CHAOTIC TRANSMEMBRANE VOLTAGE BEHAVIOUR IN HIPPOCAMPAL NEURONS

A thesis submitted in conformity with requirements for the Degree of Muter of Appiied Science

Graduate Department of Elecfncal and Computer Engineering and Institute of Biomediesi Engineering

Faeulty of Applied Science and Engineering University of Toronto

Supervisor: Profwor Berj L Bardakjiiin

@ Copyright by Maja Jeiaca, 1998

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CHAOTIC TRANSMEMBRANE VOLTAGE BEHAVlOUR IN EUPPOCAMPAL NEURONS MASTER OF APPLIED SCIENCE, 1998 MAJA JELACA DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING UNIVERSITY OF TORONTO

Abstract

A novel appnwch, based on nonluiear system analysis and chaos theory, was used to

investigate the cornplex oscillatory activity in the brain The coupled Mapped Clock

Oscillator model, with intrinsic panuneten obtained nom experimental recordings fiom

hippocampal neurons, was used for the study of oscillatory behaviour in this region of the

brain Emphasis was placed on the investigation of synaptic couphg and self-feedback

rnechanisms between the oscillators. It was found that the chemical synaptic coupling

and self-feedback mechanisms between the neurons greatly increases the functionally

chaotic region in the topological coupling map of the hippocampal neurons. ln addition,

the region of non-osciliatory behaviour and oscillator death is greatly decreased. It is

bel ieved that synaptic coupling and sel f-feedback mechanisms offer great flexibility to

the system and provide for a more robwt operation in the chaotic region, without

experiencing oscillatory mode transitions that may lead to different pathoIogica1 states.

Acknowledgements

Using tIiis opportunity I wouid like to extend my gratitude to Rof Berj L. Bardakjian for

convincing me that 1 can do anythmg. This thesis was more than a positive experience.

Speciai ttiank you to Ed Vigmond and AmK Mitchell. 1 don't think I would have dow

half of these things without them.

A big th& you to Aaron Courville for doing a million things to help.

1 am more than grateful to my whole family, Marko, Sandra and Dragana for their

constant s u p p a understanding and making yet another degree more fun

Table of Contents

List of Symbols List of Figures List of Tables

1 introduction 1.1 Nonlinear Analysis of Biological Systems 1 -2 Coupling in Neurons

1 -2.1 Gap Junctions 1.2.2 Ephatic Coupling 1 -2.3 Chernical Synapses

1.3 Epilepsy 1.4 Hippocarnpus 1.5 Moâels of Epilepsy

1. S. 1 Zero Calcium Mode1 1.6 Scope of the Thesis

1 -6.1 Thesis Hypothesis

2 Theoretical Background 2.1 Nonlinear Dynamical Systems

2.1.1 2.1.1 .Analysis of Dyaamical Systems 2.1.1.1 Qualitative Characteristics Analysis 2.1.1 -2 Quantitative Anaiysis

2.1.1.2.1.1 Lyapunov Exponents 2.1.1.2.1.2 Lyapunov Fractaï

Dimension 2.2 Nodinear Oscillators

2.2.1 Uncoupled Mkpped Clock Osciiiator 2.2.1.1 Clock 2-2.1 -2 Transformer

vi vii ix

3 Methods 3.1 Mode1 Description

3.1.1 Coupting conngurati011~ 3.2 Cornputer Simulations

4 Resdts 4.1 Unidirectional Couphg

4.1.1 Coupling via Portal PI 4.1.2 Coupling via Portal P, 4.1.3 Coupiing via Portals P4 and P,

4.2 Bidirectional Coupling 4.2.1 Chaotic Regions Depending on Synaptic Strwgth

4.2.1.1 Effects of Self-feedback 4.2.2 Non-oscillatory Regions

5 Discussion 5.1 General 5.2 Information Coduig in Chernical Synapses 5.3 Synaptic Strength and Self-feedback

Mechanisms in Synaptic Transmission

6 Couclasions and Recornrnendations for Future Work

Bibliography

List of Symbols

TCM

Long Term Potentiation

Mapped Clock Oscillator

Topoiogicai Coupling Map

Clock variables

Transformer output variable, transmembrane voltage

Common coupling factor

Specific coupling factors

ûscillator input portais

Comrnon feedback factor

Specific feedback factors

Figures List of

Figure 1.1. Figure 1.2. Figure 1.3.

Gap jlmction diagram. From Kandel et. al. [18] 6 Chernical synapse diagram. From Kandel et ai. [18] 9 EEG discharges, extracellular and intnicellulai ElCtivity associated

with PDS. From Iohnstoa et al. [20] 11 Figure 1.4. The structure of the hippocampal formation From Brown et. al 181 13

Figure 2.1. Singularities obsewed in a two dimensionai system. From Seydel 14 1 1 20 Figure 2.2. Stable and unnable limit cycle. From Parker et. al. [33] 20 Figure 2.3. Quasiperiodic tom. From Parker et, al. 1331 2 1 Figure 2.4. Lorenz strange attractor. From Seydel[4 11 23

Figure 3.1. Schematic diagram of a four portal MC0 mode1 configuration with self-feedback loops 34 Figure 3.2. Flow chart describing computer simulation steps 38

Figure 4.1. Changes in a) signal amplitude, b) resting level and c) frequency due to an increase in c+ 41 Figure 4.2. Changes in frequency of MC01 due to an increasing q at different levels of c, 42 Figure 4.3 Changes in frequency of MCOl due to an increasing c+ at different arnounts of self-feedback applied 43 Figure 4.4. Changes in signal a) amplitude, b) resting level and c) frequency due to an increme in c, 45 Figure 4.5. Changes in resting level of MC01 due to an increasing c, at different levels of c4 46 Figure 4.6. Changes in resting level of M a l due to the changes in c, at incrursing levels of self-feedback from Bfl.0 to p m . 4 . 47 Figure 4.7. Changes in resting level with negative coupling appiied through input portal Pp. 48 Figure 4.8. Effet of increasing stimulus fkquency on resting level of MC0 l 49 Figure 4.9. Effect of increasing stimulus fiequency on frequency of MC01 50 Figure 4.10. Efféct of inmeasiog stimulus fiequency on amplitude of MC01 51 Figure4.11. TCMform.1 54 Figure4.12. TCMforfH.2 54

Figure 4.13. TCM for $ 4 . 3 55 Figure 4.14. Percent of region exhiiituig chaotic behaviour depending on the amount of seiGfeedback applied 55 Figure 4.15. TCM when ody B, is activated 57 Figure 4.16. TCM when oniy PI is activated 57 Figure 4.17. TCM when ody f& is activated 58 Figure 4.18. TCM when $,+ and $, are acîivated 58 Figure4.19. Percent of investigated couphg region exhibiting chaotic behaviour depending on which self-feedback portal is activated.

59 Figure 4.20. TCM for common coupling factor c= 1.125 and no self-feedback 60 Figure 4.2 1. TCM for common couphg &or ~ 1 . 2 5 and no self-feedback 60 Figure 4.22. TCM for comrnon couphg factor FI. 125 and P=U.2 61 Figure 4.23. TCM for common coupling factor ~ ~ 1 . 2 5 and P 4 . 2 61 Figure 4.24. Percent of investigated couplhg regions exhibiting akhaotic and b) non-oscillatory behaviour depending on coupling and self-feedback amount. 62

Figure 5.1. Postsynaptlc depolaridon in response to changing fiequency stimulus fiom presy~ptic terminal. (a) presynaptic stimulus; @) presynaptic Frequency profile; (c) changes in postsynaptic resting potential. (From Stevens[35]). 67 FigureS.2. Relationship between PSP amplitude and the quantity of trammitter released. (Frorn Stevens [3 5 1). 68 Figure 5.3. Calcium role in the activation of non-NMDA receptors and LTP maintenance 72

List of Tables Table 2.1. Characterization of oscillatory modes for a dynamical system using Lyapunov Exponents. 24

Chapter 1

Introduction

Rhythmicai activiîy in biologicai systems is one of the most important Me SuStaining factors.

Even brief intemrptions of this activity can lead to serious i l k and even death A great

nurnber of biologicai pmceses has been found to osciilate with 24 h o u cycles, Lilre

susceptiiility to dmgs or even travel tbrough time zones. Higher tkquency rhythms can be

associated with signai such as electrical signals that control the contractions of h w smooth

and skeletal muscles, or production and secretion of insulin fiom the pancreatic islet ceh

[6]. In addition, electrical mythmic activiîy plays a key d e in the noornial bctioning of the

brain-

Continuous electricai oscillatory activity can be observeci fiom the potential of the brai.

recorded h m the siirhace of the head Recorded ripples in electrical potentials are r e f d to

as brain waves, while the themplete record of such oscillations consti~es the

ele~f~~ncephalogram (EEG). The intensity of the recordeci signals and the paaenis they

exhiiit can be snidied to detennine an overd excitation kvel in the brain ûverall excitation

d t s from the action of a network of newons and nuciei extending throughout the brain

stem. Br* waves d l y exhiibit a high degree of irreguiarity and a#iuatnes rangiag fmm

0.5 to 100 Hz 161. Howwer, braia waves can becorne m e l y reguiar, such as in the cases

of pathological abnormalities like epiiepsy [18]. Regular b h wave activity associatexi with

epilepsy d t s h m entrained activity of a large number of coupied neurons. Dependhg on

the size of the &ected area of the brain, this entrained activiîy cm lead to devastabng seinme

attacks. The exact causes underlinhg the onset of such entrainment are not wmpletely

known

Since a single neuron can produce ody a d exûaceiiular ammt, a large number of

neurons must be synchronized in their actiMty to produce regular electrid patterns recorded

h m the cerebral surhce of epilepsy patients. Synchrony requires simdtaneous activation of

the neurons at the same fkquency with no phase diffefence. however, a phase lag is

obse~ed, even though the neurons fire action potentials at the same fkquency, they are said

to be enfrained

A new approach in the study of dynamics of neuronal activity and entrainment is based on

the coupled nonlinear oscillaton used to mode1 the oscillatory electncat activity of the brain

cells. Coupled osciliators can be used to mode1 both morphologicd and fiinctiod

cormectivity of the nemm and provide deeper understanding of the variety of coupling

conditions tb t can be observed in the brain Such a wnlinear dynamicaI system cm be used

to midy a number of different oscillatory modes, Uicluding megular pancms observed in the

r e ~ r d i n g s of the brain finictio~ r e f d to as chsroshsros Chaos is defineci as irrepuiar, aimost

d m like, d a t o r y activity arising in the de&mb&ic ~ynem [33]. Nonlinear dynamics

and specifically chaotic beliaviour are of great importance in dmloping a redistic mode1 of

neufonai activity.

1.1 Nonlinear Analysis of Biological Systems

Majority of the physiological processes have usually beai descr i i as behg govemed by

homeostasis. The human body is thought to sustain a steady state despite numerous

pemrrbatiom fiom its environment. When a stimulus dimntK the steady date mauaained in a

human organism, the body responds in a snmmr that te& to restoce the equilibnum.

Pruiciples of homeostasis have emerged £hm the negetive feedback systems theory which is

d l y applied to linear systems. However, sources of aonlineafities in biological systems

are numerous. Kinetic process commonly exhibit nonlinear behaviour as a result of different

feedback mechanimis anâ cooperative interactions. Positive feedback was found to have an

essential role in the oscillatory behaviour of a number of biochemical proases [25]. These

and similar nonlinearities point out that complex dyoarmcal models must be employed in

order to be able to accrwtely d e m i the behaviour of biologicat systems. In addition, a

saidy of complex dyoamical systems may provide some insight into the changes beoHeen

various behaviour modes that may lead to diffèrent pathological conditions. Pathological

States may be characterizRd by a range of behaviow patkms which may be caused by a

change in one or more controllmg parameters in the operational modes of the dymrmcal

system.

The question of wbcarr chaos signifies a heaithy or a pathologicai state of an organism has

been arMressed a aumber of thes [13,15,42]. Some &ts ise chaos to define "'dyaamical

diseases" that result h m the changes h m order to disorder in the bioIogicai systems [lq.

Thus, thqr associate chaos with pabiologkal circumsfst~lces~ On the d e r band, some studies

associate chaos with hdthy conditions in the body- It is h m that physiological systems

exhibit a great degree of variabiuty and inegulanty when Mthy. It is thought that the

chactahics of chaotic activity aüow a healthy physiologicai system to adapt to the

constantly changing environment. Furthemore, Mes discovered t h t the degree of

variabilitydecreaseswitbdiseeseandaging~isthe*isewithelectr~cardio~signals

mrded h m patients suffiring h m hart disease [34,42]. Electrid signais h m the heart

becorne regular when the heaxt is no longer capable of responding to environmental changes.

Similady, brain signais cecordai from epileptic or schizophrenic patients exhi i significantly

lower degree of chsioticity [38,40].

1.2 Coupiing in Neurons

Coupliog mechanisms between the neurons play a crucial mie in both normal and

pathological hctioning of the brain As such, they are some of the key factors in the

epileptogmesk process or the @on of epilgmform activity. Three coupiing

mechanisms are assumecl to oontriiute to the mediation or enfrainment of the elecfncai

activity betweea nemns. They are (1) excitatory or inhibitory chernical synapses, (2)

eiectrotonic coupling through gap junctions, (3) q h & c hractions or elestrical M d &kas

baweai ueighboring murons. Electrical intetactions are iikely to play a d e in the orset of

entrainment spiles, but Ït is @y beiieved that the chemicai sriiapses are primaniy

responsible for the onset of the epileptifbm bmting Since the mle of chemical coupting

through the synapses is the main focus of ttUs thesis t h y wiil be considered m more detail at

the end of this section, in 1.23.

Gap Junetions

Gap junctions are me&- stmtms formeci by pore-forming proteins that aliow ions and

low molecuiar weight compounds to pass klybetwem distinct cells They are characterized

by hi& degree of symmetry and dows for the bi4.rectionai traasmission between neuons.

This movernent of the ions and rnolecuies mediates intracellular signahg at elecbical

coupiing jurtctions. Gap junctions Save as low resistance pathways for ion and cment flow

between neurons [21, 251. Coupling via gap jullctions is ofkm referred to as electrotonic

couplhg since the tmmmkion can be charactenzed by cable-like properîies Electrical

transmission through the gap jimctions is fhster than the chemical one Mlitated by the

synapses- In addition, transmission dirough gap jmctions exhibits both stmctud and

fimctiond symmetq due to the bidirectional quaiity of the information ttansfèr between the

cells. Figure 1.1 shows a diagram of the gap jrmctioa stnacane.

Figure 1.1. Gap junction diagram (Fmm Kendel et ai. [2 1))

It has been suggested that gap junctions provide a robust mode of transmission and

intraceiiuiar communication Howiever, hvestigations mealed t h . tk transmission through

gap junctiotls can be modulated aImost to the same extent as the transmission h u g h

chernid synapsessynapses Tn addition, it was f o d ttiat a large nurnber of neuronal pairs are

connected by both chernical synapses and gap junciions allowing for one type of couplhg to

be modulateci by ttse otha [W.

1.2.2 Ephatic Caupling

Ephatic interactions or field enecri between neuruns are generated wtnn a cuneiit passing

througti one excÏted cell fiows across the membranes in close proximity and excites the

neighboring neurws withouî the preseme of gap j d o m or chemicai synapses Such

interactions have been identifid in the pymmîdal cells of the hippocampus propa both in

vitro and in vivo [8, 211. The extem of exkacellular space in this area of the braia k very

smd. If this space is relabvely srnail, resistance to extracellular current flow is increased

and more current ffows wittiin the intraceliular spaœ of the nemn In this case the sire@

of epbaî~c coirpiing is incnased In additions, these iateractions may play an important role in

an onset and spread ofepileptifonn activity since eptietic depolariraton causeù by the

generated spikes may trigger action putentials and enfrairrment in the adjacent nemm [35].

1.23 Chernical Synapses

Chernical synapses are specialized iuterceildar jUIlCtions at which unidirectional signal

transmission oaws. Signals are transmitted from one celi to the other by means of

nemtransrnkîers and n e u f o n i o d ~ r s . Synaptic tmnsmkion requires high levels of

specializaton in both presynaptic and postsynaptic neurons together with very precise

alignment of the two two. ALso, synapric tnmsmkion allows for a high level of

flaobility and modifiabihty which underiines the main dynarruc pmperties of the synaptic

architectute. Synapses may be excitatory or uihriitory. Excitatory synepses have an ability O

reduce the ciifference between the restuig level of a postrynaptic neuron h m its firing

threshold, wtiile the activity of the inhiiitory synapses incr*is+s this difference. Although

there exists a wide variety of synapses in the brain, they al1 share the same s t i and

hctional features. General characteristics are descrr'bed here:

in the synaptic vesicles b o d to t&e membnm of the presynaptic tpmunal They are

released through a pocess of calcium reguked exocytosis. Action @al in the

pcesynaptic cell initiate the opening of the voltage gatd calcium c h e k causing rapid

influx of calcium into the cytoplasm. This inmase in calcium concentdon serves as a

second messenger in the signal transduction [9].

N- are finthex released into a m w space betwea the pre and postsynaptic

ce11 dled the synaptic d e f i . Neurotransnitters then mach the psr;synaptic terminal and

activate the membrane receptors. Density of membrane receptors may determine synaptic

coupliag strength and speed Postsynaptic membranes exhibit folding, allowing for an

increased surfiace area and an augmentation in the number of meptors scposed to secreted

transmitter~~ The action of the synapses is largely detennined by the type of associated

neurotransmitters and postsynaptic tecepfors. Recepiors determine the dynamics of the

specific ion influx and efflux through the activation ofdifferent ion channels [9].

TraRSmitters are released in quanta allowing for the spatial and temporal nmunation of the

presynaptic action potentials at the postsyraptic terminal and fiequency coduig [43].

Frequency coding implies an onset of depolarking or hyperpolanPng response in the

postsynaptc neurm proportioml to the hqueacy of die presyoeptc action potentials. These

properties will be d e S c n i in more detail in Section 5.2. Impulses arriving at the presynaptic

terminal cause postsynaptic membrane âepolarization with each quantum of neufotransmitter

releesed Hence, postsynaptic response is a h dependent on the fhquency and duration of

the presynaptic anion potentds. With repeeted or longer lasting action potnitids h m the

Presynaac ce& larger quanta of neurotransmitter are released [43]. umeased intracelluiar

calcium concentration may also play an important role in this process [21]. However, the

postsyaapa'c response also exhi'bits ann nation p o p d e s for strong euough preSynaptic

stimulation. For this reason, the magnÏtude of the postsynaptic membrane potential, as a

fiindon of w p t i c impulse hquency, has o h been descn'bed as a sigmoid fimction

m a s e [7]. In this work, synaph'c action will be modeled so as to incoprate the impulse

fhquency difference between the pre and posayFaptc teminai in the associated changes in

resting level and amplitude of the stirnulated cell. In addition, limiting effects of cellular

feedbiick wiU be exammed A detailed description of tk mode1 wiil be given in Section 3.1.

Main elements and fimctions of a chernical synapse are shown in figure 1.2. -wwic r r i a i m

Figure 1.2 Chernical syoapse diagram (Fnw Kaadel et. ai. [21])

1.3 Epilepsy

Epilepsy does not r e f a to a spscific pathological condition bui ratba to a number of

symptoms that may rem.& h m a number of ciiffernit causes 1371. The main characteristics of

epilepsy include sudden, mcurrent dkturbces of mentai W o n and body movement

resuiting front excessive discherges h m a large nimiber of neufonal ce&. The symptoms of

epitepsy may be static or Progresive. Most cornmon symptorns invoive muscle spasms and

nippessed awareness of apparenty pirposeful actions and partid or mmplete loss of

consciousness [12].

Epileptic are characterized according to the location of the initial dûchar- and the

amomt of their spread. Main classification distinguishes between partial (focal or

localizafion related) and generalued seinires 1121. Partial seinires are initiated unilateraly,

in specific loci in the cortex They are c h a m % e k d by a number of symptorns m@ng nom

sensation disordem and convulsive movemenîs to povntial loss of consciousness if the

seinires becorne generabd. ûeneralized seiarres involve both hemispheres of the brain In

this case consciousness may be impaired and motor manifestations are bilateral. Neuronal

discharges are widespread and affect a large area of the bbram Partial seizirres result h m

excessive excitation or inhibition of specifk groups of neufons, ~ferred to as an epileptic

focus in the case of generalized seizures, excessive excitability or inhiiition affect the

complete cortical d e .

Nomially, tbere exists a balance between excitatory and inhiitory synapPic actions between

neurons throughout the epilept~genic regions of the brain However, recordings fkom a single

n e m tr;uimiembrane voltage reveal that this balance can be disnirbed Excessive

excitability in epiteptic foci may result nom the blockage of yamuio butyric acid (GABA)

inhibition GABA is one of the main nemtransmiitiers released by inhi'bitory neufons to both

somas and dendrites in the pyramidal cells of the hippocampus[24]. In addition, a number of

non-synapbic mechanisms may trigger excessive excitation leading to encrainment of

neuronal elecûicai activity and the formation of the epilepfic foci [37.

Characteristic pattern, calleci proximal depolanzation shift (PDS), is observeci in the EEG

recordings of almost al1 epilepsy patients. PDS refers to prolonged membrane depolarization

of about 30 mV with superimposexi high firesuency action potnm'als PDS was observed both

in experimentally induced epileptic lesions and in a human epileptic cortex As epiieptifom

activity develops, PDSs occur with Uicreased fresuency Unitil the onset of the seizure.

Neuronal excitability demases after a seizure. EEG discharges, extracellular and

intraceUuiar activity recordeci h m an experimemal epileptic lesion are illustmted in figure

Figure 1.3. EEG dischargs. extraceuular and intrace11ular actMty assockted with PDS ( F m Jolmston et al. [20])

1.4 Hippocampus

Hippocampus is a corticai structure believed to be of key importance for memory and

leanUlIg h humans, hippocampus is fomd in both medial temporal lobes. It exhibits a

conid sbape forming a semickie a m d the thdnmus with its longitudinal axk It is built

fiom a number of transversely oriented l a m e k and a number of longitudinal pathways

perpeadicular to the lamellae and pamllel to the longitudinal axk. Main elements of the

neuronal structrire of the hippocampus are shown in Figure 1.3. Hippocampal formation

consists of a hippocampus proper, denderitare gyrus, subicuium and entorhinal cortex

Hipgmampus proper is divided into four regions, Qesignated by CA1 to CA4. CA1 and CA3

are cmsidered to be the two main subdivisions comaining very well defined neuronal

path-~s-

Hippocampai structure is coasidered to play a major role in the transmission of epileptic

activity tbrough the neuronal circuitry of mosçy fiben, CA3 pyramidal cells and inhibhy

baska cells. Inhibitory pd~ways between pyramidal and basket cells are

GABA In addition, hippocampl cells exhibit bursting activity predisposition

them especially nisoeptile to rhythmic discharges leadmg to possible

entraimnent and the onset of epileptiform activity [8].

- -

Figure 1.4. The structure of hippocampai formation (From Brown et al. [8])

1.5 Models of Epilepsy

M&is of epilepsy bave d t e d nom a reductionalist expxhentai avproach. Such shdies

provide deeper understanhg of disease mechanisrns by examinhg ody a specific part of a

cornplex system. EpileptZorm activity can be i n d d artificially which aiiows for the

mation of many models used in a sardy of physiology and couplhg mechanisms involved in

such pathoiogical p r ~ c e ~ ~ e ~ ~ Brain Peparatiom bakd in Merent extracellular media are of

key importance in these d e s . Such preparations allow for a study of only d l neuronal

networks and tbey can not be & for d e s of all types ofepiieptic disorders. However,

Epilepsy models can be categorized h o a number of p u p s accordmg to their main

characteristics In this study only in vitro models wilI be discussed since such brain slice

prepambions will be used to obtain the intnnsic data for the OSCiUafOm

In vitro models are based on brain slice preparatiom and allow for the studies of specific

processes; such as the blockage of inhi'bitory pwtsynaptic potentials. This can be achieved

by the addition of Merent agents, ntch as peiiicillin, to the M n g solution which block the

GABA recepton on the pyramidal celi membranes. Other models can be pmduced by

chging the concentration of the extracellular iom. These inclt.de low rnagnesium, high

potassium, low calcium or zero calcium modeis [12].

1.5.1 Zero Calcium Model

Ka hippocampal slice is studied in a medium similar to the physiologicai extracellular fluid,

it often shows no rhythmic electrical activity. However, if the expeziments are carrieci out in

zero calcium bath, iarge populations of neurons begin to oscillate exhi'biting epilepbform

activity. This activity is i n i t i a by a disturbance in the Mance of excitatory and inhibitory

processes. The absence of calcium in the extraceiluiar medium reduces the syriaptic

transmission between the neurons and changes their discharge activÎty. InitialIy slow

dischargiag nemm depolanEe and exhibit binstiag activity. Hence non-synapb:c

mechanûms are involved in the onset of the entrained peziodic behaviour. In In thesis data

recordeci h m a hippocampai slice, bathed in rno calcium medium, wili be rsed to

Mamine the inhinsic parameters of the oscillators. Under these wnditions the mtrinsic

parameters are hown for the case of Nppressed synaptic activity and the effectr of synaptic

coupling *ui be studied as its strength is gradually increased fkom the uitrinsic state.

1.6 Scope of the Thesis

The motivation for this thesis is the analysis and exphnation of dynamitai behaviour of

neuronal electrical activity taking both e1emica.I and chernid coupling modes into

consiéeration. The main premise is that the electrical signals fiom the brain are chaotic mder

normal conditions, while pathological States are marked by perïodic and quasiperiodic

activity.

Ma@ dock oscillator mode1 will be used to examine nonlinear dyoamics and behaviour

of coupled netuons Emphasis will be placed on the & ' i of changes in coupiing

mechanisms, specifically chemicai coupling The hypouiesis is that the mode1 will

qualitatively and quantitatively repduce the oscillatory modes occurring in the cuupled

neunir In addition, the hypthesis is that the pesence of chemical syaaptic coupimg and

self-feedback loop activation increase the region in which the system &'bits chaotic

behaviour and infiuencg the region of no~i-oscillatory activity.

The mode1 consists of two bidkdonaUy coripled osdators wim foin inpib portais The

oscillator dyriimud actntity is d e s c r i i by a fourth order systw of differentid equations

[i, - The project consists of chaniçteri2arion of synaptic couphg foUowed by localkation

and identifidon of differenî oscillatory modes in the system. Emphasis will be placed on

the regions in which the system demonstmtes cht ic behaviour. Effects that coupLing

through chexnical synapses and activafion of self-feedbiik rnechaaisms have in these regions

will be examined in d e t d The hope is tbat this work wi i i produce a more realïstic

mathemâtid mode1 of neuronal actnnty and potentiaily enable studies to gain more insight

into the onset ofepileptlform activity.

Chapter 2

Theoretical Background

2.1 Nonlinear Dynamical Systems

Traditionally, a dynamical systern is described by a set of differential equations:

U== f(ü,P>

Here ù is an N dimensional vector describing tbe -te of the system at any given time.

is a vector field describing the dynamics of the system. Fis the panuneter vector. If the

parameters are time dependent, that is = p( t ) , the system is referred to as non-

autonornous. Otherwise, the system is considered to be autonomous. Each set of

panuneters constitutes a point in the parameter space. Parameter vector and the vector

field f together determine evolution of the system in -te space.

A sequence of states the system passes through in t h e is refmed to as the trajectory. We

can determine ira.jectories of the system if we dlow Y(f)to evolve in time given a

specific set of initial conditions U(0) = ü, . In a given systwi ail trajectories may tend to a

point or a closed curve which constitutes an attractor for the system By examining the

state space in the neigtiborhood of a given attractor we can detemine the basin of

attraction for that amactor. Basin of attraction consists of al1 near by trajectoiies that tend

to the same amactor as they evolve with the . A given system can exhibit muitiple

amactors in its state space. Characteristics of the attractors and the associated basins can

be greatly dependent on specific parameter values and initiai conditions [4L].

2.1.1 Analysis of Dynamical Systems

Generally, behaviour of a dynamitai system can be classified according to its major

charactenstics. In cases of nonlinear dynamical systems behaviour amibutes can be often

understd. by applying lineariration around a given operating point Linearization cm

tum out to be helpful for the analysis of the systern in a limited operation range.

However, in more complex nonlinear dynamicd systems, more complete numerical

integration may be required to gain deeper insight in the qualitative and especially

quantitative characteristic of the system.

2.1.1.1 Qualitative Characteristics Analysis

In order to understand qualitative characteristics of behaviour for a noniinear dynamical

system, one of the key initial steps is the analysis of the behaviour around system

eqdibrium, or stationary, points and their stability. The equiliirium points of

autonornous dynamical systems can be found as solutions to the following equation:

u'=Y(ii.P)=0

Once singular points are identifie4 linear stability of the system around these points can

be examined This c m be achieved by determiring the characteristics of the Taylor

Series expansion of the vector field 1 around the singular point x, and the resulting

Jacobian yx,) . When the Jacobian mathv is obtained, associsrtd eigenvalues must be

calculated in order to be able to classify the singuianties as stable, unstable or non-stable.

This clsssincation can be obtained provideci that the real parts of the eigenvalues

se(& ) are non-zero. If the r d parts of the eigenvaiues are negative ( %e(A ) <O) the

singuiaity is considered to be stable. Stability of a singular point implies that al1

bajectories that start in a neighborhood of the singular point will tend to that singular

point as they evolve. in the case of an uIlStabIe singuiar point, which is characterized by

positive real parts of the eigenvalues ( '%(A, ) >O), nearby trajectories evolve away fiom

the singularity. In cases where the system is characterized by some positive eigenvalues

and some negative at a singularity, the equilibnum point is considered to be non-stable.

Nearby trajectones are attracted to the singular point in m e direction M e they are

repelled in the other directions. Saddle point in a two dimensional system is a Srpical

exampIe of a non-stable singularity. Figure 2.1. illustrates different singulanties obmed

in a two dimensional systemstem

Figure 2.1. Singularities observed in a two dimensional system a) stable node;b)uflsfable node c) saddle point . (Frorn Seydel[41]).

In general, the behaviour of a system cm be well understood by following the evolution

of one point in state space h u g h time. As discusscd above a point in space and the

trajectory it defines rnay tend to a singular point. In other cases trajectories rnay converge

to a closed curve called a lirnit cycle. The existence of a limit cycle is characteristic of

nonlinear dynamical systems that exhibit periodic behaviour. Figure 2.2 illustrates stable

and unstable - - - - lisnit-cycle. - -

a) (W

Figure 2.2- (a) Stable and @) unstable limit cycle. (Fmm Mer and Chua [33]) .

A number of systems can be found to exhibit quasipenodic behaviour. Quasiperiodic

oscillatory modes are characterized by wavefom that are sums of periodic waveforms

whose fkequencies are linear combinations of k linearly independent fiequencies. Here, k

is a finite size integer. For example, consider a system which exhibits oscillatory

behaviour of two independent fiequencies fi and fi. If the ratio of the two frequencies

fJf2 is either rational or irrational number, Fourier spectnmi contains &O fiindamental

peaks and a number of others given by afi+bf2. Here a and b are different integer

numbers. In a «ise of a system that shows quasiperiodic behaviour, associated attractor

typicaily exhibits a toroidal shape [41]. Figure 2.3 showsa typical quasiperiodic toms.

Figure 2.3. Quasiperiodic toms. (From Parker and Chua [33]).

A more compücated type of trajectory is a mange limit set, fond in nonlinear systems

with dimensions higher than two [33]. niese sets may be asymptoticdiy attractive to

nearby trajectories, in which case they form strange attractors. Certain tmjectories may

stilI be l d y divergent within the amacting set Such attractors are associateci with

chaotic behaviour in a systedïhe main characteristic of chaotic oscillatory mode is its

random-like, noisy appearaiice in time, despite the deteministic properties of the stem.

Chaotic modes exhibit macked sensitivity to the initial conditions of a Vstem. A system

is said to have sensitive dependence on initial conditions if two of its trajectories, that

start off close to one another, drift apart as they evolve wiîh t h e niis property leads to

unpredictability in the system's behaviour over a longer pend of tirne.. Cbaotic systems

may prove to be extremely complicated in their behaviour [33,41].

A classical example of a strange amactor is foimd in a solution of Lorenz equations, used

to describe convection of a thin fluid layer with a temperature gradient between a top and

a bonom suface. Formation of certain cloud patterns, drift of continents and granulation

on the surface of the Sun have been connected to similar phenornena Lorenz equations

applied in a weather forcasting cootext give one of the most fsunous illustrations of

unpfedicatability in chaotic systems; such as the "butterfly effect" which suggests that a

buttedy flapping its wings in one part of the world caws a thunderstom in another, far

away 1ocation.The Lorenz attractor, shown in Figure 2.4. is now considered a classical

example of a chaotic, strange attractor.

Figure 2.4. Lorenz strange attractor. (From %del14 11).

2.1.1.2 Quantitative Analysis

There are severai methods of quantitative classification for oscillatow modes of a

nonlinear dynamical system. In this thesis two methods were used for this purpose. These

are caicdation of Lyapunov exponentg and Lyapunov fracta1 dimension for given

attractors.

2.1.1.2.1 Lyapunov Exponents

Calcuiation of Lyapunov exponents provides a quantitative description of the oscillatory

modes in a nonliwar dynarnicai system. Lyapuwv exponents can be viewed as a

generalization of the eigenvaiue caldation for the Jacmbian matrix at a singular point

For an N-dimensionai attractor N real numbers, Ai, A*, .-. .kN. can be calcuiated to

characterire the expansion or contraction of an attractor dong a particular direction.

Lyapmov expnents cm be understood as time averages of eigenvalues of a vector field,

f , linearized dong the trajectones [33].

Calcuiation of Lyapunov arponents involves an N-dimensiod hypercube whose vertices

define a set of initial conditions. The system is then imegrated and the evolution of

Lyapunov exponent values in tirne are followed to determine the shrinkage or expansion

of the obsewed attractor.

Lyapunov exponents provide a convenient mahod for classification of dynamical

systems and identification of chaotic attractors. Chaotic attractors are characterized by at

least one positive Lyapunov exponent In annition, in a case of autonomous amactors,

one Lyapunov exponent musî be zero. Others are either zero or negative. Furthemore,

the surn of Lyapmov exponents must be negative for a dissipative system to ensure a

general decrease in the amactor volume ad, therefore, the stability of the system [26,

331. Characterization of dynamcai behaviour for a system based on Lyapunov exponent

caIculations is summarized in Table 2.1.

OJcillatory Mode

Equifibrium Point

Periodic

Quaszperiodic

Chaotic

,

Lyapunov Exponents

O > A, 2 A, r . . . . ~ A,

.v 4 =O;& < O

i r l

N

a, = 4 =...= 4 = o;C/5 < O i=I

N

4 >O;& =o;CA, < O i= 1

1 Table 2.1. Characterization of oscillatory modes for a dynamical system using Lyapunov exponents,

2.1.1.2.2 Lyapunov Frachl Dimension

The concept of Euclidean dimension is a weU bown method for classification of

different geometrical objects. It is defined as the minimum number of coordinates needed

to uniquely specifi any point in space. The nonlinear dyluunicai system dimension is the

minimum number of state variables that must be used to successfully describe the

evolution of the systern in time. To descnk a dimension of an attractor, a dimension

meanne must d o w for non-integer values. This is achieved with the use of hctai

dimensions. A set that exhibits non-integer dimension, like most strange attracton, is

called a f'ractaî [33]. Fnictal dimension is w d to quant@ a complexity of the system, or

the 'Cstrangeness77 of a strange attractor. In addition, hetd dimension can be used to

differentiate between chaotic and d o m s i m .

A number of different hctal dimensions, like conelation, capacity or information

dimension, have been defined Lyapunov fkactai dimension had been used in this work to

allow for M e r classifications of the nonlinear systems behaviour.

The Lyapunov dimension comects Lyapunov exponents dculated for a particula.

attractor with the notion of the attractor dimension. Let hl, )c2,..hN be the Lyapunov

exponents calculated, such that h,2 ic,2 .. . è ic,. In addition let k be an integer such that

kl+h2+- .- +w O. The Lyapunov dimension can then be defined as follows 1331:

For example, attractos of a fourfh order system can be described in a following manner

based on the Lyapunov dimension classification:

Point attractor mi

Limxt Cycle ~ = I - o ;

Qwis ~petiodic tom 1.W & < 2.0;

Strmge atlroctor 2 . 0 ~ DL < 4.0.

2.2 Nonlinear Oscillators

A number of mathematical models have been employed to descn'be oscillator dynamics

in a physiological systen Initially, Van der Pol oscillator was w d [34]. However, this

mode1 is limited in that it can only produce symmetrical output wavefoms. A number of

rnodels was investigated in order to improve on the Van der Pol Oscillator. However,

later models exhibiteci a number of superior properties. For example, oscillators based on

Hodglan - Huxley dyaamics have been developed [8]. Hodgkin - Huxiey oscillators are

able to produce a number of more redistic, complex wavefoms. However, parameter

estimation for the oscillators based on Hodgkm - Huxley dynamics involves a cdculation

of membrane capcitance, ionic conductances and equilihium potentials. This makes the

mode1 hard to use and inappropnate in the studies imrolving networks of oscillators. In

these cases, relationships between the mode1 and the physiologicai parameters are more

invoived and difficult to determine.

The use of transfomtional synthesis approach [3] proved to be useful in deveioping a

more general mode1 of oscillator dpamics. Inmally, the Synthesized Relaxation

OsciIiator (SRO) [5] was developed This work was later expandecl to develop Mapped

Clock Oscillator model (MCO) [SI. The MC0 model is especially useful due to its ability

to produce any arbitrary output waveform. The MC0 model has already been

successfully implemented to simulate elecnical activity of the smooth muscle tissue in

the gut [4] and some of the key elements of the electricai activity in the hippocampus [l].

2.2.1 Uncoupled Mapped Clock Osciiiator

The Mapped Clock Oscillator consists of two main components: (1) the clock wbich

determines the fnquency of the signal and (2) the transfomer which shapes the output

wavefom The cbck is represented by a second order system of ciiEerentia.1 equations

and represents a dynamic nonlinearly. The transformer is considered to be a static

nonlinearly as it represents the mapping from the state variables to the obsewable output.

The transformer is assumed to be a representative of the ionic transmission mechanisms

thougfi a ce11 membrane.

2.2.1.1 Clocsk

The clock is the basis of rhythmiciw in the oscillator model. One of the simplest

representations of rhythmic behaviour in the system is the motion through a continuous

sequence of states in a circle. This can be matkm&ically represented by the following

second order differential equation system:

Here, r and @ are state variables in a the polar coordinates. The solution to the above

systern in (r, 4) plane is the limit cycle on the unit circle. Dynamics cm be represented in

the rectanguiar coordinates (u,, uz) by setting ui-*$ and u2=rcoM which gives:

Here, ul and u2 are state variables and O is the intrinsic frequency of the oscillator.

2,t.l.Z Transformer

The transformer, as it was noted earlier is a static nonlineariy mapping the state variables

into an observable output. It can be represented in the following mannet?

Here,

is the output variable of the oscillator,

is the harmonie index;

is the intriasic resting IeveI;

are tbe Fourier coefficients of intrinsic oscihtïons and

T ( ), U ( ) are the kth Tchebychev plynomials of the first type and the second type

respectively .

An uncoupled oscillator dways has an intrinsic wavefonn This can be used to obtaùl the

Fourier coefficients, 4's and h's, which are then used to calculate an output.

Tchebychev polynomiais reduce to a Fourier senes for the uncoupled oscillators. in this

case, Fourier coefficients and the intrimic fnquency are used to fully define the innuisic

waveform of an uncoupkd oscillator- The oscillator output is often representative of a

transmembrane voltage in studies involving excitable celk.

2.2.2 Coupled Mapped Clock Oscillators

Uncoupled oscillators exhibit only periodic waveforms. In order to be able to simulate

more complex physiological waveforms, oscillators mut be coupled to one another or

allowed to receive extemal stimuli. Coupled rnapped clock oscillaton ailow modeling of

a wide variety of outputs and oscillatory modes. Oscillaton can be couplad in a number

of different ways. Connections can be made between two clocks, two transfomers a

clock of one oscillator and a transformer of the other, or a combination of the above

connections.

To represent coupled oscillators mathematically we must retum to equations (2.1) and

(2.2). These equations represent amplitude and phase of each oscillation respectively. It

is possible, however, to change the amp1itude, phase, or the neqUency of the signais by

applying appropriate stimuli. This can be represented in the following mauner:

Here, Sa is a stimulus that inflwnces the amplitude of the signal. It is delivered to the

stimulated oscillator unit through an input portal Pa. S+ is a stimulus that affect the

frequency of the signal and it is delivered through an input portal Po.

I f n oscillators are coupled in this manner, dynamics of the nh oscillator can be

represented by the following set of ciifferentai equations in rectanguiar coordinates:

Here, and S,= are stimuli delivered through input portal P,.

The transformer equation for the nh oscillator is given as foilows:

Oscillator stimuli can be delivered from anywhere in an extemal environment However,

here it is assumed that the n& oscillator is stimulated by the outputs fiom its neighboring

oscillaton. Clearly, there are numerou configurations that can provide stimulation for

the 8" oscillator. Here, only the following coupling conditions will be considered

Stimulus to the nh osciliator through Pa and Po input portals is the outprd of the m&

osciIlator. Hence, sa = Y&* and

s*= q n m YJ%

Stimulus through the P, input portal of the n' oscillator is the dock output of the mm

oscillator. Hence, &ln\ulJ& and

s&=+ukJS,

In the above equations c- cm and +, are the couphg factors. They are used to

determine what portion of the rn& oscillator output serves as a stimulus to the nh

osciUator through Pa, PI and P, input portah respectivety. 4 and &, are normalization

factors. 6, is a static dock aomalization variable set to 0.1. an is the amplitude of the

intnnsic wavefom of the nm oscillator given by:

The suantities S,, S* and S, represent total stimuli delivereci to Pa, P,+ and P, portah

respectively. Coupling factors cm, cm and represent relative influences of nearby

oscillators on the stimulateci one. Interactions of ody two coupled oscillators wiil be

investigated in this work. Hence, each of the two oscillators will be influenced by ody

one neighbor. Exact configuration of the two coupled oscillators mode1 and the addition

of the fourth input portal, P, and feedback loops wili be discussed in Section 3.1.

Chapter 3

Methods

3.1 Mode1 Description

The Mapped Clock Oscillator model (MCO), describeci in Section 2.2. was modifieci to

enable modeling of both electrica.1 and chernical coupling. In additions, modifications

were made to examine feedback effects between coupled neurons. Effeects of self-

feedback were studied on three of the four input portal. Specificdly, these were Pa, P4

and P, No self-feedback was placed on the P, input portal since there are at ieast two

separate feedback pathways associateci with this portal and they may be difficult to

distinguish from one another. When input portal P, and P self-feedback loops are

included in the cartesian coordinate model, the dock equations becorne:

Under these conditions, the transformer equation becomes:

During earlier studies, biological data recordeci from a rat hippocampal slice was used to

obtain the intrinsic parameten orthe oscillators A slice was place in the chamber and

bathed in an oxygemted, artificial cerebrospM fluid with zero calcium concenûaîion~

Patch clamp recordings were obtained nom the pyrariuchi ceils of the CA3 hippocampai

region Recordings demo~l~bated rhythmicity in the bust and spike characterized

behaviour. fou rie^ andysis of the averaged recorded wavéfonns was performed to obtain

the values for the amp1itude, resting level a . frequency as the intrinsic parameter values

for two uncoupled oscillators [Il .

The modei used in this thesis consists of two rnappeà clock oscillators. The panuneters of

the first oscillator, MCO,, remained set, while the parameters of the second oscillator,

MCOZ, were allowed to change to enable more detailed modeling of different coupling

configurtionk. For a schematic diagram of the mode1 configuration please see figure 3.1.

Tramducers (abbreviated Trm) included in the diagram serve to exaact rdevant

parameter fiom the output of the oscillators. For example, transducer for portai P,

extracts 4 information.

Figure 3.1. Schematic diagram of a four portal MC0 mode1 configuration with feedback I o w

3.1.1 Coupling Configurations

Two oscillators can be coupled either ~directiondly or bidirectiody. In a

unidirectional configuration only one oscillator is allowed to stimulate the other, while in

a bidùectionally coupled system both oscillators an allowed to stimulate one another.

Four portal oscülators provide the means for a large number of different coupiing

configurations. In addition, the amount of feedback allowed between different portals

further iacreases a widc range of coupling conditions that cari be investigated. Portals

may be completely closeci, by seaing the couphg factors to zero. Coupling may be

graddly increa~ed untll the input portais are M y open d e n coupling factors are set to

one. Similady, the e&cs of increming amount of f-k may be studied by changing

the feedback factors, fi, fîom zero to one.

In a bidirectionally coupled system two main conf3gurations that can be studied are

symm&c and m e t r i c coupling. Symmetric coupling is achieved by sening the

corresponding coupliag facto= of both oscillaton to the same value. In a case of

asymmetnc coupling the vaiues of coupling factors are not equd. In this work, only the

effects of sywaetric coupling were investigated. In this case the following relations hold

for the conesponding couphg &ors:

C ~ Z I = C ~ ~ ~ = C * C ~

Here, c-, c*, and c,, are referred to as effective coupling factors. C is the

common coupling factor and cas c+ $ and cp are specific coupling factors.

In this thesis the &ecf~ of unïduectiod couplkg in the syçtem were investigated by

activating two input portals. h particular, following configurations have been studied.

p coupling with f3, feedback;

4 coupling with Po feeùback and

p and 4 coupling with B, and fi+ feedback.

Other unidirectional configurations have been prwiously investigated and the emphasis

of this work was placed on characterizing the propeaies of P, inpuî port&

The investigation of the bidirectionally wupled system has been carried out wiîh the

emphasis on the effects of p coupling and feedback on the oscillatory modes

demonstrated by the system. Al1 four portal have been opened simultaneously with the

following feedback loop configurations:

only Bo active;

ody p4 active;

only p,active;

&, and p, active and

Ba, P+ and B,

3.2 Cornputer Simulations

As mentioned eariier, the system of two coupled oscillators can be described

rnathematically as a fowth order differentid equation system. To investigate the

behaviour of the system, simulations were carricd out on a SUN SPAEX 10 and ULTRA

1 UNIX work stations.

Simulations were carried out using a nonlinear analysis commercial package iNSITE

[33]. The package solves differential equations systems by implementing the BSODE

based integration routine, based on a Gear method [36]. In order to impiement BSODE

integration rnethod, subroutines that calculate vector field, 7, and the Jacobian matrix,

3( f ) , must be provideci by the user. Cdculation of the Jacobian can be very involved

since it requires evaluation of dl elements of a square matnx d o s e dimension depends

on the order of the investigated system. Each row of a Iacobian matrix contains a

derivative of an elemem of the vector field with respect to each of the state variables. In a

case of the forth order system, representing two coupled MCOs, Jacobian matrix contains

16 elements. For listing of the two required subroutines in the C programming language

please see Appendix A. First part of the 2ippendix defiaes variables introduced in the

derivation of the Jacobian.

INSITE, used for detailed system simulations, is an interactive, graphically oriented

software package. INSITE provides two integraton routine algorithrns: Gear methoà

based BSODE and the Rmge Kima algorithm. INSITE accommodates the calculation of

Lyapunov exponent and Lyapunov nactal dimension with an algorithm based on Wolf s

mahod [45]. In addition, the package provides tcmk for v i s ~ t i o n of trajatorïes and

attractors of the system.

Topoiogicd coupling maps (TCM) presented in the study were obtaùied using the

Irnaging Toolbox, provideci in a commercial software package MATLAB [29]. Steps

involved in simulation processes and calculation of associateci results are outlined in

figure 3.2.

Figure 3.2. Row chart descriiing the cornputer simulation steps

Chapter 4

Results

4.1 Unidirectional Coupling

Unidirectionally coupled oscillators were studied to examuie the properties of the input

portals with respect to their influence on frequency, amplitude and resting level of the

signal. For the unidirectional coupling configuration, the output of the second oscillator

(MCO?) was fed into the input of the first oscillator (MCO,) as a stimulus. The output of

MCOl was monitored for the effects of the unidirectional coupling arrangements.

Properties of unidirectional coupling through Pa and P, input portais were investigated in

the earlier studies [Il. The emphasis of this work was placed on modeling of the synaptic

coupling. To achieve this, the properties of input portals Po and P, were shidied in detaif.

Furthemore, effects of self-feedback were examined even in the unidirectional coupling

anangement to enable rnodeling of molecdar seIf-feedback mechanisrns present in

chernical coupling between neurons.

Simulations were initiated with closed portals (effective couplhg set to zero). Couplkg

mengths were gadually incraseci until the ponals were fully open (effective coupling at

one). The intrinsic frequency of MCO, was kept coosiant at 1.275 Hz, while fiequency of

MCOZ was alIowed to change tiom 0.5 Hi to 10.5 Hz Effécts of changing the stimulus

fiequency were studied to examine the accuracy of the synaptic mupling mode1 through

the activation of Po and P, input portais, offset and intrinsic resting level was kept

constant for both oscillators since the effkcts of changing these parameters was

investigated in previous midies [l]. Reliminary runs with changing the offset of the

çtimulating signal that were carried out as a part of this thesis only confimed earlier

4.1.1 Coupling via Portal P+

Two oscillaton were initially coupled only through P+ input portal. Intrinsic fiequencies

of both MCOl and MCOz have been kept constant at 0.4587 Hz and 0.128 Hz for this

initial study. The wmmon coupling factor c was kept at 1.0, while specific coupling

factor c+ was changed fiom 0.0 to 1 .O.

Figure 4.1. shows the effect of increasing 4 coupling on the amplitude, resting level and

Frequency of the stirnulated oscillator signai. As expected from portai definition, 4

coupling has the most profound influence on the frequency of the stimulated oscillator.

These results are similar to the ones stated in a previous study [l]. Starting fiom the

intrinsic fkquency, 0.128 Hz, the frequency of the stimulateci oscillator MCO,

experiences a linear &op with the inmeashg 4 coupling strength. The oscillations are

finally blocked at approxùnately q=û. 17. As couplhg is increased beyond this point, the

MCO, fresuency increased again The trend of linear incresse in fkquency dws not

change for M e r increase in coupling strength.

Cphi

Figure 4.1. Changes in signal a) amplitude, b) resting level and c) frequency due to an

ui the next part of the study similar experiments were c h e d out with the addition of p

wupting Figure 4.2. shows the changes in fkequency of MCOl due tu increasing @

coupling at three different ievels of p coupling. Obsewed frequency changes showed

similar trends as in the previous part, with zero p coupling applied lncrease in p coupling

produces only a slight shift dowaward for the 'frequency vs. c+ 'cuwe. tt is important to

note that the addition of the P, input portal did not change the properkies of the P+ input

portal*

Effea of changing Cphi on frequency

Cphi

Figure 4.2. Changes in fiequency of MCO, due to an increasiug c4 at diflerent levels of p coupling applied.

In M e r experiments the addition of self-feedback loops was examined Figure 4.3.

shows the changes in fkquency of the oscillator MCOl si@ at self-fdback b e l s

ranging from &û.O to 6+=0.4. With increasing seIf-f-k, the trend in fiequency,

under inmeasing coupLing, remained largely the same. It was, however, noted that the

rate of change of fiequency generally deereared with the iacreasing self-feedback This

is an expected effect of the incfe8sing amount of negative ~e~feedback in the system

Figure 4.3. Changes in Frequency of MCOI signai due to increasing c* at different levels monts of self-feedback applied

4.1.2 Coupling via Portal P,

Two oscillaton were coupled only through the P, input portal. Again, intrinsic

fresuencies of MCO, and M C 4 were kept constant at 0.4587 Hz and 0.128 Hz

respectively. Common coupling htor,c, was kept at 1.0, f i l e specifc couphg c, was

changed fiom 0.0 to 1 .O.

Figure 4.4. shows the eff- of increasing c, strengîh on amplitude, resting level and

frequency of the stimulated signal from oscillator MC0 I. As expected nom the design of

the input portal P, the most profound effect of increasing c, in the system was observed

in the changes of resting level. The resting level of the stimulated oscillator gradually

decreases for increasing c, This trend continues for ail values of c, coupling factor.

A sirnilar study was carrieci out at different levels of c+ Figure 4.5. shows the changes in

resting level of MCOl with incrtasing c, and at three different leveis of c+. The addition

of c4 produced a slight shift upward of the 'resting level vs. c,' curve, but the general

trend of decreasing resting level remained the sarne.

Figure 4.4. Changes in signai a) amplitude, b) resting level and c) frequency due to an increase in c,

Coupling via portal P, was further examinecl for the effects of increasing ~el~feedback

applied. Figure 4.6. shows the changes in resting level of the stimulateci oscillator MC0

under increasing c, and self-feedback levels from &û.O to Pp=û.4. Changes in the

resting level under Uiaeasing c, remained iinear at different self-feedback levels.

However, the rate of change of resting level increases with an increasing amount of self-

feedback applied By examining the design of the input portal P, we can conclude that

the dope of the line 'resting level vs. c,' changes from negative to positive when the

following condition is satisfied: C,(&-/?'~A) = -1.

Effea of Crho on reçting level

Figure 4.5. Changes in resting level of MCO, due to an increasing c, at different levels of c4 appiied

In additio~ experiments were carried out allowing for the negative values of specific

coupling factor c, as it was gradually decreased fiom 0.0 to -1.O.The intrinsic

frequencies of the oscillators were kept constant at the sarne values as in the first piirt of

the experiments, at 0.4587 Hz and 0.128 for MCO, and M C 9 respectively. Changes

observed in the resbng level of the stimulated oscillator are shown in Figure 4.7. Figure

4.7. also depicts the effects of introducing self-feedback in the system with negative p

coupling as f3 was changed fiom 0.0 to 0.4. It was observed that the resting level

depolarized for f H . 0 as c, becornes more and more negative. Wi?h increasing self-

feedback, the resting level of the stimulated oscillaior mis hyperpolarized

Effeb of Cmo on resüng ievel with self-feedback -5 1 1 1 D I 1

Figure 4.6. Changes in resting level of MCO, due to the changes in c, at increasing levels of self-feedback fiom p,=0.0 to Pp=û.4.

4.1.3 C i p b g via PoBals P+and P,

Experiments were carrieci out to examine fkquency and amplitude coding known to be

present in synaptic coupling The simulations were carried out with both P4 and P, input

portais open. Common coupling factor c was kept at 1 .O, while the specific coupiing

factors were set at c4 = c, -0.5. The kequency of the stimuiating oscillator MCO2 was

changed from 0.5 Hz to 10.5 Hz. The intrinsic fkquenicy of the MCO,, whose output was

monitored, was kept constant at 1.275 Hz. With this setup, the second oscillator can be

Effed of changing Crho on Resting Level with self-feedback

Figure 4.7. Changes in resting b e l with negative coupling applied through input portal PP

assumed to mode1 a presynaptic neuron, while the first oscillator acts as a posfs~naptic

one. By being able to change the fkequency of M C 4 this setup helps examine the

influence of increasing stimulus fffquency and some aspects of modeling the fiequency

amplitude coding present in qmaptic coupling.

Figures 4.8 to 4.10. show the effkct of increasing stimulus frequency on frequency,

amplitude and resting level of the stimulated oscillator at self-feedback levels ranging

from f$,=&,=O.O to P,+=P,=0.5. Changes in the resting level resembled the changes

change in resting ievel Mth stimulus frequency

Figure 4.8. lnfiuence of changing stimulus fiequency on the resting level of MCQ. Coupling factors are set to ~ ~ 4 . 5 and cp=û.5 while the amount of self feedback was varied fiom B=0.0 to 0.4

observed with increasing rho coupling. With increasing stimulus Frequency, the resting

level of MCOl decreases lineady. Rate of change becornes les negative with an

increasing level of self-feedback applied

Frequency of the stimulated osdlator exhibited an dmoa Iinear de- with increasing

stimulus fiequency, dl the osciüations finaily stopped for the stimulus fkquency of 7.0

Hz, in the case for B 4 . 2 Fiirther increase in stimulus iiequency produced 1:l

entrainment for fiequencies of 7.5 Hz and 8.0 Hz For 84.4, fkquency decreased until

the stimulus frwtuency of about 1.1 Hr Further increase in fkequency again produced 1 : 1

entraimnent at stimulus fkquencies fiom 1.5 to 3 H z If the stimulus fkquency was

increased beyond these entrainment point, fiequency MCOl decreased, and f i d y

increased almosî lineariy. It is important to note thrit no fiequency changes for MCO,

were incraseci 4.0 Hz due to the effects of 4 coupling, but later stayed constant for

increasing stimulus frequency.

Figure 4.9. Innuence of changing stimulus fiequency on the frequency of MCO,. Coupling factors are set to c+l.5 and c 3 . 5 while the amount of self feedback was varied fiom P 4 . 0 to 0.4

change in frequency with stimulus frequency

Amplitude changes with increasing stimulus fiequency exhibited slight changes around

8

7

6

p- U 5 4 - 3 w g3-

2 -

1

O

constant level which decreases with an increase in the amount of féedback. Amplitude

I I I rp 1 w

- ------ beta=O .2 . . - - - beta=0.4 . ' . . . - - . . u . . *

f 'e ; * ' 0 ; /

;. Y . i /

, . : / ; . .

% 0 ; : i - . r; ;. i '. '-- -. / \ ----* / - . S . 1 / * .*..a-

l*- .* #'

I . -*. . .. .- \ .y * - - y , -/ '-. - - -.. : ,.---- \ -- - .--. - - a - --.

1 l , *-: 1 l &

varies around 40 mV, 35 mV and 30 mV for P chmghg nom O to 0.4. Amount of

O 2 4 6 8 10 12 frequency [Hz]

deviation from a constant level increased where 1:l entrainment was observed between

the oscillaton. Amplitude falls to a minimum at a stimulus fiequency of 7.0 H z , for

8=0.2, for which the oscillations were observed to be blocked

Figure 4.10. Influence of changing stimulus fiequency on the amplitude of MCO,. Coupling factors are set to ~ ~ 4 . 5 and cp=û.5 while the amount of self feedback was varied from p=û.O to 0.4

4.2 Bidirectional Coupling

Oscillaton were studied in a biduectionally wupled anangement to anable modeling of

coupled bct ional neuronal groups in hippocampus through the action of rnostiy

inhibitory intemeurons. The emphasis in the study of bidirectionaily coupled oscillators

was placed on the regions in which the system exhibits chaotic bebaviour. In pticular,

the effm of coupling via portal P, and introduction of self-feedback between the input

portais was examinai in detail. Analysis was initiated at a point where chaotic behaviour

was observed in a previous study [l]. Coupting factors associated with the three ponals

included in an earlier investigations were set at the following values: c,=O. 100, c4=0. 1 18

and ~ 4 . 1 0 0 . The inthsic fiequmcy of the first oscillator was set at 1 -275- while the

fiequency of the second oscillator was varied between 0.6375 and 1.1925. In this manner,

the difference between intrinsic fiequemies in the system ranged from -50% to +50%.

Couplhg was in a symmetrical arrangement for ail the simulations.

Oscillatory modes in the system were identifiai by calculating Lyapunov exponents and

detennining the Lyapunov dimension. In al1 cases the positive Lyapuwv exponent,

characteristic for chaotic behaviour, was distinguished from the negative ones if it had a

larger absolute value than the negative Lyapunov exponent with smallest absolute value.

The Lyapunov exponent with the smallest absolute value was considered to be the zero

exponent for the system. This distinction between the positive and negative Lyapunov

exponents values was consistent with the characterization of attractors by their Lyapunov

dimension I f the system was found to have a positive Lyapunov exponent, Lyapunov

dimension of the attractor was found to be larger than 2.0, which is characteristic for

chaotic, strange attracton.

4.2.1 Chaotic Regions Dependence on S p a ptic Strength

Simulations were Camed out with increasing wfic coupling factor c, from 0.0 to 0.5,

with a step size of 0.05. Common coupling factor c was kept coostant at 1.0. Other

specific coupling factors were kept constant at values listed above. The topologicd

coupling map (TCM) was obtained to characteriz oscillatory behaviour of the system in

different coupling regions. It was generally observed that behaviour changes from

periodic to quasiperiodic and then to chaotic as the coupling via portal P, was increased-

Furthemore, the degree of chaoticity increased with an increase in c, strength. It is

important to note that the points where chaotic behaviour was observed fuseci together to

form distinct regions. In the systern with no synaptic coupling or self-feedback chaotic

behaviour was observed only at three distinct points, as it was noted in [Il .

4.2.1.1 Effects of SeKfeedback

ERect of self-feedback on the size of the regions in which the system exhibits chaotic

behaviour was investigated next to gain more uisight into associateci changes between

different oscillatory modes. TheTCMs were generated for increasing c, at different levels

of self-feedback applied Common self-feedback parameter ce was kept at 1 .O, while

specific self-feedback faqors were changeci nom P,=P+=B,= 0-0 to Pa=p4=&= 0-3-

Results are shown in figures 4. 1 1 to 4.13.

Chaotic regions depending on çynaptic strengLh (beta=O.l

Figure 4.11. The TCM for P=O. 1 C o i o ~ gives the value of the Exponent

Chaotic regions depending on synaptic çn-ength (beta=0.2 --

Figrne 4.12. The K M for 84.2. Colorrrbar gives the value of the Exponent

Figure 4.23. The K M for 8=0.3. Colourbar gives the value of the higbest Lyapunov Exponent

Figure 4.14. Percent of region exhibitiag CMC behavior depending on the amount of self-feeciùac k

The TCMs show vahies of Lyapunov exponents for chmgmg qaaptic streogth identified by increasing cm and the Merence h e e n i n h i c fresuencies of the two oscillators.

Result~ are summarued in figure 4.14. that illushates percent of the investigated region

where chaotic behaviour was observed It is clear that the extem of the chaotic region

increased with increasing the self-feedback Level in the system.

Fufthennore, the effect of self-feedback through individual portais was investigated. The

resdting TCMs are showa in figures 4.15 to 4.17. In addition, simulations were carried

out with self-feedback to P+ and P, portals activatd The r d t i n g TCM i s show in

figure 4.18. It is clear that the P4 self-feedback loop i n c d the extent of chaotic

region the most. In the case when only P, was activated chaotic behaviour was observed

only at six distinct points. Points of chaotic oscillatory modes were fbsed together and

somewhat increased for p, self-feedback pathway activated Results are summarized in

figure 4.19. which illustrates percent of the investigated couplhg region where chaotic

oscil 1 ations were observed depending on whic h sel f-feedback loops were activated.

4.2.2. Non-oscillatory Regions

In an eariier study [Il it was found that as the coupling strength increased, the system

exhibited non-oscillatory behaviour. This phenomenon was investigated by increasing the

Y

ttic regions depending an synaptic Sength (beta-alpha=

Figrne 4.15. The T m when only $, is activateci. Calombar gives the d u e of the highesî Lyapuuov Exponent

Chaotic regi on . depending on çynaptic strengin (be?ashi =O -2)

Figure 4.16. The T a when ody is activated- CoIolrrbar gives the vdue of t&e highest Lyapwv Exponent

Chaotic regions depending on ~yrmptic strengtn (betatartK>=0,2)

Figure 4.17. The TCM when ody p, is activated. ColaUrbar gives the value of the highestLyapu~>vExponent

Chaotic regions deosnding on synaptic strength (beta=O-2)

Figure 4.18. The TCM when fi, and $, are activateci, Colourbar gives the d u e of the hi- Lyapunov Exponent

Extent of chaotic mgion dependhg on feedbaclc; advation

Figure 4.19. Percent of hvestigated coupling region exhibiting chaotic behaviour depending on which ~e~feedback portal is activated.

comrnon couplùig factor c to 1.125 and 1.25. Specific coupling factor c, was once again

increased from 0.0 to 0.5. Resuiting TCMs are given in figures 4.20 and 4.2 1. In addition,

the effeçt of activating self-feedback loops in the system on the non-oscillatory regions

was investigated by setting the common self-feedback factor ce to 1 .O and specific self-

feedback factors to pa=p4=P,,= 0.2. TCMs for these coupling conditions are show in

figures 4.22 and 4.23. As illustrated, non oscillatory behaviour, identified by al1 negative

Lyapmov exponents, was exhibiteci in the system if no self-feedback pathways were

activated With self-feedback loops open, the nonsscillatory regions disappeareci.

Furthemore, it was noted that in the system with no self-fdack, the extent of non-

oscillatory regions decreased with increasing coupling. Also, with no self-feedback loops

open, chaotic behaviour was not observed Chaotic regions were identified in the systern

with self-feedback pathways activated These chaotic regions were found to decrease

Figure 4.20. Tbe TCM for common couphg tktor ~ 1 . 1 2 5 and no self-feedback Coloiirbsr gives the vdue of the bighest Lyrpmov Exposent

Chgotic regions depending on çynspric çtrength (beta=O.O.c= t 15)

Figure 4.2i.The TCM for cornmon coupling factor c=L250 and no ~el~feedback Colourbar gives the valw of the highest Lyapunov Exponent

Fig the

O

Chaotic regions depending on synapric aength (beta=0.2.~=1.125)

pre 4.22. The TCM for common couphg fmor ~ 4 . 1 2 5 and fHI.2. Colourbar gives vaiue of the highest Lyapunov Exponent

Chaotic regions depsnding on çynapfic çtrength (Deta=O2,c=i 25)

-- O.? O -2 O .3 O -4 0.5

Figure 4.23. The TCM for common coiipliiig f'actor ç=I -250 and $4.2. Coloinf,ar @es the value of the highest Lyapmov Exp0nem.z

Figure 4.24. Percent of investigated coupling region exhibiting of a) chaotic and b) non- oscillatory behaviour depending on coupling and self-feedback amount.

Chapter 5

Discussion

5.1 General

A mode! consisting of two mapped dock oscillaton was used to mode1 the electrical

activity of coupled neurons. The fint oscillator was used to represent a neuron in the

CA3 hippocmpal region, white the second one enabled modeling of a neighboring

neuron. In addition, the intrhk properties of the second oscillator could be modified

when necessary to completely investigate a number of different conditions of interactions

between hippocampai neurons and their neighbors. It was possible to use coupled

mapped dock oscillators to reproduce al1 oscillatory modes observed in biological

recordings fiom the brain slices. Periodic, quasiperiodic, chaotic and non-oscillatory

activity were detected. Different oscillatory modes were produceci by changing the

Vitnnsic fiequencies of the oscillators and coupling conditions. It was possible to

investigate the regions of transitions between diflerent oscillatory modes. However, the

emphasis of this work was placed on the investigation of the regions where the system

exhibited chaotic behaviour. In particular, îhe effects of adding input portal P, and self-

feedback loops to the system had on these regions was studied in detail.

Researchers have oRen relied on the analysis of bime series from the EEG to gain insight

into dynamical behaviour of the brain [19]. Nodinear analysis methods have ofien been

used to determine dimensionality of the recordecl time series signai and find out a degree

of chaoticity of the signal. However, there are several problems associated with an

application of these procedures. In order to successfully detennine dimensionality of the

time senes signal, long recordings are needed and they are often unavailable. The

Mapped CIock Oscillator model is a parametric mode1 completely defined by a system

of difierential equations. Intrinsic parameters for the oscillators were obtained from

biological expenmental &ta to ensure greater agreement of the modeled and

neurophysiological processes. Parameûic rnodels prove to have a number of major

advantages in modeling of neurological events. Une of the major ones is that since a

parametric model is represented by a system of diflerentiai equations, simulations cm be

carried out for any necessary length of time.

The Mapped Clock Oscillator mode1 with four input portals has been designed to enable

modehg of both chernical and electrical coupling modes between neurons. The design

relies on the following properties confirmed in earlier studies:

1 ) Frequency and amplitude can be selectively altered without affecting one another.

This is achieved through the activation of input portals P1 and Pa respectively. Earlier

studies confirmeci that a nimiber of pharmaceutical agents have this effect on the

brain, changing either fiequency or amplitude of the bmin signai, independently from

one another.

2) Earlier investigations revealed that the input porial P, exhibits a number of functional

characteristics similar to those of the gap junctions.

3) Functional characteristics of the synaptic coupling were inwrporated into a design

of input portai P, Both gap junctions and chemical synapses have been reporied in

the CA1 and CA3 regions of the hippocampus [8]. They are both found to play an

essential role in the omet of different oscillatory modes in neuronal activity. In

particular, a large number of intemewons bas been identified in the hippocampus,

synapsing between somas of neighhring neurom. Intemeurons mostly serve an

inhibitov h c t i o n

4) Selfifeedback loops are introduced in the mode1 as it is known that a number of

molecdar feedback mechanisms play a key role in modulations of neuronal activity.

These include ionic rnechanisms, receptor inactivation processes and others.

OsciIlatory modes of wupled oscillators are determined both by intrinsic oscillator

properties and coupling arrangements between them Entrainment properties of the

coupled oscillator system were stuâied in earlier investigations (11. In a study performed

with three input portal oscillator entrainment properties were midied in detail, but

chaotic behaviour was observed oniy at one specific coupling configuration. Specific

coupling factors were set at the foiiowing values: ~ ~ 4 . 1 , ~ ~ 4 . 1 1 8 and $4.1 . Chaos

was M e r identified at three points distinguished by intrinsic fiequency difference. It is

assumed that healthy neurobiological systems operate in a chaotic mode in a quite robust

manner. Coupling anangement stated above was used to initidly bring the mapped dock

oscilkitor system into this operathg mode. It was observed that the addition of syMptic

coupling portal and feedback pathways significantly increases the extent of the chaotic

operation area, ensuring that the system c m maintain desired operation mode. Inclusion

of the input portal P, and increasing the amount of seIf-feedback between the portals,

bring additional oscillating forces into the system and provide the means for necessary

fluctuating activity. Complex dynamic behaviour may result from oscillatory paths

shanng common instability generating feedback pathways. Complex behaviour modes

mise when oscillatory rnechanisms of similar significance borne coupled in parallel

contiiguration Importance of these rnechanisms is especially underlined in

neurotransmitter controlled cellular interaction, such is the case in chernical synaptic

coupling.

5.2 Information Coding in Chernical Synapses

The basic characteristics of synaptic transmission enable this type of neuronal coupling

to specializ for information receiving and coding. These characteristics can be examined

by obsewing simultaneous recordings from presynaptic and postsynaptic terminais. It is

observed that impulses of changing frequency constantly arrive to the presynaptic

terminal. At the postsyaaptic terrninal, depolarization, or hyperpolarization, proportional

to the frequency of the presynaptic stimulus is recordeci at every instant.

These properties are illustrated in figure 5.1.

Figure 5.1. Postsynaptic depolarization in response to changing fiequency stimulus from presynaptic terminal. (a) presynaptic stimulus; (b) presynaptic ftequency profile; (c) changes in postsynaptic resting potentiai. (From Stevens[43]).

The nerve impulse arriving at presynaptic terminal, after a short delay, causes a

postsynaptic membrane polarization, referred to as the postsynaptic potential (PSP).

Nente impulse stimuli cause newotmsmitter release into the synaph'c clefk It has been

shown that neurotransmittea are released in small quanta. This proves to be one of the

most important properties undediring synaptic transmission and its characteristics The

number of neurotransmitters released determines the magnitude of PSP response.

Synaptic transmission properties such as graded response, temporal or spatial sumation

depend on the amount and concenidon of the released neurotransmitter. Graded

responsiveness nfers to the fact that the magnitude of PSP response is proportionai to the

quantity of neumtransmitter released, as opposed to the all-or-none response associated

with action potentids. This relationship may be observeci by loolring at the peaù PSP

response as a hinaon of the quantity of traasmitter released plotted in figure 5.2.

Figure 5.2. Relationship between PSP amplitude and the quantity of transmitter released (From Stevens (431).

Graded responsiveness also allow for the addition of two simultaneous stimuli at the

postsynaptic terminai. Incoming impulses have a property to add to the remainder of the

previous signal and produce an increasing PSP response. This property is referred to as

temporal summation. Similady, graded responsiveness to releared trammitter allows for

spatial summation of stimuli from a number of neurons synapsing ont0 a single

postsynaptic neuron [43].

It is important to distinguish between excitatory and inhibitory lypa of p o s t ~ ~ ~ ~ ~ t i c

potentials. Excitatory postsynaptic potential produces a depolarization of a postsynaptic

membrane, and therefore may elicit the omet of an action potential. On the other han4

inhibitory postsynaptic potentials produce membrane depoiarizations and tends to

prevent an onset of action potentials. Except for the fact that they produce opposing

polaization of the postsynaptic membrane, excitatory and inhibitory postsynaptic

potentials have essentialIy the same propties. Inhibitory postsynaptic potentiais can

sum up together to produce an average hyperpolarization proportional to the presynaptic

impulse freswcy [43]. Iohibitory synaptic pathways are especiaily important in the

hippocampal region, as most of the interneurons that are abundant in this region, exhibit

inhibitory properties [8]. Importance of proper f'unction of the inhibîtory pahways in this

area of the brain has already been discussed in Section 1.3.

Using an MC0 mode4 synaptic coupling fiinction could be rnodeled through an

activation of input portals P, and P+ and of ~e~feedback loops. Properties of temporal

and spatial sumrnation were not examined in the two oscillator model. However, effects

of increasing stimulus fiequency were exarnined in a unidirectionai coupling

arrangement Perhaps these effects are best seen as one examines changes observed in the

resting level of the stirnulated oscillator. In the case of positive coupling via portal P, the

resting level decreases propomonally to the increase in stimulus fiequency. Hence, a

hyperpolarization proportional to the presynaptic fiequency is observed. Oppsite

changes are noted when the sign of coupling factor c, is revened Changes in amplitude

were expected to show aimost a sigrnoidal response. Although the increasing phase of the

amplitude coding was not observed, it is important to note that the amplitude did not

exhibit unbounded growth with increasing stimulus frequency. Since an increasing phase

of an amplitude response was not observe& it is assumed that in order to capture al1 the

properties of synaptic transmission, membrane properties must be rnodeled in more

detail. Transmitter release and presynaptic fuing propetties largely depend on the

presynaptic action potential frequency which is limited by refractoriness of the

p r e ~ y n a ~ c membrane. This fesrture of synaptic coupling was not included in the current

model. It is believed that incorporation of rehtorines and pperties of the membrane

in the model will make the modehg of synaptic coupling mechanisms more accurate.

5.3 Synaptic Strength and Self-feedback Mechanisms in Syna ptic Transmission

Chernical synapses govern some of the most important fiuictions of the brain such as

memory and Ieaming. As d e s c n i in Section 1.2.3, the main characteristics of synaptic

transmission include release of n e u r o ~ t t e r s nom a presynaptic neuron, binding of

the transmitter to receptors on a postsynaptic membrane and the omet of

pstsynapticresponse.

Ionic mechanisms involved in release of neurotransrnitters were investigated in detail by

a number of researchers [14]. Absence of K+ did not have a significant innuence on the

postsynaptic activity. ~ a + was found to be important only in achieving sufficient

presynaptic depolarization that allows transmitter to be released. In later studies [XI, it

was found tbat ~ a * plays an essentiai role in transmitter release enhancement, vesicle

fusion and exocytosis and even in modulation of probability that a quantum of

neurotransmitter is released. Within a postsynaptic terminal, a number of cytoskeletal

elements maintah the synaptic structure. Endoplasmic reticuia are among the commonly

found elements that are believed to be involved in removai and release of ~ a * ions. As

such, these elememts allow for the communication between pre and postsynaptic

teminals and the establishment of feedback pathway betwéen the two [9]. In addition,

Ca" plays an important role in the modification of a postsynaptic signal as a second

messenger involved in the processes of protein kinase phosphordation [2 1 1.

Second rnessenger systems, calcium regdation and feedback prove to be important in

synaptic plasticity which is related to l e d n g Self-feedback mechanisms regulated by

calcium signaling and synaptic strength modulation are especially significant in the

excitstory synapses of the hippocampus as they govem the long term potentiation (LTP)

process [2 1,251.

LTP is a phenornenon that occrns in respoose tu brief, high fkquency stimulation that

produces a long lasting increase in synaptic strength [26]. High fiequency, or tetanic,

stimulation of sufficient magnitude and fiequency produces a depolarhtion of the

postsynaptic membrane which proves to be critical in the onset of the LTP response. In

addition, binding of excitatory aansmitters is required for synaptic activation and

induction of LTP [31]. Hippocampal neurons contain both N-rnethyl-D-aspartate

(NMDA) and non-NMDA receptors that use glutamate as a transmitter. Normally,

NMDA receptors are biocked by M ~ * ions. However, when the postsynaptic terminal

becomes strongly depolarized, they become unblocked and activatecl, Removal of ~ g *

ions dlows for the flow of ~ a * ions into a postsyn;rptic c d . ~ a * innwr proves to be

critical in induchg the LTP response. Calcium was found to enhance synaptic

transmission in this process through a feedback loop. The feedback pathway is

established through activation of two calcium dependent protein kinases involved in

second messenger processes. It has been hypothesized that phosphorylation of the kinases

influences non-NMDA receptors to i n c m the synaptic response and maintain long

l&g synaptic transmission [27l. In addition. Ca* dependent feedback was shown to be

important in the reIease of retrogaie mesengen from a postsynaptic terminal [21].

These were shown to effect the release of second rnessengers that have the ability to

increase neurotnmsmitter release and maintain the LTP response. These two processes

involving feedback are show in figure 5.3.

Figure 5.3. Calcium role in the activafion of non-NMDA reœptors and LTP maintenance (From Kandel et al. [22])

From the above examples of rnolecular feedback mechanisms estabiished between

neurons and synaptic strength modulation, it is clear that these processes piay an

important role in the normal fimctioauig of hippocampai neurons. Feedback, together

with proper synaptic couphg, must be included as an integral part in a redistic mode1

of hippocampd neuronal circuits.

From the results of this midy, it is evident that the addition of feedback pathways and

synaphc portal in the MC0 mode1 significantly influences the behaviour of the system.

One of the most important points to notice is that the MC0 model allows for a wide

range of modifications of synaptic coupling, from hyperpo1a.g to depolarking effects.

These can be achieved both through the alterations in synaptic strength and feedback

effects, as it was pointed out in Section 4.1.2. Secondly, the addition of synaptic coupling

and the consideration of feedback mechanisms in the model increased the operational

area of the system. Figures 4.15 to 4.19 clearly support this statement and illustrate an

increase of chaotic operational region under the influence of synaptic coupling and the

activation of feedback pathways. Furthemore, once the system operates in this area,

signifiant changes in coupting arrangements are needed to push the system out of diis

oscillatory mode. If normal functioning of biological systems is assumed to be in a

complex oscillatory behaviour region., the operating point m u t be robustly maintained in

this region, away from pathological boundaries that may induce entrainecl activity.

Chapter 6

Conclusions Future Work

and Recommendations for

A mode1 consisting of two coupled mapped dock oscillaton was used to examine the

interactions of oscillatory activity beîween hi ppocarn pal neurons. The mapped cloc k

oscillators can successfully reproduce different oscillatory modes in a nonlinear

dynamical systern. In addition, mapped clock oscillators can be used to gain more insight

into transitions between the oscillatory modes. It was illustrated h t transitions among

different modes of electrical activity can be produced by changes in coupling

configurations and intrinsic frequencies of the oscillators.

Research suggests that the brain operates in aeas of chaotic bebaviow to achieve a high

degree of fiuictionai variability and increase its abiiity to adapt to environmental

fluctuations. It was of particuiar interest in this midy to examine the effects of chemicai

synaptic coupiing on the extent of the chaotic operational area of the systern. In addition,

the role of molecular feedback mectianisms in the onset of chaotic behaviour was

examined The input portal, P, dong with portai P+, were used to simulate the synaptic

coupling between the neurons. Self-feedback pathways were also established between the

portals. It was concludeci that the addition of synaptic coupling greatiy increases chaotic

operational axa, as compared to the one observed in prwious investigations [I l . Synaptic

coupling causai fusion of points where cbticity was identifid In addition, an applied

level of negative feedback greatly affected the extent of the chaotic region; this extent

increases as the amount of applied feedback is i n c d It is believed that the addition

of synaptic couplhg and self-feedback loops gives a hi@ degree of fiexibility to the

system. This is evidenced by the fact that coupling mechanisms can be significantly

influenced by the amount of self-feedback applied as well as coupling strengths. For

example, synaptic coupling can be changed from hyperpolarizing to depolarizing through

an increase in the amount of feedback and an increase in synaptic strength. Furthemore,

synaptic coupling and feedback mechanisms influence the non-osciliatory region Earlier

investigations reveded that the system exhibits non-oscillatory behaviour as coupling

strength is i n c r e d Wiîh the activation of ~e~feedback pathways this region virtuaily

disappean.

Following recommendatiom can be made for friture d i e s using the mapped dock

oscillator modet :

Refiactory properties of a membrane shodd be included in the mode1 to achieve

greater accufacyufacy Refhctoriness of presynaptic membrane is found to have an

important role in the proper functioning of chernical synapses as it b i t s the extent

of the action potential frequency. Thus, it should be considered in order to ensure

proper rnodeling of the coupling mecbanisms.

An attempt should be made to improve the description of the synaptic couplhg

portais to d i e easy modifications between excitatory and inhiiitory synapses.

This would enable modeling of syaaptic couphg in different areas of the brain

where synapses are not predominantIy inhibitory or excitatory.

An attempt shouid be made to improve the transformer demipion to indude the

properties of dyaamic membrane transport phemmena.

Additional, more detaied, experiments of coupling codigurations shouid be

preformed to hvedgate the transitions between oscillatory modes in the system

and gain deeper insigbt into reiated processes that may play an important role in the

onset and spread of epileptifonn advity.

Techniques of control and anti-control of chaos should be included in the study

using the new model. Control of chaos techniques are applied in order to change

chaotic oscillations to stable, periodic modes with an application of proper stimuli.

Anti-contol rnethods prove to be interesting in the search of novel therapies for

eptiepsy, as they involve changing penodic osciUations to chaotic ones. These

should reved the possibiiities for appiication of Merent stimuli to achieve de-

synchronizaîion of entrained electricai activity.

Static normalization factors should be replaced by dynamic ones to take dyaamical

changes of osciliator output into account.

Considerations shoufd be made for different kinetic characteristics of coupling

mechmisms. Greater speed of electrotonic and fieled coupling, as compareci to the

chernical one, shouid be taken into account in deîemhing the properties of

different couplhg configrwtions.

The mapped dock oscillator mode1 proves to be a vaIuable tool in the investigation of

parameter changes in the brain Ieading to different pathological states, such as epilepsy.

It may prove to be usefbi in the deveIopment of "the pacemaker for the brin" which

stands as a promising application of both neuroscientific and bioengi~eering research.

Appendix A

The following is the listing of C program for the simulation of the coupled MC0 system

for use with INSITE software package. The code is written in accordance with the

requirements given by MSITE. For more information Please see software documentation

provided in Parker and Chua [26] and as part of the installation documentation for the

package. Procedures f and df are given in bold They include equation expressions and

Jacobian. Elements of Jacobian matrix are given in the z array as - z[0] to -z[15].

/* File: mco-f Ib3 .C

Purpose: to calculate and display the trajectories; to calailate Lyapunov exponeas and dimension

compile using Malcefile. bso or M a k d e . rkf in neurodmco l/traj/CA3 for trajectories or neumdmco 1 AyexpKA3 for Lyapunov exponents and Dimension

Can use lookup table or Tchebyshev polynomids now with lookup table

*/ /****************************************************************/ /**********************************t*****************************/

/******************************+**********************************/ /**********************$**t8*********88****8*8*****4********4******/

#if EXT extern doubIe sgn(doub1e); aden double ipow(double, int);

#dse extern double sgnl), ipow();

#endif

#define LOOKUP 1 #d&e RAND-MAX 32767 /* File S a p e Variables */ /* */ static double ave 1, ave2, one-over-ave 1, global~one~ovcr~ave2; staric double a1 -+Il, a2-+Il, a1 0, a20; static double b 1 wLOCHARM+ 1 1, b2pMXHARM+ 11; static int numHarm;

static double -alpha, q h i _gama -ho; striric double -betaghi, beta-rho. -beta-alpfia;

static double c . -c-beta; static double -3, -t2; natic double -nampD;

static double w a v e w S ) ; static doubLe dwave(MAXPTS1; static char fhc-30], table_name[30], dtable_name[3 O]; static double delta-XI, one-over-deltatan. delta-n-over-2; static double mean- 1, -meaaOfiS&; M c int unidir-fbg static double deltal, d m ; int counw;

void updategarms(); void updateqarms 10; void updateqanns20;

stabc char *gW =

"c", "alpha", "phi", "gama". "rho", "ifl ". "fz", "aampI2". " - meanOflSet2". " c-ka". "beta_alphaa, "betaqhi", 'beta-rho". Nuchar)

1;

FILE *@;

iq(f$4qm(fn, accessCCeSSmode)) = MJLL)

Fprlnqstderr. "Error opening fiie Yi with acfeu mode %s\nR-ogram exitsb". f9 afc*ls mode); - exit(- 1 );

1 m m @;

1

void readg8fmso

printqn&ading tabIes\nM); Q = open-tile("aaj.optr". "ru); fscanf(@, "%*s %s ", fIp3efQ; f M @ , "%*s %s ", table-me); fkax@fp, "%*s %s ", dtable-me); fscHfp, "?/a% %F, &deha 1); fsCanqfp. "%*s %Lf", &delta2); fiau@@. "%*s %If", &meanonset 1 ); fscanf(@, "%*s %du. &unidufinP);

printqwdelta 1 O/&", delta1 ); printf("delta2 %bu, de la ) ; printq "rneanûfkt 1 %hm, mean- 1 ); p- "unidir-flag Yod\n". unidirflag); qunidir-flag==O)

printqwBidMonally coupling\nW); else q u n i d i r - f l y l )

p~"Unidirectionaiiy coupIing\n\nm); else I

pm"\n\nERROR: unidVflag=O/od is set incotfectlyh"); &t( 1 );

1

p r i n t f ( " d %s h", fhcoeff); printqMtable %s Li", table-name); printqmdtable %s W. dtabte-me);

#if LOOKT.JP prUrtqln Wnh Lookup TabIe\n");

#else prinflm\nWith TchebychevLin);

#endif fc~ose(fp);

/* Read in a0 */ ~fi,*iiO/oIf\n",&al O); a1[0] = alOtmeanOflkt1; /* Read in rest of intrinsic waveform */ avel = 0.0;

/* sets the file position hdicator to the */ /* begining of the file (only if oscl and */ /* ose2 have exactly the same waveform)

/* second oxiliaror */ fScanqfp,"%d %*lf",humHann); /* 5.8 */ pm"numHarm = WU,numHam); /* R e d in a0 */ fsCanq@, "O/oiAnn ,&a20); a2[0] = a2W-meanoffset2; P Read in rcst of intnnsic waveform */ avP3 = 0-0;

int is-dimete()

void setgar(int n, double x)

switch (n)

case O:

case 1:

case 4:

case 9:

break; case 10:

f switch (n)

rctum (Ma-alpha); case 1 t :

doubk phi& phitt, gamr21, gamalX rilpba21, alphall, rho21, rhol2; double ornegai, omega2; double bctrPbi, W p b a , B o ;

double m[1001,n[100]; double ull, ul2, dl, n22, ultgrl, PIS@, -rl, -1-2: doable uldotl, oidotl, ddotl, atdot.2, sqmagl, squtag2; double twou21, twou22, prodaTl, prodaT2, ptodbU1, prodbUR double sqteml, sqtermt, proidTI[3], prordTt[31; don blc pbi-dotl, phi-dot2; double XI, X2;

double mull, mul2, mu21, m922,twomu21, twomu22 ,mulsqrl, mulsqd; double mtl2sqr1, matsqr?; double sqtcrm~OSC+lJ,protdT~OSC+1]~3 ]@CO WU~OSC+iJ[S]: double chT1 -+Il, ch'f2pfAXEUM+l],

cbUIpvwumRM+1], chU2FIAXflARM+I]; double mal [ N O S C + I ~ ~ ~ ~ ~ [ N O S C + ~ ~ , ~ O ~ ~ ~ O S C + ~ ~ , ~ P ~ ~ ~ ~ O S C + ~ ] :

double mu2sq~OSC+l~,ûymul~OSC+1 ),dynm2~OSC+lj; dou blc one~over~avet;

/* prinw" in fO\nW); */ iflunidir-fhg=û) {

/* Bidir. couphg */ alpha21 = alplml2 = alphan-c: phi21 = phi12 = _phi*_e; gama21 = gamal2 = *ma*-c: ho21 = dl012 = -rtio8-G

1 tisc iqnnidir--1)

If Unidu. coupihg */ aipiml2 = phi12 = pnmal2= rhol2 = 0.0; alpha2 1 = -dph.*-~ phi21 = qi i f - c ; g u n d l = m a * - c : ho21 = -rilof-c:

1

u-; Pbecawe fimt inda of the un passeci in is Of l

l* fi& osdatoi. */ u l l = u[ll; u2l = u(21; u ldotl=-udot(01; ~2dot ï=~udot( l~; ulsqrl= ull*ull; uZsqrl= u2lgu21: sqtuml= sqrt(ulsqrl+u2sqrl): sqmagl=u~sqrl+u2sqrl;

a(sqtcml= 0.0)

printf("\nall and 1121 = 0.0inW); uit(1);

1

XI = sqterml*(intcrpd(w.ve, tlll, R2l).ril0)+ (al O+mcanOffJetl)*(l+(rho2Yomegril)*(phi,dot2-bct.Rho*phi~dot 1)); X2 = sqtetm2*-nuxipl;2*(MterpoI(ww~ a12, ~22)-do)+ (~W-m~nOnset2)*(l+(rho1Uomega2)*(pbi-dot 1- betaRho*pbi,dot2));

l*printf("V~Ma",pbi-doe-phi-dot 1 * betPRho); */

#clse /* Tchebychev polynomiais */

m u I l = nll/sqtcrml: mu21 = u2l/sqttrml; mulsqrl = mull*mull; m d q r l = mu21 *mu21; twomu21r: muZl+mu21;

If Compute the f i two temu of the Tchtbphtv potyaomtib f / l* and the k t two tems of the fvst derivatives Y chT1[l) = mu21; chTl (21 =. Z.O*mPZsqrl- 1.0; cbUl(1) = 1.0; chU1[21= twomu2l;

P Compute the Tscbebysbev poiynomhh ushg initd vrhics */ /* and cornputcd funciions rbove */

/* Compnte the f i two trrms o f the Tchebysbcv polynomaiïs */ /* and the first two terms of the fvst derivatives */ cbT2[1) = ma22; chT2[2] = 2.0*mu2sqr2 - 1.0; cbU211) = 1.0; chlJ2[2J = twomn22:

/* Compute the Tschebyshev polynomiab uid its f i */ /* partiai derivatives (dyT,dyU) using inital vdues */ /* and compated fmctions above */

/* Compute tbe product tenus Ak*Tk and Bk*Uk-l

- udot [O)--1 *alUddtaI + omcgrtle((l + ~-bttriPbi*X1)*pb~1*oae~m~~avtl)*rir1 +

n l l * ( l + W-bctnAlphP*X1)*aïphr31*oae1~er~avcl - alsqrl - ii2sqrl j);

return (TRUE); 1 /*n***nt******nnnete~nn**n*lt~r~***n*********a****nnnee*n*~**n**n**/

dl( doubk -z[l,doubk *u,double t)

doable phal, plil2, g-1, gamal2, dpba21, aipb.12, rhol2, rholl; double betaAipha, beâaPhi, -0;

doubk orne@, om@:

doubk ult, ~12,1121,1122, ulsqrl, ulsqx2, u2sqr1, ntsqr2; d o u h twou21, twou22, prodaTl, prodaT2, prodbul, pradbU2; double sqtcrml, sqterm2, p d T l [ 3 ] , proadT2[3), pmWUl[31, pmbdU2[3]; doubk sqmrgl, *ma@; double sqsql, sqsq2;

double mull, mal& m d l , mu22,twomu2l, twornu2.2 malsqtl, mulsqi2; double mu2sqr1, ma2sqGdymall, dymnl2,dymu21, dymu22;

doiiblt XI, X2 . Xdotl 1 , X d ~ t l a Xdottl, Xd0t22; doubie yl, y2, yldotll, yldot21, yldotl2, yldot22; double y2dot11, y2dot21, y2dot12, y2dot22;

doubk denyl, dcnyz doublt constyl, consty2; doabk chtbl, cheb2;

double Id-dwavel, IOCi10Cildwave2: donMe W-intcrpdl, kal-interpotZ., double one~ovu~ave2;

double DER, DER-11, DER-21, DER-12, DER-=; double dcrl, dtr2, der3, der44 doubie derl-11, dci.1-21, derl-12 d ~ l - t 2 ; doubk der2-Il, dtr2-21, der2-12, der2-22: double der3-11, der3-21, det3-12, d e r 3 3 double dcr4-11, dcr4-21, dcr4-12. dtr4-22;

double AlO, A20;

int ÿ,k,jml jm2J;

FILE 'fpt:

Xdot 12 = ( Id-dwavt 1 *(on 1 1) + kalOcalintetpoll *u2l)lsqterm 1;

/* Compute the first two t trms of the Tchtbyshev polynomicrls */ /. and the fita two tenns of the first derivative */ cbTl[l] = rnu21; cb'ï'lp] = 2.O*mu2sqrl- 1.0; cbU1[1] = 1.0; chU1(2]= twoma21;

P Compate the Tschebyshev poipomirls and its fvst */ P putPl derkatives (dyT,dyU) asing hitd vducs */ P and cornputcd fanctions above */

for (j=3;jt= numFlnim; j*)

jml = j-1; jmt = j-2; chlr Uj= twomuZ1 *chTl~ml]-ehT1 fjm2); chUllj] = ~omri21*chUl[jml]-~hUl~m2];

/* Compute the product ttrms Ak8Tk and Bk*Uk-1

for (j=l;j<= numHnrm;j++) t p d T l = prodaTl + chTl [j}*al fi]; prodbUl= prodbUI + chUlU]*blu]; pmdTl(1J = prordT1[1] + âyTIlj]jl]*af lj]; proadTl (21 = p ~ o ~ d T t 121 + dyT1 u) (2) *a 1 Ejl; probdU1[1] = probd'tll[ll+ dyUl(j]jl)*bl[j]; probaUl[2] = probdU1[2] + dyU1~][2~*bt [j]; 1

/* FiIn- computt X[t) and Xdot[tl */ X I = A10 + (prod.Tl+mul l * p d b U l ) * sqterml: Cîlebl=Xl;

Xdotll = mull*prod.Tl+sqtcrml*proodTl[l~ + prodblJl+rill*probdUl[l);

Xdotl2 = mrirl*pradrTl+jqtcrm1*pmadT1[2~ + alluprobdU1[2J;

P Computc tbe fint two t lws of the Tcbeôyshev pdpnomhh */ /* and the f u ~ t two terms of the first derivatives */ chT2[1] = mu22; cbT2[2J = 20*moZsqr2 - 1.0; chUt(1) = 1.0; chU2[2] = twomu22:

/* Compute the Tscbebysbev pdynomi.ls and its first */ P partul derivatives (dyT,dyU) using initai vrlrrcs */ /* and cornputcd fanctions above */

/* Cornpute tbe prodact teirm AkeTk and Bk*-1

prodin = 0.0; prodbU2 = 0.0; proadT2[l] = 0.0; proadT2 (2) = 0.0; pro bdU2IlJ = 0.0; prabdU2[21 = 0.0;

for (j=l;j<= numHana;j++) I prodPT2 = prodaTZ + chT2[j]*a2IjJ; prodbU2 = prodbU2 + chU2tj]*b2(j]; prordT2[ll= pmadT2[1] + dyT2bj (1]*a2(j]; proad'ï2[2] = proadT2[2) + dyT2[jf[2]*a2[i]; probdU2[1] = prabdU2[1] + dyU2[j][l]*b2(j]; pmbdU2[21= probdU2[2] + dyU2[j][2J*b2[j); 1

/* Fîmaity campute X[t] and Xdot[t] */ X2 = Mû+ (prodaT2+mal2*prodbU2)*sqtcrm2*-nrimpl2; cheb2=X2;

/* FinaMy, compute yl[tl and y2[t] and the derivatives wiîh respect to uII, d l , nl2 and u22 */

dcrl-1 l~u22*sqmrgl-Ze~l l*ul ltri22)hqsql; dtr1~21~-2*~11%22*~21~sqs~I; derl-124.0; der1_22=ul Ihqmagl;

DER-I 1=(((A10*rho2l)/omega1)*((gamal2*dcr3~1 lfddtrrtp (gum&21*bet.Rbofdtrl-f l/dehrl-f 2*der4-1 l/deita2)- (gama21 *bttaRbo8dert_l I/dtltal))H((AZO*rhol2)/omcgriî:)* ((gama2l*dcrl-1 l/ddt.I)+(g~ma12*ktrRho*dcr3~11/ddta2~ (ganta2 1 * d e c l l/deital)-(gamatl2*dcr4_11 *WaRho/delta2)));

DER-

yldotl l=duiyl *@dot1 1+(AIO*rb02t/omcg.1)*(0mtgiZ* ( - p h i 1 2 ) * ~ b i * o n e e w t r ~ a v e U c o n ~ 2 * d o t l I+DER-11)- beîaRhoaomegal *phi21 *bctaPbi*0ne~ov~~~avel/~0nsty2* (Xdotl l+DER-1 I F garnal2*(derS_l Ider4-1 l)lddta2+ gnma21 *bctriRlio*(derl-1 l-der2-11 )/ddtal));

fprinti(rp1,"deltal fp~tt(fp1,"ddtaZ fprinflfp1,"alO fprintf(fpl,"a20 fprintT(fpl,womegnl fprintf(fp1," omega2 fprinqfp 1, "avel fprin~fpl," me2 fprinqfp l,"alpbal2 rprinatp 1," aipiIr21 fpriLiqfp1,"philZ fp~tr(fpl,"pbi21 fpiZntf(fp l,wgama 12 fprintf(tpl,"gama2l fprintf(Cpl,"rhol2 fpriatf(fp1,"rhoZl fprintf(fp1," DER fj~rintf(fpl,"derl @riiitfCfpl,"der2

/",z[O] */ -z[O~~megrrl*(~lnphi21*~2dofll-t>ttnWi*y ldotl I)*ont~ovcrdavel+

I+alpha2l *ull*one-ovctCtave~2dotl 1-betaAIpha8y1dot11)+ alpha21 *(y2-ktaAlphrnyl)*one_overfavel - S*ulsqrl- uZsqtl);

/*-21 Il*/ z ~ l ] = o r n ~ l + ( l + phitl*~orZl-bcuPhigyldot2I)*~1*oue~~er~avel+ph2l* -

(y2-betaPhiny l)*oatt0vererave1+ul l*dpha21* ~dot21-bttlAlpba*yIdot7ll)~0ne~wer~~el-2*ul l*u21);

/*-z[2j */ z [ 2 ~ ~ a ï l / d e i t a l + omcgril "((y2dotf Z-betaAlpha*p Idotl2)*rlpha2f *n 11 *one-over-wel+ -

(y2dotl2-betaPhiny ldotI2)*phi2 1%2l*one~ovcr-ave 1); /*-z[3) */ z [3 ]~mtgal *(@Zdot22-bct iAlpbeyld~~)*afph~1l 1 *one~over~avel+ -

(ytdot22-betaPhiny ldot22)*phi21nu21*one~wer~avel); /*-z[4 */ -z[4l=omcgri 1 *( *2-betrPni*yl)*pbi21*one~overerwe1-1-

ull*phi21*(y2dotl 1-betrPhiayldotl l ) * o n c _ o I + aZl*alpha21*(y2dotl l-WpheyldotlI)*one-wtr_rvel-2*ul l*uZI):

I * , z [~ */ -z[~..orncgaln(-(u 1 1 *phi2 1*~2do~1-barWi*y1dotZ1)*onee~er~we1)tl +

d p W l n ~ - ~ p h . * y l ) * o n e ~ o v t r _ r v e l + dphrrll*~2dotZl-bttrAlphi*y1do~I)*~l *one-over-avel - ulsqrl - 3%2sqrl);

/*,zfq*/ z[6)=~mcyl*(-(ul l'phi21 a~ot12-bctlPhi*yld~t12)nonte~vtr~avcl) + -

~y~otl2-bctrrAlpha*yldotl2)*dp~I*a2l*one~o~er_rvel); /*-z[7) */ z[~"gim&l/ddial +amwLg(41y2dott2-bctilWi*yldot22)* -

phi2Pull *one~wererivel~Mo~ZIbttPAlp~*yldo~2)* alpha21 * a l *one-wer-=el);

1 /* end of dfO */

{ double t h e

/* interpolates the waveform 6om the lookup table, returns the tfansforrner output

*! double interpoI(double *local-wavqdouble u 1, double u2)

double indexvai, flmrwave; int fioorindex, ceüindex;

indd=dindex(ul, u2); /* exact theta */ flwrindex = (mt) floor(mdexvaI); fioorwave = Id-wave[flooriadtx]; ceilindex = rit) ceiI(indewal);

/* interpolates the derivative wavefom h m the lookup table, retums the transformer output

'1 double dinteqml(double *Id-dwave, double u 1 , double u2)

t double indemai, flootwave; int floo~dex, ceilindex;

indexval=dindex(u 1, u.2) - deltanover-2; /* exact theta shifted by */

double didex(double u 1, double u2)

double theta;

void read-tables() t

FILE *Qi; im waveindex, max, dmax; double dummy',

/* read table uifonnation */ /* */ printq"Rcading m table %su, table-me);

@=open-&(table-name, "r"); S!check-sic-fom(@i I , 1, 1))

exit( 1 1; /* wrong file format! ! */ waveindex = 0;

wave[max]=wave[O] ; /* to close the period circie.. . */ pNitq"\tmax=O?'d~n". max); fcIose(fi>i);

prùitf("Done reading in table %s\nw, table-me);

break;

/* read wave file with dcrivatives */ /* */

printqmReading in dtable %", &le-me); fpi=openpenfile(dtablefilname, "r");

if(!checkkfife-fom?~t(f@, 1, 1, 1)) 1; /* wrong file formst!! */

waveindex = O;

wavemd*; 1

dmama~veinddex; dwave[dmax ]=dwavefO]; /* to dose the period circIe.. . */ p~fl"\t*/od\n~, dmax); /*iqdmax != niax-1)

myerror("Error in read-tables(): files tabie or &le are wrong!!\ dmax must be max-1 !");*/

fctose(Qi); p m N D o n e reading in table O/as\nn, dtabie-me);

delta-n = (double) (M-2PVdmax); one_ovér_delta_n = Ifdeita-n; delta-n-over-24&-n/Z.O;

/****************************+**************/

void tnyerror(ctuU enor-tcxtQ)

/* void exho; */

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