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�

a p p o r t

d e r e c h e r c h e

9 -6 3 9 9 IS R N IN R IA /R R -- 6 1 6 1 -- F R + E N G

Thème BIO

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

Jonathan Touboul

N° 6161

March 4, 2008

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

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

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

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:

(

d

v

d

t = F (v) − w + I

d

w

d

t = a(bv − w)

where

a, b

^and

I

^{arerealparametersand}

F

^{isarealfuntion12.}

Inthisequation,

v

^{representsthemembranepotentialoftheneuron,}

w

^{istheadaptation}

variable,

I

^{representstheinput intensity of theneuron,}

1/a

^theharateristi time ofthe adaptationvariableand

b

^{aountsfortheinterationbetweenthemembranepotentialand}

theadaptationvariable 3

.

This equationis averygeneralmodelof neuron. Forinstane when

F

^{is apolynomial}

of degreethree, weobtain aFitzHugh-Nagumo model, when

F

^{is apolynomial of degree}

1

Thesamestudyanbedone for aparameter dependentfuntion. More preisely,let

E ⊂

R

n

bea

parameterspae(foragiven

n

^)and

F : E ×

R

→

Raparameter-dependentrealfuntion.Alltheproperties showninthissetionarevalidforanyxedvalueoftheparameter

p

^.Further

p

-bifurationsstudiesanbe doneforspei

F (p,· )

^.

2

Therstequationanbederivedfromthegeneral

I

^-

V

^{relationinneuronalmodels:}

C

^d

^V

d

t = I − I 0 (V ) − g(V − E K )

^where

I 0 (V )

^istheinstantaneous

I

^-

V

^urve.

3

Seeforinstanesetion2.2wherethe parametersoftheinitialequation (2.2 )arerelatedtobiologial

onstantsandwhereweproeedtoadimensionlessredution.

F

Gerstnermodel[5℄. However,inontrastwithontinuousmodelsliketheFitzHugh-Nagumo

model[8℄,thetwolaterasesdivergewhenspiking,andanexternalresetmehanismisused

afteraspikeisemitted.

Inthispaper,wewantthislassofmodelstohaveommonpropertieswiththeIzhikevih-

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

thefuntion

F

^{. Therstassumptionisaregularity}assumption:

Assumption(A1).

F

^{isatleastthree times}ontinuouslydierentiable.

Aseondassumptionisneessarytoensureusthatthesystemwouldhavethesamenumber

ofxedpointsastheIBGmodels.

Assumption(A2). The funtion

F

^{isstritlyonvex.}

Denition1.1(Convexneuronmodel). Weonsiderthe two-dimensionalmodeldenedby

the equations:

(

d

v

d

t = F (v) − w + I

d

w

d

t = a(bv − w)

^(1.1)

where

F

^satisestheassumptions (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

if

F (v) = v log(v)

^{,itwillnot. For}

F

^{funtionssuhthat}

F(v) = (v ^1+α )R(v)

^forsome

α > 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 ^∗

^{, a spikeis emitted, and} subsequently we havethe followingresetproess:

( v(t ^∗ ) = v r

w(t ^∗ ) = w(t ^∗− ) + d

^(1.2)

where

v r

^{is theresetmembranepotentialand}

d > 0

^{arealparameter. Theequations(1.1)}

and(1.2),togetherwithinitialonditions

(v 0 , w 0 )

^{giveustheexisteneanduniquenessofa}

solutiononR

+

.

Thetwoparameters

v r

^and

d

^{areimportantto understandtherepetitivespikingprop-}

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

isoutsidethesopeofthispaper,andwefoushereonthebifurationsofthesystemwith

respetto

(a, b, I )

^{,in ordertoharaterizethe}subthresholdbehavioroftheneuron.

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 := v 7→

F ^′ (v) − 1 ab − a

(1.3)

Thexedpointsofthesystemsatisfytheequations:

( F(v) − bv + I = 0 bv = w

(1.4)

Let

G b (v) := F (v) − bv

^{. From(A1)and (A2),weknowthat thefuntion}

G b

^isstritly

onvexandhasthesameregularityas

F

^{. TohavethesamebehaviorastheIBGmodels,we}

wantthesystemtohavethesamenumberofxed points. Tothispurpose,itis neessary

that

G b

^{hasaminimumforall}

b > 0

^{. Otherwise,theonvexfuntion}

G b

^{wouldhavenomore}

thanone xed point, sinea xedpointof the systemis theintersetion ofan horizontal

urveand

G b

^.

This means for the funtion

F

^that

inf

x ∈

^R

F ^′ (x) ≤ 0

^and

sup

x ∈

^R

F ^′ (x) = + ∞

^{. Using the}

monotonypropertyof

F ^′

^{, wewritetheassumption(A3) :}

Assumption(A3).







x →−∞ lim F ^′ (x) ≤ 0

x → lim + ∞ F ^′ (x) = + ∞

Assumptions(A1),(A2)and(A3)ensureusthat

∀ b ∈

^R

^∗ + , G b

^{hasauniqueminimum,}

denoted

m(b)

^{whihisreahed. Let}

v ^∗ (b)

^{bethepointwherethisminimumisreahed.}

Thispointisthesolutionoftheequation

F ^′ (v ^∗ (b)) = b

^(1.5)

Proposition 1.2. The point

v ^∗ (b)

^{and the value}

m(b)

^are ontinuouslydierentiable with respetto

b

^.

Proof. Weknowthat

F ^′

^{isabijetion. Thepoint}

v ^∗ (b)

^{isdenedimpliitlybytheequation}

H (b, v) = 0

^where

H (b, v) = F ^′ (v) − b

^.

H

^{is a}

C ¹

-dieomorphism with respet to

b

^{, and}

the dierential with respet to

b

^{nevervanishes. The impliit funtions theorem (see for}

instane [7, Annex C.6℄) ensuresus that

v ^∗ (b)

^solutionof

H (b, v ^∗ (b)) = 0

^is ontinuously dierentiablewith respetto

b

^{,and sodoes}

m(b) = G(v ^∗ (b)) − bv ^∗ (b)

^.

{ (I, b); I = − m(b) }

ofthe system(see gure1 ):

(i). if

I > − m(b)

^{then thesystemhas noxedpoint;}

(ii). if

I = − m(b)

^{then the systemhas aunique xedpoint,}

(v ^∗ (b), w ^∗ (b))

^{, whih isnon-}

hyperboli. Itisunstableif

b > a

^.

(iii). if

I < − m(b)

^{thenthe dynamial system has twoxedpoints}

(v ₋ (I, b), v + (I, b))

^suh

that

v ₋ (I, b) < v ^∗ (b) < v + (I, b).

The xed point

v + (I, b)

^{is a saddle xed point, and the stability of the xed point}

v ₋ (I, b)

^{depends on}

I

^{andonthe signof}

(b − a)

^:

(a) If

b < a

^{then thexedpoint}

v ₋ (I, b)

^isattrative.

(b) If

b > a

^{,there isauniquesmoothurve}

I ^∗ (a, b)

^{denedby the impliit equation}

F ^′ (v ₋ (I ^∗ (a, b), b)) = a

^{. This urve reads}

I ^∗ (a, b) = bv a − F(v a )

^where

v a

^{is the}

uniquesolution of

F ^′ (v a ) = a

^.

(b.1). If

I < I ^∗ (a, b)

^{the xedpointisattrative.}

(b.2). If

I > I ^∗ (a, b)

^{the xedpointisrepulsive.}

Proof. (i). We have

F (v) − bv ≥ m(b)

^{by denition of}

m(b)

^{. If}

I > − m(b)

^{, then forall}

v ∈

^Rwehave

F (v) − bv + I > 0

^{andthesystemhasnoxedpoint.}

(ii). Let

I = − m(b)

^{. Wehave already seenthat that}

G b

^{is stritly onvex,}ontinuously dierentiable,andfor

b > 0

^{reahesitsuniqueminimumatthepoint}

v ^∗ (b)

^{. Thispoint}

is suh that

G b (v ^∗ (b)) = m(b)

^{, so it is theonly point satisfying}

F (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

.

Itsdeterminantis

0

^{sothexedpointisnonhyperboli(}

0

^{iseigenvalueoftheJaobian}

matrix). Thetraeof thismatrixis

b − a

^{. Sothexedpoint}

v ^∗ (b)

^{isattrativewhen}

b > a

^{andrepulsivewhen}

b > a

^{. Thease}

a = b, I = − m(b)

^{isadegenerateasewhih}

wewillstudy morepreiselyinthesetion1.3.3.

(iii). Let

I < − m(b)

^{. Bythestrit onvexityassumption(A2) of thefuntion}

G

^together

with assumption(A3), weknowthat there areonly twointersetionsof theurve

G

to alevel

− I

^{higherthan itsminimum. These two}intersetionsdene ourtwoxed points. At the point

v ^∗

^{the funtion is stritly lowerthan}

− I

^{so the two solutions}

satisfy

v ₋ (I, b) < v ^∗ (b) < v + (I, b)

^.

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

^{,isadereasingfuntionof}

v

^{andvanishesat}

v ^∗ (b)

sodet

(L(v + (I, b))) < 0

^{andthexedpointisasaddlepoint(the Jaobianmatrixhas}

apositiveandanegativeeigenvalue).

Forthe otherxed point

v ₋ (I, b)

^{, the} determinantof theJaobianmatrixis stritly positive. SothestabilityofthexedpointdependsonthetraeoftheJaobian. This

traereads:

F ^′ v ₋ (I, b)

− a

^.

(a) When

b < a

^{,wehaveastablexedpoint. Indeed,thefuntion}

F ^′

^{isaninreasing}

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 of dynamis around the xed point}

v ₋

^{depends on the}

inputurrent(parameter

I

^{). Indeed,thetraereads}

T (I, b, a) := F ^′ v ₋ (I, b)

− a,

whihisontinuousandontinuouslydierentiablewithrespetto

I

^and

b

^,and

whihisdenedfor

I < − m(b)

^{. Wehave:}







I →− lim m(b) T (I, b, a) = b − a > 0

I →−∞ lim T (I, b, a) = lim

x →−∞ F ^′ (x) − a < 0

Sothereexistsaurve

I ^∗ (a, b)

^denedby

T (I, b, a) = 0

^andsuhthat:

ˆ for

I ^∗ (b) < I < − m(b)

^,thexedpoint

v ₋ (I, b)

^isrepulsive.

ˆ for

I < I ^∗ (b)

^,thexedpoint

v ₋

^{is attrative.}

Toomputetheequationofthisurve,weusethefat thatpoint

v ₋ (I ^∗ (b), b)

^is

suhthat

F ^′ (v ₋ (I ^∗ (b), b)) = a

^{. Weknowform thepropertiesof}

F

^{thatthere is}

auniquepoint

v a

^{satisfyingthis equation. Sine}

F ^′ (v ^∗ (b)) = b

^,

a < b

^and

F ^′

^is

inreasing,theondition

a < b

^impliesthat

v a < v ^∗ (b)

^.

Theinputurrentassoiatedsatisesxedpointsequation

F(v a ) − bv a +I ^∗ (a, b) = 0

^,orequivalently:

I ^∗ (a, b) = bv a − F (v a )

Thepoint

I = I ^∗ (a, b)

^{will bestudied in detailin thenext setion,sineit is a}

bifurationpointofthesystem.

Figure 1representsinthedierentzonesenumeratedintheorem 1.1andtheirstability

intheparameterplane

(I, b)

^.

2 3 4 5 J 6 7 8 9 10

K 6

K 5

K 4

K 3

K 2

K 1 0 1 2 b

No fixed point

Two fixed points:

: One saddle point : One repulsive point Two fixed points:

: One saddle point F One attractive point

I = -m(b) g

f I = b va - F(va) I

Figure1: Numberofxedpointsandtheirstabilityintheplane

(I, b)

^,fortheexponential adaptivemodel.

Remark. Inthis proof, weused thefat that

F ^′

^{is invertibleon}

[0, ∞ )

^{. Theassumption}

(A3) ensuresus that it will bethe ase, and that

F

^{has auniqueminimum. Assumption}

(A3)istheweakestpossibleto havethisproperty.

1.3 Bifurations of the system

Inthestudyofthexedpointsandtheirstability,weidentiedtwobifurationurveswhere

thestabilityofthexedpointshanges. Thersturve

I = − m(b)

^{orrespondstoasaddle-}

node bifuration, andthe urve

I = I ^∗ (a, b)

^{to an} Andronov-Hopf bifuration. These two urvesmeetinaspeipoint,

b = a

^and

I = − m(a)

^{. Thispointhasadouble}

0

^eigenvalue

andweshowthatitisaBogdanov-Takensbifurationpoint.

Letusshowthatthesystemundergoesthesebifurationswithnomoreassumptionthan

(A1), (A2) and (A3) on

F

^{. We also prove that the system an undergo only one other}

odimensiontwobifuration,aBautinbifuration.

1.3.1 Saddle-node bifuration urve

In this setion we haraterize the behavior of the dynamial system along the urve of

equation

I = − m(b)

^{andweprovethefollowingtheorem:}

Theorem1.2. The dynamial system (1.1) undergoes asaddle-node bifuration along the

parameterurve:

(SN) : { (b, I) ; I = − m(b) } ,

^(1.6)

when

F ^′′ (v ^∗ (b)) 6 = 0

^.

Proof. We derive the normal form of the systemat this bifuration point. Followingthe

works of GukenheimerHolmes [9℄ and Kuznetsov [19℄, we only hek the transversality

onditionstobesurethat thenormalformat thebifurationpointwill havetheexpeted

form.

Let

b ∈

^R

⁺

^and

I = − m(b)

^{. Let}

v ^∗ (b)

^{betheuniquexedpointof thesystemforthese}

parameters. The point

v ^∗ (b)

^{is the unique solution of}

F ^′ (v ^∗ (b)) = b

^{. At this point, the}

Jaobianmatrix(1.3)reads:

L(v ^∗ (b)) =

b − 1 ab − a

This matrix hastwoeigenvalues

0

^and

b − a

^{. Thepairs of right} eigenvaluesand right eigenvetorsare:

0, U :=

1/b 1

and

b − a, 1/a

1

Itspairsoflefteigenvaluesandleft eigenvetorsare:

0, V := ( − a, 1)

^and

b − a, ( − b, 1)

Let

f b,I

^{bethevetoreld}

f b,I (v, w) =

F (v) − w + I a(bv − w)

.

Thevetoreld satises:

V ∂

∂I f b,I (v ^∗ (b), w ^∗ (b))

= ( − a, 1) · 1

0

= − a < 0

So theoeient of thenormal form orresponding to the Taylorexpansion alongthe

parameter

I

^{doesnotvanish.}

FinallyletusshowthatthequadratitermsoftheTaylorexpansioninthenormalform

doesnotvanish. With ournotations,thisonditionreads:

V

D ² _x f _{b, − m(b)} (v ^∗ (b), w ^∗ (b))(U, U )

6

= 0.

Thispropertyissatisedin ourframework. Indeed,

V

D _x ² f b, − m(b) (v ^∗ (b), w ^∗ (b))(U, U )

= V (

U ₁ ² ∂ ² f 1

∂v ² + 2U 1 U 2 ∂ ² f 1

∂v∂w + U ₂ ² ∂ ² f 1

∂w ² U ₁ ² ∂ ² f 2

∂v ² + 2U 1 U 2

∂ ² f 2

∂v∂w + U ₂ ² ∂ ² f 2

∂w ²

)

= V ( 1

b ² F ^′′ (v ^∗ ) 0

)

= ( − a, 1) · 1

b ² F ^′′ (v ^∗ ) 0

= − a

b ² F ^′′ (v ^∗ ) < 0

Sothesystemundergoesasaddle-nodebifurationalongthemanifold

I = − m(b)

^.

Remark. Note that

F ^′′ (v ^∗ (b))

^{an vanish only ountably many times sine}

F

^{is stritly}

onvex.

1.3.2 Andronov-Hopfbifuration urve

Inthissetionweonsiderthebehaviorofthedynamialsystemalongtheparameterurve

I = I ^∗ (b)

^{andweonsiderthexedpoint}

v ₋

^.

Theorem1.3. Let

b > a

^,

v a

^{be the uniquepoint suhthat}

F ^′ (v a ) = a

^and

A(a, b)

^dened

bythe formula:

A(a, b) := F ^′′′ (v a ) + 1

b − a (F ^′′ (v a )) ² .

^(1.7)

If

F ^′′ (v a ) 6 = 0

^and

A(a, b) 6 = 0

^{,thenthesystemundergoes an}Andronov-Hopfbifuration atthe point

v a

^{,alongthe parameter line}

(AH) := n

(b, I) ; b > a

^and

I = bv a − F (v a ) o

(1.8)

This bifuration issubritialif

A(a, b) > 0

^andsuperritial if

A(a, b) < 0

^.

Proof. TheJaobianmatrixat thepoint

v a

^reads:

L(v a ) =

a − 1 ab − a

Its trae is

0

^{and its} determinant is

a(b − a) > 0

^{so the matrix at this point has a}

pair of pure imaginary eigenvalues

(iω, − iω)

^where

ω = p

a(b − a)

^{. Along the urve of}

equilibria when

I

^{varies, the} eigenvalues are omplex onjugates with real part

µ(I) =

1 2 T r

L v ₋ (I, b)

whih vanishesat

I = I ^∗ (a, b)

^.

Wereallthat from proposition1.2,thistraevaries smoothlywith

I

^{. Indeed,}

v ₋ (b, I)

satises

F(v ₋ (I, b)) − bv ₋ (I, b) + I = 0

^andisdierentiablewithrespetto

I

^{. Wehave:}

∂v ₋ (I, b)

∂I (F ^′ (v ₋ (I, b)) − b) = − 1

Atthepoint

v ₋ (I ^∗ (b), b) = v a

^,wehave

F ^′ (v a ) = a < b

^sofor

I

^losefromthisequilibrium point,wehave

∂v ₋ (I, b)

∂I > 0

Nowlet us hek that the transversalityondition of anAndronov-Hopf bifuration is

satised(see[9,Theorem3.4.2℄). Therearetwoonditionstobesatised: thetransversality

ondition d

µ(I)

d

I 6 = 0

^andthenon-degenerayondition

l 1 6 = 0

^where

l 1

^{is therstLyapunov}

oeientatthebifurationpoint.

Firstofall,weprovethatthetransversalityonditionissatised:

µ(I) = 1

2 T r(L(v ₋ (I, b)))

= 1

2 (F ^′ (v ₋ (I, b)) − a)

d

µ(I)

d

I = 1

2 F ^′′ (v ₋ (I, b))

^d

v ₋ (I, b)

d

I

> 0

Letusnowwrite thenormalformat thispoint. Tothis purpose,wehangevariables:

( v − v a = x w − w a = ax + ωy

The

(x, y)

^{equationreads:}

( x ˙ = − ωy + (F (x + v a ) − ax − w a ) =: − ωy + f (x)

˙

y = ωx + _ω ^a (ax − F(x + v a ) + w a − I) =: ωx + g(x)

^(1.9)

AordingtoGukenheimerin[9℄,westatethat theLyapunovoeientofthesystem

atthispointhasthesamesignas

B

^where

B

^{is denedby:}

B := 1

16 [f xxx + f xyy + g xxy + g yyy ] + 1

16ω [f xy (f xx + f yy ) − g xy (g xx + g yy ) − f xx g xx + f yy g yy ]

Replaing

f

^and

g

^bytheexpressionsfoundin (1.9),weobtaintheexpressionof

A

^:

B = 1

16 F ^′′′ (v a ) + a

16ω ² (F ^′′ (v a )) ²

= 1

16 F ^′′′ (v a ) + 1

16(b − a) (F ^′′ (v a )) ²

= 1 16 A(a, b)

Hene when

A(a, b) 6 = 0

^{, the system undergoesan} Andronov-Hopf bifuration. When

A(a, b) > 0

^{, the bifuration is subritial and the periodi orbits generated by the Hopf}

bifuration are repelling, and when

A(a, b) < 0

^{, the bifuration is} superritial and the periodi orbitsare attrative(the formula of

A

^{hasbeenalso introdued by Izhikevih in}

[16,eq.15(p.213)℄).

Remark.Thease

A(a, b) = 0

^{isnottreatedinthetheoremandisalittlebitmoreintriate.}

Wefullytreatitinsetion1.3.4andshowthataBautin(generalizedHopf)bifurationan

ourifthe

A

^{-oeientvanishes. Sinethethirdderivativeisapriori}unonstrained,this aseanourandweprovein setion3that thisistheaseforasimple(quarti)model.

1.3.3 Bogdanov-Takens bifuration

We have seen in the study that this formal model presents an interesting point in the

parameterspae,orrespondingtotheintersetionofthesaddle-nodebifurationurveand

theAndronov-Hopf bifurationurve. Atthis point,weshowthatthesystemundergoesa

Bogdanov-Takensbifuration.

Theorem1.4. Let

F

^{bearealfuntionsatisfyingthe}assumptions(A1),(A2)and(A3). Let

a ∈

^R

^∗ + , b = a

^and

v a

^{theonly pointsuhthat}

F ^′ (v a ) = a

^{. Assumeagain that}

F ^′′ (v a ) 6 = 0

^.

Then at this point and with these parameters, the dynamial system (1.1) undergoes a

subritialBogdanov-Takens bifuration of normalform:

( η ˙ 1 = η 2

˙ η 2 =

8F ^′′ (v a ) a I 1

(a+b 1 ) ³

−

2(2 b 1 a+I 1 F ^′′ (v a )) (a+b 1 ) ²

η 1 + η ² ₁ + η 1 η 2 + O ( k η k ³ )

^(1.10)

where

b 1 := b − a

^and

I 1 = I + m(a)

^.

Proof. TheJaobianmatrix(1.3)atthispointreads:

L(v a ) =

a − 1 a ² − a

This matrixisnon-zeroandhastwozeroeigenvalues(itsdeterminantandtraeare

0

^).

The matrix

Q :=

a 1 a ² − a

is the passagematrix to the Jordan form of the Jaobian

matrix:

Q ^{− 1} · L(v a ) · Q =

0 1 0 0

ToprovethatthesystemundergoesaBogdanov-Takensbifuration,weshowthatthenormal

formreads:

( η ˙ 1 = η 2

˙

η 2 = β 1 + β 2 η 1 + η ² ₁ + ση 1 η 2 + O ( k η k ³ )

^(1.11)

with

σ = ± 1

^{. Theproofofthistheoremonsistsin(i)provingthatthesystemundergoesa}

Bogdanov-Takensbifuration,(ii)ndingalosed-formexpression forthevariables

β 1

^and

β 2

^{and(iii)provingthat}

σ = 1

^.

Firstofall,letusprovethatthenormalformanbewrittenintheformof (1.11) . This

is equivalent to showing sometransversalityonditions on thesystem (see forinstane in

[19,Theorem8.4℄).

Tothisend,weentertheequationatthispointandwritethesystemintheoordinates

givenbytheJordanformofthematrix. Let

y 1

y 2

= Q ^{− 1} _w ^{v −} ₋ ^v _w ^a _a

,atthepoint

b = a + b 1 , I =

− m(a) + I 1

^{. Weget:}

( y ˙ 1 = y 2 + ^b _a ¹ (ay 1 + y 2 )

˙

y 2 = F (ay 1 + y 2 + v a ) − w a − m(a) + I 1 − a ² y 1 − ay 2 − b 1 (ay 1 + y 2 )

^(1.12)

Letusdenote

v 1 = ay 1 + y 2

^{. TheTaylorexpansionontheseondequationgivesus:}

˙

y 2 = F (v 1 + v a ) − w a − m(a) + I 1 − a ² y 1 − ay 2 − b 1 (ay 1 + y 2 )

= F (v a ) + F ^′ (v a )v 1 + 1

2 F ^′′ (v a )v ₁ ² − w a − m(a) + I 1 − a ² y 1 − ay 2 − b 1 (ay 1 + y 2 ) + O ( k v 1 k ³ )

= (F (v a ) − w a − m(a)) + I 1 + (F ^′ (v a ) − a)v 1 − b 1 v 1 + 1

2 F ^′′ (v a )v ₁ ² + O ( k v 1 k ³ )

= I 1 − b 1 (ay 1 + y 2 ) + 1

2 F ^′′ (v a )(ay 1 + y 2 ) ² + O ( k y k ³ )

^(1.13)

Letusdenoteforthesakeoflarity

α = (b 1 , I 1 )

^{andwritetheequations(1.12)as:}

( y ˙ 1 = y 2 + a 00 (α) + a 10 (α)y 1 + a 01 (α)y 2

˙

y 2 = b 00 (α) + b 10 (α)y 1 + b 01 (α)y 2 + ¹ ₂ b 20 (α)y ₁ ² + b 11 (α)y 1 y 2 + ¹ ₂ b 02 (α)y ² ₂ + O ( k y k ³ )

(1.14)

theoeients

a ij (α)

^and

b ij (α)

^.

Letusnowusethehangeofvariables:

( u 1 = y 1

u 2 = y 2 + ^b _a ¹ (ay 1 + y 2 )

Thedynamialsystemgoverning

(u 1 , u 2 )

^reads:

( u ˙ 1 = u 2

˙

u 2 = (1 + ^b _a ¹ ) − b 1 a u 1 + ¹ ₂ ^{a 3 F} _a+b ^{′′ (v a )}

1 u ² ₁ + ^{a 2} _a+b ^{F ′′ (v a )}

1 u 1 u 2 + ¹ ₂ ^aF _a+b ^{′′ (v a )}

1 u ² ₂

The transversality onditions of a Bogdanov-Takens bifuration [9, 19℄ an easily be

veriedfrom thisexpression:

(BT.1). TheJaobianmatrixisnot

0

^.

(BT.2). With thenotationsof (1.14),wehave

a 20 = 0

^and

b 11 (0) = aF ^′′ (v a ) > 0

^so

a 20 (0) + b 11 (0) = aF ^′′ (v a ) > 0

^.

(BT.3).

b 20 = a ² F ^′′ (v a ) > 0

^.

(BT.4). Weshowthatthemap:

x :=

y 1

y 2

, α :=

I 1

b 1

7→ h

f (x, α),

^Tr

D x f (x, α

,

^Det

Dxf(x, α) i

isregularatthepointofinterest.

Fromthe tworst assumptions, weknowthat thesystem anbe put in theform of

(1.11). Gukenheimer in [9℄ proves that this ondition an be redued to the non-

degenerayofthedierentialwithrespetto

(I 1 , b 1 )

^ofthevetor

^β _β ¹

2

oftheequation

(1.11).

Inourase,wean omputethese variables

β 1

^and

β 2

^{followingthealulationsteps}

of[19℄andweget:

( β 1 = ^8F _(a+b ^{′′ (v a ) a I 1}

1 ) ³

β 2 = − ^{2(2 b 1} _(a+b ^{a+I 1} ₁ ^F ₎ ^{2 ′′ (v a ))}

(1.15)

Hene thedierentialofthevetor

β 1

β 2

withrespetto theparameters

(I 1 , b 1 )

^atthe

point

(0, 0)

^reads:

D α β | (0,0) =

8F ^′′ (v a )

a ² 0

− 2 ^{F ′′} _a ^(v 2 ^{a )} − 4/a

!

Thismatrixhasanon-zerodeterminantifandonlyif

F ^′′ (v a ) 6 = 0

Therefore we have proved the existene of a Bogdanov-Takens bifuration under the

ondition

F ^′′ (v a ) 6 = 0

Letusnowshowthat

σ = 1

^{. Indeed,thisoeientisgivenbythesignof}

b 20 (0)

a 20 (0)+

b 11 (0)

whih inouraseisequalto

a ³ F ^′′ (v a ) ² > 0

^{sothebifurationisalwaysofthetype}

(1.10)(generationofanunstablelimityle)forallthemembersofourlassofmodels.

The existeneof aBogdanov-Takensbifuration point impliesthe existeneof smooth

urveorrespondingtoasaddlehomolinibifurationinthesystem(see[19,lemma8.7℄).

Corollary 1.3. Thereis aunique smooth urve

(P )

orresponding toa saddle homolini bifuration in the system (1.1) originating at the parameter point

b = a

^and

I = − m(a)

denedby the impliitequation:

(P ) := n

(I = − m(a) + I 1 , b = a + b 1 ) ;

I 1 =

− ²⁵ 6 a − ³⁷ 6 b 1 + ⁵ ₆ p

25 a ² + 74 b 1 a + 49 b 1 2 a

F ^′′ (v a ) + o(( | b 1 | + | I 1 | ) ²

^(1.16)

and

b 1 > − I 1 F ^′′ (v a ) 2a

o

Moreover,for

(b, I)

^inaneighborhoodof

(a, − m(a))

^{,thesystemhasauniqueandhyper-}

boli unstableyle for parametervalues inside the region bounded by the Hopf bifuration

urveandthe homolini bifuration urve

(P )

^{,andno yleoutsidethis region.}

Proof. As notied, from the Bogdanov-Takens bifuration point, wehave theexistene of

thissaddlehomolinibifurationurve. Letusnowomputetheequationof thisurvein

theneighborhoodof theBogdanov-Takenspoint. Tothispurpose weusethe normalform

wederivedintheorem1.4andusetheloalharaterizationgivenforinstanein[19,lemma

8.7℄forthesaddlehomoliniurve:

(P) :=

(β 1 , β 2 ) ; β 1 = − 6

25 β ₂ ² + o(β ₂ ² ), β 2 < 0

Usingtheexpressions(1.15)yields:

(P) := n

(I = − m(a) + I 1 , b = a + b 1 ) ; 8F ^′′ (v a )aI 1

(a + b 1 ) ³ = 24 25

(2 b 1 a + I 1 F ^′′ (v a )) ²

(a + b 1 ) ⁴ + o( | a 1 | + | I 1 | )

and

b 1 > − I 1 F ^′′ (v a ) 2a

o

We an solvethis equation. There are twosolutions, but the only one satisfying

I 1 = 0

when

b 1 = 0

^{. Thissolutionistheurveofsaddlehomolini}bifurations.

InthestudyoftheAndronov-Hopfbifuration,weshowedthatthesuborsuperritialtype

of bifuration depended on the variable

A(a, b)

^{dened by (1.7). If this variable hanges}

sign when

b

^{varies, then the stability of the limit yle alongHopf bifuration hanges of}

stability. Thisanourifthepoint

v a

^{satisestheondition:}

Assumption(A4). For

v a

^suhthat

F ^′ (v a ) = a

^,wehave:

F ^′′′ (v a ) < 0

Indeed,ifthishappens,thetypeofAndronov-Hopfbifurationhanges,sinewehave:







b lim → a ⁻ A(a, b) = + ∞

b → lim + ∞ A(a, b) = F ^′′′ (v a ) < 0

InthisasetherstLyapunovexponentvanishesfor

b = a − (F ^′′ (v a )) ² F ^′′′ (v a )

Atthispoint,thesystemhastheharateristisofaBautin(generalizedHopf)bifuration.

Neverthelesswestillhavetohektwonon-degenerayonditionstoensurethatthesystem

atuallyundergoesaBautinbifuration:

(BGH.1). Theseond Lyapunovoeientof thedynamialsystem

l 2

^{, doesnotvanish at this}

equilibriumpoint

(BGH.2). Let

l 1 (I, b)

^{betherstLyapunovexponentofthissystemand}

µ(I, b)

^{therealpartof}

theeigenvaluesoftheJaobianmatrix. Themap

(I, b) 7→ (µ(I, b), l 1 (I, b))

isregularatthispoint.

Inthisasethesystemwould beloallytopologiallyequivalenttothenormalform:

( y ˙ 1 = β 1 y 1 − y 2 + β 2 y 1 (y ² ₁ + y ₂ ² ) + σy 1 (y ² ₁ + y ² ₂ ) ² ,

˙

y 2 = β 1 y 2 − y 1 + β 2 y 2 (y ² ₁ + y ₂ ² ) + σy 2 (y ² ₁ + y ² ₂ ) ²

WereduetheproblemtothepointthathekingthetwoonditionsofaBGHbifuration

beomesstraightforward.

Let

(v a , w a )

^{the point where the system undergoes the Bautin bifuration (when it}

exists). SinewealreadyomputedtheeigenvaluesandeigenvetorsoftheJaobianmatrix

alongtheAndronov-Hopfbifurationurve,weanuseittoreduetheproblem. Thebasis

whereweexpressthesystemisgivenby:



 

  Q :=

1 b

ω ab

1 0

!

x y

:= Q ^{− 1} _w ^{v −} ₋ ^v _w ^a _a

Letuswritethedynamialequationssatisedby

(x, y)

^:

( x ˙ = ωy

˙

y = ^ab _ω F v a + ¹ _b x + _ab ^ω y

− w a − x + I a − ay

To ensure that we have a Bautin bifuration at this point we will need to perform a

Taylorexpansionuptothefthorder,soweneedtomaketheassumption:

Assumption(A5). The funtion

F

^issixtimesontinuouslydierentiableat

(v a , w a )

^.

Firstletusdenote

v 1 (x, y) = ¹ _b x + _ab ^ω y

^{,theTaylorexpansionreads:}

˙ y = ab

ω F(v a ) − w a + I) + ab

ω [F ^′ (v a )v 1 (x, y) − ay] + 1 2

ab ω

F ^′′ (v a )v 1 (x, y) ² + 1

6 ab

ω F ^′′′ (v a )v 1 (x, y) ³ + 1 4!

ab

ω F ⁽⁴⁾ (v a )v 1 (x, y) ⁴ + 1

5!

ab

ω F ⁽⁵⁾ (v a )v 1 (x, y) ⁵ + O ( k x

y

k ⁶ )

This expression together with the omplex left and right eigenvetors of the Jaobian

matrix allow us to ompute the rst and seond Lyapunov oeients and to hek the

existeneofaBautinbifuration.

Nevertheless, we annot push the omputation any further at this level of generality,

but, fora given funtion

F

^{presenting ahange in the sign of}

A(a, b)

^{, aneasily bedone}

throughtheuseofasymboliomputationpakage. Theinterestedreaderisreferredtothe

appendixAforhekingtheBautinbifurationtransversalityonditions,wherealulations

aregivenforthequartineuronmodel.

1.4 Conlusion: the full bifuration diagram

Wenowsummarizetheresultsobtainedin thissetioninthetwofollowingtheorems:

Theorem1.5. Letusonsider the formal dynamial system

( v ˙ = F(v) − w + I

˙

w = a(bv − w)

(1.17)

where

a

^isaxedreal,

b

^and

I

^{bifuration parametersand}

F :

^R

7→

^{Rarealfuntion.}

If thefuntion

F

^{satisesthe following}assumptions:

F

(A.2).

F

^{isstritlyonvex,and}

(A.3).

F ^′

^{satises theonditions:}







x →−∞ lim F ^′ (x) ≤ 0

x lim →∞ F ^′ (x) = ∞

Then the dynamial system (1.17)shows thefollowing bifurations:

(B1). A saddle-nodebifuration urve:

(SN ) : { (b, I) ; I = − m(b) } ,

where

m(b)

^{is the minimum of the funtion}

F(v) − bv

^{(if the seond derivative of}

F

doesnotvanish atthis point)

(B2). An Andronov-Hopfbifurationline:

(AH) := n

(b, I) ; b > a

^and

I = bv a − F (v a ) o

where

v a

^{is the unique solution of}

F ^′ (v a ) = a

^{, and if}

F ^′′ (v a ) 6 = 0

^{. The type of this}

Andronov-Hopfbifuration isgiven bythe sign ofthe variable

A(a, b) = F ^′′′ (v a ) + 1

b − a F ^′′ (v a ) ² .

If

A(a, b) > 0

^{then the bifuration is subritial and if}

A(a, b) < 0

^{, the bifuration is}

superritial.

(B3). ABogdanov-Takensbifurationpointatthepoint

b = a

^and

I = − m(a)

^,if

F ^′′ (v a ) 6 = 0

^.

(B4). AsaddlehomolinibifurationurveharaterizedintheneighborhoodoftheBogdanov-

Takenspointby:

(P ) := n

(I = − m(a) + I 1 , b = a + b 1 ) ;

I 1 =

− ²⁵ ₆ a − ³⁷ ₆ b 1 + ⁵ ₆ p

25 a ² + 74 b 1 a + 49 b 1 2 a

F ^′′ (v a ) + o(( | b 1 | + | I 1 | )

and

b 1 > − I 1 F ^′′ (v a ) 2a

o

Theorem1.6. Consider the system (1.1)where

a

^{isagiven real number and}

b

^and

I

^are

realbifurationparameters and

F : E ×

^R

7→

^{Rbeafuntion satisfyingthe} assumptions:

(A.5). The funtion

F

^{is sixtimes}ontinuouslydierentiable (A.2).

F

^{isstritlyonvex,and}

(A.3).

F ^′

^{satises theonditions:}







x →−∞ lim F ^′ (x) ≤ 0

x lim →∞ F ^′ (x) = ∞

(A.4). Let

v a

^{bethe uniquereal suhthat}

F ^′ (v a ) = a

^{. Wehave:}

F ^′′′ (v a ) < 0

If wehavefurthermore:

(BGH.1). The seondLyapunovoeientof the dynamial system

l 2 (v a ) 6 = 0

^;

(BGH.2). Let

l 1 (v)

^{denotetherstLyapunovexponent,}

λ(I, b) = µ(I, b) ± iω(I, b)

^theeigenvalues of the Jaobian matrixinthe neighborhoodof the pointof interest. The map

(I, b) → (µ(I, b), l 1 (I, b))

^{isregularatthis point.}

Then the systemundergoesa Bautin bifuration at the point

v a

^{for the parameters}

b = a − ^F _F ^{′′ ′′′ (v} _(v ^a _a ^{) 2} ₎

^and

I = bv a − F (v a )

^.

Remark. Theorem1.5enumeratessomeof thebifurations thatanydynamialsystemof

thelass(1.1)willalwaysundergo. Togetherwiththeorem1.6,theysummarizealltheloal

bifurations the system an undergo, and no other xed-point bifuration is possible. In

setion3weintrodueamodelatuallyshowingalltheseloalbifurations.

2 Appliations: Izhikevih and Brette-Gerstner models

Inthissetionweshowthattheneuronmodelsproposed byEugeneIzhikevihin [14℄and

BretteandGerstnerin[5℄arepartofthelassstudiedinsetion1. Usingtheresultsofthe

later setion, we derive their bifuration diagram,and obtainthat they show exatlythe

sametypesofbifurations.

2.1 Izhikevih quadrati adaptive model

Weproduehereaompletedesriptionofthebifurationdiagramoftheadaptivequadrati

integrate-and-remodel proposed by Izhikevih[14℄and [16, hapter 8℄. Weuse herethe

dimensionlessequivalentversionofthismodel,withthefewestparameters:

( v ˙ = v ² − w + I

˙

w = a(bv − w)

^(2.1)

F(v) = v ²

F

^{is learly stritly onvex and}

C ^∞

^.

F ^′ (v) = 2v

^{so it satises also the ondition (A3) .}

Furthermore,theseondderivativenevervanishesso thesystemundergoesthethreebifur-

ationsstatedin theorem1.5

(Izh.B1). Asaddle-nodebifurationurvedened by

(b, I) ; I = b ² 4

.

For

(I, b) ∈

^R

²

^{,thexedpointisgivenby}

(v ^∗ (b) = ¹ ₂ b, w ^∗ (b) = ¹ ₂ b ² )

^.

For

I < ^b ₄ ²

^{,thexedpoint(s)are:}

v _± (b, I) = 1 2 b ± p

b ² − 4I

(Izh.B2). An Andronov-Hopf bifurationline:

n (I, b) ; b > a

^and

I = a 2 (b − a

2 ) o ,

whosetypeisgivenbythesignofthevariable

A(a, b) = 4 b − a

Thisvalueisalwaysstritlypositive,sothebifurationisalwayssubritial.

(Izh.B3). ABogdanov-Takensbifurationpointfor

b = a

^and

I = ^a ₄ ²

^,

v a = ^a ₂

^.

(Izh.B4). A saddle homolini bifuration urve satisfying the quadrati equation near the

Bogdanov-Takenspoint:

(P ) := n (I = a ²

2 + I 1 , b = a + b 1 ) ; I 1 = a

2

− 25 6 a − 37

6 b 1 + 5 6

q

25 a ² + 74 b 1 a + 49 b 1 2

+ o(( | b 1 | + | I 1 | )

and

b 1 > − I 1

a o

ThegureFig.2representsthexedpointsofthisdynamialsystem,andtheirstability,

togetherwiththebifuration urves.

0 10

b 20 10 5 v* 0 –5

–200 –150 –100 –50 0 50 I

Figure 2: Representation of the

v

^{xed point with respet to the parameters}

I

^and

b

ⁱⁿ

theIzhikevihmodel. Theredomponentisthesurfaeofsaddlexedpoints,theblueone

orrespondstotherepulsivexedpointsandthegreenonetotheattrativexedpointsThe

yellowurveorrespondstoasaddle-nodebifurationandtheredonetoanAndronov-Hopf

bifuration.

ron

Inthissetion westudy thebifurationdiagram oftheadaptiveexponentialneuron. This

model has been introdued by Brette and Gerstner in [5℄. This model, inspired by the

Izhikevih adaptivequadrati model, anbetted to biologialvalues, takesinto aount

theadaptationphenomenon, andis ableto reproduemanybehaviorsobservedin ortial

neurons. Thebifuration analysiswederivedin setion1allowsus tounderstand howthe

parametersofthemodelanaetthebehaviorofthisneuron. Weshowthatthismodelis

partofthegenerallassstudiedin 1andweobtainthexed-pointsbifurationdiagramof

themodel.

2.2.1 Redution of the originalmodel

Thisoriginalmodelisbasedonbiologialonstantsandisexpressedwithalotofparameters.

Werstredue thismodelto asimplerform withthefewestnumberofparameters:

Thebasiequationsproposedintheoriginalpaper[5℄read:



 



 

 C

^d

^V

d

t = − g L (V − E L ) + g L ∆ T exp

V − V T

∆ T

− g e (t)(V − E e ) − g i (t)(V − E i ) − W + I m

τ W

^d

W

d

t = κ(V − E L ) − W

(2.2)

First,wedonotassumethatthereversalpotentialofthe

w

^{equationisthesameasthe}

leakagepotential

E L

^{,andwritetheequationfortheadaptationvariableby:}

τ W

d

W

d

t = a(V − V ¯ ) − W

Nextweassumethat

g e ( · )

^and

g i ( · )

^{areonstant(intheoriginalpaperitwasassumedthat}

thetwoondutaneswherenull).

Aftersomestraightforwardalgebra,weeventuallygetthefollowingdimensionlessequa-

tionequivalentto(2.2) :

( v ˙ = − v + e ^v − w + I

˙

w = a(bv − w)

^(2.3)

wherewedenoted:



 

 

 

 

 

 

 



 

 

 

 

 

 

 



˜

g := g L + g e + g i

τ m := ^C _{˜ g} B := ^κ _{˜ g}

E L

∆ T + log( ^g _{g ˜} ^L e ^{− V T /∆ T} ) v(τ) := ^{V (τ τ} _{∆ T} ^{m )} + log

g L

˜

g e ^{− V T /∆ T} w(τ) := ^W _{g∆ ˜} ^{(τ τ m )}

T + B

a := _τ ^τ _W ^m b := ^κ _{g ˜}

I := ^{I m +g L E L} _{g∆ ˜} ^+g _T ^{e E e +g i E i} + log( ^g _{g ˜} ^l e ^{− V T /∆ T} ) + B

(2.4)

andwherethedotdenotesthederivativewithrespetto

τ

^.

Remark. These expressions onrm the qualitative interpretation of theparameters

a

^,

b

and

I

^{ofthemodel(1.1). Indeed,}

a = ^τ _τ ^m _w

^{aountsforthetimesaleoftheadaptation(with}

themembranetimesaleasreferene),theparameter

b = ^κ _{˜ g}

^isproportionaltotheinteration betweenthemembranepotentialandtheadaptationvariableandinverselyproportionalto

thetotalondutivityofthemembranepotential. Eventually,

I

^{isananefuntionofthe}

inputurrent

I m

^{andmodelstheinputurrentoftheneurons.}

2.2.2 Bifuration diagram

From equation (2.3) we anlearly see that the Brette-Gerstner model is inluded in the

formallassstudiedin thepaperwith:

F (v) = e ^v − v.

Thisfuntionsatisestheassumptions(A1), (A2)and(A3). Furthermore,itsseondorder

derivativenevervanishes.

Theorem1.5showsthat thesystemundergoesthefollowingbifurations:

(BG.B1). Asaddle-nodebifurationurvedened by

{ (b, I) ; I = (1 + b)(1 − log(1 + b)) } .

So

v ^∗ (b) = log(1 + b)

^{. For}

I ≤ (1 + b)(1 − log(1 + b))

^{,thesystemhasthexedpoints:}







v ₋ (I, b) := − W 0

− _1+b ¹ e ^{1+b I} + _1+b ^I v + (I, b) := − W ₋ 1

− 1+b ¹ e ^{1+b I} + _1+b ^I

(2.5)

where

W 0

^{is the prinipal branh of the Lambert's}

W

^{funtion4 and}

W ₋ 1

^{the real}

branh ofLambert'sW funtionsuhthat

W ₋ 1 (x) ≤ − 1

^,denedfor

− e ^{− 1} ≤ x < 1

^.

4

TheLambertWfuntionistheinversefuntionof

x 7→ xe ^x

^.

{ (b, I) ; b > a

^and

I = I ^∗ (a, b) = (1 + b) log(1 + a) − (1 + a) }

at the equilibrium point

(v a = log(1 + a), w a = bv a )

^{. The type of} Andronov-Hopf bifurationisgivenbythesignof thevariable

A(a, b) = F ^′′′ (v a ) + 1

b − a F ^′′ (v a ) ² = (1 + a) + 4

b − a (1 + a) ² > 0

Sothebifurationisalwayssubritialandthere isnotanyBautinbifuration.

(BG.B3). ABogdanov-Takensbifurationpointatthepoint

b = a

^and

I = log(1 + a)

^.

(BG.B4). Asaddlehomolinibifurationurvesatisfying,neartheBogdanov-Takenspoint,the

equation:

(P ) := n

(I = (1 + a)(log(1 + a) − 1) + I 1 , b = a + b 1 ) ;

I 1 =

− ²⁵ ₆ a − ³⁷ ₆ b 1 + ⁵ ₆ p

25 a ² + 74 b 1 a + 49 b 1 2 a

(1 + a) + o(( | b 1 | + | I 1 | )

and

b 1 > −

1 + 1 a

I 1

o

IngureFig.3werepresentedthexedpointsoftheexponentialmodelandtheirstability,

togetherwiththebifuration urves,in thespae

(I, b, v)

^.

3 The riher quarti model

Inthissetion,weintrodueanewspei modelhavingariherbifurationdiagramthan

thetwo models studied in setion 2. It is as simpleasthe twoprevious models from the

mathematialand omputational pointsof view. Tothis end, wedene amodelwhih is

partofthelassstudiedin setion1,byspeifyingthefuntion

F

^.

3.1 The Quarti model: Denition and bifuration map

Let

a > 0

^{a xed real, and}

α > a

^{. We} instantiate the model (1.1)with the funtion

F

^a

quartipolynomial:

F (v) = v ⁴ + 2av

Remark. The hoie of the funtion

F

^{here is just an example where all the formulas}

are rather simple. Exatly the sameanalysis an bedone with any

F

^{funtion satisfying}

F ^′′′ (v a ) < 0

^andthetransversalityonditionsgivenin theorem1.6. Thiswouldbethease forinstaneforanyquartipolynomial

F (v) = v ⁴ + αv

^for

α > a

^.