neuron based time optimal controller of horizontal saccadic eye movements

2nd Reading

May 9, 2014 11:42 1450017

International Journal of Neural Systems, Vol. 24 (2014) 1450017 (19 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0129065714500178

A NEURON-BASED TIME-OPTIMAL CONTROLLEROF HORIZONTAL SACCADIC EYE MOVEMENTS

ALIREZA GHAHARI and JOHN D. ENDERLE∗

Department of Electrical EngineeringUniversity of Connecticut, 260 Glenbrook Road

Storrs, Connecticut 06269-2247, USA∗[email protected]

Accepted 1 April 2014Published Online 9 May 2014

A neural network model of biophysical neurons in the midbrain for controlling oculomotor muscles duringhorizontal human saccades is presented. Neural circuitry that includes omnipause neuron, premotorexcitatory and inhibitory burst neurons, long lead burst neuron, tonic neuron, interneuron, abducensnucleus and oculomotor nucleus is developed to investigate saccade dynamics. The final motoneuronalsignals drive a time-optimal controller that stimulates a linear homeomorphic model of the oculomotorplant. To our knowledge, this is the first report on modeling the neural circuits at both premotor andmotor stages of neural activity in saccadic systems.

Keywords: Saccade; neural network; burst firing; circuit model; time-optimal controller; oculomotor plant.

1. Introduction

The control mechanism of the human binocularvision is staggering in its complexity, and hasstunned many neuroscientists in their quest to matchits functionality to a greater or lesser degree. Sac-cades are fast eye movements during which a target istracked by registering the image of that target on thefovea. They are categorized into two different modesof operation: small (ranging from 3 to 7) and large(above 7).1–3 The differentiation between these twomodes is based on the fact that when the saccadesize increases, more active motoneurons are firingsynchronously to form the agonist neural input forsmall saccades. For large saccades, however, the num-ber of active motoneurons firing maximally remainsunchanged, and the agonist pulse duration is directlyrelated to the saccade magnitude.

The saccade neural network requires involve-ment of a series of neurons designed to imitate thebehavior of actual neuronal populations in the hori-zontal saccade controller. Considerable research has

been concerned with developing a generic neuronmodel that is capable of offering a framework uponwhich to base the development of other neurons. Thewidespread use of spiking neural networks (SNNs)lies in leveraging efficient learning algorithms to thespike response models.4,5 A novel Multi-SpikePropalgorithm was adapted in a multi-SNN model thatenabled pulse-encoding spike-train communication.6

For epilepsy and seizure detection, the classifi-cation accuracy obtained from the Multi-SpikePropalgorithm was notably higher than that of a back-propagation training algorithm for the single-spikingSNN model. A spike pattern association neuron iden-tified five classes of spike patterns associated withnetworks of 200, 400 and 600 synapses, with successrates of 96%, 94% and 90%, respectively.7 A hybridanalog-digital circuitry was laid out to implementan SNN that outputs the postsynaptic potential byintegrating the filtered action potentials.8 A neu-ral system comprised of a persistent firing sensoryneuron, a habituating synapse and a motoneuron was

∗Corresponding author.

1450017-1

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

http://dx.doi.org/10.1142/S0129065714500178

2nd Reading

May 9, 2014 11:42 1450017

A. Ghahari & J. D. Enderle

developed to illustrate the spike-timing dependencyof the working memory.9 The persistent firing neu-ron stems from the Izhikevich10 neuron model, thehabituating synapse is a conductance-based model,and the motor neuron captures the essence of theHodgkin–Huxley (HH) model.11 These studies pro-vide abundant evidence that an SNN is well suited toevoke the properties of the firing patterns of the pre-motor neurons during the pulse and slide phases of asaccade. However, none of the studies have presenteda demonstration of the neural circuits reproducingelectrophysiological responses in a network of neu-rons at both premotor and motor levels. To encom-pass all of the desired neural behaviors, a neuralcircuitry is used to match the firing rate trajectoryof the premotor neurons.3 We model the saccade-induced spiking activities at the premotor level withan HH model for the bursting neurons, and with amodified FitzHugh–Nagumo (FHN) model12 for thetonic spiking neurons.

A neural controller justifies the relationshipbetween neural firing rates and eye orientation dur-ing saccades.1 Time-optimal control theory of thehorizontal saccade system establishes the fact thatthere is a minimum time required for the eyes toreach their destination by involving thousands ofneurons. Conjugate goal-directed horizontal saccadeswere well characterized by a first-order time-optimalneural controller.3 It is important that this new,more complex time-optimal controller ascertains thatthe firing rate of the motoneurons does not changeas a function of saccade magnitude during the pulseinnervation of the oculomotor plant.

In this paper, we focus on neural control of hor-izontal human saccades. A neural network modelof saccade-related neural sites in the midbrain isfirst presented. We next characterize the underly-ing dynamics of each neural site in the neural net-work that needs to be treated in the case of spikingneurons. In consequence, to match the dynamics ofthe neurons and the synapses, a saccadic circuitry,including omnipause neuron (OPN), premotor exci-tatory burst neuron (EBN), inhibitory burst neuron(IBN), long lead burst neuron (LLBN), tonic neuron(TN), interneuron (IN), abducens nucleus (AN), andoculomotor nucleus (ON), is developed. Finally, themotoneuronal control signals drive a time-optimalcontroller that stimulates a linear homeomorphicmodel of the oculomotor plant. We abbreviate the

“conjugate goal-directed horizontal human saccade”with the term “saccade” throughout the paper. Theterms “motoneurons” and “agonist (antagonist) neu-rons” are also substitutable in this paper.

2. Neural Network

Neurophysiological evidence and developmentalstudies indicate that important neural populations,consisting of the cerebellum, superior colliculus (SC),thalamus, cortex and other nuclei in the brain-stem, are involved in the initiation and controlof saccades.1–3,13–15 The studies also provided evi-dence that saccades are generated through a parallel-distributed neural network. Zhou et al.1 illustrateda comprehensive compendium of this neural net-work programmed to move the eyes 20. Such net-work topology is updated herein by inclusion ofIN between TN and motoneurons. A block dia-gram representation of the network of involved neu-ral sites is depicted in Fig. 1.3 The two sides ofthe symmetric network in Fig. 1 are known as theipsilateral side and the contralateral side. The ipsi-lateral side exhibits coordinated activities in the ini-tiation and control of the saccade in the right eye,while the contralateral side simultaneously coacti-vates with the ipsilateral side to generate a saccadein the left eye. Each neuron fires in response to otherneurons to stimulate the final motoneurons on bothsides of the network to execute a saccade. The neu-ral populations on each side of the midline excite andinhibit one another sequentially to ensure that thiscoactivation leads to the coordination of movementbetween the eyes. Neural coordinated activities of theSC and the fastigial nucleus (FN) of the cerebellumare identified as the saccade initiator and termina-tor, respectively. The FN is stimulated by the SCand projects ipsilaterally and contralaterally to theLLBN, IBN and the EBN on the opposite side ofthe network. The cerebellar vermis (CV) retains thecurrent position of the eye by registering the informa-tion on the proprioceptors in the oculomotor musclesand in an internal eye position reference. The CValso keeps track of the dynamic motor error usedto control the saccade amplitude in connection withthe nucleus reticularis tegmenti pontis (NRTP) andthe SC.

In the context of the neuroanatomical connectiv-ity structure in Fig. 1, the saccade neural network

1450017-2

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017

A Neuron-Based Time-Optimal Controller of Horizontal Saccadic Eye Movements

LGNRetinalError

LGNRetinalError

Midline

Contralateral

Ipsilateral

T0

SC

-25

T0

S. Nigra

-40

T0

S. Nigra

-40

T0

SC

T0

NRTP

T0

Vermis

T0

Vermis

-20

T0

NRTP

-20

T0

LLBN

T0

Fast. Nucl.

-20

T0

Fast. Nucl.

-20

T0

LLBN

-20

T0

IBN

T0

EBN

T

OPN

-10

T

EBN

-8

T

IBN

-8

(a)

Fig. 1. A functional block diagram of the saccade neural network model.3 Solid lines are excitatory and dashed linesare inhibitory. Each block represents the neural activity of the corresponding neural site. Saccade initiates at time zero,and terminates after T seconds. The negative time for each neural site refers to the onset of the burst before saccade: (a)Neural pathways from the formation of the lateral geniculate nucleus (LGN) retinal error to the EBN and the IBN. (b)Neural pathways from the EBN and the IBN to the rectus muscles in both eyes.

1450017-3

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


T

OPN

-10

Contralateral

Ipsilateral

Rig

htE

yeP

lan

t

Late

ralR

ectu

sM

uscl

eM

edia

lRec

tus

Mus

cle

Lef

tEye

Pla

nt

Med

ialR

ectu

sM

uscl

eLa

tera

lRec

tus

Mus

cle

00

T0

IBN

T0

EBN T

TN

-5

T

O. Nucleus

-5

T

A. Nucleus

-5

T

O. Nucleus

-5

T

A. Nucleus

-5

T

TN

-5

T

EBN

-8

T

IBN

-8

T

IN

-5

T

IN

-5

(b)

Fig. 1. (Continued)

1450017-4

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


includes neuron populations to imitate the behaviorof actual neuronal populations in the initiation, con-trol and termination of the saccadic burst generator.Here, we indicate the characteristics of the premotorneurons in the paramedian pontine reticular forma-tion (PPRF) and the IN. The synaptic properties ofall the other neural sites are explained in Ref. 3.

2.1. Premotor neurons in the PPRF

The PPRF encompasses neurons that show domi-nantly increasing burst frequencies of up to 1000Hzduring the saccade and remain inactive during theperiods of fixation. The LLBN and the medium leadburst neuron (MLBN) are the two types of burst neu-rons in the PPRF. The LLBN forms an excitatorysynapse to the IBN and an inhibitory synapse to theOPN.

There are two types of neurons in the MLBN:the EBN and the IBN. The EBN serves as one ofthe vital excitatory inputs for the saccade controller.The primary inputs to this neuron are the excita-tory input of the SC and the inhibitory input fromthe contralateral IBN and OPN. This neuron formsexcitatory synapses to the TN and the AN. The IBN,on the other hand, controls the firing of the EBN aswell as the TN, both of which are on the oppositeside of the network to the corresponding IBN. It alsoinhibits the ON and the IN on the same side as itself.This neuron receives excitatory inputs from the FNof the cerebellum on the opposite side and the LLBNon the same side and an inhibitory input from theOPN.

2.2. Interneuron

Many excitatory and inhibitory INs in the centralnervous system stimulate and control motoneurons.The cerebellum aggregates most of these INs whosefunctionality depends on the anatomical aspects andproperties of their membranes. The IN receives theexcitatory and inhibitory inputs from the corre-sponding TN and IBN, respectively. It consecutivelyprovides the step component to the agonist andantagonist neural controllers. As with the TN, theutility of the modified FHN model under the tonicbursting mode exhibits this particular neural spik-ing activity.12 Section 3 characterizes the under-lying dynamics of each neural site in the neuralnetwork.

3. Firing Characteristics of EachType of Neuron

The saccade generator investigated herein is builtupon the extant research.1–3,13,14,16,17 The model isfirst-order time-optimal; that is, it does not dependon the firing rate of the neurons to determine the sac-cade magnitude. In this section, we demonstrate thefeatures of the neural dynamics of the saccade neu-ral network to highlight the underlying neurologicalcontrol implications.

3.1. Neural activity

The structure of the saccade neural network lever-ages a neural coding so that burst duration istransformed into saccade amplitude under the time-optimal condition. Such coding manifests activities,including the onset of burst firing before saccade,peak firing rate and end of firing with respect to thesaccade termination, for each neuron on the basisof the physiological evidence. These characteristicsare provided for the neural sites3 as a framework forour simulations. Table 1 summarizes the activities ininitiation, control and termination of the burst firingin the neural network, generating a saccade in theright eye.

3.2. Burst discharge mechanism

As motoneurons receive excitatory input from theipsilateral EBN, the burst discharge in them duringa saccade is adequately similar to the EBN bursting.Such burst discharge in the motoneurons is respon-sible for the movement of the rectus muscles duringa saccade. The firing rate trajectory of the EBN is ofprime importance in control of such a saccade. Thepresented EBN model3 showed a constant plateauof bursting during the second portion of the burstbefore the decay occurs.18 Figure 2 shows the EBNbursting rate as a fit to the data in Ref. 18 with(A), and the data in Ref. 19 with (B), for threesaccades. Note that the interval [0, T1] depicts thesmallest possible interval required for EBN burstaccording to physiological evidence. The interval [T1,T2] represents the duration of the second portion ofthe burst, by the end of which the EBN drives themotoneurons to move each eye to its destination.The gradual decay in firing occurs in the intervalfrom T2 until the EBN stops firing. This intervalindicates the time it takes for the OPN to resume

1450017-5

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


Table 1. Firing activity of neural sites during an ipsilateral saccade.3

Burst onset Peak firing Burst end w.r.t.Neural site before saccade (ms) rate (Hz) saccade end

Contralateral SC 20–25 800–1000 Almost the sameIpsilateral LLBN 20 800–1000 Almost the sameOPN 6–10 150–200 (before and after) Almost the sameIpsilateral EBN 6–8 600–1000 ∼ 10 ms beforeIpsilateral IBN 6–8 600–800 ∼ 10 ms beforeIpsilateral TN/IN 5 Tonic firing (before and after) Resumes tonic firing when

saccade endsIpsilateral AN 5 400–800 ∼ 5 ms beforeIpsilateral FN 20 Pause during saccade, and a

burst of 200 Hz near theend of the saccade

Pause ends with burst ∼ 10 msbefore saccade ends; resumestonic firing ∼ 10ms aftersaccade ends

Contralateral FN 20 200 Pulse ends with pause ∼ 10msbefore saccade ends; resumestonic firing ∼ 10ms aftersaccade ends

Ipsilateral CV 20–25 600–800 ∼ 25 ms beforeIpsilateral NRTP 20–25 800–1000 Almost the sameIpsilateral Substantia Nigra 40 40–100 Resumes firing ∼ 40–150 ms after

saccade ends

Firing Rate Firing Rate Firing Rate

Time Time Time

17 0

0 T1 T100 T1

20

T2 T2T2

(a)

Firing Rate Firing RateFiring Rate

TimeTimeTime

207 10

0 00T1 T1 T1T2 T2 T2

(b)

Fig. 2. Block sketch of EBN firing rates for 7, 10 and 20 saccades drawn3 to match the data in Ref. 18 shown in (A),and in Ref. 19 shown in (B).

its inhibition of the EBN. The mechanism for intro-ducing this decay in firing into the axon model is toreduce the firing rate linearly by modifying the chan-nel equations,3 as described later. Note that the onlydifference between the three saccades is the dura-tion T2–T1. In other words, the saccade magnitudeis governed only by the duration of time that the

ipsilateral EBN bursts in the interval [0, T2]. Wemodel the EBN firing rate by applying the firingrate trajectory shown in Fig. 2(B), where a slow lin-ear reduction in firing rate is assumed in the inter-val [T1, T2]. This slow drop in firing rate has beenreported to be attributed to the IBN inhibition ofthe LLBN.3 We also consider this trajectory for the

1450017-6

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


contralateral SC stimulation of the ipsilateral LLBN,which accords with the different simulations in exam-ining the effects of several depolarizing stimulus cur-rents in the EBN axon (specifically, see Fig. 2.10 inRef. 3). It should be emphasized at this point how theSC contributes to the optimal control of the saccadesby driving the LLBN. The movement fields withinthe SC are indicators of the number of neurons fir-ing for different small and large saccades (see locusof points on a detailed view of the SC retinotopicmapping in Fig. 2.14 in Ref. 3). It is implied thatthe number of cells firing in the LLBN is determinedby the number of cells firing in the SC as long asthere is a feedback error maintained by the CV.3 Thenumber of the OPN cells firing after inhibition fromthe LLBN determines, in turn, how many EBN cellsare released from inhibition. Finally, the number ofEBN cells firing determines the number of motoneu-rons driving the eyes to their destination. However,the movement field is known to be almost constantfor saccades above 7.3

3.3. Sequence of neural firing

The saccade completion involves the evolution ofsome events in an orderly sequence in the neuralsites. The output of each block in Fig. 1 depicts therelative firing pattern at each neural site manifestedduring the saccade: saccade initiates at time zero,and T represents the saccade termination. The neg-ative time for each neural site refers to the onsetof the burst before saccade (see Table 1). The neu-ral activity within each block is represented as pulsesand/or steps, consistent with the described burst dis-charge mechanism, to reflect the neural operation astiming gates.3 Finally, motoneurons innervate rectusmuscles in both eyes at the end interaction level ofthe block diagram.

The following description outlines eight stepsrequired to implement the saccade control strategyin the context of Fig. 1. It represents the sequenceof events accounted for in Ref. 3, with modificationsmade in steps (iv)–(vii) to indicate the function oflocal neural integrators (TN and IN) in providingthe step of innervation to the motoneurons:

(i) The deep layers of the SC initiate a saccadebased on the distance between the currentposition of the eye and the desired target.

(ii) The ipsilateral LLBN and EBN are stimulatedby the contralateral SC burst cells. The LLBNthen inhibits the tonic firing of the OPN. Thecontralateral FN also stimulates the ipsilateralLLBN and EBN.

(iii) When the OPN ceases firing, the MLBN (EBNand IBN) is released from inhibition.

(iv) The ipsilateral IBN is stimulated by the ipsi-lateral LLBN and the contralateral FN ofthe cerebellum. When released from inhibi-tion, the ipsilateral EBN responds with a post-inhibitory rebound burst for a brief period oftime. The EBN, when stimulated by the con-tralateral FN (and perhaps the SC), enables aspecial membrane property that causes a high-frequency burst that decays slowly until inhib-ited by the contralateral IBN. The burst firingactivity of EBN is integrated through the con-nection with the TN. The IN follows closelythe same integration mechanism as that of theTN.

(v) The burst firing in the ipsilateral IBN inhibitsthe contralateral EBN, IN and AN, as wellas the ipsilateral ON.

(vi) The burst firing in the ipsilateral EBN causesthe burst in the ipsilateral AN, which thenstimulates the ipsilateral lateral rectus mus-cle and the contralateral ON. With thestimulation of the lateral rectus muscle by theipsilateral AN, and the inhibition of the ipsi-lateral medial rectus muscle via the ON, a sac-cade occurs in the right eye. Simultaneously,the contralateral medial rectus muscle is stim-ulated by the contralateral ON, and with theinhibition of the contralateral lateral rectusmuscle via the AN, a saccade occurs in the lefteye. Hence, the eyes move conjugately underthe control of a single drive center. During thefixation periods, the INs provide the steady-state tensions required to keep the eyes at thedesired destination.

(vii) At the termination time, the CV, operat-ing through the Purkinje cells, inhibits thecontra-lateral FN and stimulates the ipsilateralFN. Some of the stimulation of the ipsilateralLLBN and IBN is lost because of the inhibi-tion of the contralateral FN. The ipsilateral FNstimulates the contralateral LLBN, EBN andIBN. The contralateral EBN then stimulates

1450017-7

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


the contralateral AN. The contralateral IBNthen inhibits the ipsilateral EBN, TN, andAN, and the contralateral ON. This inhibitionremoves the stimulus to the agonist muscle.

(viii) The ipsilateral FN stimulation of the contralat-eral EBN allows for modest bursting in thecontralateral EBN. This activity then stimu-lates the contralateral AN and the ipsilateralON. Once the SC ceases firing, the stimulusto the LLBN stops, allowing the resumptionof OPN firing that inhibits the ipsilateral andcontralateral MLBN, hence terminating thesaccade.

The advances in computational neural modelinghave supplied us with abundant information at dif-ferent structural scales, such as the biophysical,4–7

the circuit3,8 and the systems levels.9 The followingincludes our modeling of the premotor and motorneurons at the circuit level. We introduce a neuralcircuit model that can be parameterized to matchthe described firing characteristics of each type ofneuron.

4. Neural Modeling

A typical neuron embodies four major components:cell body, dendrites, axon and presynaptic terminals,as shown in Fig. 3. The neural cell body encompassesthe nucleus and is similar to the other cells. Dendritesact as the synaptic inputs for the preceding excita-tory and inhibitory neurons. Upon this stimulation ofthe neuron at its dendrites, the permeability of thecell’s plasma membrane to sodium intensifies, andan action potential moves from the dendrite to theaxon.16 The transmission of action potential alongthe axon facilitates by means of nodes of Ranvier inthe myelin sheath. At the end of each axon, there are

Dendrites

Cell Body

Axon Hillock

Axon

Node of Ranvier

Myelin Sheath

Presynaptic Terminals

Fig. 3. A schematic presentation of the different com-ponents of a neuron.16

presynaptic terminals, from which the neurotrans-mitters diffuse across the synaptic cleft.

A complete understanding of the properties ofa membrane by means of standard biophysics, bio-chemistry and electronic models of the neuron willlead to a better analysis of membrane potentialresponse. A neuron circuit model is desired toquantify the neural stimulations meticulously, thusreflecting the physiology linked to the dendrite, cellbody, axon and presynaptic terminal of each neuron.Such a model is sketched in this section, togetherwith the description of the modifications on it,required to populate a neural network for control ofsaccades. The saccade neural network includes eightneuron populations at premotor and motor levels:

(i) Long lead burst neuron (LLBN),(ii) Omnipause neuron (OPN),(iii) Excitatory burst neuron (EBN),(iv) Inhibitory burst neuron (IBN),(v) Tonic neuron (TN),(vi) Interneuron (IN),(vii) Abducens nucleus (AN),(viii) Oculomotor nucleus (ON).

The saccade circuitry underlies the dynamicsof the above eight distinct neurons, each of whichcontributes to the control mechanism of the sac-cade. Except for the OPN, the proposed parallel-distributed neural network accommodates two ofeach of the other neurons in the network. The den-drite model delineated below is adjustable to thestimulation mechanism of all eight neurons. Theaxon model for all spiking neurons, except the EBNand OPN, adheres to the HH model. The EBNand OPN are neurons that fire automatically whenreleased from inhibition — these neurons are mod-eled using a modified HH model.3 The TN integratesits input and is modeled with a FHN model underthe tonic bursting mode.12 The presynaptic termi-nal elicits a pulse train stimulus whose amplitudedepends on the membrane characteristics of the post-synaptic neuron.

4.1. Dendrite model

The dendrite is partitioned into a number ofmembrane compartments, each of which has a pre-determined length and diameter. Each compart-ment in the dendrite has three passive electrical

1450017-8

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


Cm Cm Cm

REQ

vm2 vmnvm1

Ra Ra

VTH

REQ REQ

VTHVTH

is (t)

Fig. 4. The dendrite circuit model with n passive compartments: is(t) models the stimulus current from the adjacentneurons to the dendrite. Each compartment has membrane electromotive, resistive and capacitive properties — VTH , REQ

and Cm in the second compartment are noted. The batteries in the circuit, VTH , are the Thevenin equivalent potential ofall the ion channels. The axial resistance Ra connects each compartment to the adjacent ones (remains unchanged amongthe neurons). Appropriate values for the membrane resistance and capacitance of the dendrite model are found to matchthe firing characteristics of each type of neuron.

characteristics: electromotive force (emf), resistanceand capacitance, as shown in Fig. 4. Axial resistanceis used to connect the dendrite to the axon.

The presynaptic input to the dendrite is modeledas a pulse train current source (is). The node equa-tion for the first dendrite compartment is

Cmdvm1

dt+

vm1 − VTH

REQ+

vm1 − vm2

Ra= is, (1)

where vm1 is the membrane potential of the firstcompartment and vm2 is the membrane potential ofthe second compartment. The membrane resistanceREQ, capacitance Cm, and the emf VTH characterizeeach compartment. Ra is the axial resistance.

For all intermediate dendrite compartments,there are two inputs: the input from the previouscompartment’s membrane potential and the inputfrom the next compartment’s membrane potential.The node equation for the second compartment is

Cmdvm2

dt+

vm2 − VTH

REQ+

vm2 − vm1

Ra

+vm2 − vm3

Ra= 0, (2)

where vm3 is the membrane potential of the thirdcompartment.

The last dendrite compartment receives just oneinput from its preceding compartment. The corre-sponding node equation is

Cmdvmn

dt+

vmn − VTH

REQ+

vmn − vm(n−1)

Ra= 0,

(3)

where the membrane potential vmn is related tothe preceding compartment’s membrane potential(vm(n−1)) through the axial resistance Ra.

The dendrite model of each neuron is realized byexperimental tuning of the parametric capacitanceand resistance properties of the above-defined den-drite model. This parametric adaptation allows forthe accommodation of the synaptic transmission inthe neural network as required to stimulate eachpostsynaptic neuron. Each neuron’s dendrite risetime constant determines the delay to emulate thepostsynaptic potential propagation along the den-drite, consistent with the initiation of firing withrespect to the saccade onset provided in Table 1.Table 2 includes the membrane resistance and capac-itance of the dendrite compartments for each neuron.

Initial condition of the capacitor is set to VTH atsteady-state. Computational efficiency accrues whenthe minimum number of compartments in the den-drite model is required. We chose to include 14compartments in the dendrite to achieve the desiredmembrane properties in each type of neuron. Forillustration, the EBN dendritic membrane poten-tial across the first, second, third and last compart-ments is shown in Fig. 5. The farther the compart-ment is along the dendrite, the smoother its potentialresponse to the pulse train current source.

4.2. Axon model

The HH model of the axon serves as the basis forthe neurons modeled here — only the EBN and theOPN are based on a modified HH model. As elab-orated later, this modification leads these neurons

1450017-9

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


Table 2. Parametric realization of eight distinct neurons in terms of dendritic, axonal and synapticbehaviors in the proposed neural circuitry.

Dendrite Axon Synapse

Capacitor Resistor Firing threshold PulseNeuron (µF) (kΩ) voltage (mV) Coefficient amplitude (µA)

LLBN 0.5 3.75 −45 18,000 20OPN 1.0 6.3 −60 1800 45EBN 0.45 3.1 −60 35,000 75IBN 0.35 4.5 −45 15,000 65AN 0.35 5.5 −45 17,000 55ON 0.45 4.0 −45 17,000 55TN 0.35 4.5 NA NA 10IN 0.4 4.5 NA NA 10

100 105 110 115 120 125 130 135 140 145

-65

-64

-63

-62

-61

-60

-59

-58

Time (ms)

Dendritic P

ote

ntial (m

V)

0 20 40 60 80 100 120 140 160 180 200-85

-80

-75

-70

-65

-60

-55

Time (ms)

Den

driti

c P

oten

tial (

mV

)

1 compartment 2 compartments 3 compartments 14 compartments

Fig. 5. The EBN dendritic membrane potential acrossthe different compartments. The corresponding intervalof burst firing is emphasized. The membrane parame-ter values are: VTH = −60mV, Cm = 0.45 µF, REQ =3.1 kΩ and Ra = 100 Ω.

Cm

+

Vm

-

RKRNa RCl

EK EClENa

Im

Outside

Inside

Fig. 6. The circuit model of an unmyelinated portionof squid giant axon.3 The variable active gate resistancesfor Na+ and K+ are given by RK = 1/gKN4 and RNa =1/gNaM3H , respectively. The passive gates are modeledby a leakage channel with resistance, Rl = 3.33 kΩ. Thebattery is the Nernst potential for each ion: El = 49.4 V,ENa = 55 V and EK = 72 V.

to fire automatically at high rates after releasingfrom inhibition, given minor stimulation. The HHmodel describes the membrane potential at the axonhillock caused by conductance changes. The circuitdiagram of an unmyelinated portion of squid giantaxon is illustrated in Fig. 6. The node equation thatexpresses the membrane potential Vm as a function ofstimulus current Im from the dendrite and voltage-dependent conductances of the sodium and potas-sium channels is3

gkN4(Vm − Ek) + gNaM3H(Vm − ENa)

+(Vm − El)

Rl+ Cm

dVm

dt= Im, (4)

wheredN

dt= αN (1 − N) − βNN,

1450017-10

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


dM

dt= αM (1 − M) − βMM,

dH

dt= αH(1 − H) − βHH,

gk = 36 × 10−3S, gNa = 120 × 10−3 S.

The coefficients in the above first-order system ofdifferential equations are related exponentially to themembrane potential Vm, i.e.

αN = 0.01 × V + 10

e( V +1010 ) − 1

ms−1,

βN = 0.125e( V80 ) ms−1,

αM = 0.1 × V + 25

e( V +2510 ) − 1

ms−1,

βM = 4e( V18 ) ms−1, αH = 0.07e( V

20 ) ms−1,

βH =1

e( V +3010 ) + 1

ms−1, V = Vrp − Vm mV,

(5)

where the resting potential Vrp is −60mV.The neural firing rate of all the bursting neu-

rons has been adjusted to meet the peak firing raterequirement in Table 1. This adjustment intendsfor each neuron to contribute to the generation ofthe saccade by mimicking the required physiologicalproperties.3 To this end, the right-hand side of theN , M and H differential expressions in Eq. (4) ismultiplied by appropriate coefficients to achieve thedesired peak firing rates. For instance, the requiredcoefficient for the EBN has been 35,000; thereby itpresents a peak firing rate at 1000Hz. Note that theabove equations of the basic HH model of the axonhave been used for all the bursting neurons, exceptfor the EBN and the OPN. For these latter neurons,the modified HH model is used to change the thresh-old voltage from −45mV to −60mV. Enderle andZhou illustrated experiments in which this variationcaused EBN to fire autonomously without the exis-tence of any excitatory stimulus.3 From the descrip-tion of the dominant effect of the sodium channelcurrent on the changes in the threshold voltage at thebeginning of the action potential3, the threshold volt-age in the EBN axon model is changed by modifyingthe αM equation to

αM = 0.1 × V + 10

e( V +1010 ) − 1

ms−1. (6)

The OPN axonal threshold voltage of firing has beenadjusted following the same modification by Eq. (6).

This alteration of the threshold voltage for the EBNand the OPN enables them to fire spontaneouslywithout any significant depolarization from periph-eral current stimuli. Table 2 lists the firing thresh-old voltage and the coefficient required to adjust thepeak firing rate for each bursting neuron.

The axon transfers an action potential from thespike generator locus to the output end of the synap-tic mechanism. The transmission along the axon thusamounts to introducing a time delay, after which theaction potential appears at the synapse.

4.3. Synapse model

When the action potential appears at the synapse,packets of neurotransmitter are released. This ismodeled by excitatory or inhibitory pulse train stim-uli to stimulate the dendrite of the postsynaptic neu-ron more realistically. Current-based synapse modelsoffer significant analytical convenience when describ-ing how a postsynaptic current pulse is triggered byan action potential in very large SNNs.20 As thesemodels disregard the voltage-dependent property ofthe postsynaptic currents, for the networks with boththe interspike intervals and the burst onsets of theneurons uniformly distributed, they are preferredto the conductance-based synapse models. Follow-ing the concepts of the current-based synapse mod-els, the amplitude and width of each single pulse arechosen experimentally to provide the desired postsy-naptic behavior in the interconnected neurons. Fig-ure 7 shows a number of action potentials and thesynaptic current pulses of the EBN toward the endof the burst firing interval. Note that the time delaybetween each action potential and the correspondingcurrent pulse is evident.

In addition to the transmission time delay alongthe axon, all chemical synapses introduce a smalldelay before the generation of postsynaptic poten-tials from an input excitatory or inhibitory pulsetrain. This delay accounts for the time required forthe release of neurotransmitters and the time it takesfor them to distribute through the synaptic cleft.This small synaptic delay was taken into effect byincreasing the rise time constant of the followingpostsynaptic dendritic compartments.

As indicated, the amplitude and width of synap-tic current pulses for each neuron are uniquely chosenin order that the postsynaptic neurons exhibit thedesired behavior. Table 2 includes such amplitude

1450017-11

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


146 147 148 149 150 151

0

Axo

nal P

oten

tial (

mV

)

Time (ms)146 147 148 149 150 151

50

Syn

aptic

Cur

rent

(µA

)Fig. 7. A train of action potentials (dashed) and cur-rent pulses (solid) reflecting the synaptic mechanism ofthe EBN. Each current pulse shows a time delay withrespect to the corresponding action potential due to thetransmission delay along the axon.

of the synaptic current pulses. This table summa-rizes all the differences (dendritic, axonal and synap-tic) among eight distinct neurons whose realizationis important in the neural circuitry for time-optimalcontrol of the saccade.

We next describe the time-optimal control of alinear homeomorphic muscle model that captures thenonlinear properties of the muscle, namely, force–velocity and length–tension relationships.

5. Linear Homeomorphic Modelof Muscle

The time-optimal controller model was investigatedto obtain the saccadic eye movement model solutionthat drives the eyeball to its destination for differentsaccades.3,17 Here, we describe that the saccadic eyemovement model solution is characterized by realiza-tion of the agonist and antagonist controller models,thereby providing the active-state tensions as inputsto a linear homeomorphic model of the oculomotorplant.

5.1. Muscle neural stimulation

The first-order time-optimal controller model isdefined by two complementary controllers: theagonist controller model and the antagonist con-troller model. These models describe how the neuralinnervation signals are converted to the active-state

tensions to drive the agonist and antagonist mus-cle during the saccade. The active-state tensions aredefined as the low-pass filtered neural innervationsignals as follows.

5.1.1. Agonist controller model

The agonist controller is a first-order pulse-slide-step neuronal controller that describes the agonistactive-state tension as the low-pass filtered neuralstimulation signal.3 The neural stimulation signalis the firing rate of the ipsilateral AN and that ofthe contralateral ON. The slide is meant to modelthe transition between the pulse and the step expo-nentially. The expression of low-pass filtering of theneural innervation input to the agonist controllermodel is

Fag =Nag − Fag

τag, (7)

where

τag = τgac(u(t − t1) − u(t − t2))

+ τgdeu(t − t2), (8)

and Nag represents the agonist neural innervationinput from which the agonist active-state tension,Fag, is generated. The agonist time constant τag isexpressed by two step functions dependent on theagonist activation time constant, τgac, and the deac-tivation time constant τgde. t1 indicates the saccadelatent period, and t2 is the start of the transitionslide interval for the agonist controller. It is note-worthy that the activation (deactivation) time con-stant in the model accounts for the different dynamiccharacteristics of muscle upon increasing (decreas-ing) stimulation.

5.1.2. Antagonist controller model

The antagonist muscle is unstimulated by a pauseduring the saccade, and remains fixed by a step inputto keep the eyeball at its destination. To serve thispurpose, a first-order pause-step neuronal controlleris defined.3 The neural stimulation signal to the con-troller is the firing rate of the ipsilateral ON andthat of the contralateral AN. The antagonist active-state tension can be expressed as the low-pass filteredpause-step waveform:

Fant =Nant − Fant

τant, (9)

1450017-12

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


where

τant = τtde(u(t − t1) − u(t − t3))

+ τtacu(t − t3), (10)

and Nant denotes the antagonist neural innervationinput, and the Fant is the antagonist active-statetension generated. The antagonist time constant isdescribable by two step functions, introducing theantagonist deactivation time constant, τtde, and theactivation time constant τtac. t1 takes into accountthe latent period, and t3 is the onset of the changeto the step component necessary to keep the eyeballsteady at its destination.

The time-optimal controller has been found tobe reasonably consistent with the characteristics ofthe main-sequence diagrams.3 In what follows, a lin-ear homeomorphic oculomotor plant that capturesthe nonlinear properties of the muscle in the saccadesystem is described.

5.2. Oculomotor plant

A linear homeomorphic muscle model that capturesthe nonlinear properties of the muscle, namely, force–velocity and length–tension relationships,3 is inves-tigated. The proposed oculomotor plant is shown inFig. 8. Note that one Voigt passive element (withviscosity element Bp and elasticity element KP inparallel) and two horizontal rectus muscles [agonist(ag) and antagonist (ant)] have been realized. Foreach rectus muscle, there is an elastic element Kse,

Fig. 8. The mechanical components of the oculomotorplant3 that include two rectus muscles [agonist (ag) andantagonist (ant)] connected to the eyeball through nodes1 to 4.

in parallel with a viscous element B2, connected toa series active-state tension generator F in paral-lel with a length–tension elastic element Klt and aviscous element B1. θ represents the angle of rota-tion of the eyeball, and x denotes the length of arccorrespondingly traversed. The displacements fromequilibrium for the stiffness elements in each rectusmuscle are represented by x1 to x4. The displace-ments from equilibrium at the primary position forx1 to x4 quantities are denoted by xp1 to xp4. Themoment of inertia of the eyeball is represented byJp. Consequently, the tension contracting the ago-nist muscle is Tag, and the tension that stretches theantagonist muscle is Tant.

The linear behaviors in the model of theeye movement system render a series of rigorousequations3 that are tractable to the derivation of alinear differential equation, expressing saccades as afunction of θ. After writing the node equations forthe corresponding free body diagrams of the system,the third-order linear differential equation to solvefor the response for a saccade is

δ(B2(Fag − Fant) + Kse(Fag − Fant))

=...θ + P2θ + P1θ + P0θ, (11)

where

J =Jp

πr2× 180, B =

Bp

πr2× 180,

K =Kp

πr2× 180, Kst = Kse + Klt,

B12 = B1 + B2, δ =180

πrJB12,

P2 =JK st + B12B + 2B1B2

JB12,

P1 =2B1Kse + 2B2Klt + B12K + KstB

JB12,

P0 =KstK + 2KltKse

JB12,

and r is the radius of the eyeball. Note that Eq. (11)shows that the inputs to the muscle model arethe agonist active-state tension Fag and the antag-onist active-state tension Fant, obtained from thedescribed neural controllers.

The analytical solutions for both Fag and Fant

were yielded in the previous study,3 and it was foundthat the three different saccade characteristics arevery well matched to those of the experimental data.

1450017-13

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


The estimation routine3 in that study involved esti-mation of 25 parameter values. No empirical param-eters are involved herein other than the parametersof the investigated oculomotor plant for human (seep. 47).3 The simulation specifications and resultsfollow.

6. Simulation Results

We have investigated three large saccades: 10, 15

and 20. At the neural circuit level, each neuronconsists of 14 dendrite compartments with mem-brane properties included in Table 2. The determi-nation of the rise time constant for each neuron’sdendrite plays a vital role in the integration of cur-rent pulses at the synapse. Analyses of the den-dritic membrane potentials were performed with theNI Multisim circuit simulation suite, and the neu-ral network and oculomotor plant were simulatedin the MATLAB/Simulink environment. The mod-ular programming and test of each individual neu-ron were achieved to constitute our Simulink modelof the system of neurons at the highest level ofthe hierarchy. The saccade-induced spiking activi-ties at the premotor level are modeled with a HHmodel for the bursting neurons.3 The tonic spikingbehavior of the TN/IN is implemented by a mod-ified FHN model as well.12 Transmission along theaxon introduced a delay after the presence of actionpotential at the axon hillock, after which an actionpotential appears at the synapse. Synaptic connec-tions between functionally modeled neuron popula-tions are modeled following a current-based synapsescheme20 (see Table 2 for differences in the mem-brane parameters among the neurons). The onsetdelay before saccade, peak firing rate and burst ter-mination time for the different neuron populationsare chosen according to Table 1.

Duration of burst firing is set to evoke the desiredsaccades with pulse train synaptic stimuli slightlybefore the onset of ipsilateral or contralateral sac-cades. According to the saccade duration-saccademagnitude characteristic of the main-sequencediagrams,3 this duration was chosen 50ms for the10 saccade, 56ms for the 15 saccade and 65ms forthe 20 saccade. Notice that the latent period is notzero in our simulations. The saccades start at 120ms,and they terminate solely after the duration of theburst under the time-optimal control strategy.

For sample illustrations, the plots of den-dritic membrane potential (first column), axonalmembrane potential (second column) and synapticcurrent pulse train (third column) for the burst neu-rons and the IN of the ipsilateral side in generationof the 10 saccade are shown in Fig. 9. Recall thatthe train of action potentials is converted to a trainof the current pulses in the presynaptic terminal ofthe neuron to provide excitatory or inhibitory inputto the succeeding neurons based on the neural con-nections in Fig. 1. This current pulse flows throughthe postsynaptic dendritic compartments of the lat-ter neurons, thus providing the smooth postsynap-tic potentials to prime the axonal compartment. Itis evident that upon the increasing of the stimuluscurrent pulse amplitude, the depolarization of thepostsynaptic membrane intensifies.

Notably, the burst onset and offset for each pre-motor neuron in Fig. 9 agrees with its place withinsaccadic circuitry’s hierarchical processing sequencein generating the final motoneuronal signals. It alsoappears that the synapse propagation raises differ-ent excitatory and inhibitory postsynaptic potentialsin the dendritic compartments of each postsynapticneuron (shown in the first column of Fig. 9). Onecan realize that, in view of the trajectory of changesin the membrane potential among the compartments,each postsynaptic neuron, in turn, can either becomecloser to firing an action potential chain, or inhibitedfrom firing.

It is clear that the LLBN membrane responseis different from the rest, since it is stimulated bythe contralateral SC current pulse (not a pulse trainstimulus). Note that the EBN serves as the fun-damental excitatory input for the analysis of thesaccade controllers. This neuron makes excitatorysynapses to the TN and AN. It is also noted thatthe IN firing rate follows that of the TN and isdetermined based on the current position of the eyebefore the completion of the saccade. In this context,the agonist and antagonist active-state tensions dur-ing the periods of fixation are found as functions ofeye position at steady-state (see p. 47).3 Obviously,the burst-tonic firing activity of the AN [Fig. 9(q)]reflects the burst firing of the EBN and the tonicfiring of the IN.

We next present the ipsilateral agonist (first row)and antagonist (second row) burst-tonic firing rateswith their respective active-state tensions, based on

1450017-14

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


020

4060

8010

012

014

016

018

020

0-7

0

-60

-50

-40

-30

Den

dri

tic

Po (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

(r)

(s)

(t)

(u)

ten

tial

(m

V)

020

4060

8010

012

014

016

018

020

0-8

0

-60

-40

-200

Axo

nal

Po

ten

tial

(m

V)

020

4060

8010

012

014

016

018

020

005101520

Syn

apti

cC

urr

ent

( µA

)

020

4060

8010

012

014

016

018

020

0-6

6

-64

-62

-60

020

4060

8010

012

014

016

018

020

0-1

00-50050

020

4060

8010

012

014

016

018

020

00204060

020

4060

8010

012

014

016

018

020

0-9

0

-80

-70

-60

-50

020

4060

8010

012

014

016

018

020

0-1

00-50050

020

4060

8010

012

014

016

018

020

0020406080

020

4060

8010

012

014

016

018

020

0-9

0

-80

-70

-60

-50

020

4060

8010

012

014

016

018

020

0-1

00-50050

020

4060

8010

012

014

016

018

020

0020406080

020

4060

8010

012

014

016

018

020

0-9

0

-80

-70

-60

-50

020

4060

8010

012

014

016

018

020

0-5

000

500

1000

020

4060

8010

012

014

016

018

020

00510

020

4060

8010

012

014

016

018

020

0-6

0

-50

-40

-30

-20

020

4060

8010

012

014

016

018

020

0-1

00-50050

020

4060

8010

012

014

016

018

020

00204060

020

4060

8010

012

014

016

018

020

0-1

00-80

-60

-40

-20

Tim

e (m

s)0

2040

6080

100

120

140

160

180

200

-100-50050

Tim

e (m

s)0

2040

6080

100

120

140

160

180

200

0204060

Tim

e (m

s)

E B

N

A NI N O N

I B N

O P

N

L L

B N

Fig

.9.

The

den

dri

tic

mem

bra

ne

pote

nti

alin

mV

(firs

tco

lum

n),

axonalm

embra

ne

pote

nti

alin

mV

(sec

ond

colu

mn)

and

the

synapti

cpulse

curr

ent

train

inµA

(thir

dco

lum

n)

of

each

neu

ron

ina

10

ipsila

tera

lsa

ccade

neu

ralco

ntr

oller

:(a

)–(c

)LLB

N(d

)–(f

)O

PN

(g)–

(i)

EB

N(j

)–(l

)IB

N(m

)–(o

)IN

(p)–

(r)

AN

(s)–

(u)

ON

.E

ach

neu

ron

fire

sin

harm

ony

wit

hth

eoth

ers

ingen

erati

ng

this

sacc

ade.

For

exam

ple

,E

BN

burs

tact

ivity

start

sat

112

ms

and

last

sfo

r50

ms

(see

Table

1).

1450017-15

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


0 20 40 60 80 100 120 140 160 180 2000

500

1000F

iring

Rat

e (H

z)

Time (ms)0 20 40 60 80 100 120 140 160 180 200

0

1

2

Act

ive-

stat

e T

ensi

on (

N)

0 20 40 60 80 100 120 140 160 180 2000

500

1000

Firi

ng R

ate

(Hz)

Time (ms)0 20 40 60 80 100 120 140 160 180 200

0

1

2

Act

ive-

stat

e T

ensi

on (

N)

(a) (b)

0 20 40 60 80 100 120 140 160 180 200 2200

500

1000

Firi

ng R

ate

(Hz)

Time (ms)0 20 40 60 80 100 120 140 160 180 200 220

0

1

2

Act

ive-

stat

e T

ensi

on (

N)

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

Firin

g R

ate

(Hz)

Time (ms)0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

Activ

e-st

ate

Tens

ion

(N)

(c) (d)

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

Firin

g R

ate

(Hz)

Time (ms)0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

Act

ive-

stat

e Te

nsio

n (N

)

0 20 40 60 80 100 120 140 160 180 200 2200

50

100

150

Firin

g R

ate

(Hz)

Time (ms)0 20 40 60 80 100 120 140 160 180 200 220

0

0.2

0.4

0.6

Act

ive-

stat

e Te

nsio

n (N

)

(e) (f)

Fig. 10. The ipsilateral neural stimulation signals for the agonist (first row) and antagonist (second row) neural controlinputs (dashed) and the corresponding active-state tensions (solid) plotted on the same graph: 10 saccade [(a) and (d)]15 saccade [(b) and (e)] 20 saccade [(c) and (f)]. The agonist and antagonist controller models provide the active-statetensions to the linear homeomorphic model of the oculomotor plant.

the agonist and antagonist controller models, inFig. 10. It is of interest to note that the firingrate of each AN in all scenarios does not vary as afunction of saccade magnitude, thus proving that the

proposed time-optimal controller is well capable ofmimicking the physiological properties of the saccadeby just changing the duration of burst. The obtainedagonist–antagonist firing patterns fairly well match

1450017-16

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

Time (ms)

Po

siti

on

(D

egre

es)

0 20 40 60 80 100 120 140 160 180 200-2

0

2

4

6

8

10

12

14

16

Time (ms)

Po

siti

on

(D

egre

es)

(a) (b)

0 20 40 60 80 100 120 140 160 180 200 220

0

5

10

15

20

Time (ms)

Po

siti

on

(D

egre

es)

(c)

Fig. 11. The ipsilateral control simulation results for the eye position generated by the proposed first-order time-optimalneural saccade controller used in a third-order linear muscle model: (a) 10 saccade (b) 15 saccade (c) 20 saccade. Notethat the saccade onset is 120 ms for all cases, but the end time of each saccade differs from the others.

the estimated wave-forms based on the system iden-tification approach (see Fig. 1.19 in Ref. 3).

The ipsilateral control simulation results of eyeposition for the three different saccades under thetime-optimal control strategy are demonstrated inFig. 11. The parameterized saccadic oculomotorplant for human has been used (see p. 47).3 Fromthe corresponding saccade velocity profile, the peakvelocity is found 223 s−1 for the 10 saccade,277 s−1 for the 15 saccade and 322 s−1 for the20 saccade. It is noteworthy that the investigatedoculomotor plant does not considerably influence themain-sequence diagrams, as envisioned.1

The entire neural stimulation signals and motiontrajectories (position, velocity and acceleration) on

the contralateral side were in close agreement withtheir corresponding ipsilateral signals for all of thesaccades.

7. Discussion

The simulation results show remarkable agreementwith those provided by analytical descriptions ofthe agonist and antagonist neural inputs, and thecorresponding active-state tensions (see Fig. 1.19 inRef. 3). From these results, it follows that the agonistburst duration uniquely controls the saccade mag-nitude under the time-optimal control strategy. Theburst duration is found to be correlated to the MLBNduration of burst firing from the extracellular single-unit recordings.21

1450017-17

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


As evident by different firing rate trajectories ofthe EBN, this neuron has tightly coupled character-istics to the saccade.3 For the three saccades exam-ined herein, the initial duration of the EBN firingremained constant among them. However, the dura-tion of the second portion of the burst discharge(gradual drop) varied among them based on theentire duration of the burst firing. As indicated inTable 1, the EBN firing lags behind the saccadeby 6–8ms, whereas the AN starts burst firing 5 msbefore the saccade (see Fig. 9). Finding the dendriteparameters for both of these neurons in meeting therequired onset time delay was experimentally chal-lenging. Moreover, the AN peak firing rate at thebeginning of the pulse period showed dependency onthe EBN peak firing rate, necessitating the use ofcorresponding coefficients to change the initial firingrate of the basic HH model (see Table 2).

Implementing the OPN dendrite and synapsemodels in order that this neuron stops inhibiting theEBN about 10 ms before the saccade and resumes itsinhibition almost when the saccade ends, was subjectto numerous parameter tunings (see Table 1). With-out this coordination in timing of the burst firing inthe EBN, this neuron can show the rebound burstfiring activity. This rebound burst, in turn, causesthe saccade to deviate from the normal characteris-tics. It also was vital that the end of the IBN inhi-bition of the antagonist motoneurons coincides withthe resumption of tonic firing in them such that nodeviation from the normal saccade is present.

While the midbrain coordination mechanism ingenerating saccades is qualitatively studied,13,15,22 acomplete neural circuitry that includes both the pre-motor and motor neurons in quantifying the finalmotoneuronal command to eye muscles has not yetbeen attended. The utility of SNNs to the biophys-ical modeling of interconnected neurons4–7 eluci-dates broad insights to modeling at higher structuralscales, such as the circuit3,8 and the systems levels.9

The computer simulations of neural circuitry hereinallows for synaptic stimuli to propagate throughthe saccade pathways so that the motoneurons ulti-mately drive the oculomotor plant.

A time-optimal neuronal controller for humansaccadic eye movements was first proposed basedon experimental data analysis.23 Exactly how thereis a one-to-one relationship between the firing ratein agonist neurons and the saccade magnitude is a

matter of controversy in the literature. For refer-ence, firing rate-saccade amplitude dependent con-trollers were proposed.19,24 These studies lackedthe use of a homeomorphic oculomotor plant, andnone of the investigated controllers thereof offeredthe feasibility of a time-optimal control strategy.We exploited the first-order time-optimal controller3

that includes the activation and deactivation timeconstants in agonist and antagonist controller mod-els. This controller has been proven to agree withthe experimental findings.23,25 The set of agonist–antagonist controllers of the oculomotor plant sup-ports the time-optimal control theory in that themotoneurons’ firing rate does not determine saccademagnitude. It is noteworthy that the duration of ago-nist burst discharge solely determines the saccademagnitude based on Fig. 10.

The eye position results in Fig. 11 substantiatethe time-optimal controller due to the close agree-ment between them and the analytical solutionsof saccade characteristics.3 The saccade duration-saccade magnitude characteristic3 as well corrobo-rates our simulation results. These observations giverise to the accuracy of the experimentally foundmembrane parameters in the modeling of each neu-ron listed in Table 2.

The insight from neural modeling at the biophys-ical and the circuit levels in the saccade generatorwill be a stepping stone for other studies, such ascontrolling a robot’s eye movements and diagnosingmild traumatic brain injury.

8. Conclusion

We investigated the neural control of three large sac-cades. A parallel-distributed and hierarchical neuralnetwork model of the midbrain was first presented.To develop the quantitative computational modelsthat establish the basis of this functional neural net-work model, we next described the saccade burst gen-erator dynamics. A neural circuit model was thendemonstrated and parameterized to match the firingcharacteristics of eight neuron populations at boththe premotor and motor stages. The obtained neuralcircuitry permitted the cycle of action potential gen-eration by synaptic transmission of pulse train stim-uli among the neurons. Finally, the saccades werewell characterized by integrating a time-optimal con-troller to a third-order linear homeomorphic model

1450017-18

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.

2nd Reading

May 9, 2014 11:42 1450017


of the oculomotor plant. The proposed saccadic cir-cuitry is thus a complete model of saccade genera-tion since it not only includes the neural circuits atboth the premotor and motor stages of the saccadegenerator, but it also uses a time-optimal controllerto yield the desired saccade magnitude. The saccadecharacteristics were found to be well correlated tothose found by analytical descriptions and experi-mental data.

References

1. W. Zhou, X. Chen and J. D. Enderle, An updatedtime-optimal 3rd-order linear saccadic eye plantmodel, Int. J. Neural Syst. 19(5) (2009) 309–330.

2. J. D. Enderle, Neural control of saccades, InJ. Hyona, D. Munoz, W. Heide and R. Radach(eds.), The Brain’s Eyes: Neurobiological and Clin-ical Aspects to Oculomotor Research, Progress inBrain Research, Vol. 140 (Elsevier, 2002), pp. 21–50.

3. J. D. Enderle and W. Zhou, Models of HorizontalEye Movements. Part 2: A 3rd-Order Linear SaccadeModel (Morgan & Claypool Publishers, San Rafael,CA, 2010), 144 pp.

4. S. Ghosh-Dastidar and H. Adeli, Spiking neural net-works, Int. J. Neural Syst. 19(4) (2009) 295–308.

5. S. Ghosh-Dastidar and H. Adeli, Improved spikingneural networks for EEG classification and epilepsyand seizure detection, Integr. Comput.-Aided Eng.14 (2007) 187–212.

6. S. Ghosh-Dastidar and H. Adeli, A new supervisedlearning algorithm for multiple spiking neural net-works with application in epilepsy and seizure detec-tion, Neural Netw. 22(10) (2009) 1419–1431.

7. A. Mohemmed, S. Schliebs, S. Matsuda and N.Kasabov, SPAN: Spike pattern association neuronfor learning spatio-temporal spike patterns, Int. J.Neural Syst. 22(4) (2012) 1–16.

8. J. L. Rossello, V. Canals, A. Morro and J. Verd,Chaos-based mixed signal implementation of spikingneurons, Int. J. Neural Syst. 19(6) (2009) 465–471.

9. K. Ramanathan, N. Ning, D. Dhanasekar, L. Guoqi,S. Luping and P. Vadakkepat, Presynaptic learningand memory with a persistent firing neuron and ahabituating synapse: A model of short term persis-tent habituation, Int. J. Neural Syst. 22(4) (2012)1250015.

10. E. M. Izhikevich, Simple model of spiking neurons,IEEE Trans. Neural Netw. 14 (2003) 1569–1572.

11. A. L. Hodgkin and A. F. Huxley, A quantitativedescription of membrane current and its application

to conduction and excitation in nerve, J. Physiol.117 (1952) 500.

12. R. T. Faghih, K. Savla, M. A. Dahleh and E. N.Brown, Broad range of neural dynamics from atime-varying FitzHugh–Nagumo model and its spik-ing threshold estimation, IEEE Trans. Biomed. Eng.59(3) (2012) 816–823.

13. J. D. Enderle, (1994), A physiological neural networkfor saccadic eye movement control, Air Force Mate-rial Command, Armstrong Laboratory AL/AO-TR-1994-0023, 48 pp.

14. J. D. Enderle and E. J. Engelken, Simulation of ocu-lomotor post-inhibitory rebound burst firing usinga Hodgkin-Huxley model of a neuron, Biomed. Sci.Instrum. 31 (1995) 53–58.

15. O. A. Coubard, Saccade and vergence eye move-ments: A review of motor and premotor commands,Eur. J. Neurosci. 38 (2013) 3384–3397.

16. J. D. Enderle and J. D. Bronzino, Introduction toBiomedical Engineering, 3rd edn. (Elsevier, Amster-dam, 2011), 1253 p.

17. J. D. Enderle, Models of Horizontal Eye Movements.Part 1 : Early Models of Saccades and Smooth Pur-suit (Morgan & Claypool Publishers, San Rafael,CA, 2010), 149 pp.

18. D. A. Robinson, Models of mechanics of eye move-ments, in Models of Oculomotor Behavior and Con-trol, ed. B. L. Zuber CRC Press (Boca Raton, FL,1981), pp. 21–41.

19. G. Gancarz and S. Grossberg, A neural model ofthe saccade generator in reticular formation, NeuralNetw. 11 (1998) 1159–1174.

20. W. K. Wong, Z. Wang and B. Zhen, Relationshipbetween applicability of current-based synapses anduniformity of firing patterns, Int. J. Neural Syst.22(4) (2012) 1250017.

21. D. L. Sparks, R. Holland and B. L. Guthrie, Sizeand distribution of movement fields in the monkeysuperior colliculus, Brain Res. 113 (1976) 21–34.

22. M. M. G. Walton, D. L. Sparks and N. J. Gandhi,Simulations of saccade curvature by models thatplace superior colliculus upstream from the localfeedback loop, J. Neurophysiol. 93 (2005) 2354–2358.

23. M. R. Clark and L. Stark, Time optimal controlfor human saccadic eye movement, IEEE Trans.Automat. Contr. AC-20 (1975) 345–348.

24. C. A. Scudder, A new local feedback model ofthe saccadic burst generator, J. Neurophysiol. 59(5)(1988) 1455–1475.

25. J. D. Enderle and J. W. Wolfe, Frequency responseanalysis of human saccadic eye movements: Estima-tion of stochastic muscle forces, Comput. Biol. Med.18 (1988) 195–219.

1450017-19

Int.

J. N

eur.

Sys

t. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by D

r. J

ohn

End

erle

on

06/1

0/14

. For

per

sona

l use

onl

y.