform

for further spindle research and as a robust peripheral element in higher level biorobotic

models.

4.2.2 Approach

This article describes the design and performance of a biorobotic, structural muscle

spindle model in which the biological behavior is captured through both the

performance characteristics of mechatronic hardware and the modeling algorithms of

the control software. In the Methods section we describe the design and

implementation process by which we integrated three robotic subsystems into a

structural model of the muscle spindle. Technical engineering details of the robotic

subsystem design, implementation and performance are described elsewhere[63].

The tuning and validation process was divided into two independent stages. First we

tuned the model parameters against five data sets obtained from the literature describing

the cat muscle spindle’s response to a ramp and hold position input. The performance

of the robotic muscle spindle in each of these tuning studies is presented in the first half

of the Results section. We then validated the fully tuned robotic muscle spindle against

five additional experiments also obtained from the cat muscle spindle literature. These

validation studies are presented in the second half of the Results section.

In the Discussion section we evaluate the model’s successes and limitations as revealed

by the tuning and validation studies. We also comment on the significance of the model

42


including use of the biorobotic modeling technique and potential contributions to

biological theory raised through the modeling process.

4.3 Methods

4.3.1 Design

4.3.1.1 Conceptual Design

4.3.1.1.1 Modeling Approach

In conceptualizing the robotic muscle spindle, we abstracted three core functions from

physiological behaviors intrinsic to the muscle spindle for hardware implementation: (a)

the mechanical filtering produced by intrafusal muscle contractile tissue, (b) the neural

transduction from strain to receptor potential, and (c) the encoding of receptor potential

as an action potential spike train. The medium for implementing each of these

functions was selected from the repertoire of available engineering technology using the

selection criteria that it must (a) meet performance specifications derived from

biological studies on the analogous physiological system, and (b) be miniature enough

to viably mount the full robotic muscle spindle in parallel to a human biceps muscle.

Once the technologies were selected, the specific robotic systems were designed and

implemented to capture as much of the physiological functionality as possible in the

mechanical and electrical behavior of the hardware itself. Aspects of the muscle

spindle’s behavior not intrinsic to the electrical and mechanical design were

implemented in control software using an adaptation of the structural mathematical

model developed by Schaafsma et al.[28].

4.3.1.1.2 Model Framework

The conceptual framework for the model consists of a contractile element in series with

a linear elastic sensory element. External position inputs are applied as a strain across

the whole system. The strain is then unequally distributed between the contractile

element and the linear elastic sensory region. The contractile element’s force

43


production is a complex function of its length, velocity and contraction level, while the

sensory element’s force production is a simple linear function of length. The resulting

instantaneous variations in the mechanical properties of the two elements result in the

mechanical filtering behavior of the muscle spindle in which the strain across the

sensory region is different from that applied across the whole muscle spindle.

The model output is then generated as a function of the sensory region strain. The

receptor potential of the muscle spindle model is calculated as a linear function of strain

across the sensory element. This reproduces the neural transduction function of the

muscle spindle. Finally, the model’s output signal, Ia firing frequency, is calculated as

a function of the receptor potential and the receptor potential’s first derivative, thereby

reproducing the muscle spindle encoder function.

The robotic muscle spindle models two fiber types: dynamic and static. These fibers

receive their sole efferent input from the dynamic and static γmn, respectively. Further,

their parameter values model the analogous intrafusal fiber: the dynamic nuclear bag

and a hybrid of the static nuclear bag and nuclear chain fiber, respectively.

4.3.1.2 Design Implementation

4.3.1.2.1 Sensory Element Model

We used published data from the experimental muscle spindle literature to create

performance specifications for the sensory element. These specifications include: (a)

absolute deflection amplitudes greater than 0.24 mm, suitable for a maximum 3:1 scale

model of sensory region deflection, (b) resolution better than 20 µm[4], and (c) a linear

response region at low deflection levels, followed by stiffening and decreasing

sensitivity at increasing amplitudes[4].

44


The resulting design, Figure 4.2, is a

pair of strain-gaged cantilevers. The

base of the cantilevers is rigidly

mounted to a nut that defines the

interface between the contractile

element and the sensory element.

The cantilevers are connected directly

to a pair of cables that provide

external strain inputs across the full

length of the robotic muscle spindle.

Strain between the cable insertion and

the cantilever base is transduced by

electronic circuitry into a millivolt

potential. This millivolt potential, representing the strain across the sensory region, is

then converted into a frequency-modulated spike train and transmitted to the computer

as the output of the sensory element. A description of engineering aspects of this

robotic length sensor is given in Jaax et al.[63]. By successfully meeting all of the

biologically-derived design specifications, this robotic sensor is able to reproduce the

strain-to-millivolt-potential transduction behavior of the sensory element directly in the

mechatronic hardware.

Functionally, the output of the sensory element serves a dual role in the muscle spindle

model. First, it represents the receptor potential that is used to calculate the muscle

spindle output. Secondly, it provides sensory information for the feedback control

algorithm that drives the contractile element. This second role will be addressed in the

Linear Actuator Control Algorithm section below.

4.3.1.2.2 Intrafusal Muscle Model

The contractile element in the model’s conceptual framework is implemented using a

5 MM

CANTILEVER

STRAIN GAGE

CABLE

Figure 4.2: CAD drawing of sensory element design. Displacement of cable with respect to cantilever base causes bending. Strain gages mounted to cantilevers transduce bending into millivolt potential change analogous to Ia receptor potential.

45


linear actuator. Muscle-like behavior is produced in the linear actuator by means of the

software algorithm controlling the actuator. Hence, the primary performance

requirement for this device is that it respond to the software controller’s commands

rapidly enough to reproduce the experimentally measured dynamics of intrafusal muscle

tissue. We used published experimental data to identify the following biologically-

motivated performance specifications: (a) a rise time for a 30mm/sec ramp stretch of 22

msec, based on optical measurements of the kinematics of intrafusal motion[4, 16] and

(b) a maximum position error of 0.3 mm during the fastest experimental trajectory, the

30mm/sec ramp and hold. The latter specification arises from the need to keep the

sensory element from exceeding its maximum deflection. We used these two

specifications to identify specific engineering design criteria and design an actuator and

controller that met the required performance specifications. The biologically-motivated

performance specifications were successfully met with the following performance

metrics: (a) the 0-90% rise time on a 30 mm/sec ramp is 21 msec and (b) the maximum

position error is 0.15 mm on a 30 mm/sec ramp. Engineering aspects of the resulting

design, Figure 4.3, are described in detail in Jaax et al.[63].

A software-based control algorithm

supplies the muscle-like behavior to

the lead screw linear actuator. A

computational muscle model

calculates the force that should be

present across the contractile element,

Fd, based on its length, velocity and γ

motorneuron firing frequency. The

sensory element measures the actual

force across it, Fa. The difference between these two forces, Fd - Fa is then used as the

error signal, E, to control the linear actuator. The computational muscle model is

described in further detail in the Mathematical Muscle Model section below. Muscle

spindle modeling aspects of the control algorithm are covered in the Linear Actuator

10 mm

MOTOR NUT

LEAD SCREW

Figure 4.3: CAD drawing of linear actuator design. Motor rotates threaded rod, driving linear travel of nut. Muscle model in the control algorithm (see text) generates muscle-like response to length and γγγγmn inputs. (top housing removed for visibility)

46


Control Algorithm section below. Engineering aspects of the position controller, which

ultimately controls the linear actuator, are described elsewhere[63].

4.3.1.2.2.1 Mathematical Muscle Model

The muscle model algorithm is adapted from an extrafusal muscle fiber model

developed by Otten[30]. It calculates force as a function of velocity, length, and γ-

motorneuron input level. In developing their mathematical muscle spindle model,

Schaafsma et al.[28] retuned the 10 parameters of Otten’s extrafusal fiber to match

intrafusal fiber dynamics by using experimental muscle spindle data as the optimization

target. In implementing this algorithm as our muscle model, we used the structure of

Otten’s muscle fiber model combined with the ten parameter values in the Schaafsma

model. The resulting equation for intrafusal force is:

≤>

+++=0,0

0,,,,,,,,

i

ieiaiiipipiqiviaia v

vFkvbFkFFFkF ( 4.1 )

where i is fiber type (1=dynamic bag1, 2= static bag2 ), ka,i and kp,i are maximum

active and passive isometric force, respectively, Fa,i is active force generated at current

length (normalized), Fv,i is active force generated at current velocity (normalized), Fq,i

is active force generated by gamma stimulation rate (normalized), Fp,i is passive force

generated at current length (normalized), bi is passive damping, vi is velocity of

contractile region, and Fe is force enhancement.

Equations defining Fa, Fv, Fq, Fp are in Otten’s muscle model[30]. Parameters were

freed and tuned when justifiable on either biological grounds or due to subsumption of

the behavior into the mechatronic device. Details regarding the new parameter values

and their justifications are included in the Results and Discussion sections.

47


4.3.1.2.2.2 Linear Actuator Control Algorithm

Figure 4.4 is a block diagram describing the algorithm used to control the linear

actuator. The force error signal, E, drives the position of the linear actuator. Force

errors arise from three sources: (a) updates to Fd, the desired force, calculated by the

muscle model, (b) updates to C, the external position input, and (c) the dynamics of the

control loop. In case (a), the desired muscle force, Fd, calculated by the mathematical

muscle model, serves to maintain continuous strain across the sensory region, adjusting

its magnitude up and down as the mathematical muscle model’s force calculation

varies. In case (b), the external position input, C, maintains a continuous stretch across

the whole spindle equivalent to the input C. As the magnitude of the position input, C,

changes, that instantaneous change, ∆C, is transmitted directly to the position controller

causing the nut to move an identical distance. Finally, in case (c), the dynamics of the

closed loop controller results in transient force errors as the negative feedback loop

works to keep the actual force, Fa, close to the desired force, Fd. A linear scaling factor

was used to tune the magnitude of the muscle model force output, Fd, to the stiffness of

the sensory region to reproduce the sensory region displacements seen in the biological

literature.

Mathe-matical Muscle Model

1/kPhysical

Plant

External position input, C

k Convert displacement

to force

Sensory Region Force, Fa

Sensory element strain, ε

Position Controller

Force Error,

E

Desired Force,

Fd

Convert force to displacement

Desired Position, x

Nut position, B γ-mn input

Compute Ia output

Ia

H Feedback

Linearization

+ - +

- Nut

Position, B + -

Controller

ENCODING TRANS-

DUCTION

Figure 4.4: Block diagram of linear actuator controller. Algorithm compares actual force, FA, to force predicted by muscle model, FD. The difference, E, is used as error signal to drive linear actuator position. E arises from three sources: updates to FD from muscle model, external position inputs (∆∆∆∆C), and dynamics of control loop.

48


4.3.1.2.3 Encoder Model

The function of the encoder, translating the output of sensory transducer into a

biologically accurate Ia action potential frequency, is accomplished in two stages. The

first stage, conversion from millivolt receptor potential to a frequency modulated spike

train, is done onboard the spindle itself to minimize distortion of the signal. The

circuitry design used to accomplish this conversion was designed by Marbot [37]. The

raw frequency signal is transmitted in the range of 1kHz-11kHz to maximize resolution

and then rescaled in the computer. The second stage uses the algorithm adapted from

Schaafsma et al.[28] to convert the raw sensory element output into a Ia signal:

iii dltpP ×= ( 4.2 )

iii PhPptrIa ×+×= ( 4.3 )

where Pi is receptor potential, ltpi is the conversion from sensory region length to

potential, di is the displacement of the sensory region beyond the zero firing length, the

length at which there is no mechanical contribution to the receptor potential in the

passive muscle spindle, ptr is the conversion from receptor potential to Ia firing rate, h

is rate sensitivity of encoding from receptor potential to firing rate, and Iai is the firing

rate of muscle spindle Ia afferent.

A 2nd order filter with a cutoff frequency of 20 Hz was implemented on the first

derivative of sensory element strain to minimize propagation of noise extraneous to the

experimental protocol[64]. The 20 Hz cutoff frequency was selected based on Fourier

analysis of the Ia signal that revealed a significant noise source in the motion of the

linear actuator mechanism at frequencies just above 20 Hz. This choice is in agreement

with the opinion stated by PBC Matthews that “frequencies above 20 Hz were not really

relevant for motor control[65].” Given that we are not examining external vibration

protocols, frequencies in excess of 20 Hz are unlikely to be due to the physiology we

are examining.

49


4.3.2 Experimental Methods

4.3.2.1 Linear Positioning Device

A linear positioning device (LPD) was designed and built to apply position inputs to the

robotic muscle spindle in a manner analogous to that used in experimental muscle

spindle studies[63]. This device has a stroke length of 19 mm, sufficient to allow a

maximum 3:1 scaling of the amplitudes used in the majority of the muscle spindle

literature[10, 58]. The resolution of the LPD’s length sensor is 0.33µm. This is within

0.1µm of the highest resolution length data available in the muscle spindle literature[10,

26, 58]. Using a 2:1 scale in our robotic muscle spindle, the resolution of the LPD

length sensor is greater than the highest resolution length data in the muscle spindle

literature.

4.3.2.2 Experimental Protocols

4.3.2.2.1 Implementing Biological Experimental Protocols

In experiments where we reproduced biological experiments, close attention was paid to

accurately implementing the biological position trajectories. In the case of trajectory

amplitude, physiologists often report stretch amplitudes in terms of the displacement

applied across the entire host muscle body. When this is the case, we assume that this

stretch is proportionally transmitted to the muscle spindle without distortion, and thus

apply the appropriate linear scaling factor to the reported amplitude. The initial spindle

length for an experimental protocol was selected by experimentally identifying the

robotic spindle length at which there was optimal correspondence between the

magnitude of the biological and robotic Ia response across multiple γmn activation

levels. These lengths are reported along with the initial length used in the biological

experiment, if available.

50


4.3.2.2.2 Scaling

For the purpose of scaling cat soleus and tenuissimus displacements into muscle spindle

strain, we define the optimal spindle length of the biological muscle spindles modeled

here as 11.5 mm, identical to the optimal spindle length in the Schaafsma model[28].

The robotic muscle spindle is a 2:1 scale model of such a biological muscle spindle,

giving it an optimal length of 23 mm. Zero length, the length at which the mechanical

effect on receptor potential is zero in the passive muscle spindle, is set at 10 mm in the

biological muscle spindle[28] and 20 mm in the robotic muscle spindle. The muscle

fiber length of the cat soleus is 42.6mm[66].

4.4 Results

Tuning and validation of the model against data from the muscle spindle literature was

performed in two independent stages. In the first stage we tuned model parameters to

five metrics from the muscle spindle literature describing the muscle spindle’s ramp and

hold response: mean Ia output during ramp, peak Ia output, dynamic index, time domain

response of Ia output, and time domain response of the physical stretching of the

sensory region. The results of this process are presented in the first half of the Results

section. In the second stage we validated the fully tuned model against five additional

experiments from the muscle spindle literature including experimental protocols and

results not used in the tuning studies. The results of these validation studies are

presented in the second half of the Results section.

4.4.1 Model Tuning Studies

This section shows the degree of similarity achieved between robotic and biological

results by tuning the model parameters to replicate these specific sets of biological data

from the muscle spindle literature. The majority of the model parameters retain the

values originally identified by Schaafsma et al.[28]. Changes from these parameter

values, Table 4.1, were justified by one of two reasons: (a) the behavior was subsumed

by the mechatronics of the robotic muscle spindle or (b) there is a biologically-based

51


reason for the new value. Further details on specific changes are included in the

Discussion section.

Table 4.1: Parameter values changed during tuning of robotic muscle spindle

Name New Value Function

Biologically Motivated K2 .4 static F-v slope h

≤>

0,pps(mV/s)00,pps(mV/s)15

1-

-1

i

i

PP

encoder rate sensitivity

Mechatronically Motivated Fx 0 FU cross-bridge rupture Fe 0 force enhancement b1 8.6x10-4 FU(mm/s)-1 bag1 passive damping b2 4.6x10-4 FU(mm/s)-1 bag2 passive damping

4.4.1.1 Ramp and Hold: Ia Metrics

Optimization of the robotic muscle spindle’s parameters focused primarily on

reproducing three metrics reported by Crowe and Matthews[67] for a biological muscle

spindle given ramp and hold position inputs: mean, peak and dynamic index. Dynamic

index is defined as the change in the Ia output between the end of the ramp and 0.5

seconds after the ramp. Figure 4.5 shows the results of this process overlaid on the

original biological data. The plots present the metrics as a function of ramp velocity as

well as γmn activation level. Figure 4.5a&b depict the mean and peak Ia response

during the ramp, respectively. Figure 4.5c depicts the dynamic index of the Ia response.

The biological metrics from the muscle spindle literature were reported as the

“approximate average for several spikes[67].” In an effort to reproduce this

methodology, we applied a 2nd order 7 Hz low pass filter to the robotic Ia output before

calculating the metrics.

52


At 5 mm/s, the mean difference

between the robotic and biological

metrics is –4.7 Hz with a standard

deviation of 12.3Hz. Across all ramp

speeds, the mean difference is 1.1 Hz

with a standard deviation of 19.7 Hz.

The few notable discrepancies occur

at high velocities. The static robotic

muscle spindle exhibits greater

velocity dependency than the

biological muscle spindle,

demonstrated by the increased peak

and dynamic index metrics at high

velocities. Also, the mean response

of the passive robotic muscle spindle

is less than its biological counterpart

at high velocities.

4.4.1.2 Ramp and Hold: Ia Time Domain

Time domain plots of the muscle spindle’s Ia response to a ramp and hold stimulus

allow its characteristic morphology to be observed and tuned. Responses to a 5mm/s

ramp and hold were overlaid in Figure 4.6 to show how closely the fully tuned robotic

model (black) matches Crowe and Matthew’s biological data from an identical

stimulus[67] (grey). Note that in the original biological data the x-sweep rate of the

recording oscilloscope was a linear function of the muscle spindle position input[67].

Accordingly, the time scale of the x-axis only applies to the hold region. We plotted the

robotic muscle spindle data with a similar x-axis distortion during the ramp (solid bar)

to allow direct comparison of the results. These data, as with all time domain Ia

response plots in this article, are filtered with a 2nd order 60 Hz low pass filter.

0 10 20 300

100

200

300

Meana.)

Ia O

utpu

t (Hz

)

0 10 20 300

100

200

300

Peakb.)

Velocity (mm/sec)0 10 20 30

0

100

200

300

Dynamic Index c.)

Figure 4.5: Model parameter tuning study. Ia output metrics during ramp and hold experiment: (a) mean response during ramp input, (b) peak response, (c) dynamic index (see text). Robotic muscle spindle response (markers with lines) closely matched cat soleus data (markers without lines, Crowe et al.[67]) for different levels of γγγγmn stimulation (‘+,’ 100 Hz dynamic, 0 Hz static, “*,” 0 Hz dynamic, 100 Hz static, “o,” 0 Hz dynamic, 0 Hz static). Displacements refer to biological host muscle. Final length in biological tissue (max. physiologic length) similar to robotic muscle spindle (24.5 mm).

53


At all γmn activation levels, the

robotic muscle spindle model

replicates the major elements of the

biological muscle spindle Ia

response. First, the accuracy of the

gain between position and Ia output

is evident in (a) the slope of the Ia

response during the ramp and (b) the

magnitude of the Ia response during

the hold. Second, the accuracy of the

gain between velocity and Ia output

is demonstrated by the offset of the

Ia response during the ramp period at

all three γmn activation levels.

4.4.1.3 Ramp and Hold: Sensory

Region Stretching

In building a structural physical model, one of our goals was to accurately reproduce the

mechanical deformations of the two regions of the muscle spindle. The ramp and hold

tuning study in Figure 4.7a depicts the displacement of the sensory region of the robotic

muscle spindle. Figure 4.7c presents for comparison data from Dickson et al.[10]

showing the displacement of a point in a biological muscle spindle 0.3 mm from the

spindle equator, just lateral to the junction between the sensory region and the intrafusal

muscle. Note that since the robotic muscle spindle is a 2x scale model, the actual robot

displacements are 2x the values presented here.

Figure 4.6: Model parameter tuning study. Comparison of Ia responses (top graph) during ramp and hold input (bottom graph). Robotic muscle spindle response (black) closely reproduces cat soleus muscle spindle response (gray, Crowe et al.[67]) under varying γγγγmn stimulation levels ((a) 0 Hz dynamic, 0 Hz static (b) 70 Hz dynamic, 0 Hz static (c) 0 dynamic, 70 Hz static). Solid bar indicates region where x axis is a function of position input, not time. See text for details. Lengths refer to displacements of host muscle. Final length in biological tissue (max physiological length) similar to robotic muscle spindle (24.5 mm).

54


The peak displacement of the robotic

and biological data match with a

value of 21 µm. Further, the ratio

between the peak passive and peak

dynamic response is similar between

the two data sets, with the robotic

muscle spindle exhibiting a slightly

larger peak passive response. Both

cases also exhibit a slow decrease of

the sensory region strain at the end of

ramp, although the time constant for

the robotic spindle is much faster

than the biological spindle showing

we do not fully replicate the slow

decay behavior. Finally, between 0

and 150 msec, the displacement of

both the passive and dynamic sensory

regions show an initial burst spike

typical of short-range stiffness.

These spikes are qualitatively similar,

though the robotic spindle’s initial

burst exhibits a steeper rising slope

than the biological spindle. As the

robotic spindle’s spike behavior is

the result of transmitting 100% of the

whole spindle’s displacement to the sensory region, this suggests that in the biological

spindle some displacement does occur across the contractile region during the initial

burst. This mechanism would also explain why the peak occurs later in the biological

spindle. If the applied strain is being absorbed by both the sensory and the contractile

Figure 4.7: Model parameter tuning study. Sensory region stretch during ramp and hold stretch applied across whole muscle spindle. (a) Robotic muscle spindle sensory region stretch, (b) Input displacement applied across whole muscle spindle, (c) Displacement of cat tenuissimus muscle spindle tissue 0.3 mm from spindle equator, just beyond sensory region (Dickson et al.[10]). For all graphs, Left column: 0 Hz dynamic, 0 Hz static γγγγmn stimulation (passive), Right column: 100 Hz dynamic, 0 Hz static γγγγmn stimulation (dynamic). Range and shape of sensory region displacement closely matches biological data. Lengths refer to displacements applied directly to biological muscle spindle. Final length in biological tissue not available to compare to robotic spindle length (24 mm).

55


region, it will take more time for the ramp position input in this experiment to apply

enough strain to achieve the displacement of the sensory region and associated force

necessary to rupture the actin-myosin crossbridges and end the initial burst.

4.4.2 Model Validation Studies

Once the robotic muscle spindle

was completed and tuned, we

validated its performance by

comparing its behavior to a

different set of five experiments

obtained from the muscle spindle

literature. No parameter values in

the robotic muscle spindle were

adjusted while performing this set

of validation studies.

4.4.2.1 Ramp and Hold

The first experiment compares the

ramp and hold response of the

robotic muscle spindle to

biological data from Boyd et

al.[68] (Figure 4.8). In both the

dynamic and static cases, the

morphology of the robotic muscle

spindle response shows a close

correspondence to the biological

data. In the passive case the

morphology is still similar, although an unusually large initial gain in the biological data

results in a large positive 45% offset that is not present in the robotic data. The

Figure 4.8: Completed model validation study (cf. Boyd et al. 1977[68]). Comparison of Ia response to ramp and hold position input (bottom row) Parameters tuned with data from Crowe et al.[67] (Figure 4.5 and Figure 4.6) and Dickson et al.[10] (Figure 4.7) applied to data from Boyd et al. Normalized robotic muscle spindle response (top row) very closely matches normalized dynamic and static response of cat tenuissimus muscle spindle (middle row), although amplitude of passive is small. γγγγmn stimulation levels: Left column: 0 Hz dynamic, 0 Hz static (passive), Center column: 100 Hz dynamic, 0 Hz static (dynamic), Right Column: 0 Hz dynamic, 100 Hz static (static). All Ia responses normalized to maximum depth of modulation in dynamic response of respective spindle, robotic or biological. Positions refer to deformations applied to host muscle. Final length data for biological muscle spindle not available to compare to robotic muscle spindle (24.4mm).

56


morphological similarities between the robotic and biological data include the position

dependency, velocity dependency and initial burst. The position dependency similarity

can be seen both in the slope of the ramps and the final value of the hold after the

transients have dissipated. The velocity dependency similarity is best seen in the

similarities between the robotic and biological offsets during the ramp. Both the time

course and magnitude of initial burst phenomenon are mimicked nicely in the static and

dynamic robotic muscle spindle data, with the passive robotic data showing an initial

burst with a slightly faster time course. Note that the data are presented with their

scales normalized to the full depth of modulation of that muscle spindle’s dynamic

response, robotic or biological, with zero set as the minimum Ia value in each individual

response. This was done to allow comparison of the morphology despite substantial

differences in the scale of the two responses. Our robotic muscle spindle had a range of

200 Hz in this study while the biological muscle spindle range was only 48 Hz.

The robotic muscle spindle ramp and hold response also matched data from P.B.C.

Matthews[69], but the normalization of Figure 4.8 was not required. The major

discrepancy in the two data sets was a small velocity gain in the robotic muscle

spindle’s passive and dynamic data sets, which results in a smaller offset during the

ramp phase of the robotic passive and dynamic response.

4.4.2.2 Sinusoidal Stretch Experiments

During a 2 mm peak-to-peak amplitude, 1 Hz sinusoidal input, the robotic muscle

spindle’s time domain Ia response closely matched data from Hulliger et al.[57] under

passive, maximal dynamic and maximal static γmn activation. Similarities included a

phase lead of approximately 80° across all γmn activation levels, dynamic γmn input

generating the maximum Ia depth of modulation, and zero Ia output in the passive

muscle spindle at lengths less than the “zero length.” Scaling of the robotic passive

response, though, was notably smaller than the biological response.

57


The fourth validation study, the effect of the amplitude of a sinusoidal position input on

the depth of modulation of the muscle spindle’s Ia response, is shown in Figure 4.9.

The lines for the static and dynamic

robotic output are very close to the

biological behavior reported by

Hulliger et al.[57]. Further, the

robot’s static and dynamic slopes

exhibit the gain compression

phenomenon. There is a steep linear

relationship between sinusoid

amplitude and Ia response at small

amplitudes, which then abruptly

decreases and stabilizes at a

shallower slope at higher sinusoid

amplitudes. In the passive case,

however, the robotic muscle spindle

output is much smaller than its

biological counterpart. To test the

origin of this, a sensitivity analysis

was done on the passive damping

parameter, b1, which had been

reduced from 9.91x10-3 to 8.6x10-4 FU (mm/s)-1 due to the intrinsic damping of the

mechatronics. Restoring this parameter to its original value only increased the

amplitude of the passive response by 5-8 Hz.

Figure 4.9: Completed model validation study (cf. Hulliger et al. 1977[57]). Comparison of depth of modulation of Ia output in response to varying amplitude of sinusoidal stretch input. Robotic muscle spindle data (dashed lines) closely matches cat soleus muscle spindle data (solid lines) during dynamic γγγγmn (“+”, 100 Hz dynamic, 0 Hz static) and static γγγγmn (“o”, 0 Hz dynamic, 100 Hz static) stimulation, while the passive response (“*”, 0Hz dynamic, 0 Hz static) is about 25% of experimental amplitude. Amplitudes refer to displacement of the host muscle. Mean length of biological spindle (1-2 mm less than physiological max) similar to robotic spindle (22 mm).

58


4.4.2.3 γ Motorneuron Performance

The final validation experiment, Figure 4.10, shows the effect of varying γmn stimulus

amplitude on the mean Ia output of the robotic and biological muscle spindle. The

robotic Ia response matches the

biological data reported by

Hulliger[70] nicely under both static

and dynamic γmn stimulation.

Indeed, all values for the robotic

muscle spindle response lie within

the standard deviation bars for the

biological experiment, which was

performed on 28 static and 20

dynamic γ mn axons[70]. The slopes

in both cases are extremely similar to

their biological counterparts, with

only a 10 Hz offset. Finally, the

saturation point to γmn input

corresponds well at approximately

100Hz. These data were collected

with the robotic muscle spindle held

at the same length for the static and

dynamic tests, 23.5 mm, reproducing

the length constraint from the biological experiment.

4.5 Discussion

This biorobotic model of the muscle spindle tests the spindle mechanism theories which

comprise it by quantitatively assessing their performance in a novel testbed, a physical

model built in robotic hardware. Further, testing and validating the model against

Figure 4.10: Completed model validation study (cf. Hulliger 1979[70]). Comparison of effect of varying γγγγmn stimulation level on Ia response. Robotic muscle spindle data (dotted lines) matches slope and saturation point of cat soleus muscle spindle response (solid lines, error bars and shading indicate std. dev.) under two different types of γγγγmn stimulation (dynamic “+” and static “*”). Muscle spindle held at constant length throughout all experiments. Biological muscle spindle length (2 mm less than physiological max) similar to robotic muscle spindle (22.5 mm). Note, robotic data exactly overlap biological data if inequality allowed between static length (23mm) and dynamic length (22mm).

59


biological data from numerous experimental protocols across multiple authors has

challenged the universality of its structure. Working to replicate these data from the

literature has given us insight into the sources of the model’s limitations and their

implications. These processes have collectively spawned new hypotheses regarding

spindle function and physiology.

We will first discuss the model tuning, examining which parameters were tuned and

why, as well as its successes and limitations. We will then evaluate the validation

studies for the model’s ability to capture key elements of muscle spindle behavior in a

more general context. Finally, we will conclude by presenting hypotheses about muscle

spindle function generated through the development and validation of this model.

4.5.1 Model Tuning

The initial parameters of the model included six determined by the mechatronics of the

system[63] and twelve intrafusal muscle model parameters, ten of which were identified

by Schaafsma et al.[28] and two of which arise from Otten’s original extrafusal muscle

model[30]. Using this initial parameter set, we compared the model’s performance

against five biological metrics characterizing the ramp and hold response: peak Ia

output, mean Ia output during ramp, dynamic index, Ia response in the time domain, and

sensory region displacement. When discrepancies arose, the responsible parameter was

identified and evaluated according to the following criteria: (a) was there evidence in

the physiology or anatomy of the biological muscle spindle to support changing the

parameter value, and (b) was this parameter duplicated in the mechatronics and the

mathematical muscle model? If either criterion was met, the parameter was freed and

tuned accordingly.

4.5.1.1 Mechatronically Motivated Parameter Changes

The software model of short-range stiffness used in the Schaafsma model was the first

term modified due to subsumption into the mechatronics. Fx, a parameter controlling

60


the force threshold above which a single cross-bridge will rupture, was set to zero,

thereby eliminating the short-range stiffness algorithm. We instead modeled it with a

physically analogous mechanism: stiction. In the biological muscle spindle, short-range

stiffness is thought to arise from persistence of bound cross-bridges until a force large

enough to rupture the bonds is placed across the muscle spindle[71]. In our linear

actuator, short-range stiffness arises from the persistence of a surface bond between the

nut and lead screw until a force large enough to rupture the bond is placed across the

robotic muscle spindle. In the active robotic spindle, approximately 33µm of whole

spindle stretch is required to generate a force error signal, E, (Figure 4.4) large enough

to break the surface bond in the linear actuator. This corresponds to 0.15% strain across

the whole spindle, compared to the 0.3% whole spindle strain at which cross bridges are

thought to rupture in the biological muscle spindle[72]. The success of this physical

model in producing an initial burst by transmitting initial displacements directly to the

sensory region is demonstrated by the sensory region displacement, Figure 4.7, and the

Ia response, Figure 4.8.

The second mechatronically motivated change was force enhancement, Fe, which had

been implemented in the muscle model in Eq. 4.1 as a discontinuous force offset term: a

positive constant in lengthening and zero in shortening. Schaafsma et al.[28] added Fe

to the Otten muscle model[30] while tuning the model for intrafusal muscles. We again

removed Fe because the discontinuity introduces significant instability into closed loop

control systems. Further, the effect of the force enhancement term is to increase the

magnitude of the force-velocity term on lengthening, which in the dynamic fiber is

already near maximum. Hence, we omitted this property from our muscle model and

have accounted for its effects elsewhere.

The muscle model’s passive damping term, bi, from Eq. 4.1 was the final change caused

by subsumption by the mechatronics. Since the mechanical plant has intrinsic damping,

the passive damping model is redundant and we reduced its value accordingly.

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4.5.1.2 Biologically Motivated Parameter Changes

Biological motivations resulted in two changes: (a) the encoder rate sensitivity, h, was

increased in magnitude and made unidirectional and (b) the slope of the static fiber’s

force-velocity relationship was decreased.

Based on biological data[14], we raised the magnitude of the encoder rate sensitivity

term, h, in Eq. (4.3) to compensate for lack of “Force Enhancement,” Fe. Increasing the

magnitude of h revealed the need for a second change in h: unidirectional rate

sensitivity. In previous models this term has always been symmetrical, driving the Ia

output up or down as the receptor potential rose and fell[27, 28]. On raising the

magnitude of h, though, we observed that falling receptor potentials, e.g. ramp

cessation, led to large sustained non-physiological Ia undershoots. Experimentation

with our model revealed that eliminating h just during falling receptor potentials

allowed the Ia output to maintain its velocity-dependent offset during the ramp, while

eliminating the large non-physiological undershoots.

We found two studies in the biological literature with data to support this theory of

unidirectional rate sensitivity in the transfer function between receptor potential and Ia

frequency. Hunt and Ottoson[14] overlaid on top of an actual Ia response a theoretical

Ia response predicted as a linear function of receptor potential. The actual Ia response

was much greater than predicted during rising receptor potentials, but corresponded

well to the predicted value during falling receptor potentials. Fukami’s data showed

similar results for snake muscle spindles[73]. Hunt and Gladden also observed in their

reviews that Ia output during stretch is proportionally greater than the receptor potential

predicts[4, 5], although neither explicitly addressed Ia output during shortening. Based

on this evidence, we postulate that the encoder transfer function is:

≤>

×+×=0,00,15

,i

iii P

PhPhPptrIa ( 4.4 )

62


where ptr = potential to rate conversion factor, Pi = receptor potential, i = fiber type and

h = encoder rate sensitivity. Additional biological experiments could further test this

hypothesis by measuring the relationship between receptor potential and Ia output under

a wider range of experimental protocols than the ramp and hold studied by Hunt and

Ottoson [14]. Such experimentation could also be used to quantify the magnitude of the

encoder rate sensitivity, h.

The second biologically-motivated parameter change was K2, the slope of the static

(bag2) fiber force-velocity curve in Eq. 4.5 below. Due to differences between the bag2

and dynamic (bag1) fiber’s parameter values, removal of the Fe term had a much

smaller effect on the positive stretch sensitivity of the bag2 fiber than the bag1 fiber.

Further, the compensatory increase in the h term was tuned to the bag1 fiber. Thus, to

restore bag2 sensitivity, we needed a parameter to selectively decrease bag2 sensitivity

during stretch. The optimal choice was the bag2 fiber force-velocity relationship from

the original Otten muscle model[30]:

( )

<×−

+−−

≥×+

−

0,max/56.71

max/1)1(

0,)max/(1

max/1

F

22

222

22

2

v

vVKv

Vvee

vVKv

Vv

( 4.5 )

where: Fv is the force due to velocity, v is velocity, Vmax2 is the maximum bag2

velocity, e2 is maximum bag2 force due to velocity, and K2 is the slope of the bag2

force-velocity curve. We targeted this relationship for several reasons: (a) it is

biologically accurate to tune F-v of the static fiber independently of the dynamic fiber,

(b) its exact value for intrafusal muscle is still unknown and (c) the available evidence

suggests extremely low viscosity in the static fiber, e.g. fast myosin isoforms[4], driving

in the nuclear chain fiber[74], and extremely small dynamic indices[67]. Based on this

biological support we increased K2 from 0.25 to 0.4, lowering the slope of the static

force-velocity curve.

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4.5.1.3 Quality of Fit

The goal of the tuning process was to match the model’s output to five different

measures of the biological muscle spindle’s ramp and hold response. The first three

measures are metrics describing the accuracy of quantitative aspects of the Ia response,

Figure 4.5. The overall closeness of the match is very strong, particularly at 5 mm/s,

the slowest ramp speed. The dynamic and passive responses are quite accurate at all

speeds, reflecting a high quality of fit for the dynamic fiber, which generates both the

dynamic and passive response. At higher velocities the static muscle spindle exhibits

too great a dependence on velocity. This is because the intrinsic damping in the robotic

muscle spindle makes it difficult to replicate the static fiber’s extremely low velocity

gain at high velocities. Sources of damping in the static muscle model, b2 and K2, were

tuned to minimize the damping. A sensitivity analysis on b2, K2, and e2, the static force-

velocity curve’s maximum value, showed that further changes would not appreciably

lower the peak and dynamic index metrics. Hence, the static muscle spindle is slightly

over-dynamic at speeds greater than 15mm/s.

The time domain tuning studies demonstrate how well the qualitative features of the Ia

response were tuned. Figure 4.6 shows that the robotic muscle spindle’s Ia output

echoes the biological Ia output almost exactly at all three γmn input levels, indicating

that it successfully reproduces the qualitative aspects of the biological muscle spindle

response. These aspects include both position and velocity gain. Out of the force-

length relationship of the muscle model comes the dependence of the position gain on

γmn activation. The dependence of the velocity gain on γmn input, exhibited by the Ia

offset during ramps, arises from the muscle model’s force-velocity relationship.

The final tuning measure was physical displacement of the sensory region.

Physiologists have long thought that the mechanical filtering of the intrafusal muscle

generates much of the muscle spindle’s behavior[6]. Optical recordings support this

theory by demonstrating that aspects of the muscle spindle’s nonlinear Ia response are

64


present in the dynamic strain of the sensory region[10, 68]. Since this concept forms the

foundation of our structural model, we included this tuning study to (a) ensure that the

major Ia response features are present in the dynamics of the intrafusal muscle model

and (b) tune the range of the sensory region displacement to match the biological data.

Figure 4.7 shows our success in fitting the model to these requirements. The range of

displacements is very similar to the biological range for both the dynamic and passive

case. Further, these graphs show that we have reproduced in our intrafusal mechanics

most of the major features of the Ia response, including both the time course and

magnitude of the initial burst.

4.5.1.4 Muscle Length

In tuning the robotic muscle spindle to match the results of multiple biological

experiments, it quickly became apparent that the initial length at which the study is

performed is an important factor in replicating the Ia response. This phenomenon arises

from several factors. First, the muscle force-length relationship is markedly nonlinear,

meaning both the initial value and the position gain of the Ia response change with

length. Second, the passive muscle spindle’s position sensitivity increases significantly

as a function of length while the active muscle spindle’s position sensitivity exhibits a

slight decrease in the robotic muscle spindle and almost no variation with mean initial

length in the biological spindle. Finally, the passive muscle spindle has zero Ia

response below its zero firing length. To accommodate this, for each experiment we

repeated the experimental protocol at 5 different initial lengths throughout the robotic

muscle spindle’s working range. We then used the relative Ia amplitudes at each of the

γmn activation levels to determine which length best corresponded to the length of the

biological muscle spindle when the data were collected. These initial lengths are

reported in the figure captions along with the approximate initial lengths reported by the

biological investigators, when available.

65


4.5.2 Model Validation

To validate our model, we obtained Ia response data from five different experiments in

the muscle spindle literature. These experiments differ in several ways from the Crowe

and Matthews studies used for Ia tuning[67]. Four studies come from different authors,

introducing variation in experimental technique. Further, three studies are new types of

experiments: two examine sinusoidal response and one looks at fusimotor response. All

studies used cat spindles: four soleus and one tenuissimus.

The key to this validation was testing the fully tuned model under novel circumstances

to examine its general applicability. Absolutely no modifications to the robotic muscle

spindle were made while performing these studies. The only variable adjusted to get

the best match to specific studies was the initial length at which the experimental

protocol was applied.

4.5.2.1 Ramp and Hold Studies

The validation included two ramp and hold studies to test the robotic muscle spindle’s

response to data from different authors and γmn input levels. In our comparison with

the Boyd et al. study[68], the robot’s static and dynamic responses are similar to the

biological data (Figure 4.8). The passive data are qualitatively similar, but the

biological response exhibits a large positive offset. In our comparison with the

Matthews study[69] there was also qualitative similarity between the biological and

robotic data.

The data in Figure 4.8 were normalized due to range differences which we suspect

result from the fact that our model was tuned to muscle spindles with larger depths of

modulation than the muscle spindles used in Boyd et al.[68]. When we compare the

robotic muscle spindle’s behavior to data[69] from P. B. C. Matthews, the same author

who published the data used for tuning[67], we find that the robotic muscle spindle’s

range is quite accurate.

66


One aspect of the Boyd et al. passive data is atypical for passive muscle spindles. The

extreme steepness of the initial force-length (F-L) relationship is inconsistent with the

shallow F-L relationship at the end of the ramp, perhaps signaling the presence of

stretch activation. We present this study despite these unusual data, though, because it

nicely illustrates the short range stiffness phenomenon.

The robotic muscle spindle exhibited many characteristic features of biological muscle

spindle Ia response in these validation studies. The static and dynamic data in Figure

4.8 demonstrate nicely the robotic muscle spindle’s ability to mimic the position and

velocity dependency of the Ia response. Further, the initial burst phenomenon is well

illustrated in Figure 4.8. The data show that our mechatronic model of short range

stiffness works well in all three γmn input levels, reproducing both the magnitude and

the time course of the initial burst under static and dynamic γmn stimulation.

4.5.2.2 Sinusoidal Studies

Sinusoidal experiments test whether the robotic muscle spindle model is complete

enough to reproduce a range of muscle spindle behaviors beyond its tuning studies. The

model’s time domain sinusoidal response has good qualitative correspondence to the

biological response. The intrafusal fiber viscoelasticity is evident in the phase lead of

all three responses, as well as in the dynamic bag1 fiber’s large response. Further, the

robotic spindle successfully mimics the passive biological spindle’s zero firing length.

The scale of the robotic passive response, though, is smaller than the biological

response.

The second sinusoidal study (Figure 4.9) was included to test our modeling of the “gain

compression” phenomenon, another manifestation of short range stiffness. Biological

muscle spindles will exhibit a “linear range” with high position-Ia gains at small

amplitude stretches while the cross-bridges are still bound. At larger stretches the

cross-bridges rupture and there is a flattening of the curve to a new lower position-Ia

gain. The robotic muscle spindle data reproduce this behavior very nicely in both the

67


dynamic and static cases, not only matching the range of the depth of Ia modulation

extremely well, but also exhibiting a distinct linear range. The passive robotic data,

however, are much smaller than the biological data with an amplitude similar to the

Schaafsma model’s passive response.

In summary, the robotic muscle spindle’s sinusoidal response is good under γmn

activation, exhibiting phase lead, gain compression, and biologically plausible Ia

amplitudes. In the absence of γmn activation, the robotic muscle spindle’s response is

smaller than the physiological response. This behavior will be commented on below.

4.5.2.3 γ Motorneuron Study

The fusimotor validation study was performed to test the response of the robotic muscle

spindle to various frequencies of γmn stimulation (Figure 4.10). The model’s response

matches the slope, magnitude and saturation point of the biological response under both

types of γmn stimulation, static and dynamic. The graph also shows that both the

robotic and biological muscle spindles are more sensitive to variation in static than

dynamic γmn input, reflecting the steeper active force-length relationship of the static

fiber. This figure, combined with the success of the active γ mn cases in each of the

other validation studies, strongly supports the accuracy of the robotic muscle spindle in

replicating the behavior of the biological muscle spindle under active γmn inputs.

4.5.2.4 Limitations

Although in 5 of the 9 measures the robotic muscle spindle’s passive response matched

the biological response quite well, in the remaining four studies its amplitude was much

smaller than the biological response, representing the only major limitation of the

model’s general applicability. We identified three possible sources for this behavior:

failure to correctly identify the initial length, a missing term in the passive model and

stretch activation.

68


The initial length theory comes from the fact that the passive Ia position sensitivity rises

as a function of spindle length while the static and dynamic spindle’s position

sensitivity decreases (robotic) or increases only slightly (biological). If we performed

these studies at a longer length we would likely be able to replicate the relative

amplitudes of the passive, dynamic and static cases. The absolute magnitude of the Ia

response would then exceed the biological data, but such variability in scaling is

observed in the biological data[75]. This explanation is appealing since only four of the

nine passive experiments exhibited low output amplitudes.

An absent term in the passive muscle spindle model is the second possibility. Careful

examination of the passive sinusoidal time domain response suggests it has insufficient

phase lead, indicative of a missing damping term. We performed a sensitivity analysis

to test the effect such a term might have. Theoretical calculations, confirmed by

experimentation, showed that increasing passive damping by a factor of 10 only

increases the passive Ia depth of modulation by 5-8 Hz during a 1 mm sinusoid. Since

this change is so slight and would have equal effect in the active spindle, we concluded

that the passive damping term was not contributing substantially to the small passive

response.

Stretch activation is the final possibility. If the prediction is true, that the act of

stretching a passive intrafusal fiber can lead to contraction[10], this could account for

the four biological experiments whose passive Ia response amplitude we were unable to

replicate. Unequivocal evidence for this phenomenon has not yet been found. The one

study that reported visual evidence of intrafusal muscle shortening on stretch used a

grip technique that damaged the muscle spindle[16]. Further experiments using

simultaneous recording of intrafusal muscle length, tension and receptor potential

during passive stretching at different velocities may be able to establish whether stretch

activation truly exists and better characterize the kinematic properties of the passive

intrafusal muscle. Perhaps the studies in which the biological Ia amplitudes exceeded

our prediction, e.g. large sinusoidal position inputs, could be used as a guide for the

69


types of experimental protocols where unaccounted for behavior such as stretch

activation might occur.

4.5.3 Summary of Contributions

4.5.3.1 First Biorobotic Muscle Spindle Model

Our device and its prototype [37, 38] are the first muscle spindle models to be built

using the biorobotic modeling technique. This technique offers several unique

advantages over traditional software modeling including (a) rigorous adherence to all

physical laws, (b) insights gained through implementing concepts in physical hardware,

(c) the ability to apply realistic inputs directly to the model, (d) educational advantages

of having students physically interact with the model and (e) having a working device

upon completion of the project.

The biorobotic modeling technique significantly enhanced the results of the robotic

muscle spindle project in several respects. First, we realized that a discontinuous force

enhancement term results in an extremely difficult system to control, suggesting that the

biological system exhibits more continuous behavior than that described in the

Schaafsma intrafusal muscle model. Second, we gained insight into the bandwidth of

our model as well as the technology with which the biological tuning data were

collected through building in-house a Linear Positioning Device to apply position

inputs. Third, since our model is physically realized in robust robotic hardware, we can

install it on a robot or prosthetic. This feature is especially significant for researchers

developing biologically accurate biorobotic models of the stretch reflex.

4.5.3.2 Potential Applications to Biological Theory

Ideally, the modeling process is closely coupled with experimentation. We have drawn

extensively upon the work of experimenters to develop and validate this model and in

this final section we hope to offer something in return. While developing this model,

two issues arose from which we wish to postulate two new hypotheses about muscle

70


spindle mechanisms. The first issue is force enhancement, implemented in the

Schaafsma model as a discontinuous term that produces a constant positive force offset

during lengthening that is absent during shortening. This type of discontinuity is

extremely difficult for a control system to accommodate and might provide similar

difficulties for the nervous system. We therefore hypothesize that, if force enhancement

does occur in the intrafusal fiber, it has a more continuous form, e.g. sigmoidal.

The second hypothesis we propose is unidirectional rate sensitivity in the encoding

process. Symmetrical rate sensitivity between receptor potential and Ia frequency led to

non-physiological large undershoots on ramp cessation. Investigation of biological data

on the encoding process[14] supports the hypothesis that this rate sensitivity is indeed

only present during increasing receptor potentials, not decreasing. We implemented

this behavior in our model and were able to eliminate the large undershoots on ramp

cessation. Hence, we hypothesize that the true encoding function exhibits only

unidirectional rate sensitivity and encourage further experimentation to test this theory.

The final element we wish to comment on is a functional implication of the relative

length sensitivities of the muscle spindle. In both the robotic muscle spindle model and

biological muscle spindles[76, 77], passive position sensitivity increases substantially as

a function of length while active position sensitivity increases only slightly (dynamic

biological), remains constant (static biological), or decreases slightly (robotic) as a

function of length. These relative effects in which the γmn input stabilized the position

sensitivity[76] made it important to replicate the initial length of the biological muscle

spindle when attempting to match the relative responses of the passive and active

spindles. Such effects may also contribute to biological phenomenon such as the

dependence of ankle joint motion sensitivity on extensor muscle length, observed in the

passive limb[78]. Further, Schafer[79] observed that prestretched passive muscle

spindles replicate the Ia response amplitudes of shorter muscle spindles under dynamic

γmn stimulation and postulated the origin to be prestretch-dependent stretch activation.

Again, the nonlinearity of the passive force-length relationship might contribute to this

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phenomenon, although it could not account for the increased velocity sensitivity.

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Chapter 5:

Fusimotor Effect on Signal Information Content of Ia

Ensemble Model Reconstructed from Dynamic Intramuscular

Strain Data

5.1 Summary

1. It is long observed that the transducer characteristics of the muscle spindle Ia

response, e.g. noise and nonlinearity, sharply limit kinematic information. Many

propose the ensemble response as a source of an accurate signal, but technical

difficulties limit experimental population size and fusimotor control.

2. We reconstruct the ensemble response of a hypothetical population of 20-28

muscle spindles from dynamic local strain data from contracting rat medial

gastrocnemius. For 18 contractions in 3 rats, individual Ia responses are

generated by a nonlinear muscle spindle model and then averaged to form

ensemble Ia response.

3. Results under dynamic fusimotor stimulation show significantly improved

correlation to linear function of whole muscle position and velocity in ensemble

vs. individual Ia response.

4. Correlation to whole muscle velocity increased with rate of homogeneously

distributed dynamic fusimotor input and proximity of initial length to optimal

length of extrafusal muscle.

5. The results support our hypotheses that the reconstructed ensemble would

reduce Ia signal nonlinearity and that homogeneously distributed fusimotor

stimulation can suppress ensemble noise and nonlinearities in a dose-dependent

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manner. Proposed mechanisms include decorrelation by intramuscular strain

inhomogeneities, fusimotor-dependent length and velocity selectivity, and

decorrelating effect of fusimotor-dependent noise and nonlinear gains.

5.2 Introduction

The noise and nonlinearity of the individual muscle spindle’s output [1] sharply limits

the kinematic information capacity of the signal produced by a single muscle spindle.

In response to this, many physiologists have looked to the ensemble response of a

population of muscle spindles as the way for the central nervous system (CNS) to get an

accurate signal from these sensors [39, 40, 45, 46]. The population encoding theory is

supported by experiments showing that firing of a single muscle spindle is insufficient

stimulus to elicit perception of motion [40].

The question then arises: What variables might be critical for increasing the ensemble’s

information capacity? Ensemble size, simultaneous recording, and an intact fusimotor

system have been shown to improve information content [41, 43]. In fact, the fusimotor

system has been implicated as the mechanism by which such effects as heteronymous

muscle fatigue [47] and ketamine application [44] can degrade ensemble information

content. Several investigators have raised the issue of decorrelating individual muscle

spindle responses as a means to improve spatial filtering of ensemble information

content. Proposed mechanisms include the fusimotor system behaving as a neural

network [43, 47], random noise introduced by the active fusimotor system [49] and

membrane firing threshold variability [50]. Such decorrelation mechanisms have only

been tested indirectly [47], theoretically [50] or in small populations [49].

The technical difficulties associated with recording the afferent response of a population

of muscle spindles have limited the availability of simultaneously recorded

experimental data to populations of 10 or fewer [41-44, 47]. Sequential recording under

similar experimental conditions has allowed large data sets to be gathered, but the

discontinuities of time, muscle and animal, e.g. 34 cats employed in measuring a total of

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47 muscle spindles [52], limits the ability to study decorrelation of an intramuscular

spindle population.

With the paucity of experimental data, a model becomes an attractive option for

reconstructing the information content of the ensemble response of a large population of

muscle spindles. Further, a model allows one to readily control variables such as

fusimotor stimulation rates across a large population, something not possible in animal

models. Muscle spindle ensemble models in the literature use simple models of

individual spindle behavior to examine limb position encoding by spindle populations

spanning multiple muscles [34, 48]. To date, no ensemble model has been developed

that offers the level of detail necessary to reconstruct the influence of physiologic

variables on suppression of the individual spindle’s noise and nonlinearities in a single

muscle body’s ensemble response.

In this study, we create such a model of the ensemble response of a large population of

muscle spindles residing in a single muscle. Because the noise and nonlinearities in the

spindle’s behavior are the very thing that limit its information content, it is essential that

a model designed to generate physiologically relevant results regarding ensemble

information content be accurate in capturing the nonlinear features of the individual

muscle spindle response. Accordingly, we employ a structural muscle spindle model

that captures the major features of muscle spindle response: position gain, velocity gain,

fusimotor response, gain compression and normally distributed noise [63, 80].

Further, we propose that local strain variation within a muscle is so relevant for

decorrelating individual spindle response that it must also be included in the model to

generate physiologically relevant data. The muscle spindle Ia response has consistently

shown itself to be a function of the local strain directly adjacent to it in studies

comparing the Ia response to maximum strain [81], velocity [82] and contraction of

motor units [83]. Further, recent studies have demonstrated that local strains vary

substantially across the extrafusal muscle’s surface and with respect to muscle origin-

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to-insertion length under a variety of experimental protocols including passive

stretching [84], active contraction [85, 86] and locomotion [87, 88]. Accordingly, to

accurately model the mechanical environments of the members of our muscle spindle

population, we use as mechanical input local surface strain data simultaneously

recorded from multiple locations on contracting muscle tissue using a three-dimensional

determination method.

Once the ensemble model is developed, we then use it to ask two questions about the

ensemble’s information content. The first question is whether the ensemble response

reduces the nonlinearities seen in the individual muscle spindle response, and if so, by

how much. This aim tests whether the sources of variability in the ensemble model

(local strain variability, fusimotor-induced random noise, and fusimotor-induced

nonlinear responses to the strain variability) decorrelate the noise sufficiently to allow it

to be spatially filtered out of the ensemble response. As such, it is an explicit test of the

widely held theory that a large population will reduce the presence of the individual

spindle’s noise and nonlinearities in the ensemble response [39, 40, 45, 46].

The second question goes on to ask whether the rate of homogeneously distributed

fusimotor stimulation improves the correlation between input trajectories and ensemble

response, i.e. if the fusimotor system has a dose-dependent effect on ensemble response.

This objective stems from the observations that an intact fusimotor system improves the

information content of a spindle ensemble. Bergenheim et al.[43] proposed that the

fusimotor system is acting as a neural network to decorrelate the output from each of

the spindles, thereby increasing the ensemble’s discriminative ability for kinematic

variables. While we concur that the neural network mechanism could produce the

observed behavior, we postulate that a simpler mechanism, the differing transducer

properties of the active spindle vs. the passive spindle, could also produce the observed

effect. We put forward the idea that, even at a fixed stimulation level across the

population, the fusimotor system could increase the correlation of the ensemble to

whole muscle position or velocity by (a) increasing the random noise and variability of

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the individual spindle response [49, 50], (b) increasing the decorrelation introduced by

local strain variability by means of the nonlinear mechanical properties ascribed to the

intrafusal fiber [28, 63], and (c) increasing the percentage of the Ia signal that responds

specifically to velocity or length, assuming exclusively dynamic or static fusimotor

input, respectively.

We therefore pose the following hypotheses:

Hypothesis 1: The response of an intramuscular muscle spindle ensemble is a more

linear function of length and velocity than the individual muscle spindle response.

Hypothesis 2a(b): Increasing the rate of homogeneously distributed dynamic (static)

fusimotor stimulation to a muscle spindle population improves the strength of the

correlation between whole muscle velocity (length) and ensemble response.

5.3 Methods

5.3.1 Collecting Local Muscle Fiber Strain Data

The methods for collecting muscle strain data are described in detail elsewhere [86] and

are briefly summarized here. Three male, 12 week old Lewis rats were anaesthetized

with sodium-pentobarbital (Numbutal®, 0.1 ml/kg BW, i.p.) after short-term (<20s)

sedation with CO2. Sodium-pentobarbital was supplemented as necessary. The local

ethical committee approved the experiments. The medial surface of the medial

gastrocnemius was surgically exposed and dissected free of fascia. Approximately 70

fluorescent polystyrene spheres (Bangs Laboratories Inc. Fishers USA) of 0.45 ±

0.05mm diameter were attached to the muscle surface in a uniform distribution with an

interdistance of ~2mm. The calcaneus bone was dissected free of the leg and fixed to a

force transducer. The femur was securely fixed to the lower traverse. Elevation of the

upper traverse allowed control of muscle length with an accuracy of 0.01 mm. Two

electrode wires wrapped around the sciatic nerve supplied pulsed electrical stimulation,

0.7-0.9 Volts at 80-90Hz, to generate maximal muscle contraction force. No increase in

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force was observed with further increases in voltage or frequency. Images of the

markers were captured at 50 fields per second with two synchronized CCD cameras and

digitized for analysis. For each 350 msec contraction, recording began 80 msec prior to

contraction and captured 600 msec of video data. 3D marker tracks were reconstructed

from the pair of digitized images. Strains were calculated for each video frame with

respect to the first image of the data acquisition. Local strains at each marker position

were calculated using the procedure adapted from Peters [89] in which a linear strain

field was assumed within a strain group, defined as a circle (r=4mm) around the marker.

As markers were on the muscle surface, strains calculated were two-dimensional

surface strains. Strain in the third-dimension, normal to the muscle surface, were

assumed to be zero. Strain groups containing less than 5 markers were excluded from

analysis to calculate reliable strain.

Seven trials of isometric contractions were recorded from three rats. The first trial was

performed at just below the muscle’s optimal length, with the remaining trials

performed at –1 mm, +1mm, -2mm, +2mm, -3mm, +3mm, respectively. Only markers

recording muscle fiber motion were included in the study. These markers were

identified by the presence of a negative principal strain aligned with the muscle’s

longitudinal axis during maximal contraction. In addition to exhibiting temporally and

spatially appropriate contractile behavior, microscopic examination of the tissue

confirmed that all markers used in the study lay either directly on the muscle (68

markers) or on the muscle’s proximal border separated from the muscle fibers by a thin

layer of aponeurosis (5 markers). Viable muscle markers were limited to those present

in the data for all seven trials in a given rat. This subset included all muscle fiber

motion markers except those at the distal and lateral edges of the muscle body, with a

total of 25, 20 and 28 markers in rats 1-3, respectively. The trial at –2mm was omitted

from further analysis for all rats due to insufficient numbers of viable markers in Rat 1.

Local muscle fiber orientation was identified as the negative principal strain axis during

peak tetanic contraction. Marker principal strains were rotated to this orientation in

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each frame to generate a timecourse of the local strain that would be experienced by

sensors lying parallel to the muscle fiber. Whole muscle strain was determined by

measuring the displacement between a marker adjacent to the proximal muscle insertion

and a marker at the musculotendonous junction with the Achilles tendon.

5.3.2 Calculating Muscle Spindle Ensemble Response

The timecourse of strain experienced by each marker was run individually through the

model of the mammalian muscle spindle described in Chapter 4. Initial spindle length

was calculated using the assumption of homogeneous strain distribution in the passive

muscle body. In the first frame of each trial the distance was calculated between

markers at the proximal and distal end of the muscle belly whose negative principal

strain during contraction was approximately collinear, giving a reference length along

the muscle fiber axis. Initial strain was then calculated by normalizing this length to the

corresponding length at optimal fiber length. All measurements from the medial

gastrocnemius were reported as strain. The optimal length of the muscle spindle model,

the length at which the intrafusal muscle generates maximal force, was used to

denormalize the strain and calculate physical displacements to apply to the muscle

spindle model.

Fusimotor activation level was constant throughout a given trial. All experiments were

repeated under eight fusimotor stimulation rates: 25, 50, 75 and 100 Hz dynamic, and

25, 50, 75 and 100 Hz static. No experiments were performed with simultaneous

stimulation of the static and dynamic γ motorneurons (γmn).

A linear positioning device applied the strain trajectories to the robotic muscle spindle

one at a time. Ia output was sampled at 1000 Hz and a 5 point moving average was

recorded at 200 Hz using a dSPACE 1102 data acquisition card and the ControlDesk

software interface. Technical details regarding this system are published elsewhere [63].

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The individual muscle spindle Ia responses were compiled into an ensemble metric by

calculating their average response as a function of time:

∑=

=m

nn tIatE

1)(

301)( ( 5.1 )

Where: E(t) is the ensemble response, n is the individual trajectory number, m is the

number of spindles in the ensemble, Ian is the afferent output of the muscle spindle for

the nth trajectory, and t is time.

5.3.3 Data Analysis: Nonlinearity of Spindle Ensemble Output.

The goal of this analysis was to test the hypothesis that the ensemble response fits the

model of a linear weighted sum of position and velocity better than the individual

muscle spindle response. To test this, a multiple regression was performed to calculate

the correlation coefficient for the following model:

ε+++= CdtdxBtAxtIa )()( ( 5.2 )

Where: Ia = Ia output (ensemble or individual), x = position input to whole muscle,

dx/dt = velocity input to whole muscle, C = offset in data, ε = residual error. Fisher’s Z

transformation was performed to obtain a normally distributed variable, Z’, describing

the correlation coefficient.

A Student’s paired t-test was calculated between (a) Z’ for ensemble response in a given

trial, rat, and fusimotor activation level, and (b) the average of all individual spindle Z’

values in the population corresponding to that ensemble response. This test was run

separately for data collected under static γmn stimulation and dynamic γmn stimulation.

All statistical computations were performed with the MATLAB statistics toolbox.

For both the nonlinearity and fusimotor studies, we limited our analysis of the static

fusimotor data to the window from 250 msec to the 600 msec to exclude a non-

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physiological spike observed on extrafusal contraction initiation (80 msec) in the static

fusimotor stimulated muscle spindle model. An additional non-physiological spike was

discovered later at ~400 msec in all rats and was not excluded. Its source and

implications on the data are described in the discussion. Because of its differing

dynamics, the muscle spindle model under dynamic fusimotor stimulation did not

exhibit these non-physiological behaviors and therefore the dynamic analysis

encompassed the full 600 msec contraction.

5.3.4 Data Analysis: Effect of Fixed Fusimotor Stimulation Rate

The goal of this analysis was to test hypotheses 2a&b: that increasing the rate of

dynamic (static) fusimotor stimulation to a muscle spindle population improves the

strength of the correlation between ensemble response and whole muscle velocity

(length). For each fusimotor rate, the correlation coefficient was calculated between

ensemble response and whole muscle velocity (length). Fisher’s Z transformation was

used to convert the correlation coefficient into a normally distributed variable, Z’.

Sources of variation in the Z’ variable were determined in JMP statistical software

(SAS Institute Inc., Cary, NC) with a repeated measures ANOVA using the following

linear model:

Z’ijk = µ + Gk + Ti + βj + εijk ( 5.3 )

Where: G=Gamma motorneuron treatment level, T= Block for run order (repeated

measure) and initial length, β= Block for rat and number of spindles in the ensemble, ε

= residuals. Significance was determined as p<0.05.

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5.4 Results

5.4.1 Local Strain Data

The negative principal strain alignment at each of the markers during peak contraction,

Figure 5.1, was used to assign muscle fiber orientation. The circle denotes muscle

marker location, and the line’s

orientation and length denote the

negative principal strain alignment

and magnitude during peak

contraction, respectively.

Aponeurosis markers are not shown.

Despite the inability to visualize

muscle curvature in this frontal view,

the alignment appears consistent

across all markers, verifying the

muscle fiber orientation assignments.

The magnitude is consistent through

the main muscle body, decreasing

rapidly at the aponeurosis boarder.

Markers at [-.5,-7] and [0,-6] are on the distal border of the aponeurosis. All data in

Figure 5.1-Figure 5.4 are from Rat 3, Trial 6.

Figure 5.1: Location of 28 markers (o) on surface of rat medial gastrocnemius muscle fibers used to reconstruct mechanical input to 28 hypothetical muscle spindles. Local muscle fiber axes ( | ) are well aligned across the marker set, suggesting the muscle fiber axes were correctly assigned as negative principal strain axis during maximum contraction. Distal end of muscle at top. All data in Figure 5.1-Figure 5.4 are from Trial 6 of Rat 3.

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The 28 markers attached to the

muscle body show considerable

diversity in the local strain time

course experienced during a typical

isometric contraction, Figure 5.2.

Although strain is similar among the

individual markers during the initial

contraction, during the hold and

relaxation periods, 100-600msec, the

strains experienced by the individual

markers vary from one to another in amplitude, velocity, smoothness and final strain.

5.4.2 Ensemble Reconstruction

The middle row of Figure 5.3 shows the 28 marker strain timecourses from Figure 5.2

as they were physically applied to the robotic muscle spindle: 1 trajectory every 1.5

Figure 5.2: Dynamic strains recorded at the 28 markers shown in Figure 5.1. Strains exhibited variation in amplitude, velocity, time course and smoothness during and after a 350 msec contraction

Figure 5.3: Sequence of 28 displacement trajectories (middle row) laid out in manner in which they were physically applied to muscle spindle model. This protocol generated 28 individual Ia responses corresponding to 28 hypothetical muscle spindles (top row, 50Hz static γγγγ mn stimulation, Bottom Row, 50 Hz dynamic γγγγ mn stimulation) which were then pooled to form ensemble response.

83


seconds. The other two rows depict the robotic muscle spindle’s response to the

individual strains for one replicate of the 50 Hz fusimotor stimulation rate, both static

(top) and dynamic (bottom). The Ia data in Figure 5.3 are low pass filtered at 10Hz to

increase visibility. No filter was applied during data analysis.

The markers in Figure 5.3 exhibit

a wide diversity of strain

trajectories. Amplitude and shape

diversity are particularly evident

in this figure. The muscle

spindle response also exhibits

variation among markers. The

static response is loosely related

to the position input while the

dynamic response closely follows

the velocity input. In the static

response, a large spike occurs at

the peak of contraction as the

input rapidly accelerates from

shortening to lengthening. All

responses exhibit the noise and

nonlinearities typical of spindle

Ia response.

Figure 5.4 illustrates the

similarities between the Ia

ensemble response (solid line) and the input trajectory (dotted line). The compliance of

the experimental set-up allowed the muscle body to shorten, resulting in the motion

seen here. The ensemble response under dynamic fusimotor stimulation mimics the

input velocity in both amplitude and phase throughout the 600 msec trial. Under static

Figure 5.4: Comparison of ensemble response to kinematic inputs. (a) Under 50 Hz dynamic fusimotor stimulation ensemble Ia response of muscle spindle population (solid line, left axis) closely parallels whole muscle velocity (dashed line, right axis). (b) Under 50 Hz static fusimotor stimulation baseline ensemble Ia response (solid line, left axis) loosely follows whole muscle position (dashed line, right axis), but is dominated by large spike at onset of relaxation ramp.

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γmn stimulation the ensemble’s correlation to input position is smaller. Although the

baseline amplitude follows the sigmoidally increasing position (175-600 msec), the

response shows a definite velocity offset throughout and is punctuated by large positive

spikes that coincide with large changes in velocity, both during shortening and

lengthening.

5.4.3 Nonlinearity of Spindle Ensemble Output.

Multiple regression analysis of the linearity of the relationship between spindle Ia

output and position and velocity inputs generated a correlation coefficient, Z’, which

quantifies the proportion of the nonlinear Ia response which can be accounted for by a

linear function of position and velocity. The resulting correlation values, Z’, are shown

for a typical case in Figure 5.5a&b. Figure 5.5a shows the response under 100Hz

dynamic γmn input, while Figure 5.5b shows the response under 100Hz static γmn

input. The 6 trials correspond to a single repetition of each of the 6 contractions

performed by a single rat. Under both static and dynamic γmn stimulation, the

ensemble response correlation (solid line) is typically higher than the averaged

correlation of the individual muscle spindle responses (dotted line). The individual

muscle spindle response’s correlation to whole muscle motion (dots) varies greatly,

with an average standard deviation of 0.12 and 0.13 in the dynamic and static trials

shown, respectively. The ensemble’s multiple correlation to position and velocity is,

for all cases, greater than or equal to the ensemble’s single correlation to just position or

velocity under either static or dynamic γmn input, respectively (dash-dot line).

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A paired t-test examined the hypothesis that the ensemble response of a spindle

population has a higher correlation coefficient than the average individual muscle

spindle’s correlation to the whole muscle’s position and velocity. Correlation

coefficients are reported as the

normally distributed coefficient,

Z’. Under dynamic γmn input the

ensemble correlation is

significantly higher than the

average individual response, p <

.0001. The mean difference is

0.22 higher; the mean ensemble

correlation coefficient is 0.62 ±

0.20(std. dev.); the mean

individual correlation coefficient

is 0.40 ± 0.11. Under static γmn

input, the ensemble correlation is

also significantly higher than the

average individual response, p <

.0001. The mean difference was

0.08; the mean ensemble

correlation coefficient is 0.40 ±

0.18; the mean individual

correlation coefficient is 0.32

±0.084.

Figure 5.5: Correlation coefficients for multiple regression on whole muscle position and velocity. Correlations are consistently higher for ensemble response (solid line) than when the 28 highly variable individual muscle spindle correlation coefficients (dots) are averaged together (dotted line). Correlation coefficients for single regression of ensemble response against single kinematic variable (dash-dot line) shows strength of fusimotor stimulation in tuning ensemble selectivity to velocity (5a) or position (5b). Fusimotor stimulation (5a, 100 Hz dynamic, 5b, 100 Hz static) constant across all six trials (x-axis) of rat 2.

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5.4.4 Effect of Fixed Fusimotor Stimulation Rate

Under dynamic fusimotor

stimulation, the effect of

fusimotor stimulation rate on the

correlation between ensemble

response and whole muscle

velocity, Figure 5.6a, is

significant, p < .0001. The mean

correlation values, Z’, increase

with increasing dynamic

fusimotor input with values of

0.469, 0.537, 0.560, 0.583 for the

25, 50, 75 and 100 Hz dynamic γ

fusimotor stimulation rates,

respectively. The effects of run

order and initial length are shown

in Figure 5.6b&c. These plots

indicate strong trends in the data,

though statistical assessment of

the effect is not possible due to

the experimental design. The

plot of run order vs. correlation,

Figure 5.6c reveals oscillations in

parallel with the oscillating initial

lengths, as well as decreasing

correlation with repeated muscle

contraction. Plotting muscle

initial length vs. correlation.

Figure 5.6: Correlation between ensemble response and whole muscle velocity under dynamic fusimotor stimulation (a) increases monotonically with increasing rates of dynamic fusimotor stimulation, (b) peaks at optimal muscle length then decreases with distance from optimal muscle length for all rates of dynamic fusimotor stimulation (o, 25 Hz, x, 50 Hz, +, 75 Hz, ∆∆∆∆, 100 Hz), (c) decreases with repeated extrafusal contraction for all rates of fusimotor stimulation.

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Figure 5.6b, reveals a consistent trend that correlation increases with proximity to

muscle optimal length. Note that in Figure 5.6b 4 of the 21 trials were omitted from

calculating mean correlations because their initial length deviated from the mean initial

length across all rats for that trial by >2%. The data from Rat 2 and Rat 3 from the trial

at –2mm, which was omitted from the statistical analysis due to insufficient markers in

Rat 1, were included in the plot of initial length correlations.

Under static fusimotor stimulation, the effect of fusimotor stimulation rate on the

correlation between whole muscle position and spindle ensemble response, Figure 5.7a,

is significant with p=.0065. The direction of the effect is opposite of what was

predicted. The highest mean correlation, Z’, occurrs in the 25 Hz case, with means of

0.281, 0.255, 0.185, and 0.187 for the 25, 50, 75 and 100 Hz static fusimotor

stimulation rates, respectively. Figure 5.7b&c show the effect of initial length and run

order, respectively. No clear trend of the effect of run order or initial length is evident

in the static fusimotor data.

5.5 Discussion

The aim of this article is to use mechanical data collected from 70 locations on an

actively contracting muscle to reconstruct the ensemble response that would have been

produced by 20-28 muscle spindles scattered throughout that muscle. In doing so, we

asked the question of whether the sources of variability in our ensemble data, strain

inhomogeneity leveraged by the nonlinear transfer function of the active spindle, and

the random noise of the active spindle, increased the information content of the

ensemble response. We further asked whether the variability in the individual Ia output

introduced by fusimotor stimulation could, when applied homogeneously across a

muscle spindle population, produce a dose-dependent improvement in the ensemble

information content.

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5.5.1 Reconstructing the

Ensemble Response

5.5.1.1 Local Strain Data

Our thesis presumes variability in

local strain across the muscle

body, and the results (Figure 5.2)

show that there is indeed

substantial variation in amplitude,

time course, velocity and

smoothness between individual

marker strain trajectories.

Further, although the strain

magnitudes seen in the bottom

three markers are small, Figure

5.2 shows that the strain

trajectories experienced by all

markers, including these three, are

typical of the strain trajectories

experienced by contracting

muscle, confirming that they were

correctly identified as muscle, not

aponeurosis, markers.

The experimental protocol

employed to get a sample of

typical local strains experienced

by a population of muscle

spindles was isometric contraction

Figure 5.7: Correlation between ensemble response and whole muscle position under static fusimotor stimulation (a) decreases monotonically with increasing rates of static fusimotor stimulation, (b) shows no apparent relation to initial muscle length for all rates of dynamic fusimotor stimulation (o, 25 Hz, x, 50 Hz, +, 75 Hz, ∆∆∆∆, 100 Hz), (c) shows no discernable change with repeated extrafusal contraction for all rates of fusimotor stimulation. All correlation values are extremely low, reflecting a nonphysiological short-range stiffness behavior in the static muscle spindle model which severely degraded the ability to reconstruct an accurate correlation value.

89


of the musculotendonous unit, which raises two issues. First, is there sufficient change

in muscle body length to test our hypotheses? Elek et al. [87] showed that in the cat

medial gastrocnemius muscle spindles signal muscle body length, which differs from

changes in origin-to-insertion length [88, 90]. Figure 5.4 supports Elek et al.’s data,

showing there was considerable variation in muscle body length as well as velocity

during our experiments. The second issue is whether fixed fusimotor stimulation, both

static and dynamic, is observed during extrafusal contraction. Studies in which

fusimotor outflow is reconstructed for volitional movements in the cat show that fixed

fusimotor levels provide the best match to experimental data during locomotion (fixed

static fusimotor input) and stretching (fixed dynamic fusimotor input), far

outperforming EMG-linked fusimotor input [56].

Several assumptions are implicit to these data. First, since data are unavailable

regarding the spindle count in the rat medial gastrocnemius, we assumed a population of

20-28 muscle spindles. Spindle counts from related muscles, rat gracilis (13-17) [91]

and cat medial gastrocnemius (46-80) [92], suggest that 20-28 spindles is a reasonable

approximation. The second assumption is that the distribution of those 20-28 spindles

across the muscle’s medial surface (Figure 5.1) is representative of the distribution in a

typical unipennate muscle. The spread across the muscle surface is consistent with the

limited data available[93-95] on the distribution of spindles in rat and cat medial

gastrocnemius. These same data suggest that few muscle spindles in the medial

gastrocnemius lie near to the muscle surface, but other data indicate that spindle output

is closely correlated to the overlying surface strain of the extrafusal muscle[81, 82].

Hence, we conclude that the distribution of our hypothetical spindle population is a

reasonable approximation for the purposes of reconstructing ensemble response.

5.5.1.2 Muscle Spindle Population Response

Our proposal that the variability in the individual response is spatially filtered out of the

ensemble response is supported by the smoothness of the ensemble Ia response (Figure

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5.4) as compared to the noise in the individual Ia responses (Figure 5.3).

Essential to the reliability of our reconstruction of the ensemble response is the

accuracy of the muscle spindle model in generating the pertinent features of the

individual Ia response. Because the questions we ask are sensitive to the nonlinear

aspects of the spindle response, we used a muscle spindle model, the robotic muscle

spindle, which accurately replicates many of the linear and nonlinear features of the Ia

response including position and velocity gain, normally distributed noise, fusimotor

response and gain compression. In tuning and validation studies with protocols similar

to the spindle inputs used in this study, e.g. ramp and hold, sinusoidal and fusimotor

response, the fusimotor-stimulated model reproduced the biological data well in 10 out

of 10 cases. Without fusimotor stimulation, the response matched the biological data in

5 of 9 cases, with non-physiologically small responses in the remaining 4, as described

in Chapter 4. Recognizing this limitation, we restricted the experiments in this study to

active fusimotor stimulation. In using this model, we make the assumption that the use

of two different animal models, the rat for extrafusal motion and the cat for muscle

spindle modeling, does not impair our ability to draw meaningful conclusions from our

reconstruction of the spindle ensemble response.

The large spike observed in the static ensemble response (Figure 5.4b) is a symptom of

a previously unrecognized limitation in the static fusimotor response of the muscle

spindle model: the short-range stiffness model allows the "cross-bridges" to rapidly

reset during fusimotor stimulation. As a result, when the muscle spindle briefly comes

to rest during the contraction plateau, the short-range stiffness model engages, causing a

large spike when the relaxation ramp begins. This type of spike is not observed

physiologically during active fusimotor stimulation [96]. Because of the spike’s large

size and the fact that the sudden cessation of α-motorneuron stimulation caused a

temporal correlation of relaxation ramp initiation across all of the local strains

(~380msec in Figure 5.2), this non-physiological nonlinearity persists in the ensemble

response. This phenomenon is not observed under dynamic fusimotor stimulation

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because the dynamics of the muscle spindle model prevent the short-range stiffness

response from resetting.

5.5.2 Effect of Ensemble on Kinematic Information Content

Under dynamic fusimotor stimulation, the results provide unequivocal support for

hypothesis 1. We find that the ensemble response is far superior to the individual

response in terms of the correlation to position and velocity. Also, the spread in

individual Ia correlations (dots in Figure 5.5a) suggests there is extensive variability

within the spindle population. This evidence supports the proposed mechanism that the

sources of variability in the model are sufficiently decorrelated to allow spatial filtering

of the individual spindle’s noise and nonlinearities.

Under static fusimotor stimulation, the results are unfortunately masked by the large

spike introduced by the muscle spindle model’s non-physiological short-range stiffness

response. All correlations are extremely low with a mean of .30 for the average

individual spindle correlation. The low correlations are exacerbated by the fact that the

very large Ia responses associated with the short-range stiffness occur at the beginning

of the relaxation ramp when length is at a minimum. We did still observe a slight

improvement in correlation in the ensemble response as compared to the individual

spindles. This is likely due to the filtering which occurred in the time windows before

and after the short range stiffness peak.

5.5.3 Dose-Dependent Effect of Homogeneous Fusimotor Stimulation

Under dynamic fusimotor stimulation, the results clearly support our hypothesis,

showing a statistically significant monotonic increase in correlation between velocity

and ensemble response with increasing dynamic fusimotor stimulation rates (Figure

5.6a). This supports our theory that the fusimotor-dependent mechanisms described

above work together to increase ensemble information content by both increasing the

spindle’s selectivity for velocity inputs and increasing the decorrelation effect. The plot

92


of correlation vs. length (Figure 5.6b) further supports this theory. It shows that

ensemble information content, defined by the correlation coefficient Z’, peaks when the

contraction is performed at optimal muscle length in a fashion similar to the familiar

active muscle length-tension relationship [30]. This behavior is also consistent with the

theory of fusimotor level increasing information content. In the muscle model used to

drive the intrafusal dynamics, all of the properties that contribute to the active tension

(length-tension, velocity-tension, fusimotor-tension) are multiplied together to calculate

the active component of the muscle’s force. As a result, the length-tension property can

act as a coefficient to modulate the other active properties in a manner similar to

increasing fusimotor stimulation. The possibility also exists that the extrafusal tissue

from which we reconstructed the ensemble response exhibited increasing decorrelation

of its local strain with proximity to optimal muscle length.

Repeated contraction of the extrafusal muscle leads to decreased ensemble information

content (Figure 5.6c). This could be a result of repeated contraction of the extrafusal

muscle increasing correlation between local strains, particularly since the muscle was

not preconditioned. We must, however, temper our conclusions with the following

caveat. Due to confounding of the experimental design, it is impossible to assess the

relative effect of run order vs. initial length (Figure 5.6b&c). The number of

contractions and the deviance of initial length from optimal length could both be having

the same depressive effect on ensemble information content.

Under static fusimotor input, the non-physiological coherent short-range stiffness

nonlinearity overwhelms the correlation between the static ensemble response and

whole muscle length. Through the convergence of two factors: (a) the size of the short-

range stiffness spike is proportional to the stiffness of the intrafusal muscle making it

increase with fusimotor input and (b) the large spike occurs at the onset of the

relaxation ramp where position is at a minimum, we see both very low correlation

coefficients for position and a decreasing correlation with increasing fusimotor input

(Figure 5.7a). Initial length and run order have no appreciable effect (Figure 5.7b&c),

93


again likely due to degradation of the correlation by the coherent short-range stiffness.

The degradation observed here is testimony to the power of a coherent noise source to

alter the ensemble’s information content. Perhaps certain types of nonlinearities which

tend to be temporally correlated, such as short-range stiffness, are useful to the CNS and

are handled by behavior-specific decoding mechanisms.

5.5.4 Conclusions

Using physiological data on the local strain distribution during an active contraction, we

reconstruct what the ensemble response might look like from a large population of

muscle spindles. We show that, given the sources of variability in our model, spatial

filtering across a population substantially improves the information content of the signal

sent to the CNS. We also show that in our model the fusimotor stimulation rate

improves ensemble information content even if applied at a fixed rate across the

population. These data support our theory that much of the decorrelation which

suppresses signal distortion in the ensemble is the product of the combined effect of

intramuscular strain inhomogeneities and the nonlinear mechanical properties of the

actively contracting intrafusal muscle.

These studies may reconcile the seemingly disparate views of muscle spindle

nonlinearities between muscle spindle physiologists, who treat nonlinearities as an

important aspect of spindle behavior[4, 5, 72], and physiologists studying higher

organizational levels, who theorize that most nonlinearities will be negligible in the

ensemble response[34]. Our results suggest that, as previously proposed[39, 45, 46, 49,

50], it is in fact the extent of the irregularity of the individual muscle spindle responses,

their decorrelation, that is essential for producing an accurate signal in the ensemble

response. This concept elegantly reconciles what at first glance appear to be

contradictory stances within the physiological community.

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Chapter 6:

Conclusions

6.1 Summary

The theme of this dissertation is to develop precision engineering hardware capable of

accurately modeling muscle spindle behavior and to use the process of building and

tuning the model, as well as experimentation with the model, to increase our

understanding of the mechanical and neurological mechanisms by which the body

measures muscle kinematics.

A three element abstraction of muscle spindle behavior was proposed and implemented

in precision engineering hardware. Engineering tests of the individual components

show that their dynamics meet performance metrics derived from the biological

literature; physiologically realistic tests of the integrated robotic muscle spindle show

that the subsystems replicate the physical performance observed in biological muscle

spindles. The transducer hardware matches the displacements observed in the spring-

like sensory region and uses this physical displacement to replicate the sensory region's

transduction behavior directly in mechatronic hardware. The encoder replicates the

biological encoder's conversion of the analog receptor potential to a frequency

modulated spike train directly in on-board circuitry, using a software-based algorithm to

add positive rate dependency. The contractile element meets the engineering

performance specifications, exhibiting fast, precise and robust linear actuation.

Physiologically realistic tests show that, when driven by a software based muscle

model, the contractile element's physical displacement closely matches the movements

of the biological intrafusal muscle. These tests collectively show that the individual

subsystems of the robotic muscle spindle accurately model the behavior of their

95


analogous physiological systems. This lays a solid foundation for investigating the

effect of these subsystem's behaviors on the transduction of muscle kinematics.

The integrated robotic system was tuned against a battery of muscle spindle data and the

physiological faithfulness of the resulting behavior was then validated against a

different set of experimental protocols and results from the biological literature. Under

fusimotor stimulation, the robotic muscle spindle replicates biological behavior well in

all experiments, including ramp and hold and sinusoidal position inputs of varying

speeds and amplitudes as well as a full spectrum of fusimotor stimulation rates, both

static and dynamic. In the passive case, the robotic muscle spindle matches biological

behavior well in 5 of 9 experiments, exhibiting smaller amplitudes than the biological

spindle in the remaining four cases. Thus, under active fusimotor stimulation, the

model enjoys wide applicability to a variety of experimental protocols, with more

limited applicability in the passive case.

During the tuning process, non-physiological undershoots on ramp cessation were

encountered. The bi-directionality of the encoder’s rate dependency was identified as a

likely cause and it was proposed that the encoder rate dependency might instead be

unidirectional. This new hypothesis, supported by data from the biological

literature[14], was implemented and indeed eliminated the non-physiological

undershoots. Hence, the process of building the model led to the proposal of an

alternative hypothesis for spindle encoding which is more consistent with biological

evidence and the systems behavior of the mechatronic model.

Employing a novel methodology, the robotic muscle spindle was then applied to the

task of reconstructing the ensemble response of a population of hypothetical muscle

spindles on the surface of a contracting muscle. Data from collaborators describing

muscle strain time courses at 28 locations on an actively contracting muscle were run

through the robotic spindle to generate the 28 Ia responses that would have been

generated by muscle spindles at each of those locations. The average of those outputs,

96


the Ia ensemble response, shows a significantly closer correlation to a linear function of

muscle length and velocity than the individual Ia response. Further, the dynamic

fusimotor input from the central nervous system improves the correlation of the

ensemble response to muscle velocity in a dose-dependent manner.

It is proposed that it is actually the decorrelation of the complexity of the individual

muscle spindle's response, the noise and nonlinearities, that transforms the individual

responses into a much easier to decipher ensemble response. The ensemble's spatial

filtering effect will minimize the influence of noise on the ensemble response if that

noise is decorrelated. The ensemble reconstruction incorporates two major noise

sources, both of which are potentially decorrelated across the population:

inhomogeneous local extrafusal muscle strain and noise whose decorrelation is

dependent on the rate of fusimotor stimulation. These effects could explain the increase

in linearity observed in the ensemble response. Further, the fusimotor system enhances

these decorrelation sources, as well as increasing the individual spindle's selectivity to

specific kinematic variables, thereby providing a mechanism for the observed dose-

dependent effect on the ensemble response. These neuromechanical hypotheses

elegantly reconcile the noise of the individual Ia response and the nervous system's need

for a decipherable signal of muscle kinematics.

6.2 Future Work

Many different aspects of this dissertation could serve as a starting point for future

work. Candidate areas include experimentating on biological muscle spindles to test

specific hypotheses generated by this robotic modeling research, investigating new

research questions using the ensemble reconstruction technique, expanding the robotic

muscle spindle model, addressing basic science questions in biorobotics, and applying

the robotic muscle spindle to biorobotics applications in prosthetics and engineering.

Experiments to test the biological hypotheses raised by the robotic models in this

dissertation is one area of future work. First, additional biological experiments could

97


test the hypothesis of unidirectional encoder rate dependency by recording the

relationship between receptor potential and Ia response under a wider range of

experimental protocols. This method could also be used to quantitatively characterize

the rate dependency. Second, the observation that the passive spindle model matched

biological data well in five cases, but exhibited the same shortcoming in the remaining

four cases, led to the conclusion that a mechanism not included in the model heavily

influences passive behavior. Additional studies could test the proposed candidates

including stretch activation and an omitted passive damping term. Finally, in the

ensemble study three different mechanisms were proposed to explain the improvement

in Ia response with large populations and fusimotor input. Further studies, such as

repeating the experiment with homogeneous local strains, could investigate the specific

effect of each of these mechanisms.

The second area for additional research is applying the ensemble reconstruction

technique to additional research questions. First, anatomical studies have shown that, in

some muscles, muscle spindles are distributed in distinct patterns, such as being

collocated with deep, oxidative fascicles[95]. Reconstruction studies could examine the

types of information coded by different distributions of spindle populations to test the

effect of this selective distribution on ensemble information content. Obtaining strains

from internal locations in the muscle would enhance the power of such an experiment,

as well as provide an interesting comparison to the results presented here. Second, this

technique could be applied to a ramp and hold protocol to examine the ensemble's effect

on specific features such as short-range stiffness. Third, the same questions asked here

could be applied to the secondary spindle response and reapplied to the static Ia

response, pending model modification to omit the non-physiological short-range

stiffness behavior. Finally, running this experiment with and without γ motorneuron

stimulation to the extrafusal fibers would allow one to test the influence of active

contraction on decorrelation of local strains. One could speculate that a passive muscle

would exhibit greater correlation allowing the nonlinear short-range stiffness to persist

98


in the ensemble response and act as an early warning against external perturbations in

the resting animal.

The third area for future work is expanding the robotic muscle spindle model.

Candidate areas include: (a) augmenting the onboard circuitry to incorporate

unidirectional rate sensitivity, (b) implementing sarcomere length inhomogeneity in the

muscle model to incorporate phenomena such as stretch activation, local contraction

foci, spread of depolarization across the intrafusal muscle, and a cross-bridge model of

short range stiffness, (c) implementing a detailed model of ion channel transduction,

although the experimental data on which such models are based are limited [35], (d)

building an additional robotic muscle spindle to allow simultaneous stimulation of the

static and dynamic fusimotor fibers and (e) modeling the secondary afferent response.

The fourth area of future work is in applying this research to basic science problems in

biorobotics. This sensor could be used as part of a biorobotic model to study the

behavior of larger neuromuscular systems, as in Chou and Hannaford [61], or as

sensory feedback to train cerebellar learning models. Alternatively, one could

miniaturize the muscle spindle model using MEMs technology and attach a large

population of the devices to a biorobotic muscle to generate a real time ensemble

response.

The final area of future work is the use of the completed robotic muscle spindle as a

sensor for engineering applications. While this model was designed with the aim of

understanding the basic science of muscle spindles, like most biorobotic models it has

obvious applications in prosthetics and engineering as well. The robotic muscle spindle

is a functioning device that can report actuator kinematics in the language of the central

nervous system. As such, it is an attractive candidate for the development of prosthetic

devices. Further, the robotic muscle spindle is an actuated sensor, a type of sensor not

currently in the repertoire of devices used to measure kinematic properties in

engineering. It would be interesting to explore this type of device as a means of

99


increasing a transducer’s range in situations where the physical displacement of the

sensing mechanism is limited, or for real-time tuning of the sensor’s output to different

kinematic variables, e.g. absolute length vs. perturbations from a desired length.

100


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112


Appendix A:

Technical Drawings

PCB Board

Circuit Diagram............................................................... 113

PCB Layout..................................................................... 113

CAD Drawings

Cantilever ........................................................................ 114

Spindle Housing.............................................................. 115

Guide............................................................................... 116

Stop ................................................................................. 117

Shim Spacer .................................................................... 118

Nut................................................................................... 119

113


Strain Gage Y

R4b100k

R1120

Strain Gage X

R4a150k

R2120

R51k

R61k

R7a470k

R7b5100k

C41nF

R8a470k

R8b5100k

C31nF

C522pF

7555

R9390k

VCC7

4

6

2

5

1

R1010k

Out

C16.8nF

LM 308

Wire Connections: • Ground to 3 pin connector • Freq. Out to 3 pin connector • 5v to 3 pin connector • F to Strain gage Y • B to C5 • A to C5 • D to dd • Dd to Strain Gage X • H to I • J to Stain Gage X • C to Strain Gage Y Jumper Connections • Put a jumper between the

2 boards when a line “comes out” of the board.

PCB Traces to Sever (marked with X) • Through-hole in R2 • Connection between R2

and LM308

C5

R1

R5

R8a&b& C3

R6

R2

R4a&b

c

d

C2

R7a&b& C4

blank

R9

R10

C1

Freq Out

Gnd

5 V

f

LM30

8

ba 7555

dd

h

i

j

114


115


116


117


118


119


120


KRISTEN N. JAAX (206) 729-8050

[email protected] 4204 NE 95th St. Seattle, WA 98115

EDUCATION

Doctor of Medicine, Anticipated Completion December, 2002 University of Washington, School of Medicine, Seattle, WA. Doctor of Philosophy in Bioengineering, June, 2001 University of Washington, School of Engineering and School of Medicine, Seattle, WA. Ph.D. Dissertation: A Robotic Muscle Spindle: Neuromechanics of Individual and Ensemble Response. Bachelor of Science in Mechanical Engineering, with distinction, June, 1994 Stanford University, School of Engineering, Palo Alto, CA.

PROFESSIONAL AND ACADEMIC EXPERIENCE

Research Assistant, 7/96-6/01 Biorobotics Laboratory. Department of Electrical Engineering,University of Washington, Seattle, WA. PI: Blake Hannaford, PhD. Projects: hardware design and manufacture of miniature displacement sensor & linear actuator; control algorithm design; printed circuit board design; computational and mechatronic modeling of muscle mechanics, neural transduction and neural encoding; mechatronic modeling of individual mammalian muscle spindle; experimental reconstruction & analysis of multi-sensor integration behavior in muscle spindle population.

Research Assistant, 6/95-8/95 Human Motion Analysis Lab, Department of Physical Medicine and Rehabilitation, University of Washington, Seattle, WA, PI: Joe Czerniecki Projects: developed gait analysis software

Biomechanical Engineering Technician, 6/93-9/93 Bone Densitometry Lab, Spinal Cord Injury (SCI) Center, Palo Alto Veterans Affairs, Palo Alto, CA. PI: B. Jenny Kiratli, PhD. Projects: analyzed mechanical loading of femur during fracture events to develop a clinical estimator of fracture risk in Spinal Cord Injury.

Biomechanical Engineering Technician, 6/92-9/92

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Rehabilitation R&D Center. Palo Alto Veterans Affairs, Palo Alto, CA. PI: Eric Sableman, PhD. Projects: spine-stabilization mechanisms for Roto-Rest trauma therapy beds; design of automated quadriplegic transfer device.

Summer Preprofessional, 6/90-9/90, 6/91-9/91 IBM Federal Sector Division, Space Station Data Management System (DMS), Houston, TX. Supervisor: Bob Brauer. Projects: translated and updated thermal control software; generated software functionality document.

Surgical Research Assistant, 6/89-8/89 Dr. Michael DeBakey Summer Surgery Fellowship, Baylor College of Medicine, Houston, TX. Supervisor: Polk Smith. Ophthalmic Photography Assistant, 6/87-8/87 Hermann Eye Center, Hermann Hospital, Houston, TX. Supervisor: Sue McCraney.

PUBLICATIONS

Jaax, KN, “A Robotic Muscle Spindle: Neuromechanics of Individual and Ensemble Response,” Ph.D. Thesis, Department of Bioengineering, University of Washington, 2001.

Jaax, KN, van Donkelaar, C.C., Drost, M.R., Hannaford, B, “Fusimotor Effect on Signal Information Content of Ia Ensemble Model Reconstructed from Dynamic Intramuscular Strain Data,” submitted to Journal of Physiology, June, 2001.

Jaax, KN, Hannaford, B, “Mechatronic Design of an Actuated Biomimetic Length and Velocity Sensor,” submitted to IEEE Transactions on Robotics and Automation, May, 2001.

Jaax, KN, Hannaford, B, “A Biorobotic Structural Model of the Mammalian Muscle Spindle Primary Afferent Response,” submitted to Annals of Biomedical Engineering, February, 2001.

Jaax, KN, Marbot, PH, Hannaford B, “Development of a Biomimetic Position Sensor for Robotic Kinesthesia,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Takamatsu, Japan, Nov. 2000. pp. 1255-60.

Jaax, KN, “Biomechanical Analysis of the Role of Wrist Guards in "Split-Top" Forearm Fractures,” Proceedings of the World Congress 2000 on Medical Physics and Biomedical Engineering and the 22nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Chicago, IL, July 2000.

Abstracts: Jaax, KN, “Developing a Robotic Muscle Spindle,” Proceedings of The Whitaker Foundation

Biomedical Engineering Research Grants Conference 2000, La Jolla, CA, August 2000. Jaax, KN, Hannaford B, “A Biorobotic Model of the Mammalian Muscle Spindle,” Annals of

Biomedical Engineering. 2000. 28(S1). pp. S-8.

122


Jaax, KN and BJ Kiratli, “Estimating Risk of Fracture During Activities of Daily Living Within the Spinal Cord Injured Population,” Journal of Investigative Medicine. 1996. 44(1) pp. 155A.

PRESENTATIONS

Jaax, KN, “A Robotic Muscle Spindle and Other Current Research in the Biorobotics Laboratory at the University of Washington,” Invited Seminar at ATR Human Information Processing Research Laboratories, Kyoto, Japan, Nov. 2000.

Jaax, KN, Marbot, PH, Hannaford B, “Development of a Biomimetic Position Sensor for Robotic Kinesthesia,” IEEE/RSJ International Conference on Intelligent Robots and Systems, Takamatsu, Japan, Nov. 2000, platform presentation.

Jaax, KN, Hannaford B, “A Biorobotic Model of the Mammalian Muscle Spindle,” Biomedical Engineering Society Annual Meeting, Seattle, WA, Oct. 2000, platform presentation.

Jaax, KN, “Developing a Robotic Muscle Spindle,” The Whitaker Foundation Biomedical Engineering Research Grants Conference 2000, La Jolla, CA, August 2000, poster presentation.

Jaax, KN, “Biomechanical Analysis of the Role of Wrist Guards in "Split-Top" Forearm Fractures,” World Congress on Medical Physics and Biomedical Engineering, Chicago, IL, July 2000, platform presentation.

Jaax, KN, and BJ Kiratli, “Estimating Risk of Fracture During Activities of Daily Living Within the Spinal Cord Injured Population,” Western Medical Student Research Conference, Carmel, CA, Feb. 1996, platform presentation.

Jaax, KN. “Developing a Mathematical Model of Human Eye Movements to Optimize Strabismus Surgery,” 2nd Annual Super! Conference, Gainesville, FL, Apr. 1990, platform presentation.

Jaax, KN. “Developing a Mathematical Model of Human Eye Movements to Optimize Strabismus Surgery,” Supercomputing ’89, Reno, NV, Nov. 1989, poster presentation.

FELLOWSHIPS AND HONORS

Paul G. Allen Foundation for Medical Research Fellowship, 2001-present Whitaker Graduate Fellowship in Biomedical Engineering, 1996-present Medical Scientist Training Program Fellowship, 1994-present ARCS Fellowship, Seattle Chapter, 1998-2000 Travel Grant, NSF Engineering Education Scholars Workshop at Carnegie Mellon, July 1999 Terman Award, Top 5% of graduating class, Stanford School of Engineering, 1994 President, Tau Beta Pi Engineering Honor Society, Stanford University, 1994 Phi Beta Kappa, 1994 1st Place, SuperQuest National Supercomputing Competition, sponsored by IBM, NSF, and the

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Cornell Theory Center, 1989

PROFESSIONAL SERVICE & ASSOCIATIONS

Ad Hoc Reviewer, Annals of Biomedical Engineering Ad Hoc Reviewer, Behavioral & Brain Sciences Admissions Committee Member, University of Washington School of Medicine, 1995-2000 Member, IEEE, Engineering in Medicine and Biology Society Member, Biomedical Engineering Society

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