Research Collection

Doctoral Thesis

Computer simulation of nerve signal transmission

Author(s): Christen, Tobias Fabio

Publication Date: 1999

Permanent Link: https://doi.org/10.3929/ethz-a-002025540

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ETH Library

Diss. ETH No. 13024

Computer Simulation of

Nerve Signal Transmission

A dissertation submitted to theSWISS FEDERAL INSTITUTEOF TECHNOLOGY ZURICH

for the degree ofDoctor of Technical Sciences

presented byTobias Fabio ChristenDip!. Informatik-Ing. ETHborn October 19, 1967citizen of Andermatt, Uri

accepted on the recommendation ofProf. Dr. W. Gander, examinerProf. Dr. D. Walz, co-examiner

1999

Acknowledgernents

This thesis was carried out in a collaboration of Ciba PharmaResearch.Basel and the ETH Ziirich during the years }\,)9:3-96. I \vclUld like tothank the former Ciba AG for tlw project grant for this thesis. I thankThomas Kn6pfel for his initial supervision. I'm deeply indebted to ProfvValter Gander v;ho provided me with the opportunity to join the In­stitute of Scientific Computing at ETH Ziirich for the purpose of thisdissertation and to Prof Dieter vValz at the Biozentrum Basel who of~

fered me to finish this interdisciplinary project under his supervision.I'm most grateful for the attention and guiding generously offeredduring the past years.

I thank ho VralH:sic for his thorough assistance specially during theimplernentation of the "l\lonte Carlo Ivlodel". J\h thanks also go to\lario Pozza and Johannes l\losbacher at the CNS department of Cibawho madc' helpful comments on earlier drafts of this thesis. FurthermoreI would like to thank Benn~' Bettier and Franco Del Principe for the1l1any fruitful discussions we had in a motivating atmosphere.

Parts of this work would not have been possible \vit hout the supportof many co]]pagues and friends at the E'TH Ziirich. In particular I wouldlike to thank Peter Arbenz, IvIichael Oettli and Florian Schlotkc forthe many matbematical and aclministratiw tips and ach-ises I got fromthem.

I'm very grateful to Cbristoph "Volf and Hans-"Verner Ott who al­\\'ays found comforting and motivating comments. I thank my parentsfor their rnotivating words in many situations during the past years.

Last but not least I \vould like to thank Taina for her ever understan­ding support and patience during the years of this work.

11

Abstract

llBSTRACT

The signal transmission between neurons OCCurs at specialized ce11-to­cell junctions, the so-called synapses. During this process a chemicalsubstance (neurotransmitter) is released from a sending (presynaptic)neuron into the synaptic cleft. The most abundant excitatory transmit­ter in the central nervous system is the amino acid glutamate. \Vithinmicroseconds the focally released glutamate spreads into the cleft bydiffusion. The response of the receiving (postsynaptic) cell is medi­ated by the binding of the transmitter to different types of receptors.Depending on the dynamics of the spread, glutamate molecules can alsobind to more distantly located receptors of the postsynaptic cdl. Onecan distinguish bet,veen ionotropic and metabotropic glutamate recep­tors. Activation of ionotropic receptors opens an ion-channel in thecell Im~mbrane, the resulting ion-flow changes the electric potential. Incontrast the metabotropic glutamate receptors (mGluR) modulate thesignal transmission via intra-cellular pathways.

\Ve developed a model of three-climensional cliHilsion in a glutama­tergic synapse to be able to estimate the importance of diflerent influ­ences on synaptic transmission. Computer simulations with this modelrevealed a very fast decay of glutamate concentration in the synapticcleft. The calculated decay time constant of T == 80 p8 allows us toconclude, that diffusion is not the rate-limiting factor for the signaltransduction of the receptors. This assumption is also valid in presenceof transmitter reuptake and possible receptor desensitization. Further­more we have computed the postsynaptic answer resulting from AMPAand NlvlDA. receptors, as well as from a metabotropic glutamate reCt:p­tor in response to the simulated glutamate concentration transients. \Veshow nncler ,yhich conditions the number of states can be reduced in thepublished A:\lPA and I\MDA receptor models.We have worked ont thefirst quantitatin> model for a metabotropic glutamate receptor which isable to approximate the ('xci tatoric postsynaptic potentials measured incerebellar Purkinje cells. lYe found the following prerequisites to be cru­cial for a successful simulation of the threshold-like character of thesepotentials: (i) spatial cooperation of simultaneously activated neigh­bOUl'ing synapses, (ii) temporal integration of repetitive stimuli, ancl(iii) multiple G protein binding sites of the effector system.

There are two fundamentallv different ,vays to model diffusion nu-

ABSTRACT III

merically. The Monte Carlo model is based on statistical mechanics andfollows single molecules on their random walk. On the other hand thediffusion equation characterizes a density function. The former modeloffers elegant handling of geometric boundary conditions, while the lat­ter model requires clearly less computational power for a comparableaveraged result.

Discretization of the three-dimensional diffusion equation results ina system of linear equations in the order of n = 105 . Onc of the maincharacteristics of the equation systems is their high sparsity. In thepresent work we examined and implemented efficient solution methodsthat exploit this sparsity. The discretization over an irregular geometrywith Cauchy boundary conditions results in a system of equations witha non-symmetric matrix; the suitable iterative method GMHES showeda very good convergence. The shortest computational time as well asthe lowest number of iterations was a,chieved if an incomplete LU fac­torization was applied as preconditioner for th~, GMHES rnethod.

iv

Zusauunenfassung

ZUSAMMENFASSUNG

Die Signahibertragung von einer Nerv(~nzelle auf eine andere Nerven­zelle findet an speziahsierten Kontaklhcllen. den Svnapsen statt. Beidiesem Vorgang wire! eine chemische ~:pbstanz (Ner;rotransmitter) vonSenderzellen (pri:isynaptische Zelle) in den sYllaptischen Spalt allsge­schiittet. Die meisten del' erregenden Synapsen des Zentralnervensys­tems benutzen die Aminosi:iure Glutamat als Tl'ansmitter. Innerhalbweniger Mikrosekunden breitet sich das anfanglich punktartig freigeset­zte Glutamat durch Diffusion im sYllaptischen Spalt aus. Die Antwortendel' Empfangerzelle (postsynaptische Zelle) werden durch die Bindungvon Glutamat an verschiedenell Rezeptortypen verursacht. In Abhangig­keit del' Dynamik diesel' Ausbreitullg binden die Glutamatrnolekiileauch an weit entfernte Rezeptoren del' postsynaptischen Zelle. Es wirelunterschieden zwischen ionotropen und metabotropen Glutam'1trezep­toren. Bei del' Aktivierung von ionotropen Rezeptoren oflnet sich ein 10­nenkanal in del' Zellmembran worauf ein Ionenstrom das elektrische Po­tential beeinflusst. Im Gegensatz dazu moe!ulieren metabotrope Gluta­matrezeptoren (mGluR) iiber intrazellulare Wege e!ie Sign'1liibertragung.

Wir entwickelten ein Modell del' dreidimension'11en Diffusion in cinerGlutamatsynapsc um die Bedeutul1g verschiedener Einfliisse abschicitzcnzu konnen. Cornputcrsimulationen mit unserem .Moclells zeigen einensehr schnellen Zerfall del' Glutamatkonzcntratiol1 im synaptischen Spalt.Die berechnete Zerfallszeitkonstante von T ::.-c: 80/1,8 erlaubt die Schlussfol­gerung, dass die Diffusion nicht geschwindigkeitshmitierencl wirkt fiir dienachfolgenden postsynaptischen Prozesse. Dies gilt '1uch in Anwesenheitvon Transmitterwiecler'1ufnahme und einer allLilligen Rezeptordesensiti-·sierung. \Veiter simuherten wir auf cler Basis cler oben erwi:ihnten Glu­tamatskonzentrationstransienten die postsynaptischcn Antworten vonANIPA, NJvIDA und metabotropen Glutamatrezeptoren. Fiir bereitspublizicrtc ANIPA uncl NMDA Rezeptormodelle zeigen wir unter welchenBeclingungen sich eine Redllktion del' Anzahl Zustande durchfiihrenEisst. Im \veiteren erarbciteten wir ein Rezcptormodell fiir eincn spczi­fischen metabotropen Glutamatrezeptor. NIit dem Modell konnten wirgemesscne erregende postsynaptische Potentiale (EPSP) von F'urkin­jezellen aus dem Kleinhirn simulieren. Dm die schwelJen'1Ttige Ak­tivierung n'1chvolJziehen zu konnen bedarf es del' folgenclen Vorausset­zungen: riiumliche Zus:nmnen'1rbeit von simultan aktivierten benach­barten Synapsen, die zeitliche Integration von wiederholten Stimuli, wie

ZUSA.JVI1'I·fENl<'A.SSUNG

aueh mehrere G-Protein Bindungsstellen am Effektorsystem.

v

Diffusion kann auf zwei unterschiedliche 'Veisen numcrisch mod­c11iert werden. Das Monte Carlo Modell basierend auf statistiseherIvlechanik verfolgt jedes J'vlolekiil auf seinem Zufallsweg, iviihrend dieDiffusionsgleichung die Verteilllllg dureh eine Diehtefunktion eharakter­isiert. Das erste lVlocle11 bietet eleganteren Umgang mit geometrischenRanclbedingungen, wiihrend das Ietzterc fiir gemittelte Losungen deut­lieh kIeineren Reehenaufwancl benCitigt. Die clreidimensionale Diskre­tisicrung del' Dift"usionsgleichung ergab GIeichungssystemc in del' Gros­senordnung n = 105 . In del' vorlicgendcn Arbeit wurclen effizientcMethodcn beziiglich Speicher und Rechenkapazitiit filr solch grosse GIei­chungssysteme implementiert und vergIichen.

Die Diskretisierung del' DitTusionsgleichung liber unregeImiissigen ge­ometrisehen Gebieten mit Cauchy Randbeclingungen erzeugt GIeiehungs­systeme mit nicht-symmctrischeT I<:oeffizicntenmatrix; das dafiir ge­eignete iterative Verfahren GJVIRES zeigte b(~i unseren SimuIationcnsehr gute Konvergenz. Die klirzeste Rechenzeit wie auch die niedrigsteAnzahl von Iterationen \vurclen mit einer unvol]stiincligen LU Faktori­sierung aIs Vorkonclitionierer fiir GMR.ES erreicht.

Contellts

Abstract

Zusanullenfassung

Preface

1 COluputer Simulations of Neurons

1.1 Introduction........

1.2 Cable Theory for Neurons

1.2.1 Steady' State Solution

1.3 Electrophysiology·......

1.3.1 :tvlembrane Potential

1.3.2 Equivalent Electrical Circuit

1..3 ..3 IGnetic processes for gNu and 09](

1.4 Intracellular 1\Iodels " . . . . . . . ..

11

IV

1

3

4

,.,,

8

8

10

12

1.1.1 COlnpartment Model for Computation oflon Home-ostasis 14

1.4.2 Homeostasis 15

vii

Vlll

1.4.3 Uptake by Exchangers

1.5 Synapse . . . . . . . . . . . .

1.5.1 Presynaptic processes

1.5.2 I::Jostsynaptic processes

CONTENTS

16

17

18

19

1.6 Integration of this ·Work into the Framework of SynapseSimulation. . . . . . . . . . . . . . . . . . . . . . . . .. 23

2 Diffusion Simulation: Choice of Approach

2.1 Introduction .

2.2 Monte Carlo Method Model

2.2.1 Derivation of Random Step Length.

2.2.2 Restricted Space and Reflections

2.2.3 Verification.. ..

2.3 Diffusion Equation Model

25

25

26

26

27

28

29

2.3.1 Discretization :30

2.3.2 Boundary Conditions 32

2.3.3 Solution Scheme . . . 36

2.3.4 Comparing the Diffusion Equation Solution withthe Monte Carlo Method Model ~17

3 Model and Results

:~.1 Introcluction and Overview

3.2 Glutamate Dynamics in Flat Cleft.

:3.2.1 The Basic Model ...

3.2.2 Flat Cleft JVlorphology

41

43

43

44

CONTENTS

3.2.3 Glutamate Dynamics.

3.3 Vari et ion of Model Parameter

IX

45

48

3.3. Aspects of Release 48

3.3.2 Release Sites . . . . 49

3.3.3 Simultanous Release in Neighboring Synapses ,50

3.3.4 Increasing Initial Concentration. 52

3.3.5 Cleft Shape

3.4 Postsynaptic Signal Transduction

52

56

3.4.1 Ionotropic Postsynaptic H.eceptors 58

3.4.2 Metabotropic Posts:ynaptic Receptors 61

3.4.3 Translation of [glu] transients into postsynapticresponses mediated by AJVfPA- and NJVIDA typeof glutamate receptors . . . . . . . . . . . . . .. 68

3.4.4 Translation of [glu] transients into postsynapticresponses mediated by metabotropic glutamatereceptors in Purkinje cells 71

3.5 Summary . . . . . . . . . . . . .

4 Solving Large Sparse Lim~ar Systems

4.1 Introduction...

4.2 Iterative rnethods

4.:3 ](ryloY Subspace Methods

76

79

79

80

83

4.3.1 Background.... 83

4.3.2 Arnoldi's Orthogonalization 84

4.3.3 Arnoldi's algorithm for the solution of linear sys-tems 86

x CONTENTS

4.4 Preconditioning.......

4.4.1 Fast Poisson Solver

4.4.2 Incomplete LD Factorization

88

89

92

4.4.3 Performance Comparison of Different Preconditioners 93

4.5 Direct Methods 94

4.5.1 Band Solver and Reordering. 94

A.4 Receptor Simulation . .

5 Discussion and Conclusions ~}7

98

98

99

100

101

104

106

106

109

110

113

115

116

of Equations .

Confidence in Model Ideas .

5.1.1 Is Diffusion the Rate Limiting Factor?

5.1.2 Simplified Release Assumptions ...

5.1.3 Hemispherical vs Flat Synaptic Cleft

5.1.4 Receptor models

Modeling Diffusion . . .

Computational Methods

Conclusions . . . ....

5.1

5.3

5.2

5.4

A Implementation

A.l Generating the Matrix

A.2 Solution of the Large Sparse

A.3 J\!lonte Carlo DifTusion Simulation.

B AMPA Receptor Moclf~ls 119

C Glossary 121

CONTENTS

Bibliography

Curriculum

xi

127

138

Preface

Computational neurobiology is a novel discipline which emerged at theinterface between electrophysiology, cell physiology and scientific com­puting. It found its main application in the computation of electricalsignal processing in neurons and clusters of neurons. This thesis takesplace at the interface of cOTnputational neurobiology and scientific com­puting. The lack of experimental techniques to directly measure theglutamate dynamics in the synaptic cleft in a sufficient spatial and tem­poral resolution, required computer simulations to gain complementaryinformation about synaptic functionality. \Vith the recent advances inthe field of iterative methods we were able to solve partial differentialequations describing the three-dimensional diffusion phenomena in thesynaptic cleft with a sufficient geometrical resolution.

Four main challenges shall bc; mastered on the way from neuroscienceto scientific computing and back. In the figure below we indicate themain stages as well as the tasks which interconnect them. The buildingprocess of an effective COnljmter simulation requires that the ITlodeler(i) has an understanding of the processes (;ccuring, (ii) isable to describe those processes differential equations, (iii) isable to formulate a discrete to the differential system, (iv) is ableto solve that on tbe computer. It \vas the aim of our computersimulation to pinpoint ilnportant features of synaptic transmission thatmay play a role at the level of neuron or even small network

\Vith this sinmlations we hope to give input for furtherneurophysiological experiments. This thesis considers only simulation ofchemical transmission: neither abstraction to higher levels nor synaptic:plasticity and autoregulation of synapses are explicitly addressed herein.

1

2

Q)

"Clo:2

DifferentialEquation &BoundaryConditions

Discrci.·cRcpresel1iltion

co

C1lCl)-Q)

()

Cl)

o

co

PREFACE

NumcricalSolution

Most of the domain specific terrninology is explained in the introduc­tory part of each chapter. Furthermore it is the aim of the additionalglossary to provide both context independent definitions and additionalexplanations for terms which have not been explicitly introduced in thetext.

Chapter 1

Computer Simulations ofNeurons

1.1 Introduction

Ever since the hieroglyph for brain first appeared in Egyptianpapyrus dated to the 17th century B.e., humans have asked questionsabout the \vay the brain \vor1;:s. Quantitative theories on aspects of neu­ronal information processing though only appeared in the second part ofthis century. Both the development of techniques for the measurementof neuronal signals as well as the power of current digital computerswere necessary to a boom in computational neuroscience startingin the mid 1980s. Computational neuroscience seeks to develop modelsdescribing how the nervous system or some part of it carries out certainoperations, such as adapting to long lasting current stimulus, computingthe direction of a moving object, learning certain motor skills etc. Thesemodels can be carried out on different levels of abstraction, eaell levelallowing to 11llderstand another of neural information processing,what high level loose in specificity they gain in insight and analyticaltractability. Only very few models for simple types of nervous systemsm;ulagecl to simulate several different lev("ls of signal processing. Amongthe ones that deserve to be mentioned is certainly the Imuprey Iuodc1

4 CHAPTER 1. COMPUTER SIMULATIONS OF NEURONS

developed by Grillner et al. [44] or the hippocampus based epilepsymodel developed by Roger Traub [107].

Quantitative modcling and computer simulation be carried outon several levels. In this thesis we started at the lowest inte-grative level: simulation of biophysical events in symq;se. To givean introduction into rnodeling of the neuron we provide an initial insightand overview of the essential quantitative formulations cm cellular andsynaptic level.

1.2 Cable Theory for Neurons

Already at the beginning of this century it became evident that a maintask of neurons is to process electrical signals. In the mid 1950s Coombs,Eccles and Fatt [22] started to analyze analyticall}. the properties ofelectrical signal propagation along neurons. It was left to \Vilfrid RaIl[89, 90] to employ the cable theory for describing the spreacl of neu­ronal activity. The cable theory was developed for calculations of thefirst transatlantic cable by \Villiam Thompson around 1855. In analogyto a cable, a dendrite is viewed as a uniform cylinder, consisting of acore conductor and a thin membrane, which separates the intracellularspace from the cytoplasm. The cable equation is a partial differentialequation describing the spread of an injected signal along a cable tree;in neurophysiology the equation is usually expressed as:

avT-'--' = o.at (1.1 )

Here :r represents the distance along the cylinder, I represents the tilne,T is the membrane time constant, and V represents the voltage differenceacross the membrane ). The length constant,A = , is the distance along the dendrite to the site where ~1:'))has decayed to 1/e of its value at .7: O. ]'he better the insulation of tllernembrane (the higher I'm and the better the conducting propertiesof the inner core ( the lower Ti is) the greater the length constant of thedendrite. In terms of dimensionless variables, X :r / A and T =0'

Equation 1.1 is expressed as:

a2v cW-1/ ._- aT o. (1.2)

1.2. CABLE THEORY FOR NEURONS 5

(1.3)

(lA)

The membrane surrounding dendrites (and axons) is idealized asa cylinder (see Figure 1.1). Both the intracellular cytoplasm and theextracellular fluid are ionic medias that conduct electric current. Theeffective membrane resistance is much greater than the longitudinal re­sistance in the core. This ensures that the current inside the core flowsin parallel to the cylinder axis and only a very minor part leaks acrossthe Inembrane,

In our one dimensional ch?scription, we assume that the longitudinalgradient of the intracellular potential under neglect of a radial potentialdistribution can be expressed as

dVich

'where 'if represent.s the intracellular core current., 1'; represents t.he uni­form intracellular resistivity per unit length (nCTn-1) and V; is the in­tracellular potential. If Equation 1.:3 is differentiated with respect to ;,;,,ve get

Di;= -1'; -;c_...

ch:Equation lA states that for a short piece of cable the difference of cur­rent flowing in from the left and flowing ont to the right is ::::; 0.;,;d'idch.With this and together with Kirchoff's law the leaking membrane cur­rent density is expressed as (see also Figure 1.1):

(1.5)

\Vith respect to an inherent membrane capacity per unit length Crn wecan unify Equations 1,1 and 1.5 to Equation 1.6 whose equivalentelectric sdlClne is indicated in Figure 1.1:

ell' '1'= -C,n +

clt: I'mT m = T'mCm, A = (1.6)

A dendritic tree can be divided treating all branches as cylinderswith uniform membrane properties; these cylinders can have differentlengths and diameters and different uniform membrane properties, Thedifferential Equation (1.1) holds for each of these cylinders; however Aas well as the reference input conductance G= 1//\1'; can be differentin each cylinder. For the computation of dendritic current under voltage

conditions, or vice versa, we need the steady state solution fordifferent conditions,

6 CHAPTER 1. COMPUTER SIMULATIONS OF NEURONS

·A

B

c

Figure 1.1: representation of diffeT'!;nt abstraction schemes of a simpli­fied dendr'dic tree, 11: two dimens'ionallwojection of a new'on>,s denclrdesand soma. B: the same dendr'ites repT'!;sented as a branched system ofeylinclr'ical segments, C: Cirellit analog of B

1.2. CABLE THEORY FOR NEURONS

1.2.1 Steady State Solution

7

For a steady state, ((lVIDt) cc::: 0, Equation 1.1 reduces from a partialdifferential equation (PDE) to an ordinary differential equation (ODE)(PVldX 2

- V = °whose general solution is well known to be of thefonn T"(X) = Cl cosh(X) + C2 sinh(X). For finite lengths of cable, wemust specify both the length L and terminal boundary condition. Thezero slope boundaTy condition ({VI dL = 0 at X = L is called sealedend. The unique solution for this problem can bc expressed as given inEquation 1.7, this solution describes an exponential decay ovcr X.

F(X)\10 cosh(L X)

cosh(L)(1. 7)

For a pair of voltage clamped boundaTy conditions \I =10 at X = 0,and \! = TlD at X L the solution is the sum of two solutions eachdescribing clamp of \I = 0 at one end while the other end is either T/oor therefore the sum of solutions is expressed as

\I(X) (1.8)

A third boundaTy condition type assumes a lealv'1ge current at X Lwhich is proportional to the value of F at X = L and is expressed asV . (;D, where (;L represents the leak conductance at X L:

V(X)/1!L cosh(L- f- (GI)Ooo) sinh(L X). (1.9)

This last type of solution has prooh~d to be very useful in the cableanalysis of dendritic trees, where we use con catenated c:ylinders of dif­feTent diameter and lengths. To obtain the mathematical solution forthe tree, the tvvo arbitrary constants in the general solution for eachbranch must be determined from the boundary conditions, these includecontinuity of V and of core current in every branch or at everv branchpoint, a specification of the boundary condition at the distal end of everyterminal branch plus a specification of the applied voltage or appliedcurrent at the electrode location.

8 CHAPTER 1. C01\1PUTER SIMULATIONS OF NEURONS

1.3 Electrophysiology

Signal processing in a CNS neuron can be abstr!Jcred and mapped ontothe three main morphological structurt;s: (i) dendritic tree col-lects and integrates signals that arrive over and inhibitorysynapses (ii) the soma (cell body) reacts to the incorning signals of thedendritic tree in an all or nothing manner this is to say that it launches asignal (action potential) if the cell body's membrane potential excessesa threshold (iii) over the axon (nerve track) the action potential propa­gates in an active manner to the successive neuron(s). In the previoussection we described the so called passive dendritic tree. In contrastto the dendritic tree, the axon possesses means to actively carry actionpotentials. The action potential is produced by ion-fluxes across themembrane through voltage-gated channels. This current is driven bydifferent concentrations of the ions on either side of the rnembrane andresults in a change of the voltage across the membrane.In the late 1930s the increase in membrane permeability during cell ac­tivity was investigated. Hodgkin and Huxley in England and Curtisand Cole in the lJnited States, measured for the first time the full ac­tion potential of an axon \vith intracellular electrodes. At the beginningof the fifties a new technique named vohage darn]) was developed bythe same teams. Voltage clamp means to control the potential across amembrane, the current required to impose a given membrane potentialis measured. In this section we give an introduction on the mutual de­pendencies of ion channels, nHlmbralw potentials, and currents, as wellas an overview of the formulations for the corresponding cell electro­physiology',

1.3.1 Membrane Potential

Electric current flow across the cell mernbrane is controlled by the open­ing and closing of ion channels in the cell-membrane. Ion channelscan be gated or non-gated. Both channels though show spontaneousopening and closing. For non-gated channels the probability of thesespontaneous alterations is not influenced by extrinsic factors, thus weapproximately describe these channels as "always open". They are pri­marily important in rnaintaining the resting membrane potential. Incontrast, gated channels, which respond to different signals are useful

1.3. ELECTROPHYSIOLOGY 9

for rapid neuronal signaling. Three major signals can gate ion channels:voltage (voltage-gated channels), chemical transmitters (ligand-gatedchannels), pressure or stretch (mechanically gated channels).

In this subsection wc describe how non-gated ion channels establishthe resting potential. The value of the resting membrane potential innerve cells is determined primarily by non-gated channels selective forJ(+, Cl-, and N a+ and different concentrations of these ions on eitherside of the rnembrane. Two basic forces act on ions on either side ofthe cell membrane, (i) a chemical force that moves the molecules downtheir concentration gradient and (ii) an electrical force that moves theion down the electrical field gradient. The specific membrane potentialfor a given ion species at which the electrical force is equal and oppo­sitely directed to the chemical driving force, so that no net movementof charge occurs, is called Nernst potential. This potential, at which ionmovement is in equilibrium, can be calculated according to the Nernstequation:

RT [X]o= In---ZF [);~];

(1.10)

\vhere is the value of the membrane potential at which ion X is inequilibrium, R is the gas constant, I' is the absolute temperature, Z isthe valence of the ion, F is the Faraday constant and and arethe interior and exterior concentration of the ion X.In most neurons at rest, the membrane potential inside the cell is about-60 mV to 70 mV more negative than outside the cell. By conven­tion, the potential outside the cell is arbitrarily defined as zero and themembrane potential m 1is defined as

In general the resting potentia.! will be closest to the Nernst potential ofthe ion with the menlbrane permeability. The permeability foran ion species is proportional to the number of open channels permeableto that ion. The potential is slightly more positive than theNernst potential of potassium and hugely more negative than Nernstpotential of sodium. The resulting ion fluxes, balance each other so thatno net current occurs at potential ,= 0, where denotesthe valence of Xl. Still these leak currents can not be allowed to continueunopposed, otherwise the ionic gradients would gradually run down.Dissipation of ionic gradients is counteracted by the energy-dependent

pump which extrudes Na+ and takes in I(+, in Section

10 CHAPTER 1. C01\1PUTER SIMULATIONS OF NETJRONS

1.4.2 this mechanism is described qualitatively and quantitatively. InTable 1.1 we summarize the distributions according 0 the text bookof Kandel, Schwartz, and .Tessel [59]. Although and K+ set the

Na.+ 50 440 +55CZ-· 52 560 -60

Table 1.1: distribution of most prominent ions

values of the resting potential, l~n is not equal to either Er< or ENa, butlies between them, the influence of each species is determined both byits concentrations inside and outside the cell and by the permeability ofthe membrane to that ion. This relationship is given by the GolclmaneqUi1tioll, where Px denotes the permeability to ion X:

(1.11)

1.3.2 Equivalent Electrical Circuit

Previously we have introduced the equivalent electrical structure for apassive dendritic tree. Below we elaborate the additional structureswe need for an active neural structure. The first step in developingthe model is to relate the discrete physical properties of the membraneto its electrical properties. The circuit model can then be made morecomplete by adding a current generator. As described above,fluxes of and K+ ions through the passive Inembrane channelsare exactly counterbalanced by active ion fluxes driven by the -K+pump, \vhich extrudes N a+ ions and pumps in ions. This energy(NIP.) dependent No+·K+ pump keeps the ionic batteries charged(not shov:n in Figure 1.2) .

In terms of electrical circuits, each ion ChanlH?1 acts as a conductorand battery in series. The potential generated this battery is equalto the equilibrium potential Ex. In analogy to the Golclman equation

1.3. ELECTROPHYSIOLOGY 11

_

_9C_I__~J~~ ~ Cm

__T~Ec, _

Figure L 2: eil'l/.'ivalent electrical cir'cuit of 0. ne'Ur'01/. at rest.

we can formulate a dependency between the membrane voltage and theion conductances:

(1.12)

So far we have given an electrical equivalent circuit for a passivedendritic tree. 'Ve described the sources of the leak currents throughnon-gatecl channels. In the nerve cell at rest the steady N a+ -currentis balanced by a steady J(+ current, so that the membrane potentialis constant. This balance changes, however, when the membrane isdepolarized past the threshold for generating an action potential. Atransient depolarization past a threshold El for generating an actionpotential causes voltage-gated sodium channels to open rapidly. Thenet influx of positive charge causes even further depolarization. Thispositive feedback cycle drives the rnembrane potential towards the N a+equilibriurn potential of +:3;)mV. 'T\vo processes cause the subsequentrepolarization. First, as the depolarization continues, the populationof voltage-gated No+ channels gradually becomes inactivated. Second,the delayed opcning of the 1(+ channels causes the J(+

efflux to increase gradually. This dela.yecl increase in J(I- dllux com­bines with a decrease in N ini1ux to produce a nct efflux of positive

from the cell, which continues until the cell has repolarized toits resting membrane potential. For a quantitative description of anaction potential \ve hence lack a quantitative description of the voltagedependency of the channels.

12 CHAPTER 1. COMPUTER. SIMULATIONS OF NEURONS

1.3.3 Kinetic processes for gNa and gJ(

On the basis of their current clamp and voltage dam[ . nwasurementsHodgkin and lIuxley developed an empirical kinetic of themacroscopic currents, without any knowledge of the molec-ular mechanisms which govern the behavior of the channel [51,52, 53, 54, 55]. Voltage clamp means to control the potential across thecell membrane, as opposed to the current clamp, where the voltage ismeasured and the current is controlled. The clamp current flo\vs thenlocally across the membrane as ionic and capacitive current and spreadslaterally to distant patches of the membrane. In contrast during voltageclamp the current required to impose or hold a given membrane voltageis measured. In their voltage clamp experiments, Hodgkin and Huxley,rneasured a linear instantaneous current-voltage relation as in Ohm'slaw therefore they postulated the quantitative description of ionic con­ductances as:

gNa = T/ gl(\ E Na

IJ(-----F - EJ(

(1.13)

where Ex is the reversal potential of the ion X or K+), Fis the rnembrane potential, and Ix is the current flowing through theconductance gx. H we assume that every ionic channel contributes afixed conductance in the open state, the conductance definecl above isa measure of the numbers of open channels. At this stage we are onlylacking a kinetic description of the gNa and gK. This description isknown today as the Hodgkin Huxley (HH) model. The conductancechanges in this model depend only on voltage and time, but no othervariables, such as current through the channels or ion concentrations.The conductance gK is therefore expressed as gJ( = rh: f(1', I.). The ideaof the HH model applied in a mathematical form on potassium channelsexpresses the potassium current as:

(1.14)

where .ifk is the maximal conductance. andn describes the probabil­ities to find the gating lnechanism in one of two possible states, lateridentified as the permissive and non-permissive state: of gating parti­cles. Hodgkin and Huxley empirically assumed four particles ina channel each with a probability n of being in the correct position toset up an open channel. The probability that all four particles are cor­rectly placed, and hence the channel is in the open state, is therefore

1.3. ELECTROPHYSIOLOGY 13

11,4. The voltage and time dependent changes of 11, can be described bya first-order reaction scheme:

(1 _. 11, (1.1;"))

If the initial value of 11, is known, subsequent values can be calculatedby solving the differential equation:

dn= iXn(F)· (1- 11,) (:in(V)' 11,

d!(1.16)

wlwre (n - 1) indicates the non-permissive state while 11, indicates thepermissive state. Transitions between these two states are possible withthe voltage-dependent rate functions etn (V) and (:in (V):

On(V)An

(1.17)

For most of the channels, which are only dependent on the membranevoltage, the scheme presented by Borg-Graham [113] (see Equation 1.17)is applicable. A.n to Ifn are constants and F is the membrane potential.The time dependence of n.(V, t) is thus given by the differential equation1.16, \vhile the voltage dependence enters through the rate functions.

The HH model uses a similar formalism to describe I Na . Here onlythree activating particles m are assumed but in addition an inactivatingparticle h joins the party. The probability that a channel is in an openstate is therefore calculated to be hand hva is represented by

EI{ ) 18)

Again 7n and h are assumed to unclergorrrst-orcler transition kinetics,which results in the already known differential equations correspondingto (1.16). }'or an excellent collection of quantitative descriptions of ratefunctions of several channels. wc refer to the ETH dissertation of H.etoQuadroni [88].

le} CHAPTER. 1. COMPUTER SIMULATIONS OF NEUR.ONS

1.4 Intracellular Models

During the quantitative descriptions presented so far, we assailed con­stant ion concentration, that is, local membrane voltage w

under consideration of rnembrane resistivity, mernbrane andionic currents only. \Vc shall keep these simplifying assumptions kn'J(+, Na+, and Cl'. Detailed modeling of neuronal excitahility howevermust take into account. the homeostasis of certain ion and transrnittertypes inside and outside the cell. Calcium is the most discussed ionin this context [63]. The dynamics of free intracellular calciurn is ofparticular interest, because the level of calcium controls e.g. potassiumchannel activation, synapsin phosphorylation, as well as the initiation ofmechanisms that underlie synaptic plasticity. \Vhat are the mechanismsthat contribute to the homeostasis of calcium in neurons? Quantitativeformulations for the four main intracellular processes are given below.Intracellular Ca2+ diffusion is discussed in Section 1.4.1. buf­fering and Cn2+ pumps are discussed in Section 1.4.2. Uptake tbroughexchangers is discussed for the glutamate/N a+ exchanger in Section1.4.3.

1.4.1 Compartment Model for Computation of IonHomeostasis

In neurons at rest, ttHe' intracellular concentration of free calciumat about 10-7 M is over four orders of magnitudes smaller than the extra­cellular free calcium concentration . Since the surface membraneof most neurons is permeable to calcium ions, particularly during actionpotentials, such a low resting can only be maintained veryefllcient systerns for removing (diffusion, buffering and pumping out)free calcium ions following their entry into the cytoplasm. Due to longterm sllstainability all calcium that enters the cell through ion channelsmust some rime he pumped out of the cell although considerable flexi­bility is introduced through the so called buffering \vhich canbind and release free intracellular calcium.

Modeling of local ion concentrations requires geometrical discreti­zation.Most often the soma is approximated by a sphere and thereforespherical coordinates are used for a refinement of the discretization.

1.4. iNTRACELLULAR MODELS 15

Neglect of tangential diffusion components allows for reduction to oneradial dimension, therefore assuming iso-concentration surfaces to be ar­ranged in onion-shell like manner. A similar idea accounts for dendriticmodeling where cylindrical shells are employed most often. ButTeringand extrusion mechanisms as described below alter local ion concen­trations. To be able to simulate the balancing distribution changes ofcalcium, the diffusion equation is solved in spherical (cylindrical) coor­dinates:

(1.19)

Discretization and solving mechanism are then straightforward and canbe taken from the ditTusion-bible of Crank [24J.

1.4.2 Ca+ Homeostasis

Intracellular exists in two forms (i) free Co,2+ which can be sub­ject to diffusion and export and (ii) bound Ca2+ associated wi th buffer­ing systems. The complex -buffering system in neurons probablyconsists of three major components: uptake into rnitochondria, uptakeinto the endoplasmatic reticulum and c.ytoplasmic binding by proteins.Several proteins are known to serve this purpose, such as the ubiqui­tous calmodulin, parvalbumin, calbindin and calcineurin \vith up tofour separate calcium binding sites [73J. From the perspective of free

, the binding occurs at a single binding site on the buffer. Theforward (et) and backward rate constants of the binding reactionfor calmodulin are assumed to be 108 1\1-1 and 1008-1 respectively(]{D 1 {Ll\1) F<x proteins with one binding site forcalcium we then have a reaction scheme of the following form:

Bo~ CaB. (1.20)

In analogy to the first order differential equations (1.16) which \ve for­mulated for the mechanisms of gatecl ion channels wederive the diflcrential equation for the butTer binding from the abovereaction scheme.

(1.21)

16 CHAPTER 1. COMPUTER SIMULATIONS OF NEURONS

Pumps

Although the caJcium-buffering system discussed above theamount of free intracellular calcium, the remaining calcium j,c)ll'; mustultimately be removed from the cell if calcium homeostasi';; is to bemaintained. Two major transport systems have been identiiied, Cl

sodium-dependent Ca2+ efflux, in which the energy required for the ex­trusion of calcium ions is derived from tllt: inward movernent of N 0+down their electro-chemical gradient, and a calcium-extrusion systemthat works independent of N 0+ and requires ATP as an energy source.The ATP driven pathway (pump) is active at much lower val­ues but has a smaller capacity. Koch and Segev [63] give a quantita­tive description for such a pump. DiPolo et al indicate in their study[29] that the pump is voltage dependent and has a time constant ofTpttmp(V) = 17.7 * exp(V/35) TT/8ce. Again the feedback introduced bythe membrane potential V prevents us from writing down an analyticaldescription.

1.4.3 Uptake by Exchangers

Competitive antagonist experiments carried out by Jacques vVadicheet al. [109] revealed the kinetics of the human glutamate transporterEAAT2 (exc:itatory amino acid transporter Hr 2). They suggest anordered binding model for ]H"e-steacly-state currents, in which a voltagedependent N a+ binding is followed by a voltage-independent glutamatebinding. Two N a+ charges are translocated per rnolecule of glutamatewith a cyc:ling time of approxirnately 70 ms. For xenopus theycalculated an average transporter density of 1439 pm-- 2 In numericalsimulations they fItted their data to a three-state model:

-to NaGlv (l

where the dissociation rate constant ](2 =:: 1. M is indepen-dent and ](1 = 9.810- 2M' exp(0.'161/F'/RI') is dependent. Herethe factor 0.46 represents the fraction of the electric field sensed thecharge, V is the membrane voltage. and R, T and F have the usualmeaning. The turnover rate at -80TnV modeled hereby is 14.6 ,theturnover rate indicates how 1nany glutamate molecules are transportedper trarlsporter and per second. The experiments were performed under

1.5. SiWAPSE 17

a saturating glutamate concentration of 1 mM. To bring this turnoverrate into relation with the diffusion driven decay, we shall conduct abrief example calculation. Let us assume that the area of the presy­naptic membrane which borders the synaptic cleft measures 4 p:m.2

,

furthermore ''le assume that this whole area has a transporter densityof the above mentioned 14;\9 /lm- 2. Under the additional assumptionof a turnover rate of 14.6 . we obtain a transporter performance ofaround 80'000 molecules per second. Considering the fact that gluta­rnate concentration in the cleft is mostly far below the saturating value,we get a clearance rate of the transporter which is significantly sloweras the expected clearance rate by diffusion.

1.5 Synapse

Nerve cells differ from other cells in the body because of their ability tocommunicate rapidly over long distances with one another. This rapidand precise communication is made possible by two signaling mecha­nisms: axonal conduction and synaptic transmission. In previous sec­tions we exa.mined the mechanism for condnction of electricalalong axons. Here we describe the basic mechanism for c1Jemica.l synap··tic transmission. At chemical synapses the pre- and postsynaptic ele­ments are separated by a synaptic cleft. How do molecular componentsfunction together as et coordinated mechanism to provide for synaptictransmission? Figure 1.3 provides a summary of the basic steps involved(numbers in the figure correspond to numbers in the text). We can breaksynaptic transmission down to the main mechanisms discussed below:(1) Depolarization: Upon entry of the action potential into the synap­tic bouton, the presynaptic membrane is depolarized. Activation ofVVH"+',cc gated Ca2+ channels: Depolarization opens channels, aninflux of Ca2+ into the presynaptic terminal is caused. This results ina very transient elevation of [Ca 2+]. (3) Vesicle PY()rvh'l~l~'

high levels of Ca2+ induce fusion of synaptic vesicles with presynap­tic cell membrane, vesicles are already docked at the plasma-lenuna.(4) Cleft mechanism: released transmitter diffuses into the cleft to acton its postsynaptic receptors. R.euptake: transmitter concentra­tion in the cleft decreases to basal level diffusion and reuptake. (6)Presynaptic modulation: the release process is subject to modulation

neuromodulators and feedback action of the released transmitter

18 CHAPTER 1. C01l1PUTER SIMULATIONS OF NEURONS

Figure 1.3: synapt'lc processes

through autoreeeptors. (7) Postsynaptic receptors and effectors: gluta­mate acts directly on channel proteins in the postsynaptic membraneas well as on receptors which take effect through G proteins.

1.5.1 Presynaptic processes

In current research a picture is emerging of the molecular rnachine bywhich vesicles are docked at release sites and membrane fusion is trig­gered in response to calcium influx. Released vesicles are slowly re­placed, with a time constant of > 1.$ by vesicles from a "backlog store"[16].

Voltage gated channels

Due to the high gradient over the presynaptic membraneenters very fast through the open channel pore and a microdomain ofhighly elevatecl is created near the channel mouth. The coloca­lization of calcium channels and vesicle fusion sites that therelevant level of Ca2+ is nmch higher than the average concentrationachieved throughout the terminal during activation of calcium current[HJ4]. Voltage gated calcium channels can be modeled in analogy to

1.5. SYNAPSE 19

voltage gated Na+ and K+ channels as described in Section 1.3.3. AnN-type calcium channel model with two states can be assumed in thecontext of presynaptic vesicle fusion [14]. This high-threshold CaH

channel is modelcd by a Golc1man-Hodgkin-Katz constant field Equation

RT(1.23)

]S an

where IN is a calcium current through a so-called N-type calcium­channeL z 2 is the valence of , E is the membrane potential, F, Rand T have the usual meaning and m is the activation variable. Theresulting elevation of intracellular is transient due to clearanceby an ATP-driven pump.

The calcium induced cascade leading to transmitter release can bedescribed as Yamada et a1. showed in [114]:

+ S S*

S* + \' 1;i*,

where 5, S* is a synapsin related calcium binding protein in its deacti­vated and activated form respectively,V* is the released vesicle.

Transmitter Diffusion

The fusion of vesicle and cell membrane reslilts in release ofthe transmitter into synaptic cleft. In chapter 2 and 3 we extensivecalculations for effects due to diffusion.

1.5.2 Postsynaptic processes

'Two of ionotropic: ,C:llnamiHC receptors can be distinguished. First,the NI\IDA (ninned after the synthetic agonist N-methyl­

cation channel vvhich is permeable for

20 CHA.PTER. 1. CO"~;[PUTER SIMULATIONS OF NEURONS

Ca2+, j{+, Na+. Second, AMPA receptor (named after the specificagonist n-amino-3-hydro:ry-5mcthyl-4-isoxazolc proprion'ic acid), alsocalled non-NMDA receptor builds a cation channel for N a+ and I(+. Inaddition to ionotropic receptors also rnetabotropic receptors are mostoften present at glutamatergic synapses. Metabotropic receptors coupleto G proteins which activate several enzymes. Generalizing we can statethat mGluR agonists have a wide variety of actions on central neurons,which are mediated by modulation of both voltage and ligand-gated ionchannels [101].

Ion-channel coupling

Fast transmission of nerve impulses at most excitatory synapses in theeNS occurs by activation of AT\IPA receptors in response to synapticallyreleased glutamate. The AMPA. component of the synaptic currenttypically rises in less than 1001's and decays with a tinw constant of0.2 and 8 ms [~), lm]. The simplest description scheme is based on thecommon two state reaction scheme:

11+ Glllk+ 1~~

1.:- 1

R. Cilu. (1.24)

The reaction scheme (1.24) is equivalent to Equati,..1n (1.25), which de­scribes temporal changes in receptor occupancy due to changes in ago­nist concentration by the ordinary differential equation:

d [RGln] .,--.---' = k+r[Glv.]([lt]/o/ _.- [RGlu]) ­dt

[RGlv.]. (1.25)

Under the assumption that the total amount of receptors isby the sum of the bound receptors [RGlli] and the free receptors [R], wereplaced the state "R+Glv" by the term --- [RGlu]). This allowsus to describe a two state scheme by only one differential equation. In

d ., 1 d [R(;ln] () l' f' I' .'a stea y state 81tuat1On, w Jere -~...-~ = ., we (cnve rom"quatlOn(1.25) the law of mass action which describes the relationship bet\veenreceptor occupancy IRGlv], concentration [Glu] , total amountof receptors [R]/o/ and dissociation constant )

[RGlu]-----[R]/ol

(1.26)

1.5. SYNAPSE 21

Equation (1.26) is of importance in steady-state receptor theory sinceit predicts the fraction of receptors occupied by a drug (e.g. Glu) atany given concentration [Glu] if the dissociation constant is known forthe drug-receptor interaction. The dissociation constant is equal to thequotient of the two rate constants: = k_I/k].

The simplest model that can fully describe the behavior of AMPA isa five-state model [108] (see Figure 1.4), where the ligand [L] is a generic

o

,13R ==::: RL ::.-:::=:: 0

I tk l \ Ik'- 4

~ I

Figure 1.'1: state diagmrn of a simple AMPA l'eceptoT model

variable for any glutamate In addition to the states from thebasic t,vo-state diagram also chapter 3.4.1) a new state betweenclosed and open as well as two desensitized states were introduced. Forexample after prolonged exposure to its own transmitter, a receptor canbecome unresponsive to later applications of the same transmitter, thisprocess is called desensitization. For a receptor model (e.g. the schemein Figun:, 1.4) we can write down a system of differential equations inanalogy to Equation (I . Each state is represented by a variable andeach reaction patlrway into this state is considered a positiveterm in the corresponding row while each reaction pathway leading outof the corresponding state is considered by a negative term.

NMDA at synapses mediate cur-rents that rise slowly to a peak (:::::: 20m.8) and then decay bi-exponentiallywith time constants of around 40 and 200ms [68, 76]. vVhen the ll1ern-brane potential is near current i10W through NMDA channels israpidly blocked 1\1 ions the pore from the extracellular

22 CHAPTER 1. COMPUTER SIMULATIONS OF NEURONS

2L+R211:+]--..._-...,....._- L+LR

k+4-;-----~ 0

Figure 1.5: state diagrarnm of NMDA r'eccptors as proposed by Lester'and Jahr' {68}

solution [2]. Figure 1.5 shows a proposed kinetic scheme description.Differential equations corresponding to this scheme can be developed inanalogy to the equations of the AMPA channeL

Activation of metabotropic receptors

In contrast to ionotropic receptors, which are an integral of an ionchannel, the metabotropic channels are coupled to cellular effectors viaG proteins. Depending on the rnGluR-subtype, the induced intracellulareffects can include activities like increased formation of IP:; or decreasedformation of cyclic AMP [62J.

Kinetic schemes for mGluR have not been described. Therefore\ve divide the mGluR effector chain into its parts that isbinding, (ii) signa.! translation by G protein metabolic processes.At least the first two parts can be modelcd to related pro-cesses. Destexhe et a1. presented a reaction scheme for GABA}3 re­ceptors which couple via G proteins to potassium channel . Section3.4.2 describes in detail the construction and reduction of a novel quan­titative description of an mGluR.

1.6. INTEGRATION OF THIS IVORK 23

1.6 Integration of this Work into the Frall1e­work of Synapse Sill1ulation

During the 1980's and the beginning of 1990's electrical aspects of thepostsynaptic side attracted most of the work done in the field of synapsesimulation. In recent articles however the terrain of synapse simulationwas significantly widened. Khanin et al [61]' Stiles et al [105] as wellas Destexhe et al [27] sinmlated asp<:;cts of the mechanisms that lead tovesicle release. Clements [17], Barbour [5] and Holmes [56] simulatedaspects of synaptic diffusion as well as postsynaptic events. 'While themost recent work of Fiala et al [36] and Destexhe et aL[2G] dealt withmetabot1'Opic signal transductioll.

\Vith this work, which covers aspects of synaptic transmitter diffu­sion and of postsynaptic signal transduction, we intended to clarify thequestions concerning which chain link acts under which circumstancesas rate limiting in synaptic transmission. \Ve consider the three dimen­sional diffusion simulation conducted with two different models and indifferent restricting geometries, as one of two core contributions of this\vork. Consequently we first implemented a basic model which coverstransmitter diffusion and then extended the rnodel in such a way thatwe were able to gather results which also allowed a comparison withother experimental) studies in this field. In this context we regardthe implernentation of the metabotropic signal transduction as the othercore contribution of this work. Other extensions, e.g. ionot1'Opic sig­nal transduction, were ma.inly necessary to put this work in comparisonwith other studies.

Chapter 2

Diffusion Simulation:Choice of Approach

2.1 Introduction

In chapter 1.5 \ve described some of the physiological processes under­lying the synaptic transmis:jion. \Ve characterized the diffusion of theneurotransmitter into and the synaptic deft as the central partof all synaptic processes. There are two radically different ways we canmodel the synaptic difTusion numerically. On the one hand we havethe possibility to solve a system of differential equations, characterizingthe distribution of neurotransmitter molecules by a density function inspace and time fiG, 112], ,11, z, t). The corresponding partialdifferential equation for diffusion in an isotropic medium is as follows:

8Ual (

" UD , +

c7;T~(2.1)

where U is the concentration of the clitlusive material.

On the other hand we can use the "]\l10nte Carlo Method", whichcharacterizes the distribution specifying the position (3;,11, z) for eachof a given number N of neurotransmitter molecules at a given timet [106]. The distinct of such a stochastic model is, that

26 CHAPTER 2. DIFFUSION SIMULATION

we can model continuous and infinite space, though we pay for thisadvantage with a tremendous computational cost if we want to get anaveraged rather than random result. Therefore the stochastic model wasimplemented only as a reference simulation for the diffusion equationmodel. From here on wc shall refer to the stochastic model as i\IonteCarlo Method model (MCM model).

2.2 Monte Carlo ]\1ethod lYlodel

The general idea of the Monte Carlo Method model is that a number ofmolecules are tracked on their random walk. In a first approximation ,vespecify 'random walk' to be a spatial step of fixed length in a. randomdirection per temporal step. Under the assumption that comparablyfew pre- and postsynaptic receptors are available for glutamate llind­ing, and therefore binding cloes not significantly influence the randomwalk experiments, we considered the binding and unbinding process ina separate computation.

2.2.1 Derivation of Random Step Length

To derive the random walk step size froIn the diffusion constant, wecompare the analytical solution of the diffusion equation 1) withthe normal distribution [60, 77]. The analytical solution for the diffusionequation can be given under the assumption that the amount Co of thediffusing Inaterial with the cliffusion constant D is deposited in anndimensional infinite space in :r 0, x E R" at t 0:

The normalized Gaussian distribll tion in one dimensional space is de­scribed as:

g(:r)

where is the variance and p is the mean. In the multivariate case :ris a vector of dimension ri, 11 is a vector of dimension 11, I; is the rI x 11

2.2. lvIONTE CARLO METHOD lvIODEL 27

covariance matrix with the variances along the diagonal and covariancesoff diagonal:

(2.4)

where IlL is the F'robenius norm of L, and J g(x)cl":r = 1. In ourJVIonte Carlo simulation all spatial directions are considered equal suchthat p. = 0 and the covariances are zero (making L a diagonal matrix).Furthermore the variances in a free difbsion experiment can be consid­creel equal for all spatial directions therefore (Tt (T2 (Tn with(T, being thEe: diagonal elements of L. In the one dimensional case thefollowing relations are given:

var(:r) ') I ')-- 11.)-] =. :cg(x)cl:r.)

(T- , (') "')~.O

while the above relation holds for p = 0, the average distance square inthe multidimensional case can be described in analogy as

< > var(x) I I + ....:r~)g(:r)cln:I: =

(2.6)Our random walk model assumes that the average distance square cov­ered in a time period of length t is described as:

< > (t) :::: t1/cl2, 1/ = jumps/", (2.7)

where cl is the length of a random step. vVe derived the following equiv-alences from Equations , (2.4) and the H.elations (2.6, 2.7):

::= 2n.Dt = tl/cl2,

which provides us with the desired relation

2nDcz2(D) :c

2.2.2 Restricted Space and Heflections

(2.8)

(2.9)

In a restricted space random leading beyond the spatial restrictionarc reflected according; to a perfectly elastic rebound, The two cleft

28 CHArTER 2. DIFFUSION SIMULiiTION

geometrit;s under observation are hemispherical cleft and flat cleft. Bothgeometries arc provided in an analytical description. In the flat cleft case,vc assumed two parallel boundary-planes that are perpendicular to tht;z-axis. Figure 2.1 shows an example for a situation where reflection isnecessary. At time t the molecule sits at p, the random step would leadto point q that is outside the modelcd cleft geometry. The trajectoryq - p is thus reflected at point c and q is the new end point of thetrajectory. Points Cl and q have common :2:- and y-coordinates, thecoordinate is computed according to

In the hernispherical cleft case (illustrated in Figun; 2.1), we find princi­pally the same geometric setup except that we use concentric spheres asspatial restrictions. The following procedure is then applied to computea reflection:

• normalize the trajectory vector d q P to r:::: d/lld!1

• compute the point c where the trajectory vector r crosseS thesphere. The equation of the straight line is I :::: p + /\ . r. Thesphere has its origin in 0 its equation is simply + -+- :::: n2

.

Setting these two equations equal results in a quadratic equation.For our geOlnetry we considered half spheres (upper half) we aretherefore only interested in the positive solution.

c

pT,. +

P + '\od

... Ft)

(2.11)

• compute the H,flected point. As the spheres have their center inthe origin, vector c is equal the normal 11 in c, we can thereforecompute the reflected point as:

q q 2n (nTd)

2.2.3 Verification

The important aspects in this stochastic sirnulation are Cl ran donIJ1\nnber generator as well as to find the rdation between the diffusion

2.3. DIFFUSION EQUATION MODEL

zq

o/

/

n

29

Figure 2.1: illu,stmtion of a nJleetion at a hor'izontal plane (left) and ata sphen: (right). Both sketches al'e Cl cr'oss-section of Cl thr'ee dimensionalsetnp.

constant and the average distant a molecule can move for ID, 2D and3D. The random number generator used in our simulations was takenfrom "Numerical Recipes in e" and is based on Knuth's suggestionfor subtractive methods. The relationship between the diffusion con­stant D = 0.55mn and the average step taken in each move resultsfrom the comparison of a phenomenological solution and the mathemat­ical solution for the random-walk. see above for the derivation of theserelations:

<<<

>=--:: 2Dt.

>=: 4Dt,

>= 6Dt.

1 dimensional space

2 dimensional space

:3 dimensional space

In Figure 2.4 we illustrate how the stochastic movement of 10'000 par­ticles match the prediction of the analytical solution of the difl'usionequation.

2.3 Diffusion Equation JVIodel

In Equation we present()d a solution for the diffusion problem ininflnite space. Although it is possible to specify other solutions (e.g. inthe form of infinite series) their application to practical problems canpresent difficulties. First the numerical evaluation of the solutions is

CHAPTER 2. DIFFUSION SIMULATION

usually not trivial Secondly, the analytical methods and solutions arefor the most part restricted to simple geometries and to constant difFu­sion properties such as the diffusion constant. 'While modding experi­mental and practical situations we therefore used mllllerical methods.The numerical solution for the diflusion equation model approximatesU;r,L for U(;r ,t) on predefined mesh-points. \Ve therefore introduce aspatio-temporal mesh with width for the spatial dimensionsand width k for the temporal dimensions. Before the numerical compu­tations can take place we need to decide on the follmving three points:

• discretization type of the partial differential equation

• description of the boundary conditions

• solution scheme for the linear equation system

2.3.1 Discretization

For the discretization of the parabolic differential 1) wechose the Crank-Nicolson scheme because of its stability and secondorder accuracy in space and time. 'I'lwse properties are obtained bybuilding the arithmetic average of two temporarily successivediflerence quotients. Before we briefly derive the three dimensionalCrank-Nicolson scheme, let us note the following discretization funda­mentals:

dUdi

cPU

where the upper index n denotes the telnporallocation in =: ok and thelower indices i, j, k d~:note in the spatial location. Furthermore

2.3. DIFFUSION EQUATION 1\JODEL 31

we introduce the usual replacements for regular interior points in three­dimensional geometry according to Figure 2.2:

Up

i+1Jk = Up

i-l,}),: =-~

U::J+U

i,J·-l,k =

U:,':;,k-l = Us

Up

:--= OF

tJIJ=[rB

UW

ON08 .

In an explicit formulation of the diffusion equation the spatial deriva-

N

w

F

s

p

E

Figure 2.2: eTl,'/I,'rneration of vo.TClsin the 3D model

hves are evaluated at. time t, and a forward difference representation isused for the temporal derivates. Resulting in Equation (2.13) if 'we as­sume the same uniform grid with mesh-size 6.. for all spatial dimensions

32 CHAPTER 2. DIFFUSION SIMULATION

and k for the time step:

2Up + Ucv_.----:-----_. + ._----,----ccc D·(Jp ­

k13)

where D is the diffusion constant and indicates the concentration Uat point P at time (t + k). Equation (2.1:3) reduces to 14):

a(UF{ + UF + + -+. Us CTN ······6U =: [T p (2.14)

Here a denotes D . k / .0. 2 . A similar forrnula tion leads to what is knownas an implicit equation for ()

Up.

The implicit formulation is stable with regard to the propagation ofrounding errors (regardless of the time-step k). \-Ve are now ready tobuild the arithmetic average of the implicit and explicit formulae, whichevaluates tll(" spatial derivates at a time level between t and t + k:

/\ . 0' (On + OF -f· +

(1 A)' a (UFf + + [TE +

+ (Ts + (TN ..- GOp) +

+ Us + 6U1' )

The so called Crank-Nicolson mid-diflerence scheme uses /\ _.. 0.5 re­sulting in:

- 1 (~. -Up + 2 . a GU1"- [hI

U· 1. (('C' TTp 2'0 IIp-un UF UE- [T

2.3.2 Boundary COllditions

In boundary-initial value problems three classes of ccm-·ditions are distinguished. Specification of the value of the dependent

l For an extensive discussion see: "Heat 'Cransfer Calcnlations Using Finite Differ·ence Equations" by Davicl R. Croft and David C. Lilley, Applied Science PublishersLtcl , London, 1977

2.3. DiFFUSION EQUATION 1\IODEL 33

variable is referred to as a Diric1l1et boundary condif;ion, U IT. Thespecification of the derivative of the dependent variable nornal to theboundary is referred to as Neumaml boundmT condition, I'.\Vhile the specification of a linear combination of the vari-able and its normal gradient along the boundary is a Cauc1ncondition, dv/dn + ellI

In our problem we find two difTerent sorts of boundary conditions:

• Along the pre- and postsynaptic membranes we assume blockageof the diffusion thus the derivative of the concentration normal tothe boundary is zero (von Neumann boundary conditioIls) .

• At the border of the discretized geometry free diffusion is approx­imated with Cauchy boundary conditions

Discretization of the von N eUlllanll boundary condition towardsthe cell membranes shall be simplified by the assumption that theboundary coincides with a plane parallel to the plane through the mesh­points .8,F,H) also [102]). \Vith the temporarily introducedexternal mesh-point E \ve Call approximate the forward difference ex­pression 2.B:

(2.17)

In the Neurnann boundary case we thus get the following Crank-Nicolsonscheme:

1 (_ ..+ 2 . (\' 6Up - UT!

12 . 0 (6U1' .. Uu (') 1><),,-,. lJ

The discretization of the diffusion equation for mesh-points adheringto the Cauchy boundary condition is analogous. The general Cauchyhoundary condition is specified for our case as:

dC"

dn~. ···DU, (2.19)

where D is the diffusion constantconsider the three dirnensional case

5.5 * 10"10 ). Let usFigure 2.2), again we assume

34 CHAPTER 2. DIFFUSION SIMULATION

that the point E lies outside of the geometric object whose bounclaryis parallel to the plane including the points (F, N, H, 8). In analogyto the Neumann condition we find the intuitive approximation of thenormal derivative

the following Crank Nicolson

- UH',._.__....__.... = - D[; 1',2h r

In the Cauchy boundary Case we thusscheme:

UE = UH'D

217.,(2.20)

'\

2UH' - Us .. liN ) . (2.21)

(;1" + 1 . Cl' ((6.. D [Tp (rH (TF -- 2U\v - (Te;2 217;1:

I ' 1 ((' D)[, ['/p--'[Y 6- .. Jp-" iF!-2 211;);

The construction of the boundary conditions is straight forward as longas we only examine the fiat cleft case, because there the tangentialplane is in every mesh-point parallel to the local surface. During thediscretization of the hemispherical cleft ive find situations where a voxelis situated in a way such that it is considered a boundary voxel alongtwo axis. Figure 2.3 illustrates this situation in a cross-section.

T

Figun~ 2.3: cTOss-sedion thTOugh a three dimensional discrete nwdelof a hemispherical cleft On the through(R, Hl,P,S. T) 'We have to Inlfill Cauchy conditions. The outer normal nImdds the angles 0 and /3 with the aTi" as indicated in thcminiatnre :3Dcoordinate system.

Mesh-point P is a boundary point x··axis as well as y-axis, the line I indicates a cross·section th,"nl·"d, the tangential plane.

2.3. DIFFUSION EQUATION MODEL 35

This double boundary situation asks for a more general approach toderive the factors for the seven-star operator occurring in the (li­agonal coefficient matrix.1. Below we give an example for a ScJ((1l1C ac­cording to which we approximated the boundaries in the ne:t111:':;')llCrlCiJcleft. The 2D form of this scheme as well as several other 2D c::;Cllnp](lsare presented in the textbook of Sclnvarz [102]. 'Ve try to appr(.)ximatethe differential expression U"1: + U yy + U zz by developing the concentra­tion function in a Taylor series at point P and its neighboring points(TV, S, F, H). This difTerential expression builds the rmv entry in thesepta diagonal coefficienl matrix .J in Equation 2.22.

P:nr

:

8:H:F:P:

;11,

u(:r - h,;II, 2);11 h,

U(:Z:,;II,2U(.T,;II, + h)

'U:r cos n cosu y cos Cl' sin ,8 + U z sin 0'

Cp

CH'

Cs

CH

Cp

Cn

With the indicated factors we build the linear combination of the aboveequations resulting in:

,y, z) + --h,y,z)+ ;11 + h,

+ CIT)

y,c + h) + Cn ---;:-_····h) +

cos n sin B

sin et +

1/,

+ ClF + Cs + CN + Crr + CF) +h 2

U". (cn cos 0' cos ,(j hCn') + U x x 2- Cl/) +

h 2

+ vYY2cS +h2

+ n z : 2

As this equation has to benecessary conditions:

v y ,/ + /L z : = 0, we get the following

Cp + ClF + Cs + Cu + Cr + Cn

Cn cos et cos _. hcn'

oo

-2' CII'

cos 0' sin (I hcs

1

o

36 CHAPTER 2. DIFFUSION SIMULATION

1

1

o

-+ Cll)2

h2

'2C5

h(Cp ~ CJ-I) -+ Cn sin 0:

1')r

IVe did not completely exhaust generality shown in Figure 2.3) andassumed 0: = 00 /3 45 0

• ,\Ve thus obtain the following values for thecoetTicients:

Cp =1

Cn =1

Cs2 2

6 2V2Cp = --") -_.

)~ 11

2.3.3 Solution Scheme

The Crank-Nicolson discretization of the diffusion equation leads to asystem of equations (here given in matrix noLatlon

(21 -+ r.J) Uj+1 (21

where l' := D k / h, J E li.n x n is a sept:l diagonal coefficient matriX,1 E Rn xII is the identity matrix, 'Uj+l is a vector containing the concen­tration values at time step (j -+ 1) and bj is a vector carrying boundaryvalue information. tor an exemplified estimatioll of n we assume a flatcleft of 1500 x 1500 x 20 nrn. IVith hornogeneous mesh-sizes of 16 nm in;r; and y direction and smaller mesh-size of 4nm across the cleft wea large system of 46080 coupled linear equations. J'vIatrix J in Equa-tion 2.22 has thus a size of 46080 x 46080. but O.L)are nonzero entries. This certainly demands for solution methods that

this high degree of sparsity.

In we can distinguish two methods for ,-"I""", Ql'd'(onl'

linear equationsof

• explicit solvers. vVe factorize the matrix A DU intoa lower triangular matrix L and an upper triangular matrix U.

2.3. DIFFUSION EQUATION AIODEL 37

We now apply in every solving step the forward and backwardsubstitutions according to Ly f) and U:r = y .

• iterative solvers. use successive approximations to obtain mo'('accurate solutions to a linear system at each step, Here wetinguish between stationary and non-stationary methods,the latter ones are relatively recent in development and usuallymuch more dIicient. They are based on the idea of sequences oforthogonal vectors.

For our problem explicit solvers should at least be able to operate onband matrices. 'While choosing an enumeration scheme for the pointsin the mesh we have to be cardul that the bandwidth stays minimal,which can be achieved looping through the axis with the smallestnumber of mesh-points first. The IoU factorization would in our caseintroduce large fill in, as the septa-diagonal system carries hundreds ofzero side diagonals between the outermost and the main diagonal. Thismakes the band matrix approa.ch very memory consuming.

Iterative solvers produce a sequence of approximate solutionsJ\Iatrix A is only involved in the context of matrix-vector multiplication.In our problem system matrix A is clue to the boundary conditions notsyrnrnetric. therefore the GivfRES iterative method was chosen as asolver chapter il).

2.3.4 Comparing the Diffusion Equation Solution withthe Monte Carlo lVlethod Model

'1'0 verify our ideas of the underlying physical processes, the mathe­Inatical approach ancl our implementation of the model we checkecl thecoherence of the numerical sol11 tion of diffusion equation and of theMonte Carlo Method rnodel. \Ve first verified whether free diffusionin both models matches the predictions of the analytical solution (seeEquation . The analytical solution tends to zero as x approachesinfinity positively or negatively for t > °and for t=O it vanishes every­where except at :c == 0, where it becomes infinite. The integral of theanalytical solution at any t from --o() to (X) equals Co. Hence the scalarfactor is included to account for the constant amount of diffusivematerial.

38 CHAPTER 2. DIFFUSION SIMULATION

In Figure 2.4 we compare the analytical solution for a fixed :r 15nmwith the result of the IvICIvI model, where x is the perpendicular dis~

tance to release. Averaging in the I\!ICIvI model can be achieved in twoways. On the one hand we can run the simulations several times andthen average the results. On the other hand we can enlarge the boxin which we sampled. For the simulations shown the sampling box hada size of 5 x 5 x 5 nn), , results were then normalized to a unity box of1 x 1 x 1 nrn. Furthermore we averaged over 4 simulation runs. The

400 600 800 1000 1200 1400 1600 1800 2000lime (ns)

Figure 2.4: Monte Carlo Method model calculated in three dinlen-sl:onal 8pace(gmy linc) compared to the theoTetical 8olntion(black line)

diffusion equation model is always computed upon a finite geOlnetric re··gion, dilfusion into the space beyond that region is handled Cauchyboundary conditions. To check for an acceptable error introduced bythese linear boundary conditions we COlnpare the analytical solution forfree dilfusion with the diIfusion equation model nm on 2 diIferently sizedcubes (1024 x 1024 x 1024 nrn resp 512 x 512 x :512 nm) both with ahomogeneous mesh-size of 161101 on all axis. In Figure 2.5 we presentsuch a comparison. The deviation between the concentration computedby the analytical solution (full line) and the solution computed in thelarger cube (dotted line) exceeds at about 6.510-'1 m,J[. Here theerror introduced by the linear approximation of the boundary condition

2.3. DIFFUSION EQUATION IvroDEL

falls beyond the diffusive supply.

103

lime (ns)

39

Figure 2.5: diffusion equation rrwdel calwlated in fncc thr'ee dimen­sional space comparcd to thc analytical solution. Full i'inc is the con­centration tTace at a distance of 20 om fmm Telcase eomp'utedby the analytical solution for diffusion in an infinitc thnce dimcn­sional space. Dotted line TepTesents the solntion calcnlated in a cubeof 1024 x 1024 x 1024, dashed line repTcsents the solrdion calcnlated in acubc of 512 x 512 x 512. Both cube calcv,lations weTe achieved as Tesnltsof a numerical solution to the diseretized diffusion equation.

Chapter 3

Model and Results

3.1 Introduction and Overview

In the previous chapter we showed how diffusion can be computed undergiven initial and boundary conditions. For the results presented herein Chapter ;3.2 and ;3.:3, we used the partial differential equation bas("c1calculations as given in Chapter 2.3. For the results presented in Chap­ter 3.4 we used the therein given models based on ordinary differentialequations.

Information processing in the central nervous system is based onfast synaptic transmission in the mi11isecond time domain. This re­quires activation of postsynaptic receptors by neurotransrnitters witha fast onset and short duration. In addition to fastsynaptic trans­mission. the same transmitter often also induces slower postsynapticsignals which encode a temporal1y integrated state of presynaptic activ-

J\Iost synapses in the vertebrate central nervoususe glutamate as their neurotransmitter. Postsynaptic responses are

hll.lvOllU,lLLL;-hClU;U icm channels of the AMPA- and NMDAtypes as \wl1 as by G-protein coupled metabotropic glutamate 1'eceptors(mGluHs).

Postsynaptic currents produced by ANIPA receptors have onset anddecay time constants as low as O.5m.s and 2-4 7n8, respectively, at ce1'-

II

42 CHAPTER 3. MODEL AND RESULTS

tain synapses [21, 92]. There is some controversy on how the fast ter­rnination of these responses can be explained. Three mechanisms aremainly considered: (i) fast clearance of glutamate from the synapticcleft by diffusion or (ii) re-uptake and (iii) receptor desensitization. Atsome synapses slower, AJVIPA receptor mediated dynamics with onsetand decay time constants in the order of 1-2m8 and 3-7m8, respec­tively, have been reported [21, 58, 49]. These slower dynamics mightresult from different parameters associated with the above mechanisms.Post synaptic responses mediated by N?vIDA receptors are slower thanthose produced by AMPA receptors. There is clear consensus that theslow NMDA-receptor mediated responses can be explained by the slowactivation/deactivation kinetics of these receptors [2, 20, 19, 68].

During the last few years, postsynaptic potentials which last 100 to1O00 m8 and vvhich are mediated by mGluRs have also been described[9, 8]. Induction of these metabotropic responses requires strong orrepetitive stimulation of presynaptic fibers. There are a number ofnot necessarily exclusive explanations for this requirement. 0) Theremight be a threshold or cooperativity in the postsynaptic transduc­tion pathway. (ii) During repetitive stirnulation glutamate spills overto perisynaptic receptors due to saturation of clearance mechanisms.(iii) Recruitrnent of many fibers with strong stinmlation results in acooperativity between relt~ase sites.

For a number of computer simulations of fast AIvIPA receptor mecli­ated postsynaptic responses at a single synapse it has been assurned thatglutamate concentration rises and falls within 1 In8 from not more thanmicromolar concentrations to millimolar concentrations [18]. The du­ration of 1171,8 also appearing in double barrel glutamate application inelectrophysiological studies is derived from the technically limited abil­ity to apply fast enough exogenous glutamate. There is no experimentaltechnique which would allow to directly measure the glutamate dynam­ics in the synaptic cleft in a sufIicient spatial and temporal resolution.It is this lack of experimentally achievable resolution which allows com­puter simulation to provide complementary information about synapticfunctionality.

In orcler to investigate the dynamics of glutamate in the synaptic,ve have developed in a first step a basic model of glutamate dif­

fusion at a high spatial and temporal resolution. In a second step weasked whether specific sets of these parameters can explain ditIerences

3.2. GLUI~4MATE DYNAMICS IN FLAT CLEFT

described for individual glutamatergic pathways. We therefore exploredthe influence of parameter variation on synaptic glutamate dynamics. Ina third step we induded postsynaptic glutamate receptor models whichtranslate the glutamate concentration into a probability wit h which theion channel of the receptor will open or to a probability with which Gprotein coupled eflector systenls are activated. At this we are ableto compare our sirnulation results to a measurable physiological read­out, e.g. the excitatory postsynaptic current (EPSe) produced by anion-channel.

3.2 G lutarnate Dynarnics rn Flat Cleft

3.2.1 The Basic Model

In this section we shall be concerned with the time course of the neu­rotransmitter glutamate in the synaptie cleft after a quantal releaseevent. The amount of transmitter released, the duration of the releast;,the number of release sites, the shape of the synaptic cleft as well assynchronous activity of neighboring synapses are likely to be the rnainfactors which determine the transmitter time course. For each of theseparameters we chose a fixed value in our basic model.We assumed that the content of a single vesicle [64] with an inner diam­eter of 30 nm and containing glutamate at a concentration of 100 mJVf[13] is released instantaneously into a disk shaped synaptic deft. To easenumerical computation we replaced the sphere representing the vesicleby a disk shaped compartment (4nrn thick, 16nm radius) bordereel bythe presynaptic membrane containing an equal amount of g1l1tc'Ll11atlc;.\Ve estimated the eliffusion constant for glutamate in a so··lution to be 5.5 . This value was obtained tJwvalues at 1°C and 25°C Longsworth [70,71] to 37°0. Hecentestimations that several thousand transmitter nlOlecules areset free, in contrast only a few dozen glutamate receptors exist on thepostsynaptic side. The ratio between the total amount of glutamatemolecules and the amount of glutamate receptors justifies the assmnp­tion that binding does not significantly reduce glutarnate concentrationor diffusion behaviour by ternporary binding events.

44 CHAPTER 3. MODEL AND RESULTS

3.2.2 Flat Cleft Morphology

An electron micrography generously provided by Dr. Pedro Grandesfrom the Universidad del Pais Vasco in Bilbao Spain is shown in Figure3.1. It corresponds to a likely excitatory synapse in the visual cortex.Arrows are pointing to synaptic specializations between a presynapticbouton with vesicles and a postsynaptic dendrite. The indicated barcorresponds to 200 nm. vVe assumed a lateral flat cleft c:xtension of r :::::500 75071,'177. In accordance with morphological studies [46, 47] the cldtwas chosen to be 2071'177 in width. The geometry of the modelecl synaptic

Figure :~.1: electnm, mic1'Ogmphy of a hkely c:r:citatory synapse

cleft was approximated as the volume between two planes 3.2).\Ve moclelecl a flat cleft of r ::::: 75071,m half side length. \Vhile discretizedgeometry was therefore finite. boundary conditions were chosen suchthat our model approximates an infinite cleft. While the size of pre-and postsynaptic specializations are Inosl to be smaller than themocleled cleft, we chose the modcled to be this large uncleI' theassumption that either the distance to the next synaptic specializationor the cleft continuation under is around 1 2/l7n.

3.2. GLUTAMllTE ITYNAMICS IN FLAT CLEFT

postsynaptic

45

Figure ~L2: schematic drawing: cleft; geometr'y appT'o:rimated by twoplanes nwdeling p1'c- and postsyn.(l,lJl'ic nlcm.bmnes, 20 nTn apa.Tt

3.2.3 Glutamate Dynamics

Following release into a flat synaptic: cleft, diffusion of glutamate resultsin a fast rise and fast of local glutamate concentration [glu] (seeFigure 3.3). At the membrane opposing the release sitea peal, is reached alread.y after 0.1118 and thereafter Iglu] decaysthree orders of magnitude 'within N 250 )1.8. If the peak Iglu] at this sitewas 10 nu\!, it will drop, therefore, within that time interval to 10111\1,which is roughly the E(750 value for lu'vIPA-type of glutamate receptors[21]. At more distant sites peak reaches a lower peak at a later timebut the decay is comparable, i.e. at a lateral distance of l' = 500 nm apeak concentration of l:'5)1.JI! is reached after 85)1.8.

Figure ~\.4 shows local Iglu] at various times following the release.After 100 !IS is quasi evenly distributed and reachedat least 10 !I,1\l

all over the area of synaptic specialization (1' ~ 200nm.), which corre­sponds to the of the Al\IPA receptor. The decay constant wascalculated to be 80 )I.oS which is approximately in the rangeClements [1 \Ve also wondered how the present simulations of gluta-mate dif1'usion in a 3D cleft compares with previous 2D models 4, 112]and free diffusion in 3D without diffusion barriers set the pre-and cell rnembranes) .As shmvn in Figure 3.5, Iglu] decaysfaster in the .3D cleft than in the 2D models but slower than in thel111constrained 3D space.

46

A 10'f

CHAPTER 3. MODEL AND RESULTS

B

~-20nm

100nm

250 nm

o ~1 ~2 03 04 ~5

time (ms)0.01 01

time (ms)

Figure ~L:): tim.e cO'ursc of simulated glutam.ate eonccntration at /:hepostsy·n.aptic rnen)bmne. A, B: concentration at uarious dis··tances from the release site as indicated in and doublelogar'dhrnic scale.

3.2. GLUTAMATE DYNAMICS IN FLAT CLEFT

-- 0.003 ms-- - 0.010 ms- 0.030 ms

..... 0.100 ms0.300 ms

10-750 0 750

distance from release site (nm)

47

Figure 3.4: pmfile of the glutanwte eoncentmtion at the postsynapticmembrane. Pro.f7:les were calculated for times after' release as indl:cated.Thc dotted horizontal line indicates the E'C50 values of AMPA TeceptoTS.

10'

0.1 0.2 0.3lime (ms)

---

004 0.5

Figure 3.5: glutamate concentnLtion at 2011.1'17 ofl thepoint SO"UTce. Free di.ffusion lnunconstr'llined 3D space (analytical so-

is indicatcd as dotted diffusion -in 3D space coT/straincd bythe fiat cleft appm:rirnation) is indicated as dash cd

diffusion in unconstrained 2D space (analyt-ical solution) isindicated as full line.

48 CHAPTER 3. MODEL AND RESULTS

3.3 Variation of Model Parameter

In the previous section we described a basic model for simulation ofsynaptic transmitter diffusion. For some biological parameters onlyapproximate values are available and sorne parameters differ betweenneuron-types. In this section we shall elaborate what influence varia­tion of these parameters has. The five subsections presented below treatthe parameters: duration of release, multi quantal release from 5 releasesites into the sanTe cleft, simultaneous release in neighboring synapses,vesicle concentration, and cleft shape.With the additional computer simulations of synaptic diffusion shownbelow we shall try to improve the confidence into our model as we]]as into our calculations. Furthermore it is useful to have these H)Sultsat hand as prerequisites when it comes to simulation of postsynaptictransduction.

3.3.1 Temporal Aspects of Release

Computer simulations of synaptic events in this thesis genera]]y assumethat release is instantaneous. to whether this assumptionis critical for the results we carried out simulations \vhere the releaseprocess was distributed over a time window of up to 20 ps. Temporalresolutions of electro-physiological and imaging measurement techniquesare yc)t too low to get an estimate of the blue constants of the releaseprocess. In a r(~cent paper Khanin et al. [61J showed numericalcomputations that diiTusion cannot provide the obs(~rved fast dischargeof nenrotransmitter from a synaptic vesicle during neurotransmitter re­lease. For their calculations assumed that the ditlusion pore withradius T' = 0 nln at t :== 0 ns is expanding with a rate of 4 nm per I))Sand found that the peak concentration at sites would betoo low, if discharge were only by diffusion.In a further experiment we replaced instantaneous release a contin­uous release event of a duration of up to 20 I/,S. Continuous releilse washere approximated by 5 consecutive instantaneous release events occur­ring at an interval of up to 5 IfS at the same release site. The totalamount of released glut.amate was chosen to be equal t.o the amountreleased in the reference simulation in Section :3.2). Apeak in t.he multi-release setup was therefore only one fifth of the

3.3. v:ARJATION OF MODEL PAJIJHvIETER 49

inal peak.In Figure 3.6 we illustrate the consequences of such a temporally dis­tributed release process. 'Ve compared the concentration traces for thisfirst order approximation of non-instantaneous release \vith the com­puter sinmlations carried out with the basic model. After 100/18 and200 /L8, respectively, the two traces sampled each at 20 mn distance fromthe release site differ by 10% and respectively. As the simulations offast postsynaptic signal transduction of these continuous release tracesdid not reveal a significant deviation to the reference model (see also thefollowing section) we could justify the uniquantal instantaneous releaseimplementation in the basic model. This decision was made under con­sideration of experimental results of several studies [83, 95, 12] whichindicate that release is indeed faster than [jO p8.

limo(ms)

Figure 3.6: slow release simulated by [) serial r"Cleases with intervals of5 p.s (full line) eompared to instantaneous full release(dashed line)

:3.3.2 JVlultiple Release Sites

Another question we raised in the course of these studies is whethera set of vesicles would alter the temporal concentration patternsignifkantly if cOInpared to the same amount of transmitter releasedfrom a single vesicle release site. In Figure 3.7 we illustrate concentra­tion distribution throughout the cleft in a cross-section just above thepostsynaptic membrane. The concentration is both height and color

50 CHAPTER 3. lvIODEL AND RESULTS

coded. In the left column a simulation with 5 release sites, one centraland 4 sites at 100 nm lateral offset, is shown. In comparison the rightcolumn shows snapshots of a simulation with a single release site, re­leasing the same amount of neurotransmitter. Already after 2() IlS thetwo simulations differ by less than 1

1 Si0.5

0 0100 100

100 1000 0 0 0

0.4 1

0.2 05

0 0100 100

100 1000 0 0 0

0.02 0.02'1

0.01 0.01 J

~.0 o' '....•.

100 100 50······· , '. y 100

1000 0 o 0 50

release site s1:mv,­transmil;ter was re-1.6 JlS row)

In mAl. The sam-

Figure 3.7: c01nparison of andlations. in both cases the same total amount ofleased. Snapshots were taken at O.cl ps (topand 50ll s (bottom row). The ordinate indicatespling s'ite was 20 nm perpendicu.lar to the release site.

~~.3.3 Simultanous Release in Neighboring Synapses

Sirnultaneous release in 4 synapses differs from the setup inthe previous subsection :3.:3.2 in the size of the lateral offset of the releasesites. In contrast to the previous subsection here \ve as,sumed a lateraloffset of 1536 nm., which sets the release sites into separate synapses.In the left half of Figure 3.8 we illustrate the setup. Synapses :\,13,C,

3.3. VilRL4TION OF MODEL PARAll1ETER SI

and D are synchronously activated, clue to the symmetry in activationthere is no concentration gradient froni one synapse to another andtherefore no diffusion..Fbr each of the four synapses therefore the innertwo sides can be regarded as diffusion barrier (blockage of diffusion)because of the loss of concentration gradient. Under exploitation of thedescribed symmetry ,"ve therefore modeled only one synapse with half ofthe circumference closed. This simultanous release scenario has recentlybeen described for a model of GABAergic transmission [26]. The loss ofdiffusion drain due to the synunetric setup and the resulting enhanced[glu] shall be termed spatial cooperation. In Figure 3.8 we show acomparison of a nonnal simulation vs. a simulation where the cleft wasclosed (with Neumann boundary conditions) at two out of four sides.vVith an assumed EC50 of 0.01 7nl\1 we get a significantly longer timeinterval where the concentration stays above the EC50 value, 0.6 ms ascompared to 0.22 7ns. Possible consequences of this spatial collaborationare explored in Chapter :3.4.4 with regard to perisynaptically situatedmeta.botropic glutamate receptors.

10'

iO'

A

c

B

D

~~

;::.:: 10·'

S

1 O~LO_·_··_·_··O:t.~,'" ·..·_· ......·O,L.2··.... ·····..····

0·J,·3'·..••·..·....·OC·.L

4.._···_·'C

o:.s

time (ms)

Figure ~3.8: concentration time course in i.hc standanlsyna]Jse and in stinl/uJated synapses(fuJl

The concentration was at a distance of 20 nFn oH the re-lease site.

52 CHAPTER 3. MODEL AND RESULTS

3.3.4 Increasing Initial Concentration

vVhat would happen to our observable if we increased the transmitterconcentration in the synaptic vesicle'? l'he analytical solution of theequation describing free difFusion (given in Equation 2.2) gives a clearanswer to this question: An increase in the initial amount of transmitterCo deposited at < xo, to > results in an equal increase in transmitter atany place at any time. The insight that this is also true for a restrictedgeometry is straight for\vard. In Figure 3.9 we show a comparison oftwo ?vlonte Carlo Method simulations in a fiat cleft, both simulationswere run over a simulated time of 100 fI8 llsing the same time step of1 n8, with the only difference that in the first case 10'000 moleculeswere set free for diffusion while in the second case 50'000 moleculeswere deposited. Eventhough a vesicle \vith an inner diameter of 30 nrncontains only approximately 1000 molecules, for the sake of an aver­aging effect we used 10'000 resp :SO'OOO rnolecules. In both cases thecounts were carried through in a cube of 5 nm edge length positioned at17.5 nm distance of the release site, the counts were then normalized tothe amount of molecules per 17m 3 , therefore wc could detect a virtualminimal concentration of (lOOS =: • ~ • . For both initial cal-

;).:). ;)

culations 5 simulations were run. Averaging over 5 simulations resultsin a minimal detected arnount of 0.0016 molecules pernm:3, 'which isindicated by the lowest grey band in Figure 3.9. To be able to visu­ally compare the two curves we multiplied the results for the low initialconcentration with a factor 5. From the different envelopes for the twoconcentration curves we can intuitively recognize that five fold increasein deposited molecules is equal to averaging over five simulations, whichbecomes evident in Figure :\.9. In this simulation we reproduced twoaspects of diflusion: (i) with a 5 fold increase in deposited rnaterial weachieve with the lVlonte Carlo model a fold higher accuracy in theprediction of average transmitter concentration, and wc reproducedthe predicted homogeneous and identical increase" of cleft concentrationif vesicle concentration is increased in a restricted geOnlet.rv

3.3.5 Cleft Shape

AMPA receptor mediated EPSC time constants in synapsesformed by mossy fib el'S on CA;j pyramidal cells in the hippocampus (2-:3

3.3. V.!lRJATION OF MODEL PARAMETER 53

103

time (ns)

Figure 3.9: an increase in the initial ammmt of transmitter' depositedat < to, Xo > results in an equal increase in tmnsmitter at a'ny place atany time. lYe plotted per time step the amount of molewles in a vO.1:elof 1 nm edge sitvatedin 20 nrn distance to the release site. Gmy dots'represent a simulation 'with 5 fold vesicle contents, black: dots represent;a simulation with vesicle contents. To case visual compa7'isonthe black dot's val'ueswere In; El.

54 CHAPTER 3. MODEL AND RESULTS

rns, [21]) are largely different from time constants of Purkinje AMPAreceptors in the cerebellum (4-8 ms [69]). Purkinje cell dendrites carryprominent spines that are the sites for synaptic contacts by both parallelfibers and climbing fibers [81]. In Figure 3.10 we illustrate Cl 90 degreesector of a spine synapse, here the presynaptic bouton is assumed to sitlike a cap on the head of the spine. In contrast to other synapse geome­tries which can be approximated as disk shaped description of ourbasic model), the geometry of spine synapses can be approximated as thevolume between two concentric hemispheres. Barbour et al. raisedthe question whether the hernispherical crossection of spinal synapsesat Purkinje cell dendrites can retard the removal of transmitter by dif­fusion. Consequently this slowed down t.ransmitter removal could thenexplain the slow AMPA EPSC on dendritical synapses.\Vith t.he below described simulations we intended to answer the fol­lowing three qnestions: first, is the [glu] dynamics in a flat cleft inher­ently different from [glu] dynarnics in a hemispherical cleft? Second, ifso, what might be an explanation? Third, if there is a difference in theconcentration profile, is there also a diflerence in the Aj'vfPA inducedEPSCs?

ckft

Figure 3.10: illu.stmtion of 0. sphc'tico.l synapsc, pre·· and vostsvnavticrnernbmneswe'tc 'modded by two concent'tic withmdiu.scnec of 20 nrv.

Our basic model. which we use here as a refen~nce. assumed infinitecontinuation of the cleft beyond the modeled disk geometry. A simi..hr setup was used for the hemispherical simulation. In 3.11 we

3.3. VARJATION OF MODEL PARAlIlETER

give a comparison of the two geometry setups: solid lines indicate thediscretized geometry while clotted lines indicate the hypothetical con­tinuation of the cleft beyond the discretized geometry. The release siteis located in the culmination point of the hemispherically shaped cleft.The boundary conditions at the diffusible outlet used for our simula­Lions are a first order approximation of infinite continuity of the cleftin perpendicular direction. These assumptions lead to a model wherethe volume beyond the discretized part of the cleft grows with 0(:1:) forthe hemispherical geometry while it grows with 0(:1:2

) for the flat ge­onwtry (see also Figure 3.11). During the investigation of the influenceof cleft shape on [glu] dynamics we simulated two special cases: first,we chose the radius of the hernispherical cleft such that we obtained('~qual discretized volumes for the flat and the hemispherical cleft. Sec­oncl we chose the radius such, that the hemispherical cleft had the sameboundary area through which the transmitter could leave the discretizedvolume. Both simulations were run under the same conditions as thebasic rnodel. A quantitative comparison of the setups is given in Table3.1.

x I-.__:L..... .r c ..

Figure 3.11: cross-section of disk-like fiat cleft; right:cross-section of hemispherical cleft

Table 3.1: for the disk like fiat cleft and two hem.£spher'icalone with the sa'me volurne and one with the sarne boundary m'ea

as the flat

56 CHAPTER 3. MODEL AND RESULTS

In Figures 3.12 and 3.1:3 we give a comparison of the glutamateconcentration time courses simulated with the two hemispherical cleftsizes. The assumption of identical boundary areas (1'1 == 740 nm) leadsto simulation results with less than difference bet,veen the two cleftgeometries after 0.5 ms. Identical volumes == 540) in contrast yieldsconcentration differences ofWith the help of the above results we found the following answers tothe initially posed questions: first, with decreasing radius of the hemi­spherical cleft there is an increasing difference in d:vnamics. Thisfinding comes to no surprise if we look at the illustrative special cases:(i) 1'1 -t 00: hemispherical cleft-tflat cleft; (ii) 1'1 ~z.:lf due to the factthat the volume l)(~yond the discretized space grows with only 0 (:r), themolecules distribute in a smaller volmne for the same average diffusiondistance and consequently the transrnitter concentration is increased.Second, the results given in Figures 3.12 and :3.1:3 illustrate our findingthat the radius (and thus the area of the outlet) of the hernisphericalcleft is the principal factor which makes [glu] decay dynamics in hemi­spherical clefts different. Third, even though we were able to simulate adifference in concentration decay dynamics, the subsequent signal trans­duction at AMPA receptors located vis il vis the release site, did notshow any difIerence in silnulated EPSCs. In Chapter 5.1.3 we shallbriefly compare our results with the of Barbour et al.

3.4 Postsynaptic Signal Transduction

In the previous section we confirmed by parameter variation the para­meters of the basic diffusion model presented in Section ;3.2. The basicmodel proved to be stable towards the tested parameter variation. Inthis section we shall use transients computed the basicmodel as input for a separate L.vpe of simubtions. In these simulationswe compute the signal transduction dynamics in dif)'ercnt glu­tamate receptor types. The corresponding simulations base on ",.,-1;,,,,,.,,

difTerential equations and can be calculated with commercial toolsMatlab), SE,e also i\ppendix A.4.

As introduced in Chapter 1, two different transduction mech-anism are distinguished for binding (i) Af\IPAand NTvIDA receptors which fornl a transmcrnbrane pore through which

3.4. POSTSYNAPTIC SIGNAL TRANSDUCTION 57

lill'li:l(ms)

FigurtC\ 3.12: .flat cleft reference simulation (full) compared to hemi­spherical cleft sim/u,lations with either the same dijJusible boundar'y areaas the .flat cleft (dotted line) or thc samc volume as the .flat cleft (dashedline). Conccntmtion iT'accs WC7'C sampled at 20 nm perpendie1J,lar to thcrelease site.

200 400 600distance (nm)

200 400 600distance (nm)

" " \

200 400 600distance (nm)

Figure 3.13: comparison of .flat cleft (full) and hemispherical cleftswith either sanw volume as flat (dashed) or sarne b01J:nrlary areaas .flat cleft (rloUed). Left . profile snapshot at t = 11'.0. MiddleFig/we: profile snapshot at t 10 pS. Right Figlwe: profile snapshot att =: 100 lIS.

58 CHAPTER, 3. MODEL AND RESULTS

selected ions can pass (ion-channel), and (ii) metabotropic glutamatereceptors (mGluR) which indirectly regulate electrical signaling by in­fluencing intracellular metabolic processes via G proteins.

Our goal in this section (3.4) is to find a minimal form of models forglutamate receptors which satisfactorily describe the experimental data.Tb this cnd states deemed unimportant are and/or severalreaction steps are assembled into one overall In a first part (3.4.1)we deal with the published models for AlvIPA. and Ni\IDA receptors.These models 'were tested against onc or several rectangular glu-pulseof a fev'! milliseconds duration as cornpared to the glu-spike emergingfrom our simulations of less than 0.25 ms. It is reasonable to neglectdesensitization of the receptor \vhen considering single gIu-spikes sincedesensitization is likely to become effective only during longer and/orrepetitive pulses. In a second part (:3.4.2) a model for the metabotropicglutamate receptor is worked out since no quantitative model exists inthe literature. In the last two sections wc present simulation results andshow to what extent the reductions made are acceptable.

3.4.1 Ionotropic Postsynaptic Receptors

Modeling and computer simulation of A.:tvIPA receptor signal trans··duction were done by numerous parties for different neurons [32. 56, 57,58, 92]. The AJVIPA receptor signal translation in neurons of the aviancochlear nucleus is principally characterized times Ti' S; 0.51tl8as well as a maximal open probability of the corresponding ion c11an­nel of 70 -- 80% and decay time constants of T1 ;:::;1771,8 and T2 ;:::; 127)),8

[91]. Comparable fast kinetics were reported for interneurons in therat visual cortex, while significantly slower kinetics apply for pyramidalneurons in the rat visual cortex [49]. Three main causes are di"cuss(,xl.in literature, to explain these dynamics differences: (i) presynaptic as-pects release) (ii) slow [glu] transients due to cleft(iii) relative abundance of AMPA receptor subunits. Mounting evidencethough is observed for the last hypothesis [41, 49,Vyklicky et al [108] provide a model for an AMPA receptor \vith abinding site Figure lA). Ill' cutting away the states repn?sentimQ:the receptor in a desensitized we obtain a three-state rnodel asshown in Equation :LL. This lnodel shows a higher activationrate than the original as desensitization falLs away and thus more

3.4. POSTSYNAPTIC SIGN.AL TRANSDUCTION 59

receptors make the transition into the open state. The model can stillsern' though as a simplified AMPA receptor model.

/':,.1L+R ~ U1 O. (3.1)

Hill [50] describes a reduction of this three-state model into a two-statemodel as given in Scheme 3.2.

L+R 0,

The reduction is valid if + » /':+1, /':.2. \Vith this presumptionthe corresponding rate constants are then calculated as:

'Test simulations which used a rectangular 1 TnAI, 1m.3 glutamate pulseindicate that the reduction of the Vyklicky model to a two-state lnodclmaintains the shape and temporal characteristics of the time curve ofthe open state. Thus ,vc consider this reduction as applicable and theresulting reduced model as useful, i.e. for the purpose of single quantalevent sirllulation where desensitization is not a dominant factor.

Haman et al. [91] and Jonas et al. [58] are the two main proponentsin the field of sophisticated A\lPA lnodels. 'The corresponding schemesand sets of rate constants are listed in Appendix B. Besides numer··ous desensitized states which, according to our policy, can he neglectedthese models assume two binding sites for glutamate. Transmitter toreceptor binding occurs between the first and second closed state as wellas between the second and third closed state (see Scheme 3.3). After thebinding is complete the receptor molecule transforms its shape therebyopening a pore. This transformation is modcled in the transition fromthe third closed to the open state.

The clifTerential eqllat1cllls describe this reduced model:

60 CHAPTER 3. MODEL AND RESULTS

k+ 1

R+2L ~RL+Lk._ 1

(3.3)

d{RL}!it

(1 ~ {RL} ) + {RL 2 }

{RL} ~ {RL}

[L]tot{RL} + [0] {RLd-

dt[0], (:3.4)

where concentrations in curly brackets represent norrnalized concentra­tions (i.e. {RL} = [RL]/[R]tor). Providing the binding of ligand iscooperative, i.e. the ilrst ligand bound facilitates the binding of thesecond ligand so that according to Hill [50]

(3.5)

holds, the three reaction steps in Scheme :Ll can be in oneoverall step. However, as discussed in the case of cooperative bind­ing is unlikely and hence not further pursued. In the case of non­cooperative binding the first transition with rate constant does notaccelerate the transition with rate constant . This presurnption is in­tuitively illustratable with a scenario where an arriving molecule bindswith probability PI to either one of the two a second moleculefinds only one site left so that it binds with probability P2 = ]Jl/2.This probability ratio is also reflected in the rate constant ratio (i.e.k+tI k+ 2 = 2) 1 which is contradictory to the restriction in 3.5.\Vhich indicates that a reduction of states in model is not feasible.

The NJ\:IDA model shown in Figure 1.5 can he reduced to the threestate model in Scheme 3.6 when we neglect the deseusitized state andreplace the last two steps by one. According to Hill . the new rate

R+2L2k+1

-,._--- '

...-- .. RL+L

3.11. POSTSYNAPTIC SIGNAL TRANSDUCTION

constants k-I-3 and k_~~ are given by:

61

(3.7)

differential equations describing the reduced model are as follows:

dt

d{Oldt

2K:+ 1 (1 --- {RL } {0 }) +{RL} -- k+3[L]tot{l~L}

[LJtot{RL} L;J{O}.

{O}

(3.8)

3.4.2 Metabotropic Postsynaptic Receptors

Metabotropic glutamate receptors (mGluRs) are not directly linked toion channels but rather activate G proteins which then regulate a varietyof metabolic processes (for a review see [62, 84]). mGluR subtypes arenumbered following the order in which their cDNAs have been cloned todate: mGluR1 to mGluR8. Based on their amino acid sequence identity,the 8 mGluRs can be classified into ~j groups. This classification is alsosupported by their respective signal transduction mechanisms. Group-Ireceptor activation results in a G protein mediated activation of phos­pholipase C (PLC) and a breakdown of membrane phospholipids intothe messengers IP:, and DAG. 11";3 releases Ca2+ from internal stores.Group-ll and Group-Ill receptors are coupled to inhibition of adenylylcyclase (AC). Depressed AC activity results in a reduced production ofcyclic adenosine monophosphate (cAMT') [62].

It has been reported [8] that tetanic stinmlation of' the glutama­tergic parallel fiber input to cerebellar Purkinje cells gives rise to a slowbut strong depolarizing synaptic potential. \Vhile it is not clear bywhich currents this potential is generated, we might speculate that thestimulation type used in these experirnents causes spatial integration ofsignals, i.e. several parallel libel'S with synapses next to each other arestimulated simultaneously. Furthermore a tetanic pulse of 6 consequentstimuli \vith an interval of 20 ms was required to evoke the correspondingpotential. It has been shown that mGluHs sit in the perisynaptic area[80] and therefore perceive low ampli tude [glu] transients,

62 CHAPTER 3. lvlODEL AND RESULTS

On the basis of the above mentioned report and commonly acceptedfeatures of mGluR we have attempted to elaborate a quantitative modelfor an mGluR signal transduction chain which leads to an EPSC asdescribed by Batchelor et al. [8]. We first give a full description of themodel (see Schemes 3.9 3.12) and subsequently reduce it to a minimalset of differential equations.

For the extensive model we assume that in a first step glutamate(L) binds to the receptor (R), leading to an activated receptor (R* L)as shown in Scheme 3.9. In a second step a G protein binds to theactivated receptor (see Scheme i3.10). In the resulting complex (GR* L)GDP bound to the G protein can be exchanged by GTP thus yieldingan activated G protein (G*). After dissociation of R* L, G* is split intothe activated a subunit with bound GTP (G~) and the subunits(Go,). Note that R* L acts as a catalyst for the activation of G*; hence,for a sufficiently long life time of R* L, several G proteins can be ac­tivated thus causing an amplification whose factor ranges between 20and 1000 times [96]. In a third step G;, binds to an effector X yieldingthe activated effector X* (see Scheme 3.11). The amplitude of [X*]is considered to represent the experimental signal. In analogy to theGABAB model of Destexhe et al. [27], we assume X to have four bind­ing sites. In a fourth and last step GTP bound to G~ is hydrolyzed(release of inorganic phosphate Pi), and the resulting non-activated asubunit (Gc,) rebinds to G;h whereby the G protein is regenerated (seeScheme :3.12).

At present there are no values for the rate constants in this modelavailable from the literature. Therefore we try to simplify the model asfar as possible aiming at the least number of rate constants to be ad­justed. As indicated in Scheme :3.9 wc reduce the two transitions leadingto R* L to onc transition with rate constants al and ,01' As explainedin Chapter 3.4.1 such a reduction can be found for a receptor with Cl

single binding site. Similar reductions of transitions are performed forthe reaction sequences shown in Schenles 3.10 and 3.11. These reduc­tions imply that the species of the interrllecliate states have negligiblysmall concentrations. The mass balances for the different species thentake the following form:

L]L]

(3.13)

(3.1'1)

3.£1. POSTSYNAPTIC SIGNAL TRANSDUCTION

(3.9)

1'+1

R+L \ =-=-..=,~~ RL

G

]"L*L

. . . . . . . . . .. .. . .L ....

0*

(3.10)

(3.11)

(1·1

Figure 3.14: model fOT rnGluR r>eceptOT' 'induced activity

(3.12)

64 CHAPTER 3. MODEL AND RESULTS

[0]101 [0] + [0/3/] [0] + [Ory] + [o~] + 4[X*]

~ [X] + [X*].

(3.15)

(3.16)

The differential equations describing the reduced model are as fol­lows:

d[R*L]dt

(3.17)

dtCt2[H*L][OTP][0] - [R*L][GDP]

-(±Ct:3[G~]4[X] + 4,83 [X*]

-k+:¥:;~] + L 3 [Pi ][Ory] (3.18)

(3.19)

d[G]dt

--Ct2[R* L][GTP][G] + [R* L][GDP][G~]

+k+ 1[G,,][GiJ,,] k._.l[G]

where [R], [GL3/'],balances.

, and can be obtained fron1 the mass

As already mentioned in Section 3.2.1 the concentration of glutamateis during most of the time of the glutamate spike much larger thanthe total concentration of the receptor, hence Equation 3.13 can besimplified to

[L] ~ (~i.21 )

Dividing the mass balances given in Equations 3.13 - ~L16 by the per­tinent total concentrations yields the normalized concentrations whichare represented by curly brackets, e.g. {R*L} = . \Vhensubstituting all concentrations except [L] in Equations 3.17 .. 3.20 bynormalized concentrations we obtain, in view of Equation 3.21,

3.4. POSTSYNAPTIC SIGNAL TRANSDUCTION

d{R*L} ::::aJlL]tot(I-{R*L}) I'3d R*I,} ,dt

- CY2[GTP][R]tor{R*I,}{G}

[GD P][R]tot. {H*I, }[G]tot{G~

-4 " [G]4. [X]tot {G*.' }4{ X} + 4r:10,3 .7 tot [G]tot a" f

J3

{G:,} + kJ[P;]{Ga },

65

dt--'02 [GTP][R]tor{R* L}{G}

[GDP][R]tot{H*L} [G]todG~}{Gpi}

-t-k+4[G]todG a }{G/3r } -. L,t{G}.

Note that {R} and {X} were replaced by relations derived from themass balances.

The reactions shown in Schemes 3.10 and 3.12 constitute a cyclicprocess for G: G is first activated to G* and split into G:, and G(3r(3.10), subsequently G~ is deactivated to GO' which binds GCr to formG again. For such a process a relation between the rate ,constants musthold (thermokinetic balancing, [Ill]):

(3.26)

Here Ko yp :::: [[GDP][Pd/[GTP] leg denotes the equilibrium constantof the reaction which has occurred upon completion of the G-cyc1e,i.e. the hydrolysis of GTFJ. It is reasonable to assume that [GTP],[GDP], and [P;] do not change appreciably over a time period of the

66 CHAPTER 3. MODEL AND RESULTS

glutamate spikes. Similarly the total concentrations Rtot , Gtot , and X t.otare likely to relnain constant. Hence, we can define phenomenologicalrate constants as follows:

0'2 [GTP] [R]to!

[GDP][G]todR]tot

[P],

k+ 4 [G]tot

(3.27)

(3.28)

(.3.29)

(3.30)

(3.~n)

With these rate constants Equation 3.26 can be reformulated as

CY2k+3k+4 KOTPcq[GT.P]. (-6.GOTP) . (-6.C.i. ATP )_ = ::::: exp . ::::: exp _..-=::-=:--fhk-3 k-4 [GDP][P;] RT RT

(3.32)where 6.GOTP is the Gibbs energy of the hydrolysis reaction. The lastapproximation in Equation 3.32 can be written because the GTP systemis coupled to the ATP systenl so that the relation 6.Gcrrp ::::: 6.GATPholds.

Values for the rate constants 'were adjusted such that the time depen­dence of X* obtained from integrating the set of differential equationsfollows as closely as possible that of the experimental data published byBatchelor and Garthwaite [10]. The follo\ving values satisfactorily fulfillthis criterion:

CYl 6000m.i\;[ -1 60s-1

CYz - :30s-·1 0.1.5 1

ls,·1 ls- 1

k+ 3') 0···1 O.OOls-14J,s

k+4 1800s-1 1s-1

The value for k:.\,4 was calculated means of Equation 3.32 using avalue for 6.GATP ::::: --52 kJ!mol and RT 2.5 kJ/mol. The mGluRcascade can function efficiently only if the life time of R* L is long enough

3.4. POSTSYNAPTIC SIGNAL TRANSDUCTION 67

to allow several turn-avers of the G-cycle. Hence, the value of f3l waschosen such tha\ for concentrations of glutamate [L]tot of the orderof Imid (cL 3.3 - :3.8), the formation rate of R* L is about100 fold larger the deactivation rate. For the performance of theG-cycle it is that an appreciable amount of [G];, is present.Therefore wc\s chosen much smaller than ();2. In fact, with the valuechosen for ,the terms comprising this rate constant in Equations:3.23 and 3.24 can even be omitted. It is most likely that the effectoris present in considerably lower concentration than the G protein, i.e.[X]tot/[G]tot « 1. In this case the terms \vith 03 and f33 in Equation3.23 can be neglected.

According to Hill the two transitions in Scheme 3.12 can be con­densed into one with rate constants

+ {G!h}k_:3k_4

+ {Gt3,}

This reduction requires that the following inequalities apply

(3.33)

(3.34)

+ 7,:+4 {GfJ"!} (3.35)

As is evident fronl the chosen values of rate constants listed above,this condition is indeed fulfilled if {Gin} < 0.1, but test simlllationshave shown that this limit can be set as low as 0.02. As a consequenceof the reduction {G,,} ~ 0, therefore {G} ~ 1 - {G~} (cf. Equation3.15 and remember that XtotlGtot « 1) and Equation 3.24 is no longerneeded. Furthermore, the term with in Equation 3.23 can be ne­glected. Strictly speaking the last two terms in Equation :3.23 shouldbe replaced by ·-o,dG;\} +- ,(J<j {G}. But, since L 3 « k+.dGfJ"!}holds, o,dG~}{G/n} {G~}. JvIoreover, /34 < 2.8.10-5 for

} > 0.02, which corroborates the neglection of the correspondingterm mentioned above. All these simplifications lead to a minimal setof differential equations (shown in Figure 3.15) which satisfactorily de­scrib~: the behaviour of the model.

G8 CHAPTER 3. MODEL AND RESULTS

{O~}4(1·- {X*}){X*}

<iiB:ll -- [I'] ~ (1,it - J 101 0 1

{R* L}{R* L})

{G~}){R* L}(l

{G~}

Figure 3.15: illnstmtion of the activation pathway of a metaboh'opicglntamate reeeptor together' with a minimal set of diJTerential eqnationsdescribing its behavioll,r. In each state the main participant is darkshaded.

3.4.3 Translation of [glu] transients into postsynap­tic responses mediated by AMPA- and NMDAtype of glutamate receptors

Our simulated [glu]-transients are faster than those assumed in mostprevious models of postsynaptic receptor dynamics. vVe explored, there­fore, to what extend these short transients could activate AMPA-typeof glutamate receptors simulated by simple kinetic models as given inSchemes 3.1 or 3.~).

In a first step we tested an ANIPA receptor model containing only onebinding site. IVe sinmlated both the translation of a transientcalculated with the diffusion model Figure 3.16A), as well as thetranslation of a rectangular 1 ms/1mM pulse Figure 3.16B). Thedotted line (in A and B) represents the activation simulated \vith Vyk­licky's model (as shown in Figure 1,4). Rate constants were taken fromthe corresponding paper. They were evidently fitted for alms /1 rn-AI[gl'LL] pulse. If fed with our simulated pulse the model does notprovide sufficient activation, as the rate constant is too small tocatch the fast transient Figure ~LI6A).

3.4. POSTSYNAPTIC SIGNAL TRANSDUCTION 69

The full and the dashed-dotted hnc, represent results obtaint~d with atwo state Inodel using two different pairs of rate constants. ChoosingCl: = 4.8 ·102 mlvI--] and (3 =, 4.1.102 and using a 1ms/1m~M

[gl-u] pulse we can approximate results obtained for theVyklicky model. On other haud values of the rate constants can bededuced from data fO\\nd in the literature. For example, given the shortduration of the measured glutamate pulse, we reasoned that the A?vIPA­receptor deactivation time constant is determined by the backward rateconstant (3 of this model. A receptor deactivation time constant of 2 TrIS

(i.e. the range of the fastest decaying measured AMPA-receptor medi­ated postsynaptic currents) corresponds to a ,B of 5 .102s,-I. Assumingthat the EC50 of the AI\1PA receptors is 10/-1.M [82] we then obtain withthe above assumed value for et 5 . 104 m,j\!-] S-I. \Vith these rateconstants, the [glu] activates the simulated AMPA-receptors to almost90% (see Figure ;).16 A), if the receptors were assumed to be situated 20nm off the release site. Furthermore we find 44% activation at 200 nmand 1% activation at 500nm distance respectively (perisynaptic). Withthis low activation at perisynaptic sites, activation of receptors in idleneighboring synapses, called crosstalk, can be rejected.

In a second step we tested AMPA receptor models containing twobinding sites. Simulations were performed with the four state model(3.3) using the following values for the rate constants: k+ 1 = 5 ' 105

ml\I-l s -l, k-- 1 = 5,103 = 0.5· k_ 1 , k_ 2 = 2· L,], k+:3 =:

2 ' 105 ,and k_:J 5 . 104 which pertain to non-cooperativebinding. In addition the full model of Jonas as shown in Appendix B,and a reduced version without desensitized states but with the same rateconstants were used. All Inodels were subjected to the simulated [glu]peak and a rectangular 1 m,s/1 mlvJ pulse, and the results are shown inFigure ;1.17A and B, rt~spectively. It is seen that the four state modelyields a high activation of the receptor, however, the I:ise is too fastas compared to that fonnd in experiments (Figure 3.17, solid lines),"Ve therefore conclude that non-cooperative binding does not apply toAIvIPA receptors. On the other hand, the Jonas models produce areasonable activation of the receptor only if subjected to the rectangularpulse (cf, Figure 3.17, dashed and dashtxl dotted lines). Note that theneglection of the desensitized states does not basically alter the kinetics.In fact, very similar curves can be obtained with both models if the rateconstants of the model without desensitized states are properly adjusted(not shown), It is worth mentioning that we found the model of Raman

70

0.9

~ 0.5o15~0.4

0.3

0.2

0.1 .

A

ms

CHAPTER 3. MODEL AND RESULTS

B

0.9

0.8

0.7

cgO.60

g05.2t5~0.4

0.3 !\J •

0.2 I \.

i \

; \\

0.1 ! .- : ....,. ...... ' ... '.-,

0--15 0 5 10 15

ms

Figure 3.16: [glu] translation bymodcls with degree of re­duC!;ion, b1d all feabwing one binding site. A: translat'ions of a con­centmt'ion trace collec!;ed at a site opposing the release site. Fall l'ine:two state rnodel (see Scheme 8.2). DoUed Line: Full model of Vykl'icky.Dashed doUed l'ine:two state model with another of rate constants(see te.rt). B: tmnslat'ions of a nxtangular 1 rns /1 mJ\I ]Jvlse. Same lineannotat'ion as left ill'Ustndion. note thaL the doUed and the dashed-doUedl'ine coinc'ide in th-is WustraLion

3.'1. POSTSYNAPTIC SIGNAL TRANSDUCTION

to behave similarly as the Jonas model.

71

In a third step we in'C'stigated the three state model for NMDAreceptors Scheme 1.[1. Values of the rate constants were adjustedso that the simulated of the receptor follows closely thatobserved in experiments" This yielded the following data set:5· 104 nul1~J ,k~J 50 ~:: 300 s-J and 7;;--2 = 105 . Thekinetics of activation for N.tvlDA receptors located at different distancesfrom the release site is shown in Figure 3.18. It is seen that the timecourse is similar for all distances but the maximal activation decreaseswith increasing distance. This is a consequence of the two reactionsteps required for activation in the model. In the first step the fast [g]u]transient is collected in the complex RL, while the activation occurs inthe much slower second step which lives of the accumulated complexIlL.

3.4.4 Translation of [glu] transients into postsynap­tic responses mediated by metabotropic glu­tamate receptors in Purkinje cells

The somas of cerebellar Purkinje cells are lined up in a very regularway along an axis. Their dendritic trees are fan-shaped and confinedtoa plane perpendicular to the axis. Bundles ofaxons called parallelfibers nm parallel to the axis and thus penetrate the dendritic trees at aright angle. A parallel fiber traverses the dendritic trees of three to fivePurkinje cells, but forms a synapse only with one of them. There are asmany as 200000 parallel fibers which synapse on one Purkinje cell whichis the largest number of synapses per cell found in the eNS. Batcheloret al [9] have reported that a brief tetanic stimulatio.n of a parallelfiber bundle gives rise to a slow but strong depolarizing potential in aPurkinje cell. This is showlI to be induced by metabotropicreceptors acting via a hardly known intracellular signal cascade. Thepeak of the potential arising about 300 - 700 inS after stimulation canbe significantly enhanced if the munber of impulses is increased from 1to 6. It is important to note that the stimulation of a bundle of fibersactivates several neighbouring synapses in the plane of the dendritictree of the Purkinje cell.

In order to sirnulate the experimental data of Batchelor et al. we fed

72 CHAPTER 3. MODEL AND RESULTS

B

0.8 , ~

j ~

0.7 ! \c . \80,6! io !: \'0 !'."is 05 r ". i.t5 ~', \\:! 004 t '. i- ! . i, '."0.3, '. \

I \

i . "0.2 i \j '., '.

0.1 ...................... -._,""'.

11 A

0.9

0.8

0.7

c80.60

~0.5 .0

t5jgOA

0.3

0.2

o --_.~-;,.:.''='~'''''''~------''''o 5 10 15

ms5

ms10 15

Figure 3.17: [glu} translation by nwdels w'ith of re­duction, b'U,t all featnring two binding sites. A: translations of a con­centration trace collected at a site opposing the release site. Fnll line:four state model (see Scheme S.S). Dotted Line: Full model of Jonas[58j. Dashed dotted line: Jonas 'smodclwithont desensitized states. B:translations of a r'eetangnlar 1 Ins /1 nll\] pulse. Same line annotatiionas in illustration A

3.'1. POSTSYNAPTIC SIGNAL TRANSDUCTION 73

Figure 8.18: upper part: r'esponse of a simple thr-ee-state NMDA re­ceptor model towards sim.ll.lated concentration tmccs collected at 20 nrn,200 nm and 500 nrn lateml distance fmrn releasc site, lower P(lT't: gluta­mate sJJ1:kcusedin the NA1DA /7wdel.

74 CHAYTER. 3. MODEL AND RESULTS

three types of glutamate transients to the model of the mGluR describedin Section 3.4.2. The results of the simulations are presented in Figure3.19 for an effector X which binds 40;, subunits (A to C) and 10;,subunit (D to F), respectively. The first type of transient is calculateclfor a single simultaneous release in four neighbouring synapses (spatialcooperation, see Section 3.:).3). The receptor was assumed to be locatedat :340 nm from a release site. Simulations obtained with this type oftransient are represented by the dotted lines in Figure 3.19. The secondtype pertains to temporal integration and is obtained by stimulatinga single release site six times at intervals of 20m,o (represented b.y thedashed lines in Figure 3.19). The receptor was located at the sameoffset. The third type comprises both cases thus combining spatialcooperativity with temporal integration (represented by the full lines inFigure 3.19).It is evident from Figure 3.19 that the kinetics of the three principalspecies R* L, 0;" and X* is quite different. As to be expected for acascade of reactions, the first species of the cascadeR* L is built up firstbut decays considerably slower than [glu]. Since R'L acts as a catalystfor the formation of O~ the latter species accumulates during the lifetime of R* L and decays according to the slow reformation of the Gprotein (see Scheme 3.12). The last species in the cascade X* follows0;\ with a delay caused by the relative low values of the rate constants6 3 and ,Cl:). It should be noted that the values of {X*} can not beinterpreted in terms of EPSC because attributing X to a given channelin the membrane would be speculative. The results of the simulationsshown in Figure 3.19C and F clearly delnonstrate that the correct timecourse of X* with a reasonable amplitude can only be achieved if spatialcooperativity and temporal integration act together on a receptor whoseeffector has 4 binding sites for O~. Spatial cooperativity is effective onthe diffusion level and produces [glu] transients which are high enough tosufficiently activate mGluR. Temporal integration accumulates enoughO~ to sufficiently activate the effector .X, and the 4 sites of Xare responsible for the sigmoidal rise and threshold like activation of X*which is also seen in the experirnental data.

3.4. POSTSYNAPTIC SIGNAL TRANSDUCTION

o

75

120 140

B

0.5

0.5

1time(s)

1time (5)

1.5

1.5

140

E

F

Figure 3.19: signal transd71ct'ion of mGlv,R; left colu'mn: cascade 'Withan effector system that has 4 G protein binding sites; 1"ight column: cas­cade with an effect01' system that has 1 G p1'Otein binding sites; bothcolll.mns: full line indicates answers in the cascade whcre the 1'ecep­1;01' sees ternpomlintcgrat'ion as well as spai;ial coope1'O,t-ion, dashed lineindicates tempomlinteqration after single synapse activity, dotted lineindicates a single st'im;ulation in. a spatially coopemting synapse

76 CHltPTER 3. MODEL AND RESULTS

3.5 SUl111nary

\Vith the present work we intended to clarify the following questions:(i) Is diffusion of neurotransmitter a rate limiting component in thesynaptic signal transduction chain? (ii) Does the shape of the cleft havea significant influence on diffusion dynamics and thus also on EPSCs?(iii) Is there a mGluR model which can reproduce the physiologicallymeasurable metabotropic EPSC?

To answer the above questions, we have worked out a basic threedimensional model for the release and diffusion of neurotransmitter in asynaptic cleft. The computed transmitter diffusion dynamics is clearlyfaster than previous one and two dimensional moclels have predicted.For the transmitter concentration decay constant, for instance, we com­puted a value of approximately 80 microseconds. On the basis of ourbasic model we can answer question (i) as follows: diffusion is not ratelimiting for any of the glutamate binding receptors.

We have also exarnined the sensitivity of the model towards param­eter variation, e.g. the variation of the number of release or theduration of the release process. \Ve observed that our basic model israther insensitive towards the examined parameter variations. Further­more we have investigated the influence of the cleft fonn on diflusiondynamics. In this context we are able to reject the hypothesis of Bar­bour et al. [5], \vhich clainls was that a prolongation of the transmitterconcentration decay constant due to hemispheric cleft forrns also leadsto a prolongation of EPSC decay constants. \Vith this we can answerquestion (ii) as follows: the computed differences in glutamate concen­tration dynamics are not sufficient to explain the physiologically mea­surable differences in EPCS from different synapses. Furthermore oursimulations emphasize the simultaneous stimulations of neighbouringsynapses. For a simultaneous activation of four synapseswe computed a significant enhancement of the concentrationdecay constant.

Furthermore, on t.he basis of simulated concentration transients, wehave computed the postsynaptic answer resulting from AMPA, NIVIDAand metabotropic glutamate receptors. For the A.lvIPA modelswe distinguished between models with one and tviO binding sites. \Vhilethe former is easily reducible, the latter shmvs that the assumption of

3.5. SUl\fMARY 77

non-cooperative binding is a too strong restriction. First we can notcomprise the two binding transitions into one single binding transitionand second the resulting rise of activation is too fast. TheANIPA receptor models of Vyklicky al. and Jonas et al. (as well asthe model of Raman et al.) failed the signal transductiondescribed in the literature if fed witl' the fast transients from our basicdiffusion model. However, by increasing the rate constants for binding,we are able to increase the fraction of activated receptors to physiologicalvalues.At present there is no quantitative model for any kind of metabotropicglutamate receptors available. Hence we have built the fIrst model whichprovides data that can be compared to measured glutamate inducedrnetabotropic EPSe (e.g. Batchelor et al. [8]). We have investigatedunder which circumstances a reduction of states is possible and arrivedat a simple six state model. With this model we were able to describethe parallel fIber induced EPSe in Purkinje cells under the followingconditions: a) neighboring synapses are activated simultaneously, b) atleast 6 subsequent stimuli \vith a frequency of 50 Hz are applied. c)the signal transduction cascade includes a channel which opens if fourG-Proteins are bound. Herewith we provide the model asked for inquestion (iii).

Chapter 4

Solving Large SparseLinear Systems

4.1 Introduction

The discretization of the three-dimensional diffusion equation results inlarge sparse systems of linear equations. Here sparse refers to the factthat most entries in the equation system matrix are zero. As pointedout in Chapter 2. the system matrix A is non-symmetric and has septa­diagonal form. Furthermore our typical n x n system matrix A. forthe diffusion equation discretized over the synaptic cleft is of the ordern = 50'000, which rneans we have a vast sparsity about 0.1%0' Theinterest in sparsity arises because its exploitation can lead to enormouscomputational and rnemory

In general we can identify two approaches to the sparse Ax b prob­lem. On the one hand we can select an appropriate direct Inethod

Figure 4.1). Direct methods involve the factorization of the coef­ficient matrix. As fill-in is very likely to occur during the factorizationprocess, direct methods are only suitable if we adapt them to exploitthe sparsity pattern of A. IIere typical schemes include reordering theunknowns and equations, the SInart use of data structures, and specialpivoting strategies that rninimize fill-in. Although direct methods are

79

80 CHAPTER 4. SOLVING LARGE SPARSE LINEAR SYSTEMS

not best suited for the solution of the three dimensional diffusion prob­lem, we will describe in Section 4.5 how they could be applied to curproblem.On the other hewd \ve can apply iterative methods. In the COl:

of an iterative method a sequence of approximate solutions { 'i 1S

generated which converges towards the desired solution ;1;. Especi;,Jlyattractive are iterative methods that involve the system matrix only inthe form of rnatrix-vector products with A or AH , a problem specificcomputational routine must then be externally provided. In Section 4.2and LU~ we will comment upon the choice of the iterative method to solvethe three dimensional diffusion equation. In Section 4.4 we will considerdifferent preconditioning schemes for the chosen GNIRES method anddescribe how they performed for our synaptic cleft diffusion problem.

4.2 Iterative Inethods

In Figure 4.1 we indicate iterative methods as an alternative to thedirect solution of a linear system. An iterative technique for solvingthe n x n linear system A:r = b starts with an initial approximation;1;(0) to the solution x and generates a sequence of vectors {;1;(;)} thatconverges to :r. One can distinguish between two widely used classes ofiterative methods. The first class contains schemes based on splittingsof the coefficient matrix (in Figure 4.1 this class is indicated under thegeneric term "classical methods"). The seconcl class, is represented bythe large number of methods based on the approximation of the exactsolution x = A-1b by sorne :r(m) E Km, where Km denotes a Krylovsubspace of dimensionrn.

Matrix splitting based methods involve a process that con­verts the systmll Ax ~c: b into an equivalent system using e.g. thesplitting A = AI N. The original system can then be rewritten to;1; = N;1; + {J, which leads to the following iterative scheme:

(4.1 )

for some n x n matrix I' and a vector c. The sequence defined byEquation (4.1) converges to the unique solution ;1; =: Tx + c preciselyif p(T) < 1, where p is the radius defined as p(T) ::c: rnaxi lAd,where Ai represents an eigenvallle of T.

4.2. ITER.ATIVE METHODS

solution methodsfor linear systems

81

direct methods iterative methods

//~classical methods Krylov subspace methods

for non-symmetric C

Arnoldiorthogonalizationlong recurrences

Lanczosbiorthogonalization

short recurrcnces

11 r 11 =mine.g. GMRES

lie 1. Ke.g. FOM

Figure 4.1: decision tree for the choice of the computational Toutineto solve the diserclizcd three-dimensional diffusion equation. NV,n!,/JC1'S

in parenthesis indicate the Section in which we comment upon the eor­respoTl.dinq decision. Bold lines indicate the decision taken.

82 CHAF'TER 4. SOLVING LARGE SPARSE LINEAR SYSTEMS

One subclass of matrix splitting based methods where neither T norC changes during the iteration process is called stationary methodsIt's oldest representatives are the Jacobi and Gauss-Seidel iterativ'methods. Both methods use the splitting A = D L --- where D ;;a diagonal matrix with the diagonal elements of L is the stridJ[\rlower-triangular part of A, and -U is the strictly upper-triangular pal"tof A. The .lacobi method assumes TJ D-1 and C.J = D-1b. Weobserve that the .lacobi method is parallel nature, as the componentsof xUl-H ) can be computed independently of each other. Hmvever, dueto its rather poor convergence properties it is often discarded in favorof the more efficient Gauss-Seidel method_ The latter method assumesTo = (D - L)-lU and Co (D - Lt1b. If A is strictly diagonaldominant, then for any b and any choice of X((l) both methods yieldconvergent sequences {;l~(k)}.

For problems, wlwre the spectral radius of To is close to unity, theGauss-Seidel method may converge slowly. One possible way of gainingspeed is to introduce the relaxation parameter u) E 17_, which leads to

(D - = [(1 - li.,)D + +wb

This method is known as und(~r-relaxationnlethod when w is chosenin (0,1). Conversely, the procedure is called successive over-relaxa­t.ion met.hod (SOR) when u) > 1. son uses a weighted average oft.he previous iterate and the cornputed Gauss-Seidel iterate (with weightw). Given the spectral radius PT of the Gauss-Seidel iteration matrixTo we could even compnte an optimal choice for the weight w. Thisis though seldom possible, since calculating the spectral radius of thismatrix requires an impractical amount of computational effort. Stilltllere are ways to find a good SOH parameter for certain classes ofmatrices, cf. [45]. In general we can say that matrix splitting basedmethods are only valuable if we have a good estimate of the spectralproperties of the system matrix.

Since the mid 1970's Krylov subspace methods became increasinglypopular. In general they proved to converge faster than matrix split-

based stationary Inethods and do not require any knowledge of thespectral properties of A. Led by this argulnent, we only considered theclass of I(rylov S11 bspace methods as possible solution schemes for ourthree dimensional diffusion problem.

4.3. KRYLOV SUBSPACE METHODS

4.3 Krylov Subspace Methods

4.3.1 Background

83

In the previous section we mentioned Krylov projection rnethods as asophisticated and potentially efficient class of iterative solvers. Thisclass has got its name because it applies projection methods on Krylovsubspaces, which raises two questions i) what is a Krylov subspace? andii) what are projection methods?

A Krylov subspace is a subspace spanned by a Krylov sequence{Ai1'(O)}, thus we write for the Krylov subspa.ce

K". - K' .(",(0).4') = (.1,(0)4' 1·(0) ... Ak-1",(0)}~k - 'k' ," 1 ," " ' ,

where A E n"xn and 1'(0) E:Rn .

Let K and L be two subspaces, a projection method for solvingthe linear system

b A:e =0, (4.2)

is then a method which seeks an approximate solution from anaffine subspace x((J) + I( by imposing the Petrov-Galerkin conditionb - ke(k) J .. L, cf. [100J. A. single iteration step of a general projectionmethod can thus be formulated as:

find E + l( such that b Ax(k).1 L. (4.3)

For Krylov projection methods we replace the generic subspace K in(4.3) by the Krylov subspace Kd1'(O), A), where the residual is definedas 1·(0) = b·· . The different versions of Krylov subspace methodsarise from different choices of the subspace L (see also Section 4.3.3).Since x(k+l) E :c(O) + , we can represent the iterates compactly as:

) .I,.

+ L lliPi :c(k) + CY"Pk ,

i=O

(4.'1)

where Pi E K:i+l ,i 5 k, are called direction H?CtOrs. These vectors shallbe constructed such that {Po, PI , ... , Pk} forms an orthonormal basisfor

84 CHilPTER ,1. SOLVING LA.H.GE SPARSE LINEAR SYSTE11i[S

There are many possible taxonomies for Krylov projection meth­ods. Of course the actual choice of the method depends first of all onproperties of the system matrix. For symmetric positive definite systern matrices the so called conjugate gradient method (CG), due toHestenes and Stiefel [48], represents the dominant iterative solver. Con­versely, there is no universal solver for non-symrnetric system matrices.In Figure 4.1 we presumed that the system matrix A is non-symmetricand therefore the ~lassical CG is not applicable.

Presuming a non-symmetric system matrix we looked at the way theorthonormal basis for the Krylov subspace is created to conduct the nextdecision in Figure 4.1. vVe can distinguish between methods based onthe Arnoldi orthogonalization and methods based on the Lanczos bi­orthogonalization. Both classes include methods which either constructx(k+l) such that the residual is orthogonal to the current subspace, oridentify X(k+l) for which the Euclidean nonn lib Ax,lk+l) 112 is minimalover JCk+] (1)(0), A). We decided to use the Arnoldi-based Gl\IRES al­gorithm for reasons of robustness and because it does not require thetranspose of the system matrix /1. An introduction to the Arnoldi basedmethods shall be given in the Subsection 4.3.2 below. The Arnoldi andGMRES related explanations below are based on the lecture on iterativemethods of W. Gander at the ETHZ.

4.3.2 Arnoldi's Orthogonalization

In order to approximate solutions in the Krylov subspace we need a suit­able basis for this subspace. For numerical reasons one usually prefersan orthononnal basis over the standarcl basis 7'(0), ' ... ,Aj-l'r(O)

for JC k . In his original paper Arnoldi [1] proposed to use th~~ Gram­Schmidt orthogonalization process to construct an orthonormal basisof the Krylov subspace, alternatively Householcler reflections could beapplied just as well. Arnoldi's method starts with PI == 7'(0) 1117'(0)112.

Assuming that we have already an orthononnal basis PJ , ... 'Pj' it 01'­

thonormalizes the .i-th expansion of the subspace by the followingcomputations:

1. compute the new vector v, \vhich we want to use for theconstruction of the new column vector Pj+l of P.

4.3. KRYLOV SUBSPACE METHODS 85

2. compute the length h i ,.i+l pT'AJ1'IO) of the projection of tl ontothe existing orthonormal basis vectors PI' ... , Pj'

3. subtract the projections Pj+l =: A.51' (O) ~ h;',.i+l Pi

In the modified Gram-Schmidt method we project A"1'(O) sequentiallyonto the existing basis vectors and subtract the projections instantly.This is mathematically equivalent but more accurate than Gram-Schmidt.In Figure 4.2 we illustrate the complete Arnoldi algorithm. The crucialidea of Arnoldi's algorithm is the replacement of hi,.i+l pTAh·(O) byh ij pTApj, this replacement can be assumed because Apj contains acomponent in the direction of l' (0).

p(1) /111'( 0 ) 112for (j 1, .... , m)

u Apjfor U 1, .... ,j)

pT1lU 'U hijPi

end

~o

P.i+1 V/hj-f-l,}else stop

end

Figure 4.2: arnoldi(m,1'(O) ,A)

The projection of the new vector u. which we want to orthononnal­ize. can be rewritten as:

}

.jPj+l =:Ap} ".- L hi}1-); ,

i=1

where is the length of the projection of Apj onto Pj+l' Equationcan then be rcf'ormulated to

:i-I- I

Apj L hijPi,i== 1

(4.6)

86 CHflPTER 4. SOLVING LARGE SPARSE LINEAR SI"STEMS

which can again be rewritten in compact form as:

(4.7)

where PI! = [Pr P2 ... Ph] and Ih'H,1! is an upper Hessenberg rnatrixwith k: columns and k: + 1 rows. If A is symmetric then also Hk,k issymmetric which means it is tridiagonal. Thus if A is symmetric theArnoldi algorithm is just the Lanczos algorithm.

4.3.3 Arnoldi's algorithm for the solution of linearsystems

In the above subsection we explained how to achiev(~ an orthonormalbasis for a Krylov subspace. \Ve shall now proceed and see how wecan use such an orthonormal Krylov subspace for the solution of linearsystems A;r ::=: b, where A is large, sparse, non-symmetric and non­singular and an initial guess :r(O) is given. \Ve use the approach:

(4.8)

where y E RI" and PI" E R"xk and :r(k) is (in some the bestapproximate in the Krylov subspace Kk (1,(0), it) of tlH~ solution :r. Asindicated in Figure 4.1, there are two ways to determine : first wecould request r(k) .L Kdr(Ol,A) second ,ve could to minimize theEuclidean norm Ilr(k) 11 over Kdr(Ol, A), see also [11, 40,43].

The first stipulation is also called the Petrov-Galerkin condition (cf.[4~m and is equivalent to

(4.9)

Inserting Equation (4.8) into Equation (4.9) results in:

(1.10)

\Ve note that Hid- ptAPh is cmnputed by the Arnoldi algorithmand therefore readily available. Together with Equation 7) we find adescription for the solution as follows:

11)

L). KRYLOV SUBSPACE METHODS 87

This full orthogonalization method (FOM) is very expensive interms of storage and is therefore often restarted after m iterations, re­placing the initial guess for the next iteration cycle with

The second stipulation is also called the minimal residual approa.ch.Saad and Schultz [97, 99] defined an extension to F'OM, the generalizedminimal residual (GMRES), \vhich gained much popularity because ofits robustness and high efficiency. The GMRES method is a projectionmethod based on taking f{ = K k and L ::::: .llKk . As mentioned abovethis method builds its orthonormal basis with the Arnoldi algorithmand then looks for an 7'(k) E K: k+1 (r(CI) , A) with minimal length. Thecorresponding Euclidean norm can be written as:

117'(k) 11 = lib AxU)!1 = lib A (:e(O) + PkYk) 11 = Ilr(C))- APkYkll·

(4.12)At this stage we will use again the result of the Arnoldi algoritlun (4.7)and rewrite Equation (4.12) as:

Ilr·(k)11 = 117'(0) Pk+1Hk+1,J,'Ykll = Iln'+J (1Ir(O)lle1 - HHl,k) lhll·(4.13)

Asn,+l is orthonormal this becomes:

lib (4.14)

\Ve can rC\vrite the above norm such that it can be minimized by solvingthe minimum least squares problem for Hk+l. k and the right hand sideIlr(O) 112ej. In GJ'ARES this least squares problem is solved efficientlyby the QR method Givens rotations in order to annihilate thesubdiagonal elements in the upper Hessenberg matrix Ih.t.l.k' As thenumber of iterates k: increases, and work load increase linearly.j\ possible remedy is to restart the GJ'dRES process after m iterates,with the tradt)ofT of a slower convergence. The dimension of the subspaceis then fixed. Below we give an outline of a restarted GMRES algorithmused in the present work, a closer description can be found in the book"BUilding Blocks for Iterative ?vlethods" [6].

88 CHAPTER 4. SOL'lING LARGE SPARSE LINEAR S'YSTEMS

Start/Restart: Compnte TO b - Axo, and V1 =

Arnoldi Process: Generate H m and v'nMinimization: :r(l)) + VmYm

with Yr" miny 11111'(0) 112e] .,. ITm yli2

Convergence test: If satisfied then stopelse set :r(O) := :r(m) and Restart.

Figure 4.3: sketch of GMRES

4.4 Preconditioning

A preconditioner denotes a transformation of the lineal' systemAx = b into one that is equivalent in the sense that it has the samesolution but has a better condition number. The basic idea is as follows.Let 111 ~ A be a given nonsingular n x n matrix, which approximates (insome sense) the coefficient matrix A of the original lineal' system (4.2).F\lrthermore we assume that }II can be decomposed to }\1 =The l{rylov nwthod is then used to solve the preconditioned linear sys­tern e:E' = d, where

c 'cC,-l '1 'I' -1 'I1\<11 j' .Lv 2 ~ ~ :r ' =

The fact that C approximates (in some the identity matrix I.offers the desired benefit. that the condition number K'.(C) is close to1. The special cases 1111 = I or /112 = I are referred to as right orleft preconditioning. The residual vector and iterate vector find theirequivalent primed quantities as:

:r,,, = 15)

The right preconditioning is usually preferred for minimal residual basedKrylov subspace methods, as with AI] = I the preconditioned residualvectors coincide with their counterparts for the original systenl. Thereare two (in general conflicting) requirements, first Mo-] should appro­ximate as close as possible such thal the algorithm converges fast.Second the preconclitioner 111 needs to be such that. the linear systemcan be solved cheaply.

4.'1. PRECONDITIONING 89

The following three preconditioners were implemented and tested onthe discretized 3-dimensional diffusion equation: a fst based Poissonsolver, a diagonal preconditioner and an incomplete LV factorization. Ashort review of the implementation of the methods is first given follmvedby a performance comparison.

4.4.1 Fast Poisson Solver

Background

As the name reveals, fast Poisson solvers are used to solve thePoissonEquation (4.16) in a direct manner. The diffusion equation extendsthe Poisson equation by the term aU/at, we anticipated that a directmethod which solves the Poisson equation fast can be used as a precon­ditioner for our diffusion problem. In the subsection below we give anintroduction to the fast sine transform based direct solver for Poissonequations and subsequently show how to construct the solver.The three dimensional Poisson equation

iJ2U U+ + = F'(:r, y,z) (4.16)

shall b(-~ discretized on a rnesh with Inesh-points at :rj

:1J.j jSy,zk = /;:(5;; therefore yielding for every mesh-pointdifference approximation of :

- is:r,a central

2(0 +P+ +Uj,J-l,k + v;,J+u,) V;,J,k-l + Uj,J,k+d

1 :Si :S v)' 1:S.7:S n y , 1 :S /;: s:; V:;, (4.

where Ct = (h/r5:r?, (3 (h/r5y)2, (h/(lz)2, and h being the leastconunon denominator of the (l:r, . 'Ve therefore assurned con-stant mesh-widths along an axis but with possibly different mesh-widthsalong different axis. The corresponding coefFicient matrix An f canbe constructed the following sum of I<:ronecker products:

+ + l 1n" (2)lny J;" (4.18)

90 CHAPTER 4. SOLVING LARGE SPARSE LINEAR SYSTE1HS

where I~,", T"" and 7~," are low rank perturbations of the appropriatelydimensioned tridiagonal matrix:

-2 1 0

1 -2 1

E R"x"

o1 -2 1

1 -2

Poisson Equation and Fast Sine Transform

Let us briefly look at the eonllt'etion betwet'll the solution of theI-dimensional Poisson Equation (4.16) and the fast sine transform (fst)method, and then derive the aIgorithrn for the three dimensional case.

We will use the spectral dissection of the Poisson coefficient matrixT" = ;~QAQT to compute the solution of T"u j. Here Q is thematrix built from the eigenvectors qj' A represents a matrix with theknown eigenvalues /\i of T" in its diagonal. For Dirichlet boundaryconditions we receive /\j 2(1 ..·· cos ). furthermore Q is orthogonal

and even symmetric (i.e. q = qT q-1). After inversion ofT" we getu = T,;"1 j, which is now

')

t/, =, '"' Q1\-1(/1.n

(1.19)

Lt ~-- (/j,Qv is of

as.fsolve the linear system T"ll'it ~ qu, where each of the QV"j',onlQ t/,

We may thereforev, t·.. (2/n)D·.. l,n,the form:

n ..-I

2:= sin vi'I)

k=1

vVe thus require for the solution of the linear system Tnt/, = .f two fastsine transforms (f s t) and one scaling.

In the three dimensional case we 1nultiply A. in (4,18) from the leftwith @ 1"'1 Q~_). With substitutions ;r1 for (I",and Y1 for (1 . I Q;l,Jy we the systern:

+ +

;1.4. PRECONDITIONING 91

The matrix in (4.21) has block diagonal form. Again we apply a leftmultiplication, this time with (In, Q;;")J (81 In,) and with the substitu-

tions XII for (In" @ Q?,~'1 In, )X I and Yn for (In" (81 Q?")J In, )YI:

(o/I~h -+- rI]h @Iny @An,):rH '!In'"

(4.22)Tn " in Equation (4.22) is tridiagonal. Thus A. is a matrix with a diagonaland t\VO outer diagonals at an offset of ±nyTl z . 'I'his system can be solvedby explicit forward and backward substitution according to :

Ly = b:

Yf = b;jdi ,

1}k = (bk o,k.k-n, n!1

Ux y:

i = 1, .... n:rny ;

) / ch" k 71;).ny , "', 'n"nyn o;

:1:f =Yf, n.1: n yn Z1 ... , n"nyno n,rny;

:rk = (1}k .... ak.k+n., :Ck-i.n"n'I)/ch, k = (n:rnyn o TI·1yn y ..·..·1), ... , 1;

Before we are ready to apply the explicit solver wp will have to computeY n from Y I and 'liT from Y by f st. After the application of the explicitsolvpr we can hence back-substitute by using again FST to get from :r; nto :r1 and then to ;r: also Figure 4.4).

1: Yl fst(y)2: Yn = fst(YI)3: A:r;n = !in4: Xl =fst5: ~1: =: f [ )

Figure 4.4: skctch of 01/7' Poisson solvel' fmmc'Wol'k

On line 1 we transforrn Cl vector with a fast sinp transform andtherefore call n",n y tinles the fst routine with a vector of length n zcomposed of 1};. On line 2 we transform a vector of lengthnyn •. IVe therefore call T!r times fst for a vector whose elements havean inherent indpx increment ofn" in the original vector. On line 3 weuse the above mentioned direct solver and on lines 4 and 5 wc use in

to the lines 1 and 2 the fast sine transform.

92 CHAPTER 4. SOLVING LARGE SPARSE LINEAR. SYSTEAIS

4.4.2 Incomplete LU Factorization

Preconditioning with an incomplete LU rnethod [98] computes an <:1,ppro­ximate factorization of the system rnatrix in a preliminary step andthen uses the usual forward and back-substitution to solve the system.By keeping the factorization incomplete this type of method preservesthe sparsity structure of the system matrix in the upper and lowerdiagonal matrices which keeps memory consumption in tlHe' bounds ofthe possibility of current workstations. For the fOI'\vard and backwardsubstitution steps we were then able to use sparse BLAS routines [33,15].A general approach to incomplete factorization takes a set of rnatrixentries S and keeps all positions outside this set equal to zero. \Ve callthe factorization incomplete as for all matrix elements A(i, j) ~ S fill-inis suppressed. \Ve chose S to be the set of all positions j) for whichA(i,j) i:: o.If the matrix is derived from central differences on a Cartesian grid, wecan describe a particularly simple pivoting and factorization. Letting i, jand k be coordinates in a regular i3D grid, pivot on grid point j, k)is only determined by pivots on points (i 1, j. k), .7 1. k) and(i,j, k .0. 1). If there are 1 points on each of n grid lines, which aresituated in Tn planes, we get the following algorithm (note that we havea priori knowledge about ch,k i:: 0):

for k =: 1: 1· n . In(h,k = ak,k

endfor i = 1 : 1· n . In

if mod(i, 1) i:: 0o.i,I+1

endifi + 1 :c; 1. n . rn

di+U +1

tmdifi + 1. n :c; 1. n . Tn

endend

Figure 4.5: o.lqorithrn Incomplete L U tn e "on 7i!fuln

,1.4. PRECONDITIONING 93

An important aspect of incomplete factorization in general is thecost of creating the factorization. The number of operations involved isat least as much as for solving a systenl with such a coefficient matrix,so the cost may equal that of one or more iterations of the iterativemethod [6]. As the same preconditioner is used for many iterationslong as the time-step remains the sanle size) the factorization costs canbe amortized.

4.4.3 Performance Comparison of Different Precon­ditioners

The previously presented fast-sine-transform (fst) based solver, Incom­plete-L1J (ILU) as well as a so-called diagonal preconditioner where onlythe inverse main diagonal of the original matrix A is used, were com­pared against each other. The test-problem to solve was the discretizecldiffusion equation in a cube of side length 1024 nm with pure Cauchyboundary conditions towards all sides, that is we used a synunetric test­matrix. The Cauchy boundary conditions for our problem are describedin Chapter 2. The temporal discretization was 8 ns while the spatialdiscretization was systematically varied between experiments. In Table4.1 we compare the different preconditioning methods for the mentionedproblem, under varying problem sizes. \Ve indicate the overall consumedtilne as well as the number of GMR.ES iterations necessary to achievean accuracy of 10-6

. By doubling the spatial resolution the problemsize was increased a factor of 8.The computational effort to construct ILU factorizations is in the sameorder as the time to perform a single time step in the test problems(~j G]VIR.ES iterations). As the same preconditioner is used for Inanytime steps, the initial effort for the construction of the preconditionercan certainly be justified. The number of iterations required to solvethe equation system relnained constant with increasing problem sizefor the ILU-, and the diagonal-preconclitioner. In contrast the numberof iterations increases for the f st based preconditioner with increas­ing problem size. Our f st based preconditioner was implement(~d (asshown above) under the assumption of Dirichlet boundary conditions,it was thus tested whether it perforrns better on a model problem whichincludes Dirichlet instead of Cauchy boundary conditions. Neither thenumber of iterations nor the execution time improved when we adaptedthe problem to match optimally the fst implementation. For a brief

94 CHAPTER 4. SOU/ING LAT1GE SPARSE LINEAR SYSTEMS

6S

lr. 1

preconditioner type111 UUlClU ILU diagonal fst based

size time [s] iter. time [s] iter. time [s] ite512 0.065 3 0.170 9 1.550 1

4096 0.510 ;} 1.500 9 16.80 2'32768 4.280 3 LU'58 9 2:31.1 S(

Table 4.1: benchmaTking of differ'ent ]J1'econditioncr with OMRES ap­plied on the model problem

discussion of these results see Chapter 5.3.

4.5 Direct l\1ethods

4.5.1 Band Solver and Reordering

Crank Nicholson cliscretization of the three dimensional diffusion equa­tion leads to a matrix with a banded structure such that Cli,:! = 0whenever i > .i + iJi or .i > i + ])2· The constants ill and ])2 are referredto as lower and upper half banchvidths. Substantial economies can berealized when solving banded systems, because the triangular factors inLU are also banded (with diagonal pivoting), that is, when A is bandedwith upper bandwidth ])2 and lower bandwidth ])1, then U has upperbandwidth P2 and L has lower bandwidth ])1.

Duff et al. [:{4] described a straightfonvard Gaussian elilninationwhich exploits the sparsity of banded matrices. In Figure ,1.6 an exampleof a corresponding band nwmory structure is shown. Here A is repre­sented in a (p + q + 1) x n array , where band entry Clij is storedin Abond (i .i + q + 1,.i).

Reordering algoritll1l1s are often used to reduce the bandwidth. 'Whilethis may be useful with irregular grids, our regular rectangular grids willnot allow a reduction of the bandwidth. The most famous reorderingscheme proposed by Cuthill and j\lcKee orders within blocks, whereblock SI contains only element 1, while block contains all neighborsof the nodes in not ~ret selected. Cuthill-McKee algorithm or-

4.5. DIRECT 1I1ETHODS 95

0.1 Cl

b1 0.2 C2

b2 (1.3 C3

b3 CLj Cl --+bel 0.5 C5

b.5 aC CG

bn (1.7

Figure 4.6: a possible data structure for' a band rncrnory struct1l7'C

del's block-wise by taking first those nodes that are neighbors of thefirst node in 8 i - 1 , then those that are neighbors of the second node in8 i - 1 so OIL While we said that we shall not be able to profit from theordinary Cuthill-NfcEee in tenns of bandwidth-reduction, George [42]found that a reversed Cuthill-l\JcKee improves the storage requirements,In our case as can be seen in Figure 4.7 the storage requirements arehalved. The corresponding execution times for the forward and back­ward substitution are a factor 1.5 faster as well.

The high sparsity inside the band rnatrix that occurs after discreti­zation of three dimensional can not be exploited by directmethods as here fill-in occurs in the factorization phase. Assuming abanclwidth of p = n: x n y and system order n n,l' x n y x n z we willneed to allocate memory for n x p elements. The consequent enormousmemor.y consumption in direct methods makes these methods inappli­cable to even medium sized :3-dimensional geometries discretized withfinite differences. An illustration of this fill-in is given in Figure 4.7.

96 CR4PTER 4. SOLVING LARGE SPARSE LINEAR SI'STEAlS

Figul'('l 4.7: effect of Ollthill MeKee on ,"'.',U:,'II,

and lower triangular matTi;;,: L. Top left: septa diagoTl,al 1I1.atr'i:cA, bottam left: lovJe'l' t'l'iangv.lar 'mah-Lr L rcslI.lting from, facto'l'ization ofA, top right: fLC1V[ (reorde'l'cdmatri:r bottom right: lower triangularlIwtri:r LClv! T'eslIlting from faetoT'izahon of iLeM

Chapter 5

Discussion andConclusions

The central point of this work is a model of the basic biophysical pro­cesses in synaptic transmission, including transmitter release, transmit­ter diffusion, transrnitter binding, and signal transduction due to post­synaptic receptor action. To be able to compare our model of synaptictransmitter diffusion with the vast amount of experimentally measuredexcitatory postsynaptic currents (EPSCs) our model comprises the sig­nal transduction of neurotransmitter time courses. Consequently weconsidered the experimental data of several different research groups[8, 26, 58, 92] as touchstone for our model.Each syna.pse model embodies a set of ideas about how the synapticcleft receives its intrinsic dynamics. The rnodel is defined only by whatwe know about the physiological structure but not by the model's actualbehavior. Today we can hardly make any assumption about whether theparameter variance among a neuron-type is because different neuronshave to achieve difFerent informatiou processing properties or \vhetberthe variance is mostly randorn and therefore the system is bound tobe tolerable towards parameter variances. 'With the help of computersimulation of a model of synaptic transrnission we intended to find anestimation of the parameter variance tolerance in synapses. In the firstsection of this chapter we shall discuss the influence of variances in bi­ological parameters on our model. In Section 5.2 we rationalize the

97

98 CHAPTER 5. DISCUSSION AND CONCLUSIONS

choice of the modeling techniquEc~ and in the last section of the chapterwe shall draw conclusions and give an outlook on possible further work.

5.1 Confidence In JViodel Ideas

5.1.1 Is Diffusion the Rate Limiting Factor'?

One question we wanted to clarify by modeling diffusion in the synapticcleft is whether diffusion is the rate-limiting factor for the activity of allglutamate receptor types under observation. Excluding the complicatedmechanisms of receptor desensitization \ve define diffusion to be ratelimiting for a certain type of receptor if Tdeactivalion < Tc/catonce, whereTe/eaTanee is the time constant for the clearance of released neurotrans­mitter off the cleft and Tdeacl.iva.tion is the deactivation time constantof the corresponding receptor. In Chapter 3 we showed that glutamateconcentration after a single vesicle release can fall below 10-2 rni\j al­ready after less than 0.2\ms in the center of the synapse. 'While itis generally accepted that diffusion is not rate limiting for Ni'vIDA. andmGluR. receptors, there is no consensus about the AMPA receptors. Thefastest AMPA receptors have deactivation time constants of 1 1.5 ms[92], which is according to our simulations at least one order of magni­tude slower than the decay of glutamate concentration in the synapticcleft. Furthermore as described in Chapter 1.4.2) we presume that glu­tamate clearance due to diffusion is 2 to 3 orders of magnitude fasterthan the reuptake time constants given by Wadiche [109]. Therefore,we suggest that translnitter reuptake has no influence on the kinetics ofEPSC. On the basis of our diffusion simulation results we can assumethat after a single vesicle release event the diffusiou driven cleft clear­ance is not a rate limiting process for Al\IPA receptor EPSC.On the other hand, as shown in Chapter 3.3.2, the release of multiplevesicles can linearly prolong transmitter presence. Assuming a fictive 5vesicle release event (current reports assurne 1-3 vesicles for evoked glu­tamate release in the central nervous system , 64, 93]) the tirne untilthe central concentration falls below /JuU could be in the order of1.25ms.In their recent paper Tvlennerick and Zormnski discuss the rate lim­iting factors for AMPA receptor EPSC. In experiments in micro-islandsynapses of rat hippocampal cells they see an indication that diffusion

5.1. CONFIDENCE IN MODEL IDEAB 99

rapidly clears transmitter under conditions of low quantal content be­fore uptake is activated. On the other hand they could shmv that decaysof evoked AMPA receptor EPSCs can reHect prolonged actions of glu­tamate even \vhen glutamate uptake and fast receptor desensitizationare intact. This lead them to the conclusion that diffusion, uptake, andreceptor desensitization can be rate limiting under enhanced releaseconditions for AMPA but not for NMDA receptors. While they dis­cuss several explanations, our simulations rule out the possibility thatenhanced release due to an increased amount of released vesicles is re­sponsible for prolonged EPSe decay rates (see below for a discussion ofthe receptor model characteristics).

Clements et al. [18] found that glutamate peaked at 1.1 mM and de­cayed with a time constant of 1.2ms at cultured hippocampal synapses.In his most recent paper Clements [17] refines these statements by thedefinition of a biphasic decay with the fast rate being less than lOO/IS.This again brings up the complicated circumstances where the gluta­mate concentration decay is only rate limiting under enhanced releasesituations. The term enhanced release is a general description for twophenomena: (i) more vesicles are released per stimulation and (ii) neigh­boring synapses are activated simultaneously and repetively. \Vhile,according to our simulations, none of the above two enhancements cansign responsible for long AMPA receptor EPSC decay constants, the lat­ter circumstances are especially interesting for perisynaptically locatedmetabotropic glutamate receptors. They can hardly bind glutamateafter a regular release event due to low peak concentration at theirdistant location. Our cornputer simulations predict that simultaneousrelease in neighboring s.vnapses increases the glutamate concentration10 fold after 0.51'118. In our rnGluR model, we used a slow on-rate

6000 mAf- J) and received ;1 10% fraction of activated recep-

tors. Here a prolonged presence of glutamate does enhance the fractionof activated receptors, but has no influence on the decay constants.

5.1.2 Simplified Release Assumptions

In Chapter ::\ we showed that smearing the release of the neurotransrnit­ter over a time window of 20/18 does not significantly change the EPSCtransduced by the ATv[PA receptors. Below we itemize t\',:o experimen­tal studies that build the qualitative framework for our approximative

100 CHAPTER 5. DISCUSSION AND CONCLUSIONS

assullrption of instantaneous release. In addition we briefly discuss twocontradicting numerical studies which together specify a quantitativeinterval for the time-period of the release process.In recent publications Pieribone [83] and Rosahl [95] show evidence fora docking region in the presynaptic bouton which contains about 10-20vesicles and which is depleted with stimulations greater than 2 Hz. It isfurther speculated that the fusion process is prepared before the arrivalof a nerve signal such that the release can occur at a high sp(~ed. Ac­cording to their findings exocytosis takes place in three stages: priming,docking and release, where rapid release relies on fast rise and fall of[C0.2+], on large number of clocked vesicles and on statistics to ensurethat a few have enough therrnal energy to complete exocytosis. In acomplementary study Bruns and Jalm [12] gave recently an upper limitof 0.26 ms for the release process of serotonin, which includes the open­ing of presynaptic calcium channels, the diffusion of the calcium, andan abrupt opening of a pre-assembled fusion pore.'While the above two studies based their reasoning on experimental work,Khanin et al. [61] calculated that the release process can not be de­scribed as a pure diffusion phenomena. In other words, if the n~lease

process takes place by diffusion out of the vesicle into the cleft (throughan expanding pore with expansion rate 0.4 , then the diffusionprocess would be too slow to allow for experimentally measured ampli­tudes of EPSC. They therefore concluded that an active mechanism isresponsible for the fast vesicle discharge. In contrast Stiles et al. [105]used for a similar model a pore expansion rate of 25nm/m.8 which al­lowed them to simulate vesicle emptying by diffusion fast enough forrealistic miniature EPSCs (mEPSCs). In further agreement with thefinding of Stiles et al., we have shown in Chapter :3.:3 that spreading therelease over a temporal interval of 20j1.8 discretely approximatedthe pore expansion process) does not alter the decay characteristics ofthe concentration traces but only reduces the maximal amplitude of thetraces and therefore does not show changes in mEPSCs decay rates.

5.1.3 Hernispherical vs Flat Synaptic Cleft

In Chapter 3.3.:3 we gaV(; a cornparison of glutamate time courses in ahemispherical cleft ancl in the flat cleft. We found that when a hemi­spherical cleft has the same open boundary area as a fiat cleft, theconcentration traces differ less than 1%. Conv(~rsely for a smaller

5.1. CONFIDENCE IN MODEL IDEAS 101

hemispherical cleft shape, where 7'1 :::: 540 nrn, we found slower [glu] dy­namics (37% higher [glu] at 20 nm off release site after 500p.,s). \Veargued that this slower [glu] dynamics is a consequence of the fact thatin a hemispherical cleft model the volume beyond the discretized partgrows only with O(:r), where :r is the distance to the release site. Thiscan be understood if the volume beyond the hemisphere is regardedas a cylinder with inner radius 1'1 and outer radius 7'2. Consequentlythe radius of the hemispherical cleft (and with it also the the outletarea) linearly influences the [glu] decay rate. However in our simula­tions the subsequent I\MPA receptor EPSC showed no alterations forboth discussed hemisperical cleft sizes. For an explanation of the AMPAreceptor EPSC resistance towards the simulated alterations of the [glu]_dynamics, we point to the fact that the AMPA receptor models sense aninitial quick rise of the [glu]. but are then rather insensitive to the [glu]decay rates, since the decay tinw (T) of the pulse is considerably smallerthan the average life time of receptor glutamate complexes, T 80/LS(see Section 3.2.3) <-< Ims (see Section :3.4.3).Barbour et al. [5] investigated whether the slower EPSC decay rates ofAMPA receptors in C(~rebellar Purkinje cells can be explained by longlasting glutamate transient which is caused by slower removal ratf~s dueto geometric limitations. Our simulations confirm the evident slowerglutamate removal rates if a synaptic cleft has smaller open boundaries.But then fail to simulat(' the change in EPSC. Barbour used in his modela completely wrapped spine and therefore received a higher :r /1/ rate,where :r is the distance to release site and F is the correspondig volumeenclosing :r. We assume his higher [glu] decay can be attributed to thislow :r/l/ rate. Still the study of Barbour et al. left it open whether anyAlvIPA rnodel would be able to translate the retarded diffusion dynam­ics into a slower EPSC. At this point our simulation come to help whichshow that this hypothesis must be rejected.

5.1.4 R.eceptor models

In Chapter 3 we presented concentration traces that are faster thanthose previously reported other studies 26]. In the first part ofthis subsection we shall try to explain the inability of existing recep­tor kinetic models to produce enough open probabilities with ourconcentration traces. For this purpose we first discuss to what data thecorresponding receptor models were fit. Trussel et al. [92] and Jonas et

102 CHAPTER 5. DISCUSSION AND CONCLUSIONS

al. fitt.ed their kinetic receptor models to data recorded in so-calledtheta tubing experiments. This type of experiments are recorded frommembrane patches in the outside-out configuration of the patch-clamptechnique. The theta-tube is hereby a double compartment pipettecontaining the agonist (glutamate) in one compartment and control so­lution in the other. The tube is then moved by micro-steps inducedthrough piezo-cryst.als in front of the membrane patch such that shortpulses of either solution can be pumped onto the patch. This allows todeliver short pulses of the agonist to the membrane patch (1 ms) \vhichhave an almost rectangular shape. 1\'ussel et al. fitted their model to6 different current curves rneasured in 6 different application protocols.R.ectangular pulses are not representative for transmitter pulses in thesynaptic cleft and in our opinion not optimally suited for fitting as theymight lead to too slow rate constants. \Ve therefore concluded thatbecause the sophisticated models of Trussel and Jonas were fit withrectangular 1 ms /1 mM pulses, these models were not able to cleliverhigh enough open rates for the fast concentration traces computed byour three-dimensional simulation. \Vith a rough adaption of the rateconstants k+ 1 and k+ 2 , we could show that it is in principal possible tosimulate AIvIPA receptor EPSC using our computer traces as input tothe sophisticated models of Jonas and Trusse!. Certainly the parameterset would have to be readjusted such that. other characteristics of themodel (e.g. fraction of desensitized receptors) are represented as well.

In his recent review article Leff' [67] rejuvenates a so-called two­state model (see Figure 5.1) which applies a cornmon idea to both theG protein coupled receptors and transmitter-gated ion chanrlC!. Thismodel of receptor activation invokes the existence of two receptora closed (or resting) and an open (or active) state and the agonistglutamate) causes an increase in the ratio of active to resting states. IVetested this model for its ability to transform transmitter concentrationtransierlts int.o "fraction of open" curves. Several conditions have tobe fulfilled when choosing parameter values (i) in a cyclic model ofthis type the product of forward rate constants must be equal to tlwproduct of backward rate constants (ii) 1'1 /1'2 ~<:: 1 since it is known thatthe ratio of activity of inactive and active Aj\!IPA receptors (R/I?*) isknown to be 10'-4, and (iii) if fed with our glutamate transients, an80% activity (R* + AR*) and a decay time constant of T :s: 2 '1118 shouldbe achieved. 'With stipulation (1i) a high-throughput pat.hway from Rto AR' via the R* is impossible. To achieve a high activation upon

5.1. CONE1DENCE IN MODEL iDEAS 103

glutamate stimulation we would therefore require high ratios for 1'6/1'5

and 1'S/r7 and at the same time a high value for 1's. This considerationleaves us only 1'3 to compensate such that condition (i) is still valid,but physical restrictions (i.e. diffusion controlled reaction) limit 1'3 tobe < 1£7,s-17/],1\1-1. \eVe thus had to realize that these stipulations arecontradictory demands, which may not all be fullfilled at the same time.In this sense we assume that this cyclic 4-state model does not work withtransient agonist peaks but requires long lasting high concentration ofagonist as is encountered after administration of pharmaca but not inin-situ.

1'1 10[8--1]

1'2 1000[8-1] flR""",

>,.R*1'3 100000[,s-17/1Ar 1]

r7lsf2

r4l1'4 10[8-1]

1'5 1000[.,--1]

1'e 1000[8'-1] f61'7 1000[s-l] AR""",

fS>,. AR*

r8 100000[8-1mM- 1]

Figure 5.1: two state model according to LetT, left: a pos8ible set ofmt.e eon8t.ont.s, right.: model st.ate diagmrn

In Chapter 3.4.2 we presented results of computer simulations ofmetabotropic glutamate receptor models. For none of the many mGlusignal transduction pathways existed yet a quantitative model. Amongthe many physiological which were recently published, the~ studyof Batchclor et al. lO] was for 11S of most interest, as they providequantitative results. With our model we could reproduce the exper­imental findings of Batchelor et aI., which showed a large EF'SP incerebellar Purkinje cells after repeated parallel fiber stimulation. To beable to simulate the threshold-like appearance of the EPSP our modelhas to comprise at least three components. First spatial cooperationinduce:s slower transients with higher concentration levels inthe perisynaptic space. This enables metabotropic glutamate receptorto be activated. Second, activated glutamate receptors induce activa­tion of G protein, which accumulates with repetitive synaptic activity.

104 CHAPTER 5. DISCUSSION AND CONCLUSWNS

Third, four binding sites at the effector system increase the steepnessin the dose response curve of the alleged effector system X. The highlythresholded activation of mGluR EPSPs requires four cooperative bind­ing sites.Our model of the poorly understood mGluR signal transduction cas­cade shows only one of the many possible explanations for thresholdedactivation. But it is a minimal model in the sense that a lot more pro­cesses infiuenc~, the mGluR cascade but not less than three componentscan reproduce the complicated response scheme shown in Batchelor'sstudy [8]. In particular we did not model a possible influence of theomnipresent but in this case poorly understood c!'cle which isactivated via class I mGluRs.

I":: 20. Modeling Diffusion

Two modeling approaches can be employed for simulation of synapticdiffusion. On the one hand the ]\lonte Carlo Method model (?vICi'vl) sim­ulates the random walk of a set of molecules. On the other hand we cannumerically solve the three dimensional difl'usion equation. Both ap­proaches find their proponents in the very recent litt'rature. Clements,Stiles et al., and Wahl et al. [17, 105, 110] use :MCiVI for their simulationof cholinergic, gabaergic and glutamatergic synapses. Holmes [56] madehis choice for a diffusion equation model (DErvI). In Table 5.1 we give abrief comparison of the three models (including our . The results

dirnensions

model-type

boundar:r" condi­tions for DEM

diffusion con-stantTC!C(II'(!ll

Table 5.1: comparison of diffusion modeling approachcs

of Clements and Stiles et a1. are in broad agreement with our results.The markedly different decay time constant of the DElv[ employed byHolrnes can be partially explainecl the lower diffusion constant. An-

5.2. 2\IODELING DlFFUSION 105

other aspect to be noted is that he employed a 2-dimensional model.The different transmitter systems and the different modeled morpholo­gies make it difficult to compare the mentioned studies closely, we thusconcentrate here on the experiences we gained with the two modelingapproaches.With the l\ilCM a geometry is fairly easy to describe through a number ofanalytic formulations. 'Ve implemented formulations for simple three­dimensional surfaces such as heInispheres and planes. A major drawback of the MCM is the limitation of the time step size. Increasing thetime step size results here in larger spatial steps, wh~Te the maximalspatial step though is limited by the cleft size. Another inherent drawback of the MCM is that we have to average over many simulations, aswe wanted to use the resulting tirne courses as input for the common,state diagram based, receptor models. Assuming a vesicular contentsof 1000 molecules (which corresponds to ~)OO mM in a vesicle with aninner diameter of 30 nIn) and 100 averaging simulations with a simula­tion time of 105 time-steps, we a single CPU usage of 100 hours ona HP 9000/770. A corresponding DEM simulation with grid resolutionof 96 x 96 x 5 with an initial time step of 4 71,8 takes only one CPU hour.This mesh-size induced a 16nnl x 16nm x 4nm voxel which sufficed forour purpose.The DElVl finds a restriction in the discretization of the open boundaTyconditions. To simulate the diffusion in an infinite cleft we approxi­mated the diffusion into the space beyond the modeled geometry 'withthe Cauchy boundary condition du/cll D·n. The error of less than0.1% after 1001'8 simnlation time was tolerable for our purposes. An­other draw back of the DErv! is certainly the efFort to generate the systemof equations on the basis of a given geometry. To generate the matrix ofthe above mentioned system of equations we implenIented a dedicatedsoftware, which required up to 1 hour on a SparcStation 10/52.In Chapter 2 we presented our decision for the DE]V1. The decision wasbast~cl on computational efficiency. It is difficult to cornpare the twomethocls as their cornputational effort depends on different variables.Aiming at a three-dimensional simulation with a predetermined levelof accuracy, we note that the determinant cost factors for the l\iIonteCarlo Method depends on the amount of molecules while the diiiusionequation model's eifort increases 'Ivith the degree of spatial meshing.

106 CHAPTER 5. DISCUSSION AND CONCLUSIONS

COlnputational Methods

As mentioned in Chapter 4, solving systems of equations of the order ofn 5.101 -105 equations requires mE:thods that exploit the sparsity ofthe system matrix. Direct solvers can prove to be faster than iterativesolvers in the case of 2 dirnensional diHusion equations. Adding a thirddimension enlarges the bandwidth of a corresponding band matrix byIl,z while the length of the band grows by a factor ofn z as well, where 'n"is the number of volume elements in the third dimension. Fill-in occur­ring during the factorization phase of a direct solver filled up the 99%of non-zero elements inside the band of the band rnatrix. The memoryconsumption of the resulting bandmatrix as well as the required com­putational power was beyond the capacity of our workstations.Iterative methods in contrast performed well. As described in Chapter4, we chose the abundantly used GMRES method to solve the non­symmetric system. In comparison to the preconditioner 111 = 1 themore sophisticated ILU preconditioners gave rise to a speed up by afactor of 4.S. Our irnplementation of a fast Poisson solver as a precon­ditioner perforrned worse than thei\1 I trivial preconditioner. Totest whether the miss-performance is caused the Cauchy boundaryconditions, we altered the problem such that only Neumann conditionsappeared, still the performance of the FFT based preconditioner wasdependent on the problem size and therefore not applicable. IVe con­clude that the fast Poisson solver based approach, to exactly invert anapproximation of the original matrix A, is in our casp clearly less suit­able as a pn~conditioner than the incomplete decomposition approachwhere an approximate inverse of the exact matrix A is construct.ed. Apossible explanation for t.his finding might be t.he strong diagonal dom­inance in our syst.em matrix and the fact that the Poisson matrixdiffers significantly on t.he diagonal from the original matrix A.

5.4 Conclusions

Through an efficient implernentation of the numerical solution of thediffusion equation we were able to solve vast thre(~ dimensional mod­els. \Vit.h this three dimensional model ,vc could show that. transmitterclearance is faster than previous models assumed. These fast clearance

5.'1. CONCLUSIONS 107

rates lead to new restrictions concerning the postsynaptic receptor mod­els. Ionotropic and metabotropic glutamate receptor models were thuscreated considering these restrictions together with further biologicalstipulations.

According to our sirnulations neither NMDA nor AMPA receptorsfind their rate limits in cleft clearance due to diffusion. The fact that dif­fusion can lower glutaruate concentration in the cleft within 250ps to avalue below 10-2 m]\1, excludes the chances of glutamate accumulationeven during very fast stimulation (f ::::; 1000Hz). Thus enhancements inpostsynaptic NMDA and AMPA responses are not expected to be dueto glutamate accumulation in the cleft, even after high frequency stim­ulations. In contrast we want to stress the importance of the paradigmof spatial cooperation. \Ve discussed this paradigm in the context ofmetabotropic glutamate receptors as well as in the context of the ques­tion whether diffusion is rate limiting for AMPA receptors. It can beexpected that this idea will be treated by future studies. Furthermorethe field of metabotropic signal transduction cascades is rapidly pro­gressing and we expect new results which stimulate refinement of theproposed mGluR model.

Appendix A

Implementation

All of the software implementations for the simulations were prototypedin :Matlab [72], which proved to be flexible and fast for most computa­tional purposes. :VlatIab is an interpretative language, which makes itinherently slower than compiled languages like C++. For simple matrixhandling routines the computational speed of Matlab routines comparedfavorably with corresponding C++ routines. Complicated Matlab rem··tines, like the GIVIRES iterative solver, were, due to the interpretativenature of Matlab, significantly slower than corresponding C++ imple··mentations. \Ve therefore re-implemented and re-designed all programsin C++. A modest user-interface was implemented in Java to providemachine independent visualization of receptor transduction.

As the implementation of the simulation software had to considermore than only numerical aspects, object oriented design and program­ming was employed instead of the imperative programming paradiglnused by FORTRAN. The disadvantage that we could not profit fromthe enormous amount of specialized and highly optimized FOlrrRANroutines, made it important to find a good basis for the implementa­tion. Jack Dongarra, Roldan Pozo and Karin Remington wrote severalobject oriented mmreric ancl linear algebra packages, which addressedtasks from sparse matrix handling, over iterative methods up to LA­PACI( and BLAS Level 3 irnplementations [30, 31, 85, 86]. Though wewere able to profit from the efficient numerical libraries provided in the

109

110 APPENDIX k IMPLEMENTATTON

IML++ and Sparselib++ packages we certainly had to accept minorreductions. In return we were able to use the freeware

programming soft\vare.

irnplementation consists of four independent major parts, whichcan executed without nmtual interaction. In Figure A.I an overviewof the three parts and their employed libraries is provided. The diffusionequation based rnodel and the Monte Carlo method based model are twoalternative methods for the computation of neurotransmitter diffusionin the synaptic deft, they are described in the Sections (A.l and A.2)and (A.3), respectively. Both methods deliver output that can be post­processed by the receptor transduction part \vhose implementation isdescribed in Section (AA).

sparse matrix classes (SparseUb++) -s::?>",linked list classes ~:;

~-- x·(") 2'

vector classes '"ro o'Harwell-Boeing matrix I/O classes ~. "m0. .0

geometry classes " ce

(j) o'sparse matrix classes (SparseLib++)

~ "< s:

vector classes '"0.0.

'i: ~iterative method classes (IML++)

preconditionsr classes~

geometry classes S:S:0 0

0. ""' -vector classes _"'-("):?>'"

random number generator (.,6-

GUI classesIJ<D

receptor models 0

'"'0integration classes ~

persistency classes 'i:~

communication cL

Figure A.I: design and depcn.dency oveTview of the implementation

A.I Generating the l\!latrix

Description

Given a description of the geometry fixed aboutboundary conditions) the rnatrix of the of is gener-ated. Discretization of the three-dimensioned diffusion equation was

A.l. GENERATING THE MATRIX 111

implemented according to our description in Chapter 2.3.1. First, thethree-dimensional space is scanned and for mesh-points belonging tothe cleft volume, an entry is made in an index table. Second, for everymesh-point in the index table an entry is made into the matrix. vVechose a linked list to hold the intermediate matrix during the creationprocess, as we have the least cost for element insertion in this type ofsparse matrix [37, 38]. A final conversion into Harwell-Boeing[:~5, 94] or compressed row storage schemes (as used in Matlab) is thendone for file-storage.

Input

• cleft shape: a description of the geometric shape of the volumein which diffusion takes place. Geometry descriptions can be pro­vided in two ways. First, in form of a series of TIFF-Files eachcontaining a quadratic black and white sketch of a volume slice.TIFF [23] is a tagged image file format and was used becausegeometry describing images from the biological domain sciencewere best exported in this format. Second, the volume can bedescribed analytically either as the volume between two planes oras the volume between two concentric hemispheres.

• mesh size: the (cubic) volume in which the synapse is situatedwas chosen to have a side length of 1500 nm. For each of the threespatial dimensions one has therefore to specify the resolution.

• sparSt~ format: indication whether the output shall be writtenin Ha.rwdl-Boeing format or in compressed row format to a file.

Output

• sparse matrix: the rnatrix for the system of linearequations is written to a tlle either in Harwdl-Boeing format orin compressed row format.

bnplemented Classes

• geornetry objects

112 APPENDIX A. IMPLEMENTATION

point: implements operations on a vector in 3D space.

implements operations like the reflection of a givenand the determination of the relative position of a given

implements operatiom like the reflection of athe determination of the relative position of a

point and the perpendicular a given point.

pLvo1: class for operations on a volume betwt~en two par­allel planes, implements operations like the determinationwhether a given point is inside the volume.

hs_vol: class for operations on a volume between two concen­tric hemispheres, implements operations like the detennina­tion whether a given point is inside the volume.

any_vol: class for volmlles which are reconstructed by theTIFF-picture reader, implements operations like the deter­mination whether a given point is inside the volume.

volume: common parent class for the above three volumeclasses.

• index table: a two-row table is the central data structure of thisclass. This class is implemented to lookup the index of a volume­element given its coordinates or to reverse lookup the coordinatesgiven its index.

• linked list: an array of linked lists was chosen to hold the columnsof the matrix. Linked list are only used as an intermediate sparse­matrix storage format.

• TIFF-reader: reads a sequence of TIFF-files and reconstructs avolume-description according to the TIFF-picture which representa cross section through a cubic volume. The resolution at whichthe cubic volume was sliced, was to be the same as theresolution of the TIFF-pictures. This class \vas first implenlCntedtogether with the any-vol class but proved to be applicable onlyin special cases, as an analytical volume description is much moreefFicient.

:Freeware External Classes

• Harwell-Boeing input-output cf. [94]

/1.. 2. SOL UTION OF THE LARGE SPARSE SlTSTEM OF EQUATIONS1l3

A.2 Solution of the Large Sparse System ofEquations

Description

The syst.em matrix and initial conditions are read. Preconditioners areinitialized and simulation is started. As soon as the residual of theGJVIRES step falls below a predefined measure the time step is doubledand the system matrix must therefore be adapted, which in return asksfor a re-initializat.ion of t.he precolldit.ioner. The solution is written tofile eit.her in Harwell-Boeing or in the compressed row sparse matrixst.orage formats.

The "fast Poisson solver" preconditioner as well as our problemspecific "ILU" preconditioner were implemented by us as additionalclasses for IML++ [~)O]. In addition the FFT++ package from Netlib(http://wWY\T , which implements part of the FORTRAN FFTpackage, had t.o be complemented by our own fast sine transfonn rou­tines. Dedicated matrix-multiplication and matrix-addition routineswere furthermore implement.ed as additional object met.hods for theSparselib++ [31].

Input

• rnatrix A: t.he matrix class (corresponding to A in t.he linearsystem A:r b) must supply the operator function "*". The ma­trices in the linear A:r == bare accessecl only throug'h the"*" operator l1lcrnber function. The return type of this operatorfunction is a vector.

• rnatrix H: this small and dense matrix is used for the upperHessenberg matrix H that is constructed during the course of theGl'vIRES rnet.hod. Its class supplies cli:A:'erent functionality than thesparse matrix class 01'..1. In particular. it must have the operator

for individual elements.

• initial condition vector: initial conditions are read from a fileand are used to construct the start. vector in the iteration processof G:,IR.ES.

114 APPENDIX it. IAIPLEMEN1AJION

• preconditioner: the name of the desired preconditioner.

• snap:31Ot tirne t;

Output

• snapshots: the concentration distribution oyer the cleft at timet; is written into a file and could then be yisualized either withMatlab or with Gnuplot [39].

• traces: the concentration as a function of time at point ::c iswritten into a file. These traces are then used as input for thereceptor transmittt,r simulation describecl in Section A.A.

Implemented Class~,s

• preconditioners

fast Poisson solver: a description of this solwr is given inChapter 4.4.1.

ILU: we rewrote the incomplete LU clecomposition to takeadvantage of the a priori knowledge of the septa-diagonalmatrix structure.

• fast sine transform: Since the FFT++ on \"etlib does not in­clude a fast sine transform, we had to complete this library withour own implemeutation.

Freeware External Classes

• Harwell-Boeing input-output class

• GIvIRES class: this class is proyidecl as of the IML++ library.

A.a. IVfONTE CARLO DIFFUSION SIMULATION 115

A.3 Monte Carlo Diffusion Simulation

Description

The Monte Carlo difIusion simulation is an alternative approach to thedifIusion equation based approach which is mentioned above. As initialcondition for the l\Ionte Carlo approach we assumed 50'()()O particlessituated at ;:eo for to = O. Per time step each particle takes a fixed lengthmove in a random direction. If this random step would move a particlebeyond the pre- or postsynaptic cleft barrier (described analytically asplanes or hemispheres), the trajectory of the Inove is reflected at thecorresponding geometric object. In several boxes located at increasingdistances otT the release site we sampled local concentrations. Nlultiplesimulations were run to be able to average the result. Visualization ofthe results was either done in Matlab or in Gnuplot.

Input

• sampling locations: distances between release site and the sam­pling boxes.

• anlOunt of rnolecules

Output

• concentration traces: collected inside the sampling boxes

Impl<c>mentecl Classes

• geornetry objeets: the classes used in the creationof the ditfusion based simulation (described above)

were such that they could be reused in this simulation.

External Class

• ranclorn rllllnber generator: this class was implemented ac­cording to the "ran3" proposition in "Numerical Recipes in C"

116

A.4

liPPENDIX A. IlvIPLEMENTATJON

R.er'['ptor Simulation

Description

The above described solver computed the concentration distribution of aneurotransmitter over the synaptic cleft in a given time interval. Duringthe computation concentrations are sampled at several distances off therelease site. These concentration traces can now be fed into the systemof differential equations describing the modeled postsynaptic receptoLIntegration is done via a fifth-order Runge-Kutta-Fehlberg algorithm[87] (using coefficients according to Cash-Karp), featuring adaptive step­size control. A small user interface to these receptor simulations as wellas the integration itself were implemented in JAVA. The interface allowsto chose the receptor type, the location of the receptor and the scalingof the plot axis.

Input

• systern of differential equations: the simulated receptors aredescribed by a system of differential equations. see beluw.

• sirnulation tirne interval

Output

• percentage of receptor activation

hnplernented Classes

• concentration trace: the concentration traces are read from afile from a URL address) ancl put into a data structure withentries for the concentration value and the timeat which the concentration \vas sampled. 'The class contains amethod to access the concentration value of a time.

AA. RECEPTOR SIMULATION 117

• receptor: for different types of receptors we irnplemented differ­ent classes, each implemented a method which computes a vectorof derivatives. These receptor classes are employed by the Runge­Kutta integrator class.

• R.unge-Kutta integrator: ThE~ implementation of the integra-tor classes in JA,'A and is based on the Cash-Kmp Runge-Kutta approach described in "Numerical Recipes in e" [87].

• vector class: this class provides basic vector operations like sum­mation, subtraction, scalar multiplication, inner product.

External Classes

• java.io.*: input and output to and from streams/files.

• java.awt. *: the abstract windowing toolkit, provides GUI ele­ments, layout managers and graphic primitives.

• java.net. *: provides among others also access to URL objects.

Appe11dix B

AMPA Receptor Models

Below we show the full receptor models published by Trussel et al. [92]and Jonas et al. . Their kinetic models are shown in Figure B.1,vhereas the corresponding kinetic constants are given in Table 11.1.The rnain clifTerence between them is the importance of desensitization.'While Trusscl Gnds an ()2 rate as high as 20'0003- 1 , the highest 0"", ratein Jonas' model is one order of magnitude lower. Trussel at cl. havefindings which propose that the of AIvlPA receptors desensitiza­tion is much lower than ANIPA rece])tor activation, thus at low levelsof glutamate there are rnany transitions from a closed stat~~ to a d('l­sensitized state. at levels of glutamate, desensitizationcould curtail AJVIPA EPSe.

119

120 APPENDIX B. 11MPA RECEPTOR 1\;JODELS

Haman and Trussel Jonas et a1nucleus tllagnocellulnris neurons pyramidal neurOns

42602.84 * 10'32601.27*10G

280 ,15.710'100

Ll' 300 4240,(3 1200 900Ll'1 1000 [8"IJ 2890PI 500 [8--1] 39.2Ll'2 2 * 104 [8-,1 ] I"")I ~

P2 14 [8- 1] 0.727Cl:3 2500 [8-1] 17.7/h 3 4.0Ll'4 1200 8.15lit 6000 130

Table B.1: 7'atc constants fOT' thc AMPA 7'cccptm' ki'netic models

o

B

fl,

o

I~

Cl" 103, I

"D2

Figure B.1: nwdels a rte,':ensill'z:n:1i/ AMPA 'l'ccepto7', /1: model ofRaman and l)"ussel. B:model of Jonas aL

Appendix C

Glossary

'I'erms Related to Biology and Neuroscience

The definition of the terms belmv is given in the context of eNS neurons,if there are sevtTal meanings only the closest match is given. Some ofthe definitions can be found in "International Dictionary of Medicineand Biology" [28].

afferent conveying inward or toward a center used fornerve impulses, blood and information)

an agent capable of stimulating a biological re­sponse by occupying cell receptors

Al\IPA o--amino";1-hydl-oxy-5rnethyl-4-i8o.Tazoleacid, a synthetic agonist for glu-­

tamatc receptors

AMPA class of glutamate receptor subtypcs \vhichshows strong affinity towards AMPA. Afl/IPA re­('pr)t-ryl'~ build ion channels permeable for monova--lent therefore excitatory

an that opposes the action of another

ATP j\dcnosine Tl'iphosphate, most important intra­cel1ular energy provider

121

122

axon

hioinformatics

cell membrane

DEMdendrite

depolarized

desensitization

domain science

double barrel <lpplication

discipline

ECso

efferent

EPSCEPSP

exocytosis

exogenous

APPENDIX C. GLOSSARY

nerve track leading from the soma of a nerve cellto SyllClptiC junctions

the general definition of "informatics for biol-is most often reduced to "processing gene­

data", as a consequence gene-database searchtechniques are regard('d as the most importantslice of bioinfonnatics

insulating lipid bilayer with pores "whose conduc­tivity depends on voltage, ion concentration orligand binding

difI'usion equation model

dendrites can be viewed as the antenna of a neu­rem. Most of the afferent synapses of a neuron canbe found on the dendritic tree. which is a tree likeaccumulation of passive conductors

the intracellular potential is shifted to more pos­itive values, thus the cell's initial polarization isdecreased

the process where the binding of a ligand to areceptor leads to the decreased eflectiveness of thereceptor

in the context of interdisciplinary work the do­main science is the scientific field where the mainscience is applied

see theta tubing

a branch of research or instruction

agonist concentration where an efFector systemglutamate receptors) shows h:1lf maximal

activation

conveying outward or away from the center

excitatoric current

E~xcitatoric postsynaptic n(',!"('lntl:ll

process where the membrane of the vesicle andthe cell membrane of the ending fuse.At the moment of the fusion the contents of thevesicle is placed outside of the site

produced outside of the or:c;arlisll1

123

G protein guanine nucleotide-binding protein, a proteinconsisting of three subunits /3, ,The G pro­tein is most often associated with Cl receptor pro­tein, npon activation of which the G-protein ac­tivates by dissociating into et and subunitswhich can then activate enzymes

glutamate an ,,:xcitatory neurotransmitter

hyperpolarized the intracellular potential is shiHed to more neg­ative values, thus the polarization is enlarged

ion channel a protein forming a pore in a lipid cell mem­brane, through which ions can move (e.g. calciumchannels)

MCM J\lonte Carlo :Method model

membrane potential a voltage difference between two compartmentsseparated by a membrane

model abstraction and generalization of reality or realprocesses

NMDA N-rnethyl-D-aspart;ate an agonist for glutamate

Nernst potential an electrical potential gradient across a mem­brane due to differences in ion concentration oneither side of the membrane. The potential canhe calculated with the Nernst equation (WaltherlIermann Nemst, chemist 1864-1941)

Nernst equation expresses the relationship between an electric po­tential between two compartments and relativeconcentrations of an ion in the two compartmentswhen the system is equilibrium:

E ,:: RT. lnzF

E is the equilibrium potential, 'T the absolutetemperature, R the universal gas constant, z theionic charge, F Faraday's constant and Cl, C2are the respective concentration of the ion in thetwo compartment

124

nenrotransmitter

patch clamping

postsynaptic

presynaptic

receptor

release site

resting potential

synapse

synaptic cleft

theta-tubing

APPENDIX C. GLOSSAIn-

any chemical substance released at a nerve termi­nal as Cl result of a nerve impulse and capable oftransmitting that impulse across a synaptic cleftbv binding to receptors in another cel1 therebyexciting/inhibiting it

a pipette of Ipm diarneter is placed onto the sur­face of cleaned nerve the membrane patchunder the opening of the pipette is physical1y andelectrical1y isolated fr01n its environment. Withthis technique it is possible to observe molecu­lar events of signal generation and transmissionof nerve cel1. Neher and Sakmann won the Nobe1price for medicine in 1991 for the invention of thistechnique.

situated or occurring elistal to a synapse

proxirual to a synapse

A trans-membrane protein \vith binding affinityfor a particular ligand and where the ligand bind­ing gives rise to some significant effect

site in the presynaptic nerve terminal where exo­cytosis of transmitter filled synaptic vesicles takesplace

membrane potential when cc]] is at rest~70mV)

a region of structural constitutinga junction between two or more neurons. Theypermit transmission of impulses from the presy­naptic neuron to the postsynaptic neuron

cleft between the structural specialization of apn?synaptic and a postsynaptic neuron.

( double barrel very fast applica-tion technique via a pipette two bar-rels in the form which resembles the Greekletter (-). or control solution is pumpedthrough either of the barrels in conjunction with apiezo-movemenL which positions the pipette suchthat either of the two barrel is in frontof the membrane patch

125

signal transmission the physiologic process by which a nerve impulsepasses across the synapse

tetanic stirnulation High frequency stimulation of Cl presynaptic neu­ron, as much as 50 to 100 action potentials persecond

transduction to convert a signal from one (energy) fonn intoanother

transmitter clearance processes that lower the concentration in thesynaptic cleft, most discussed are diffusion anduptake via pumps and exchanger into the presy­naptic button or into Glia cells

vesicle (synaptic) membraneous organelles containing high concen­trations of neurotransmitters to be released byexocytosis during synaptic transmission.

Terms Related to Computer Science and ScientificComputing

The definition of the terms is given in the context of numerical solutionfor the discretized diffusion equation, if there are several meanings onlythe closest match is given. Some of the definitions can be found in"Templates" [6].

boundary conditions constraints of the form 1/(0) ex and 1/(b)for second order differential equations (e.g.

y" .11, )/'), 0:S;:r::S; b)dij1'usion equation a parabolic partial differential equation which de­

scribes the spatial and temporal development ofa concentration in the form:

Dtt) -- t) o

direct method an that produces the solution to a, sys-tem of linear equations in a number of operationsthat is determined a priori (usually a variant ofGaussian elimination)

126

initial conditions

iterative methods

Krylov sequence

Krylov subspace

Matlab™

Monte Carlo j'vIodel

differential equation

preconditioner

sparse matrix

APPENDIX C. GLOSSARy'

the initial condition y(a) = n is necessary to spec­ify a specific solution of the differential equationVi f(v)an algorithm that produces a sequence of approxi­mations to the solution of a linear of equa­tions. Usually the iteration is stopped if the resid­llal accomplishes a termination criteria

for a given matrix A and vector ;r, the sequenceof vectors {A i

.7: L> 0, or a fini te initial part of thissequence

the subspace spanned by a Krylov sequence

high performance numeric computation and visu­alization software

simulation technique for complex or vast statis­tical problems. At every time step a randomnumber which can be mapped onto the statisticalproblem is chosen. The order of the mean error isreciprocal inverse to the square root of the num­ber of experiments.

an equation containing an unknown function andits derivative e.g. z;' = f (V)transformation of a systelll of linear equations tomake it's condition number smaller

matrix for which the number of zero elernents islarge enough that algorithms operationson zero elements payoff

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Curricultnn

Personal Data

Name:First names:Date of birth:Nationality:

Professional Training

19861987-1992

1993-199G

CURRJCUL UI\I

ChristenTobias FabioOctober 19th, 19G7Swiss

rvlatura, Typus C, in SwitzerlandStudy of computer scienceat the ETH Zurich, SwitzerlandComputer simulation projectwith Ciba AG, Basel, SwitzerlandPhD student ,vith Ciba AG, Baseland the Institute of Scientific Computing,Department of Computer Science,ETH Ziirich, Switzerland