Bifurcation analysis of a general class of non-linear integrate and fire neurons.

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integrate and fire neurons.

Jonathan Touboul

To cite this version:

Jonathan Touboul. Bifurcation analysis of a general class of non-linear integrate and fire neurons..

[Research Report] RR-6161, INRIA. 2008, pp.47. �inria-00142987v5�

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a p p o r t

d e r e c h e r c h e

9-6399ISRNINRIA/RR--6161--FR+ENG

Thème BIO

Bifurcation analysis of a general class of non-linear integrate and fire neurons.

Jonathan Touboul

N° 6161

March 4, 2008

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Jonathan Touboul

∗

ThèmeBIOSystèmesbiologiques

ProjetOdyssée

†

Rapport dereherhe n° 6161Marh4,200847pages

Abstrat: Inthispaperwedenealassof formalneuronmodels beingomputationally

eientandbiologiallyplausible,i.e. abletoreprodueawidegamutofbehaviorsobserved

in in-vivo or in-vitro reordingsof ortial neurons. This lass inludes for instane two

modelswidelyusedin omputationalneurosiene,theIzhikevih andtheBretteGerstner

models. Thesemodelsonsistina4-parametersdynamialsystem. Weprovidethefullloal

bifurationsdiagramofthemembersofthislass, andshowthattheyallpresentthesame

bifurations: an Andronov-Hopf bifuration manifold, a saddle-nodebifuration manifold,

a Bogdanov-Takens bifuration, and possibly a Bautin bifuration. Among other global

bifurations, this system shows asaddle homolini bifuration urve. We show how this

bifuration diagram generates the most prominent ortial neuron behaviors. This study

leads us to introdue a new neuron model, the quarti model, able to reprodue among

allthebehaviorsoftheIzhikevihandBretteGerstnermodels,self-sustainedsubthreshold

osillations,whihareofgreatinterestin neurosiene.

Key-words: neuron models, dynamial system analysis, nonlinear dynamis, Hopf bi-

furation,saddle-nodebifuration,Bogdanov-Takensbifuration,Bautinbifuration,saddle

homolinibifuration,subthresholdneuronosillations

∗

jonathan.touboulsophia.inria.fr

†

OdysséeisajointprojetbetweenENPC-ENSUlm-INRIA

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Résumé: Dansetartilenousdénissonsunelasseformelledeneuronesàlafoiseaes

en termes de simulation et biologiquement plausibles, 'est-à-dire apables de reproduire

une largegammede omportements observésdansdes enregistrementsin-vivoou in-vitro

de neuronesortiaux. Cette lasse inlut par exemple deux des modèles lesplus utilisés

dans les neurosienesomputationnelles: le modèle d'Izhikevih et le modèle de Brette

Gerstner. Cesmodèlesonsistentenunsystèmedynamiqueà4paramètres. Nousalulons

le diagramme de bifuration loales omplet des membres de ette lasse et prove qu'ils

présententtouslesmêmes bifurations: une variétéde bifurations d'Andronov-Hopf,une

variétédebifurationssaddle-node,unebifurationdeBogdanov-Takens,etéventuellement

une bifuration de Bautin. Parmis d'autresbifurations globales, es systèmes présentent

aussiune ourbe desaddle homolini bifurations. Nous montrons quee diagrammede

bifurationsgénèrelesprinipaux omportementsdeneuronesortiaux. Cetteétudenous

mèneàintroduireunnouveau modèle, lequarti model, apablede reproduire enplusdes

omportementsdesmodèlesd'IzhikevihetdeBretteGerstner,desosillationssousleseuil

auto-entretenues,quisontd'ungrandintérêtenneurosienes.

Mots-lés : modèlesde neurones,systèmes dynamiques,dynamique non-linéaire,bifur-

ation de Hopf, bifuration saddle-node, bifuration de Bogdanov-Takens, bifuration de

Bautin,saddlehomolinibifuration, osillationssousleseuil entretenues

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During the past few years, in the neuro-omputing ommunity, the problem of nding a

omputationallysimpleandbiologiallyrealistimodel ofneuronhasbeenwidely studied,

inordertobeabletoompareexperimentalreordingswithnumerialsimulationsoflarge-

salebrain models. The keyproblem is to nd amodel of neuronrealizing aompromise

between its simulationeieny and its ability to reprodue what is observed at the ell

level,oftenonsideringin-vitroexperiments[15,18,25℄.

Amongthenumerousneuronmodels,fromthedetailed Hodgkin-Huxleymodel[11℄still

onsideredasthereferene,butunfortunatelyomputationallyintratablewhenonsidering

neuronalnetworks,down tothesimplest integrateandremodel [8℄veryeetiveompu-

tationally,butunrealistiallysimpleandunabletoreproduemanybehaviorsobserved,two

modelsseemtostandout[15℄: theadaptivequadrati(Izhikevih,[14℄,andrelatedmodels

suhasthethethetamodelwithadaptation[6,10℄)andexponential(BretteandGerstner,

[5℄) neuron models. These twomodelsare omputationallyalmost aseientasthe inte-

grate and remodel. The Brette-Gerstner model involvesan exponential funtion,whih

needsto be tabulated if we wantthe algorithm to beeient. They are also biologially

plausible, and reprodue several important neuronal regimes with a good adequay with

biologial data, espeially in high-ondutane states, typial of ortial in-vivo ativity.

Nevertheless,theyfailin reproduingdeterministiself-sustainedsubthresholdosillations,

behaviorof partiularinterestin ortial neuronsforthepreisionandrobustnessof spike

generationpatterns,forinstanein theinferiorolivenuleus[4,23,24℄,inthestellateells

of theentorhinalortex[1, 2, 17℄ and in thedorsal root ganglia(DRG) [3,20, 21℄. Some

modelshavebeenintroduedtostudyfromatheoretialpointofviewtheurrentsinvolved

in the generation of self-sustained subthreshold osillations [26℄, but the model failed in

reproduinglots ofotherneuronalbehaviors.

Theaimofthispaperistodeneandstudyagenerallassofneuronmodels,ontaining

the Izhikevih and Brette-Gerstner models, from a dynamial systems point of view. We

haraterizetheloalbifurationsofthesemodelsandshowhowtheirbifurationsarelinked

with dierent biologial behaviors observedin the ortex. This formal study will lead us

to dene a new model of neuron, whose behaviors inlude those of the Izhikevih-Brette-

Gerstner(IBG)modelsbutalso self-sustainedsubthresholdosillations.

Intherstsetionofthispaper,weintrodueagenerallassofnonlinearneuronmod-

els whih ontainsthe IBG models. We study the xed-point bifuration diagram of the

elementsofthislass,andshowthattheypresentthesameloalbifurationdiagram,with

asaddle-node bifurationurve,an Andronov-Hopf bifuration urve,aBogdanov-Takens

bifurationpoint,andpossiblyaBautinbifuration,i.e. allodimensiontwobifurationsin

dimensiontwoexepttheusp. Thisanalysisisappliedin theseondsetiontotheIzhike-

vihandtheBrette-Gertsnermodels. Wederivetheirbifurationdiagrams,andprovethat

noneofthemshowtheBautinbifuration. Inthethirdsetion, weintrodueanewsimple

model -the quarti model- presenting, in addition to ommon properties of thedynamial

system of this lass, a Bautin bifuration, whih an produe self-sustained osillations.

Lastly,thefourthsetionisdediated tonumerialexperiments. Weshowthatthequarti

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model is able to reprodue some of the prominent features of biologial spiking neurons.

Wegivequalitativeinterpretationsof thosedierentneuronalregimesfrom thedynamial

systemspointofview,in ordertogiveagraspofhowthebifurationsgeneratebiologially

plausiblebehaviors. Wealsoshowthatthenewquartimodel,presentingsuperritialHopf

bifurations,isableto reproduetheosillatory/spikingbehaviorpresentedforinstane in

theDRG. Finally weshow that numerial simulationresults of thequarti model show a

goodagreementwithbiologialintraellularreordingsintheDRG.

1 Bifuration analysis of a lass of non-linear neuron

models

Inthissetionweintroduealargelassofformalneuronswhihareabletoreprodueawide

rangeofneuronalbehaviorsobservedinortialneurons. Thislassofmodelsisinspiredby

thereviewmadebyIzhikevih[15℄. Hefoundthatthequadratiadaptiveintegrate-and-re

model wasableto simulate eiently alot of interestingbehaviors. Brette and Gerstner

[5℄denedasimilarmodelofneuronwhihpresentedagoodadequaybetweensimulations

andbiologialreordings.

Wegeneralize thesemodels, anddene anew lassofneuronmodels,widebut spei

enoughto keepthediversityofbehaviorsoftheIBGmodels.

1.1 The general lass of non-linear models

Inthispaper,weareinterestedin neuronsdened byadynamialsystemofthetype:

(

dv

dt =F(v)−w+I

dw

dt =a(bv−w)

wherea, bândI âre^real^parametersândF îsâ^real^funtion¹²^.

Inthisequation,v^represents^the^membrane^potentialôf^the^neuron,wîs^theâdaptation

variable, I ^represents^theînput întensity ôf ^the^neuron, 1/a ^theharateristi time ofthe adaptationvariableandbâounts^for^theînteration^between^the^membrane^potentialând

theadaptationvariable 3

.

This equationis averygeneralmodelof neuron. Forinstane whenF ^is ^a^polynomial

of degreethree, weobtain aFitzHugh-Nagumo model, when F îs â^polynomial ôf ^degree

1

Thesamestudyanbedone for aparameter dependentfuntion. More preisely,letE ⊂R

n

bea

parameterspae(foragivenn⁾^andF:E×R→Raparameter-dependentrealfuntion.Alltheproperties showninthissetionarevalidforanyxedvalueoftheparameterp^.^Furtherp-bifurationsstudiesanbe doneforspeiF(p,·)^.

2

TherstequationanbederivedfromthegeneralI^-V ^relationⁱⁿ^neuronal^models:C^d^V

dt =I−I0(V)− g(V −EK)^whereI0(V)^is^theinstantaneousI^-V ^urve.

3

Seeforinstanesetion2.2wherethe parametersoftheinitialequation (2.2 )arerelatedtobiologial

onstantsandwhereweproeedtoadimensionlessredution.

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F

Gerstnermodel[5℄. However,inontrastwithontinuousmodelsliketheFitzHugh-Nagumo

model[8℄,thetwolaterasesdivergewhenspiking,andanexternalresetmehanismisused

afteraspikeisemitted.

Inthispaper,wewantthislassofmodelstohaveommonpropertieswiththeIzhikevih-

Brette-Gerstner(IBG) neuron models. Tothis purpose,let usmakesomeassumptionson

thefuntionF^. ^The^rstâssumptionîsâ^regularityassumption:

Assumption(A1). F ^is^at^least^three ^timesontinuouslydierentiable.

Aseondassumptionisneessarytoensureusthatthesystemwouldhavethesamenumber

ofxedpointsastheIBGmodels.

Assumption(A2). The funtion F ^is^stritly^onvex.

Denition1.1(Convexneuronmodel). Weonsiderthe two-dimensionalmodeldenedby

the equations:

(

dv

dt =F(v)−w+I

dw

dt =a(bv−w) ^(1.1)

whereF ^satises^theassumptions (A1)and(A2)andharaterizesthe passive propertiesof the membrane potential.

Many neurons of this lass blow up in nite time. These neuron are the ones we are

interestedin.

Remark. Notethatalltheneuronsofthislassdonotblowupinnitetime. Forinstane

ifF(v) =vlog(v)^,^it^will^not. ^ForF ^funtions^suh^thatF(v) = (v^1+α)R(v)^for^someα >0^,

where lim

v→∞R(v) > 0 ^(possibly ∞^), ^the ^dynamial ^system ^will ^possibly ^blow ^up ⁱⁿ ^nite

time.

If the solution blows upat time t^∗^, â ^spikeîs êmitted, ând subsequently we havethe followingresetproess:

(v(t^∗) =vr

w(t^∗) =w(t^∗−) +d ^(1.2)

wherevr îs ^the^reset^membrane^potentialândd >0â^real^parameter. ^Theêquations^(1.1)

and(1.2),togetherwithinitialonditions(v0, w0)^giveûs^theêxisteneândûniquenessôfâ

solutiononR

+

.

Thetwoparametersvr ândd âreîmportant^to ûnderstand^the^repetitive^spiking^prop-

erties ofthe system. Nevertheless, the bifurationstudy with respet to these parameters

isoutsidethesopeofthispaper,andwefoushereonthebifurationsofthesystemwith

respetto(a, b, I)^,ⁱⁿ ^order^to^haraterize^thesubthresholdbehavioroftheneuron.

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1.2 Fixed points of the system

Tounderstand thequalitative behaviorof thedynamial systemdened by1.1 beforethe

blowup(i.e. betweentwospikes),webeginbystudyingthexedpointsand analyzetheir

stability. The linear stability of a xed point is governed by the Jaobian matrix of the

system,whihwedenein thefollowingproposition.

Proposition 1.1. The Jaobianof the dynamialsystem (1.1) anbewritten:

L:=v7→

F^′(v) −1 ab −a

(1.3)

Thexedpointsofthesystemsatisfytheequations:

(F(v)−bv+I= 0 bv=w

(1.4)

LetGb(v) :=F(v)−bv^. ^From^(A1)^and ^(A2),^we^know^that ^the^funtion Gb ^is^stritly

onvexandhasthesameregularityasF^. ^Tô^have^the^same^behaviorâs^theÎBG^models,^we

wantthesystemtohavethesamenumberofxed points. Tothispurpose,itis neessary

thatGb^hasâ^minimum^forâllb >0^. Ôtherwise,^theônvex^funtionGb^would^have^no^more

thanone xed point, sinea xedpointof the systemis theintersetion ofan horizontal

urveandGb^.

This means for the funtion F ^that inf

x∈^RF^′(x) ≤0 ^and sup

x∈^R

F^′(x) = +∞ ^. ^Using ^the

monotonypropertyofF^′^, ^we^write^the^assumption^{(A3) :}

Assumption(A3).







x→−∞lim F^′(x)≤0

x→lim+∞F^′(x) = +∞

Assumptions(A1),(A2)and(A3)ensureusthat ∀b∈^R^∗+, Gb ^has^a^unique^minimum,

denotedm(b)^whih^is^reahed. ^Letv^∗(b)^be^the^point^where^this^minimum^is^reahed.

Thispointisthesolutionoftheequation

F^′(v^∗(b)) =b ^(1.5)

Proposition 1.2. The point v^∗(b)^and ^the ^value m(b) ^are ontinuouslydierentiable with respettob^.

Proof. WeknowthatF^′ îsâ^bijetion. ^The^pointv^∗(b)îs^denedîmpliitly^by^theêquation H(b, v) = 0 ^where H(b, v) =F^′(v)−b^. H îs â C¹-dieomorphism with respet to b^, ând

the dierential with respet to b ^never^vanishes. ^The ^impliit ^funtions ^theorem ^(see ^for

instane [7, Annex C.6℄) ensuresus that v^∗(b) ^solutionôf H(b, v^∗(b)) = 0 îs ontinuously dierentiablewith respettob^,ând ^so^doesm(b) =G(v^∗(b))−bv^∗(b)^.

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{(I, b);I=−m(b)}

ofthe system(see gure1 ):

(i). if I >−m(b)^then ^the^system^has ^no^xed^point;

(ii). if I =−m(b)^then ^the ^system^has âûnique ^xed^point, (v^∗(b), w^∗(b))^, ^whih îs^non-

hyperboli. Itisunstableifb > a^.

(iii). if I <−m(b) ^then^the ^dynamial ^system ^has ^two^xed^points(v₋(I, b), v+(I, b))^suh

that

v₋(I, b)< v^∗(b)< v+(I, b).

The xed point v+(I, b) îs â ^saddle ^xed ^pôint, ând ^the ^stability ôf ^the ^xed ^point v₋(I, b)^depends ônI ândôn^the ^signôf (b−a)^:

(a) If b < a^then ^the^xed^pointv₋(I, b)^is^attrative.

(b) If b > a^,^there îsâûnique^smoothûrve I^∗(a, b) ^dened^by ^the împliit êquation F^′(v₋(I^∗(a, b), b)) =a^. ^This ûrve ^reads I^∗(a, b) = bva−F(va) ^where va îs ^the

uniquesolution ofF^′(va) =a^.

(b.1). If I < I^∗(a, b)^the ^xed^point^is^attrative.

(b.2). If I > I^∗(a, b)^the ^xed^point^is^repulsive.

Proof. (i). We haveF(v)−bv ≥m(b)^by ^denition ôf m(b)^. Îf I > −m(b)^, ^then ^forâll v∈^R^we^haveF(v)−bv+I >0ând^the^system^has^no^xed^point.

(ii). Let I = −m(b)^. ^We^have âlready ^seen^that ^that Gb îs ^stritly ônvex,ontinuously dierentiable,andforb >0^reahesîtsûnique^minimumât^the^pointv^∗(b)^. ^This^point

is suh that Gb(v^∗(b)) = m(b)^, ^so ît îs ^theônly ^point ^satisfyingF(v^∗(b))−bv^∗(b)− m(b) = 0^.

Furthermore,this point satises F^′(v^∗(b)) = b^. ^The ^Jaobian ^of ^the ^system ^at ^this

pointreads

L(v^∗(b)) =

b −1 ab −a

.

Itsdeterminantis0^so^the^xed^pointîs^non^hyperboli⁽0îsêigenvalueôf^the^Jaobian

matrix). Thetraeof thismatrixisb−a^. ^So^the^xed^pointv^∗(b)îsâttrative^when b > aând^repulsive^whenb > a^. ^Theâsea=b, I=−m(b)îsâ^degenerateâse^whih

wewillstudy morepreiselyinthesetion1.3.3.

(iii). LetI < −m(b)^. ^By^the^strit ônvexityâssumption^(A2) ôf ^the^funtion G^together

with assumption(A3), weknowthat there areonly twointersetionsof theurveG

to alevel−I ^higher^than ^its^minimum. ^These ^twointersetionsdene ourtwoxed points. At the point v^∗ ^the ^funtion ^is ^stritly ^lower^than −I ^so ^the ^two ^solutions

satisfyv₋(I, b)< v^∗(b)< v+(I, b)^.

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Let us now study the stability of these two xed points. To this end, we have to

haraterizetheeigenvaluesoftheJaobianmatrixofthesystematthesepoints.

Weanseefrom formula(1.3)andtheonvexityassumption(A2) that theJaobian

determinant,equalto−aF^′(v) +ab^,îsâ^dereasing^funtionôfvând^vanishesâtv^∗(b)

sodet(L(v+(I, b)))<0ând^the^xed^pointîsâ^saddle^point^(the ^Jaobian^matrix^has

apositiveandanegativeeigenvalue).

Forthe otherxed point v₋(I, b)^, ^the determinantof theJaobianmatrixis stritly positive. SothestabilityofthexedpointdependsonthetraeoftheJaobian. This

traereads: F^′ v₋(I, b)

−a^.

(a) Whenb < a^,^we^haveâ^stable^xed^point. Îndeed,^the^funtionF^′îsânînreasing

funtion equalto b^at v^∗(b)^so ^Trae

L v₋(I, b)

≤F^′(v^∗(b))−a=b−a < 0

andthexedpointisattrative.

(b) If b > a ^then ^the ^type ôf ^dynamis âround ^the ^xed ^point v₋ ^depends ôn ^the

inputurrent(parameterI^). ^Indeed,^the^trae^reads T(I, b, a) :=F^′ v₋(I, b)

−a,

whihisontinuousandontinuouslydierentiablewithrespettoI ^andb^,^and

whihisdenedforI <−m(b)^. ^We^have:







I→−limm(b)T(I, b, a) =b−a >0

I→−∞lim T(I, b, a) = lim

x→−∞F^′(x)−a <0

SothereexistsaurveI^∗(a, b)^dened^byT(I, b, a) = 0^and^suh^that:

forI^∗(b)< I <−m(b)^,^the^xed^pointv₋(I, b)^is^repulsive.

forI < I^∗(b)^,^the^xed^pointv₋ ^is ^attrative.

Toomputetheequationofthisurve,weusethefat thatpointv₋(I^∗(b), b)^is

suhthat F^′(v₋(I^∗(b), b)) =a^. ^We^know^form ^the^properties^ofF ^that^there ^is

auniquepointva ^satisfying^this êquation. ^SineF^′(v^∗(b)) =b^,a < bând F^′ îs

inreasing,theonditiona < b ^implies^that va< v^∗(b)^.

TheinputurrentassoiatedsatisesxedpointsequationF(va)−bva+I^∗(a, b) = 0^,^orequivalently:

I^∗(a, b) =bva−F(va)

ThepointI =I^∗(a, b)^will ^be^studied ⁱⁿ ^detailⁱⁿ ^the^next ^setion,^sineît îs â

bifurationpointofthesystem.

Figure 1representsinthedierentzonesenumeratedintheorem 1.1andtheirstability

intheparameterplane(I, b)^.