Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator

0.52 MB·English

by  Kevin K. Lin

Entrainment and chaos in a pulse-driven Hodgkin-Huxley oscillator
Kevin K. Lin
[email protected]
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January 18,2006
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Abstract
a
J The Hodgkin-Huxley model describes action potential generation in certain types of neurons and is a

8 standard model for conductance-based, excitable cells. Following the early work of Winfree and Best,
1
thispaperexplorestheresponseofaspontaneouslyspikingHodgkin-Huxleyneuronmodeltoaperiodic

pulsatiledrive. Theresponseasafunctionofdriveperiodandamplitudeissystematicallycharacterized.
]
S Awiderangeofqualitativelydistinctresponsesarefound,includingentrainmenttotheinputpulsetrain
D
andpersistentchaos. Theseobservationsareconsistentwithatheoryofkickedoscillatorsdevelopedby
. QiudongWangandLai-SangYoung. InadditiontogeneralfeaturespredictedbyWang-Youngtheory,it
h
t isfoundthatmostcombinationsofdriveperiodandamplitudeleadtoentrainmentinsteadofchaos.This
a
preferenceforentrainmentoverchaosisexplainedbythestructureoftheHodgkin-Huxleyphaseresetting
m
curve.
[

4
v 1 Introduction
1
6
TheHodgkin-Huxleymodeldescribesactionpotentialgenerationincertaintypesofneuronsandisastan-
1
5 dardmodelforconductance-based,excitablecells[5,18,20].Thereisanextensiveliteratureontheresponse
0
oftheHodgkin-Huxleymodeltodifferenttypesofinputs[1,2,11,14,15,16,17,19,24,25],andunderstand-
5
0 ing how single neurons respond to external forcing continues to be relevantfor the study of information
/ transmission in neural systems [21, 23]. Because neurons typically communicate via pulsatile synaptic
h
t events, it is natural to investigate the response of the Hodgkin-Huxley model to pulsatile inputs. Early
a
m studiesbyBestandWinfree[3,38]examinetheresponseofaHodgkin-Huxleymodeltoperiodicimpulse
trains,chracterizingindetailthestructureofphasesingularitiesandthetransitionfromdegree1todegree
:
v
0phaseresetting.However,theirworkdoesnotsystematicallyaddresstheasymptoticdynamicalbehavior
i
X asafunctionofdriveperiodandamplitude.1
r This paper studies a spontaneously spiking (i.e. oscillatory) Hodgkin-Huxley neuron model driven
a
byperiodic, pulsatile input of fixed amplitude and period, and systematically classifies the response asa
functionofdriveperiodandamplitude. Itisfoundthat:
1. Inresponsetoperiodicpulsatileforcingoffixedamplitude AandperiodT,aspontaneouslyspiking
Hodgkin-Huxleysystemcanexhibitawiderangeofdistinctbehaviorsdependingon AandT:
(a) Entrainment: The driven system is stably periodic and its period is a rational multiple of the
driveperiodT.
(b) Transientchaos: Thesystemexperiencesatransientperiodofexponentialinstabilitybeforeen-
trainingtotheinput. ThistransientchaosiscausedbyaSmalehorseshoe[13].
1Takabe,Aihara,andMatsumoto[32]appeartohavecarriedoutsuchasystematicstudy.But,Iwasonlyabletolocateanabstract.
1
(c) Chaos: The system becomes fully chaotic: it possesses a positive Lyapunov exponent and a
mixingattractor(see[39]forareviewoftheseconcepts).
The response of the pulse-drivenneuron is approximately T -periodic in the drive period T, where
0
T istheintrinsicperiodoftheunforcedHodgkin-Huxleyoscillator. Forexample,ifthepulse-driven
0
oscillatorischaoticforsomedriveamplitudeAanddriveperiodT,thenitislikelytobechaoticwhen
drivenbyapulsetrainofamplitude AwithperiodnearT+T .
0
2. Thescenariosenumeratedaboveareprevalentinthesensethattheycorrespondtopositive-areasub-
setsofthedriveperiod-driveamplitudespace.Prevalence,togetherwiththeapproximateperiodicity
statedabove,implythateachscenariooccurswithpositive“probability.”(SeethediscussionofFig.3
in§3fortheprecisemeaningofprobabilityinthiscontext.) Therangeofresponsesandtheirpreva-
lenceareconsistentwithatheoryofnonlinearoscillatorsdevelopedrecentlybyQiudongWangand
Lai-SangYoung[33,34,35,36].
3. While chaotic behavior is readily observable, most combinations of drive period and drive ampli-
tude leadto entrainment instead of chaos. This preferencefor entrainment canbe explained by the
structureofthephaseresettingcurve(see§4)oftheHodgkin-Huxleysystem.
This paperreliesheavilyon numerical computation and the conceptual frameworkprovidedbyWin-
free’stheoryofbiologicalrhythms[38]andtheworkofQ.WangandL.-S.Youngonnonlinearoscillators
[34,35]. Phaseresettingcurves,introducedbyWinfree,playaparticularlyimportantrolehere. Thephase
resettingcurveofanonlinearoscillatorisanintervalmapwhichcapturestheasymptoticresponseofanon-
linearoscillatortoasingle,pulsatileperturbation. Becausetheyare1-dimensionalobjects,phaseresetting
curvesareofteneasiertounderstandthanthenonlinearoscillatorstheyrepresent.Theyarefrequentlyused
toinferstabledynamicalbehaviorlikephaselocking. Wang-Youngtheoryprovidesamathematicalframe-
workforusingphaseresettingcurvestoinfertheexistenceandprevalenceofchaoticbehavior.Ratherthan
numerically verify the hypotheses of their theorems, we have opted to examine the consequences of the
theorydirectly,relyingonacombinationofnumericalsimulationandgeometricreasoningtocharacterize
thespecificresponseoftheHodgkin-Huxleymodeltoaperiodicpulsatiledrive.
For the sake of clarity, parametersareselected toensure that the Hodgkin-Huxley system possesses a
unique limit cycle and no other attractinginvariantset. This correspondsto a repeatedlyspiking neuron
withanunstablereststate. Whilethescenariosstatedaboveshouldstillholdwhenthelimitcyclecoexists
with other stable invariant sets, this choice simplifies the interpretation of numerical simulations. Other-
wise,atrajectorymayjumpoutofthebasinofthelimitcycle,whichobscuresthemechanismdescribedby
Wang-YoungtheoryandwhichWinfreeandBesthavealreadyinvestigatedthoroughly[3,38].
Therestof thispaperisorganizedasfollows: Section2brieflyreviewsthe unforcedHodgkin-Huxley
equationsanditsproperties. Mainnumericalresultsaresummarizedin§3anddiscussedin§4. Section5
discussesfurthernumericalresults,addressingsomeissuesraisedinSections3and4. Section6discusses
possibleextensionsandgeneralizations.
2 Brief review of the Hodgkin-Huxley model
TheHodgkin-Huxleyequationsareasystemofnonlinear ordinarydifferentialequations2 whichdescribe
the way neurons generate spatially and temporally localized electrical pulses [5, 18, 20]. These electrical
2ThispaperdoesnottreattheHodgkin-HuxleyPDEs:spatialdependenceisnotrelevanthere.
2
pulses,calledactionpotentials,aretheprimarywayinwhichneuronstransmitinformation. Actionpoten-
tialsaretriggered bysufficiently largemembrane voltages, which can be setup by the influx of ions into
thecell. Aneuronissaidtofireorspikewhenitgeneratesanactionpotential(Fig. 1). TheHodgkin-Huxley
model describes action potential generation in terms of the membrane voltage and dimensionless gating
variables which quantify the effective permeability (or conductance) of the membrane for various types of
ions.
TheoriginalHodgkin-Huxleyequationsmodelactionpotentialgenerationinthesquidgiantaxon.This
giantaxoncontainstwotypesofmembraneionchannels. Onetypeofchannelisspecifictopotassiumions,
theothertosodiumions. Thestatevariablesofthemodelarethemembranevoltagev,theactivationnof
thepotassiumchannels,andtheactivationmandinactivationhofthesodiumchannels. Theequationsare
[18]
v˙ = C−1 −I−g¯ n4(v−v )−g¯ m3h(v−v )−g¯ (v−v )
K K Na Na leak leak
m˙ = α (v)(1−m)−β (v)m
m (cid:2) m (cid:3) (1)
n˙ = α(v)(1−n)−β(v)n
n n
h˙ = α(v)(1−h)−β(v)h
h h
where
α (v)= Ψ v+25 , β (v)= 4exp(v/18),
m 10 m
α(v)=0.1(cid:16)Ψ v+(cid:17)10 , β(v)=0.125exp(v/80),
n 10 n (2)
α(v)=0.07e(cid:16)xp(v/(cid:17)20), β(v)= 1 ,
h h 1+exp(v+30)
10
Ψ(v)= v .
exp(v)−1
Each ion channel consists of independent, identical subunits which must all open to allow ions to pass
through. Thegatingvariablesm,n,andhtakevaluein(0,1)andrepresentthefractionofsubunitswhich
are open. The term n4 enters into the potassium conductance because potassium channels consist of 4
identical subunits; analogous structures account for the m3h term in the sodium conductance [5]. The
gatingvariableequationsaremasterequationsforcontinuous-timeMarkovchainswithvoltage-dependent
transitionratesαandβ;theMarkovchainsdescribetheopeningandclosingofthecorrespondingchannel
subunits. The v˙ equation is Kirchoff’s current law. Action potentials are downward voltage spikes and a
positive I corresponds toan inflow of positively-chargedions. The voltage convention hereis thatof [18]
andoppositecontemporaryusage: themembranevoltagevisdefinedby
v=voltageoutside−voltageinside.
Actionpotentialsaregenerallyinitiatedbyperturbationstothemembranevoltage. Suchperturbations
maybecaused,forinstance,bytheflowofionsacrossthecellmembrane.Becauseneuronstransmitsignals
through spatially and temporally localized pulses, it is natural to model stimuli as impulses [31]. The
simplesttypeofrepetitive,pulsatilestimulustoaneuronisaperiodicimpulsetrain. Thismeansreplacing
thev˙ equationaboveby
v˙ =C−1 −I−g¯ n4(v−v )−g¯ m3h(v−v )−g¯ (v−v ) (3)
K K Na Na leak leak
h(cid:229) i
+A G(t−kT),
k∈Z
where G is a “bump” function such that G(t)dt = 1. For simplicity, most of this paperuses the choice
G(t)=δ(t);Section5.2discussestheresponseoftheHodgkin-Huxleysystemtoapulsatiledrivewith
R
1/t , 0 ≤t ≤t
G(t)= 0 0 . (4)
( 0, otherwise
3
0.0
)
v
(

e
g -20.0
a
t
ol
v -40.0

e
n
a
r -60.0
b
m
e
M -80.0
0.0 10.0 20.0 30.0 40.0
Time (t)
Figure1:ThetimecoursefortheHodgkin-Huxleyequationsattheparametervalues(5). Therapid“spike”
followedbyalong“recovery”periodistypicaloftheHodgkin-Huxleyequations.
Mathematically,onecanalsochoosetoperturbthegatingvariables,butsuchperturbationsarenotentirely
naturalandarenotconsideredhere.
ThispaperusestheoriginalHodgkin-Huxleyparameters[18]:
v = −115mV, g¯ =120mΩ−1/cm2,
Na Na
v =+12mV, g¯ =36mΩ−1/cm2,
K K (5)
v =−10.613mV, g¯ =0.3mΩ−1/cm2,
leak leak
C =1µF/cm2.
TimeismeasuredinmillisecondsandcurrentdensityinµA/cm2.
Figure 2 shows a bifurcation diagram for the unforced Hodgkin-Huxley equations. When I = 0, the
neuron maintains a stable rest state, corresponding to the branch of stable fixed points on the left of the
diagram.AsufficientlylargevalueofIcausesaneurontofirerepeatedly,whichcorrespondstothecreation
of a limit cycle through a saddle-node bifurcation of periodic orbits. Further increasing I destablizes the
reststatethroughasubcriticalHopfbifurcation.
Inthispaper,the injectedcurrentisalwayssettoavaluenear I ≈ 14,correspondingtoasteadyionic
currentwhichdestabilizesthereststate. Thephenomena studiedhereareinsensitive totheprecisevalue
of I aslong asitensuresthe existenceof astable limitcycle andanunstablefixed point. Asexplainedin
theIntroduction,thesepropertiessimplifytheinterpretationofnumericalsimulations. Forthischoiceof I,
theJacobianoftheHodgkin-Huxleyvectorfield(Eq. 1)attheunstablefixedpointhastworealeigenvalues
{−4.97815,−0.146991} in the left half plane, and a complex conjugate pair 0.0763367±0.61866i in the
right half plane. The fixed point thus has 2-dimensional stable and unstable manifolds. The Lyapunov
exponentsassociatedwiththelimitcycleare0,≈ −0.20,≈ −2.0,and≈ −8.3.ItsperiodisT ≈ 12.944.
0
4
20
0
Stable fixed point
Unstable fixed point
-20
V
-40
Unstable cycle
-60
-80
Limit cycle
-100
6 7 8 9 10 11 12 13 14
I
Figure2:ThebifurcationdiagramfortheHodgkin-HuxleyequationsastheinjectedcurrentIisvaried.The
lineinthemiddlemarksthevcoordinateofthereststate. Thesolidbluepartisstablewhilethedashedred
partisunstable. Solidblackdotsnearthetopandthebottomofthefigurearethemaximumandminimum
v valuesof limit cycles. Empty blackcirclesarethe maximum and minimum v valuesof unstable cycles.
The fixed point undergoes a subcritical Hopf bifurcationas I increases. This diagramis computed using
XPPAUT[7].
5
A = 5 A = 20
0 0
t t
n n
e e
n n
o o
p-2 p-2
x x
e e

v v
o o
n n
u u
p p
a a
y y
L L

st-4 st-4
e e
g g
r r
a a
L L
-6 -6
2 3 4 5 6 7 8 2 3 4 5 6 7 8
Multiple of intrinsic period (T/T0) Multiple of intrinsic period (T/T0)
Figure3: Asymptoticpropertiesofthepulse-drivenflowaredescribedbythedynamicsofthetime-Tmap
F (see Eq. 6) and its largest Lyapunov exponent λ . Entrainment corresponds to λ < 0, and chaos
T max max
corresponds toλ > 0. This figure showsλ as a function of the drive period T, with T ranging from
max max
T ≈ 13 (the intrinsic period of the unforced Hodgkin-Huxleysystem; see §2) to 8·T ≈ 101. Left: Kick
0 0
amplitudeis A = 5. Right: Kickamplitudeis A = 40. Note(i)λ (T+T ) ≈λ (T);(ii)presenceofboth
max 0 max
positive and negative exponents for strong kicks (right), and only zero and negative exponents for weak
kicks (left); and (iii) the presenceof more negative exponents than positive ones. See §4 for a discussion.
LyapunovexponentsareestimatedbyiteratingF for1000stepsandtrackingtherateofgrowthofatangent
T
vector.
6
1.0
Rotations
0.8
Sinks
y 0.6
t
bili
a
b
o
Pr 0.4
Chaos
0.2 Unknown
0.0
10.0 20.0 30.0 40.0
Drive amplitude
Figure 4: The probability of different response types, as a function of the drive amplitude A. For each
driveamplitude A,thefractionofdriveperiodsT ∈ [T ,8T ]forwhichλ (F ) > 0,etc.,iscomputedby
0 0 max T
samplingfromauniformgridin[T ,8T ]. Itisnaturaltoequatethesefractionswithprobabilitiesbecause
0 0
the Lyapunov exponentsareroughlyperiodicfunctions of T (andbecome moresoas T → ∞), asshown
inFigure3andexplainedin§4. Empiricaldefinitions: Letλˆ denotetheestimatedLyapunovexponentandǫ
theestimatedstandarderror. Then“chaos”isdefinedasλˆ > 3ǫ,“entrainment”λˆ < −3ǫ,and“rotation”
λˆ <ǫ/3.
(cid:12) (cid:12)
(cid:12) (cid:12)
7
3 Main numerical results
Lyapunov exponents provide a convenient way to characterize the asymptotic dynamics of Eq. 3. Let
φ : R4 → R4 denote the flow map generated by the unforced Hodgkin-Huxley equations, T the drive
t
period,and Athedriveamplitude. ThePoincare´ map
F (v,m,n,h)=φ (v+A,m,n,h) (6)
T T
takesaHodgkin-Huxleystatevector(v,m,n,h),appliesapulseofamplitude Atothemembranevoltage,
thenevolvesitfor time T. Iteratingthe map F thusgivesastroboscopic recordofthe stateof ourpulse-
T
drivenHodgkin-Huxleysystembeforethearrivalofeachpulse. Thelong-termdynamicalbehaviorofthe
pulse-driven Hodgkin-Huxley oscillator can be deduced from the asymptotic dynamics of F , which is
T
characterizedbyits(largest)Lyapunovexponentλ [13]:
max
λ <0 ⇔ F hassinks ⇔ kickedflowisentrainedtoinput
max T
λ =0 ⇔ F isquasiperiodic ⇔ kickedflowdriftsrelativetoinput
max T
λ >0 ⇔ F chaotic ⇔ kickedflowischaotic
max T
(Sinksrefertostablefixedpointsandstableperiodicorbits.) NotethatofthescenariosgivenintheIntro-
duction, transientchaosalonedoesnotappearinthislist: Lyapunovexponents, beinglong-time, average
quantities,cannotdetecttransientchaos.
Figure 3 shows the Lyapunov exponents of F as a function of T/T , where T is the period of the
T 0 0
Hodgkin-Huxley limit cycle. Different colors correspond to differentvalues of A. The periodicity of the
response as a function of T is apparent. Because the response type as a function of T is approximately
identicalovereachperiod[nT ,(n+1)T ],itmakessensetospeakoftheprobabilitythatarandomlychosen
0 0
driveperiod T willelicita particularasymptotic behavior, forexamplechaos. Moreprecisely, periodicity
ensures that the fraction p of drive periods T in [0,nT ] for which λ > 0 converges to a well-defined
n 0 max
limitasn→ ∞. Similarstatementsholdforλ <0andλ = 0.
max max
Figure4showstheseprobabilitiesasfunctionsof A. At A = 10,theprobabilityofobtainingapositive
exponent is roughly 20% and the probability of obtaining a negative exponent is roughly 70%. Thus, if
oneweretopickT randomlyoutofaninterval[NT ,(N+1)T ]forlarge,fixedinteger N,theprobability
0 0
thatλ (F ) > 0 is about 20%. Figure 4 shows thatwhen A is small, the most likely type of behavior is
max T
rotation-likebehavior.ThispossibilitybecomeslesslikelyasAincreases.Atthesametime,sinksandchaos
both become more likely, with sinks dominating the scene. One feature of Figure 4 specific to the pulse-
drivenHodgkin-HuxleyflowisthatwhenAislarge,thesystemprefersentrainmentoverchaosinthesense
thatentrainmenthashigher probability. Thispreferenceismorepronounced as A increases. Note thatin
computingLyapunovexponentsnumerically,weonlyhaveaccesstofinitetimeinformation. Inprinciple,
thismeansitisvirtuallyimpossibletodistinguishpersistentchaoticbehaviorfromtransientchaoscaused
bya“large”horseshoe(butsee§5.1).
In all numerical simulations shown in this paper, Eq. 1 is integrated using an adaptive integrator of
Runge-Kutta-Fehlbergtype,withanerrortoleranceof 10−6 (inthesup norm)[30]. ThelargestLyapunov
exponentλ of F is computed in a straightforward manner, by choosing a nonzero unit vector w ∈ R4
max T
andestimatingtherateofgrowthof ||(DF )nw||. Thematrix-vectorproduct(DF )nw iseasilycomputed
T T
viathevariationalequations x˙ = H(x),w˙ = DH(x)·wfortheHodgkin-Huxleyvectorfield H (DHisthe
Jacobianmatrixof H;see[10]fordetails).
8
4 Discussion
4.1 Response to asingle pulse: phaseresetting curves
Thissectionreviewsphaseresettingcurves. SeeWinfree[38],GlassandMackey[11],andBrown,Moehlis,
and Holmes [4] for more details and applications, and Guckenheimer and Holmes [13] for background
informationondynamicalsystemstheory. See[7,8,9,37]forfurtherdiscussionsofphaseresettingcurves.
Letφ : Rn → Rn be a flow generated bya smooth vector field with a hyperbolic limit cycleγ. Such
t
a limit cycle represents a stable nonlinear oscillator. The basin of attraction of γis denoted B(γ). The
hyperbolicityofγguaranteesthatpointsinB(γ)convergetoγexponentiallyfast.(Itisconvenienttouseγ
torefertoboththetrajectoryγ:R→Rn andtheinvariantpointsetitdefines.) Animpulsiveperturbation
(“kick”) to the nonlinear oscillator can be defined by specifying a kick amplitude A and a kick direction
Kˆ :Rn →Rn anddefiningafamilyofkickmaps
K (x)= x+A·Kˆ(x), (7)
A
so that kicks send each point x ∈ Rn to K (x). For what follows, K should be smooth and satisfy
A A
K (B(γ))⊂ B(γ).
A
The Hodgkin-Huxley system with the value of I given in §2 is a nonlinear oscillator whose basin
B(γ) is an open subset of R4. The kick map corresponding to an instantaneous voltage spike is simply
K (v,m,n,h)=(v+A,m,n,h).Asin§3,itisconvenienttointroducethetime-Tmap
A
F =φ ◦K , (8)
T T A
where◦denotesfunctioncomposition. IteratingF givesastroboscopicrecordofthesystemstatebeforethe
T
arrivalofeachkick, andthusdescribesthelong-time dynamicsofthe flowφ underrepeated, T-periodic
t
kicks.
Becausethephasedimensionnmaybelarge,thedynamicsofF :Rn →Rnmaybedifficulttoanalyze.
T
Winfreeobservedthateverypointnearthelimitcycleγmustconvergetoγast →∞,sotheflownearγis
dominatedbytherotationalmotionalongγ. Thus,onecanreducethedimensionofthephasespacefrom
nto1,atleastheuristically. Todothis, firstdefinethephasefunctionθ:γ→ [0,T ) byfixingareference
0
pointx ∈γandrequiringthatφ (x ) = xforallx ∈γ. Byconstruction,θsatisfies d (θ(γ(t))) =1,0≤
0 θ(x) 0 dt
t < T . The functionθcanbe extendedtoafunctionθ: B(γ) → [0,T ) byprojectingalongstrong-stable
0 0
manifolds3: if yisapointinthebasinofγthenθ(y)isdefinedtobeθ(x),wherexistheuniquepointsuch
that y∈W (x).Thisdefinitionofphasepreservesthepropertythat d (θ(φ(x))) =1.
ss dt t
Considerthelimit[12]
F¯ = lim F . (10)
T n→+∞ T+nT0
Themap F¯ iswell-definedonthebasinofγandretractsthebasinontoγ,i.e. F¯ (x) ∈γforallx ∈ B(γ).
T T
Thus, F¯ induces an interval map f : [0,T ) → [0,T ) which, given the current phase of the system,
T T 0 0
yieldsthenewphaseafterkickingandevolvingthesystemfortimeT. Thatis, f (θ(x)) =θ(F (x))forall
T T
x ∈ B(γ).
3Thestrong-stablemanifoldW (x)ofx∈γistheset
ss
Wss(x)=(cid:26)y∈Rn:n∈Zl,inm→+∞φnT0(y)→x(cid:27). (9)
Whenthevectorfieldgeneratingφtissmooth,thestrong-stablemanifoldsare(locally)smoothsubmanifoldsofRn.Thestrong-stable
linearsubspaceE (x)isthetangentspaceofW (x)atx.See[12,13].
ss ss
9
Driveamplitude A Prob. ofsinknearplateau Prob. ofλ <0
max
5 41% 48%
10 58% 62%
20 68% 70%
30 76% 78%
Table 1: Estimates of the probabilityof obtaining sinks near the plateau, as a function of A. The datafor
thistableiscomputedbytryingabout40valuesof T foreachchoiceof A andexaminingthegraphofthe
first return map to the interval [4,10] (chosen to coincide with the “plateau”)and its intersection(s) with
thediagonal.
The map f isthe phaseresetting curve4, or more preciselythe finite phase resettingcurve(infinitesimal
T
phaseresettingcurves[4,7]arenotneededhere).Byconstruction,ithasthepropertythat
fT+δ(t)= fT(t)+δ(modT0). (11)
Thus,thefamilyofmaps{f }isperiodicinT.
T
PeriodicityindriveperiodT. Theapproximateperiodicityofλ (F )seeninFigure3iseasytounder-
max T
standheuristically: kickingtheoscillatoreveryTsecondsandkickingiteveryT+T secondsshouldyield
0
the same asymptotic response because the oscillator simply traversesγatfrequency 1/T between kicks.
0
Onecanrestatethisusingphaseresettingcurves: ifthedriveperiodT issufficientlylargeandθ(x ) = t ,
0 0
thenthe f -orbit t , f (t ), f2(t ),... shouldcloselyfollowthephases θ(x ),θ(F (x )),θ(F2(x )),... of
T 0 T 0 T 0 0 T 0 T 0
thecorrespondingF -orbit. Since f = f ,thissuggeststhatλ (F )≈λ (F ).
(cid:0) T T+(cid:1)T0 T max T+(cid:0)T0 max T (cid:1)
Preference for entrainment. Figure 5 shows phase resetting curves for the Hodgkin-Huxley equations
forvariousvaluesofdriveperiod T anddriveamplitude A. Forsufficientllysmallvaluesof A,thephase
resettingcurvesarecirclediffeomorphisms: eithertherearesinks(i.e. stablefixedpiontsorstableperiodic
orbits), or the map is conjugate to a rotation on a circle and the response of the kicked oscillator drifts
relativetotheperiodicdrive. AsAincreases,thegraphof f ratherquicklyfoldsoverandacquirescritical
T
points. AstrikingfeatureofthegraphsinFigure5isthe“plateau,”aphaseintervaloverwhich f varies
T
veryslowly. Another striking featureisthe “kink” aroundθ≈ 9.8. These featuresarediscussed inmore
depthin§5. Fornow,noticethattheplateauprovidesasimplemechanismforcreatingsinks: changingthe
kickperiodTshiftsthegraphof f vertically. Wheneverthegraphintersectsthediagonalwithaderivative
T
f′ <1,thenastablefixedpointiscreated.
T
ThismechanismcanbeusedtoverifytheresultsofFigure4: computethegraphofthefirstreturnmap
(cid:12) (cid:12)
o(cid:12)f f(cid:12) toanintervalaround the plateau, then shiftthe graphverticallyusing a number of differentvalues
T
of T andestimatethefractionofT’sforwhich f hasastablefixedpoint(seeFigure6). Table1showsthe
T
results. For A = 10, the 58% probability of sinks corresponds fairly closely with Figure 4. It is unclear
whether the ambiguous exponents in Figure 4 reallyrepresentpositive or negative Lyapunov exponents.
If a significant fraction of the ambiguous exponents are really negative, then they must come from small
sinks.
4WangandYoungrefertophaseresettingcurvesassingularlimits.Phaseresettingcurvesarealsosometimescalledphasetransition
curves[11].
10