The Slow Passage Through a Hopf Bifurcation: Delay, Memory Effects, and Resonance
Author(s): S. M. Baer, T. Erneux, J. Rinzel
Source: SIAM Journal on Applied Mathematics, Vol. 49, No. 1 (Feb., 1989), pp. 55-71
Published by: Society for Industrial and Applied Mathematics
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?
SIAM J. APPL. MATH.
Vol. 49, No. 1, pp. 55-71, February1989
1989 SocietyforIndustrialand Applied Mathematics
003
THE SLOW PASSAGE THROUGH A HOPF BIFURCATION: DELAY,
MEMORY EFFECTS, AND RESONANCE*
S. M. BAERt, T. ERNEUXt,
AND
J. RINZELt
Abstract.This paper explores analyticallyand numerically,in the contextof the FitzHugh-Nagumo
an interesting
model of nervemembraneexcitability,
phenomenonthathas been describedas a delay or
memoryeffect.It can occur when a parameterpasses slowlythrougha Hopf bifurcationpoint and the
system'sresponsechangesfroma slowlyvaryingsteadystateto slowlyvaryingoscillations.On quantitative
observationit is foundthatthe transitionis realizedwhenthe parameteris considerablybeyondthe value
predictedfroma straightforward
bifurcationanalysiswhichneglectsthe dynamicaspect of the parameter
variation.This delay and its dependenceon the speed of the parametervariationare described.
The model involvesseveralparametersand particularsingularlimitsare investigated.
One in particular
is the slow passage througha low frequencyHopf bifurcationwherethe system'sresponsechangesfroma
slowly varyingsteady state to slowly varyingrelaxationoscillations.We findin this case the onset of
oscillationsexhibitsan advance ratherthana delay.
This paper shows thatin generaldelays in the onset of oscillationsmay be expectedbut thatsmall
amplitudenoise and periodic environmental
of near resonantfrequencymay decrease the
perturbations
delay and destroythememoryeffect.This paper suggeststhatboth deterministic
and stochasticapproaches
willbe important
forcomparingtheoreticaland experimentalresultsin systemswhereslow passage through
a Hopf bifurcationis the underlyingmechanismforthe onsetof oscillations.
Key words.delayed Hopf bifurcationtransition,memory effect,resonance, FitzHugh-Nagumo
equations,nerveaccommodation
AMS(MOS) subjectclassifications.C34, 92
1. Introduction.
In mathematicalstudiesof bifurcation,
it is customaryto assume
thatthe bifurcationor controlparameteris independentof time.However,in many
as bifurcation
thatare modeled mathematically
experiments
problems,thebifurcation
parametervariesnaturallywithtime,or it is deliberatelyvariedby the experimenter.
Typically,thisvariationis slow or is forcedto be slow.
The recentinterestin the effectsof slowlyvaryingcontrolparametersarises in
physical,engineering,biological, and mathematicalcontexts.The physical interest
arisesfromthefactthattheresultsoflong-timeexperiments
maydependon parameters
thatare slowlyvarying.For example,catalyticactivitiesin chemicalreactorsare slowly
decliningdue to chemicalerosionand are decreasingthereactorperformance
[1], [2].
The effectsof slowlyvaryingparametersare not always undesirable.They may also
lead to smoothtransitionsat bifurcationpointsand mediatea gradual change in the
systemto a new mode of behaviorbeyond the bifurcationpoint.This idea has been
studied for quite differentproblems such as thermal-convection [3], [4], laser
instabilities[5], [6], and developmentaltransitionsin biology[7].
From a modelingpoint of view, we expect that a slow variationof the control
parametercan be useful for the experimentalor numericaldeterminationof the
bifurcationdiagramof the stable solutions.Also, to understandcertaincomplicated
of
multi-scaledynamicphenomena[8], it is usefulto studythe bifurcationstructure
thefastprocesseswiththeslow variablestreatedas slowlyvaryingcontrolparameters.
* Received by the editorsJune3, 1987; accepted forpublicationNovember13, 1987.
t MathematicalResearchBranch,NIDDK, National Institutesof Health, Bethesda,Maryland20892.
t Departmentof EngineeeringSciencesand Applied Mathematics,Northwestern
University,
Evanston,
Illinois60208.This workwas supportedby theAir Force of ScientificResearchundergrantAFOSR85-0150
and the National Science FoundationunderGrantDMS-8701302.
55
56
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
In such cases, it is importantthatwe have detailed knowledgeof the transitionnear
the bifurcationpointwheretransientsare veryslow.
From a mathematicalpoint of view, these problemsare formulatedby nonto solve.The studyoftheseproblems
equationsthatare difficult
autonomousdifferential
has led to new and interestingmathematicalissues [9]-[11]. References[9]-[11]
investigatethe slow passage througha steadybifurcationor a steadylimitpoint.An
studyof the effectsof a slowlyvaryingparameteron a Hopf bifurcationis
interesting
givenforthe slow passage throughresonance[26] [27].
In this paper, we concentrateon the slow passage througha Hopf bifurcation.
thiscase is quite different
froma steadybifurcationor limit
shall demonstrate,
we
As
point. Our resultsfor the Hopf bifurcationraise a series of new questions on the
instabilities.We shall considera specificmodel problemforthe
controlof bifurcation
ofa slowlyvaryingparameter
because ourgoal is to exploretheeffects
Hopfbifurcation
phenomenonhas been
bothanalyticallyand numerically.For example,one interesting
describedas a delay or memoryeffect.It can occur when a parameterpasses slowly
througha Hopf bifurcationpoint and the system'sresponse changes froma slowly
varyingsteadystateto slowlyvaryingoscillations.On quantitativeobservations(see
is realizedwhentheparameteris considerably
Fig. 1(a), (b)) we findthatthetransition
bifurcationanalysiswhichneglects
beyondthevalue predictedfroma straightforward
the dynamicaspect of the parametervariation.We describethisdelay and its dependence on thespeed oftheparametervariation.Also, we showthatthedelayis sensitive
of nearresonant
perturbations
to smallamplitudenoise and to periodicenvironmental
of bifurcation
maybe helpfulin theaccuratedetermination
frequency.This sensitivity
points. The model involves several parametersand particular(singular) limitsare
featureson theslow passage through
These limitsrevealotherinteresting
investigated.
the bifurcationpoint.
We employthe specificproblemof the FitzHugh-Nagumoequations as a model
to describethe mathematicaland qualitativefeaturesof the slow passage througha
Hopf bifurcation.Many of thesefeaturesoccur forothermodels [25].
2. Formulation.
2.1. The FitzHugh-Nagumoequations.In the early 1950s, Hodgkin and Huxley
[12] proposed a model that describesthe generationand propagationof the nerve
system
impulsealong thegiantaxon ofthesquid. The model consistsof a four-variable
equations. Subsequently,Nagumo et al. [13] and
of nonlinear partial differential
FitzHugh [14] developed a simplertwo-variablesystem,which describesthe main
qualitativefeaturesoftheoriginalHodgkin-Huxleyequationsand whichis analytically
more tractable.The so-called FitzHugh-Nagumo (FHN) equations for the space
clamped (i.e., spatiallyuniform)segmentof axon have the form
(2.1a)
dv= -f(v) - w+ I(t),
dt
(2.1b)
dw= b(v - yw),
dt
whereb and y are positiveconstantsand f(v) is a cubic-shapedfunctiongivenby
(2.1c)
f(v) = i,(v - a)(v - 1),
0< a
<2.
at time t across the membraneof the axon
Here v(t) denotesthe potentialdifference
and w representsa recoverycurrentwhich,accordingto the second equation (2.1b),
respondsslowly,when b is small,to changesin v. The firstequation (2.1a) expresses
THE
SLOW
THROUGH
PASSAGE
A HOPF
BIFURCATION
57
Kirchhoff'scurrentlaw as applied to the membrane;the capacitive,recovery,and
instantaneousnonlinearcurrentssum to equal the applied current,I(t). The applied
currentis our controlor bifurcationparameter.In this section,we considereither
constantintensitiesor slowlyvaryingintensitiesof the form
(2.1d)
I(Et) =
Ii + et,
O0<E <<1.
Frombiophysicalconsiderations,it is reasonableto restricty so that
(2.2)
y<3(1-a+a2)-1.
This insuresthat(2.1) with? = 0 have a unique steadystate.The steadystate(v, w) =
(vs(I), w(I)) satisfiesthe conditions
(2.3)
WS= v,/Y,
I =f(v,) + vdY.
of the formv = v, +p eAtand
To analyze its stability,we considersmall perturbations
the followingcharacteristic
to
This
w = w,+ q eA where IpI<<1 and q(<<1.
leads
I
equation forA
A2+AA+B=0
(2.4a)
where
A =f'(vs(I)) + by,
(2.4b)
(2.4c)
B
=
b [1 + yf'(vs(I))I
The steadystate is stable (unstable) if A> 0, B> 0 (A <0 and/or B <0). From the
conditionsA = O, B > 0 we findtwo Hopf bifurcationpoints I - I. They satisfythe
conditions
(2.5)
(2.6)
V(I) = vi= [a + 1 i (a2+1 - a -3by)12
2=_ b(l
- by2) > O.
When I < IL or I> I (I_ < I < I+), the steadystateis stable (unstable). To analyze
the response of the systemnear IL or I+, the approach of bifurcationtheoryis
particularlyuseful.When I> IL or I < I+, the transitionto the oscillationscan be
smooth (supercriticalbifurcation)or hard (subcriticalbifurcation).Details of the
bifurcationanalysisare givenin [15]-[17].
2.2. Responseto the slowlyvaryingparameter.We now considerthe effectof a
slowlyvaryingparameter.We assumethatthesystemis initiallyat a stablesteadystate
i.e., Ii < I- Figure1 illustratestheresponseto the slow,linearlyrisingcurrent(2.1d);
in Fig. 1(a), v is plottedversus t and in Fig. 1(b), v is plottedversus I. For these
parametervalues, the Hopf bifurcationat I_ is supercritical.From the bifurcation
structure(Fig. 1(b)), one mightexpectthatthe responsewould approximatelytrack
theslowlyvaryingsteadystate(v, w) = (v,(I), w,(I)), and then,as I increasesthrough
IL, the responsewould switchto the large amplitudeoscillations.Such a switchis
seen, but the value I = Ij at which it occurs is considerablydelayed beyond I.
Moreover,the amountof delay increaseswithdistancethatIi is fromIL (Fig. 1(c)).
we determinea new
To understandthisdelay,we executethefollowingstrategy:first,
of the steadystate(v, w)
(slowlyvarying)basic referencesolutionas a perturbation
(vs(I), ws(I)). Then, we analyze its stabilitywith respect to the fast time of the
oscillations.We show thatloss in stabilityoccurswell beyond I.
58
S. M. BAER, T. ERNEUX,
J. RINZEL
AND
()0.69
V 0.3v o a
-0.3~
X
--JsvIt
0
1000
t
(c)
0.4(b)
__
_
_
_
_
__
_
_
_
_
_
0.9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~9
0.2
I
0.6.
.4
0.0 -~~~~~~~~~~~~~~~~~~.
o.b Ij
0.2
1-
U.6
0.4
0.2
0.0
0.4
fromthe
(a) The transitionto slowlyvaryingoscillationsis computed
FIG. 1. Delay or memoryeffect.
numericalintegrationof (2.1) for currentI(Et)=Ii+Et, whereI =0.05. Trajectory(SV) shows that the
membranepotentialvariesslowlyin responseto a slowlyrisingcurrent.The onsetof oscillationsis indicated
firstcrossesthe horizontaldashed line v = 0.4, at t = 880. Curve (S) is thesteadystate
whenthe trajectory
valuesofI. Solid denotesstableand dasheddenotesunstable
solutionto (2.1) forincreasing(time-independent)
to
pointI_ = 0.273, whichcorresponds
steadystatesolutions.A stabilitychangeoccursat theHopfbifurcation
analysis,
loss estimated
fromtheHopfbifurcation
timet_= (I_ - I)/ E = 446. Comparedto thetimeofstability
delayed.(b) The slowlyvaryingresponse(SV) forslowlyincreasingI;
theonsetof oscillationsis considerably
dependence
branchofperiodicsolutions(P) fortheparametric
and thesteadystatesolution(S) and bifurcating
a Hopfbifurcation
occursat Ij = 0.490,wellpastthevalueI_ = 0.273predictedfrom
on I. Theonsetofoscillations
usingAUTO
envelopecomputed
theamplitudeofoscillationscontinuetotrackthebifurcation
analysis,however,
of Ij for manyvalues of Ii. Label b refersto cases (a) and (b) above. The
[19]. (c) Numericaldetermination
(dashed) are thepredictedvaluesofIj, fromthenumerical
delayincreasesas (L - Ii) increases.Superimposed
withrespecttothefasttime.Thisillustrates
solutionlosesstability
of (3.5), at whichtheslowlyvarying
integration
Parametervalues are a = 0.2, b = 0.05, y = 0.4, and E = 5 x 10-4.
effect.
thememory
a solutionof (2.1) of
The "slowlyvaryingsteadystate" is foundby determining
the form
(2.7)
v(, ?) =
00
00P
Z E Vj(T),
(T,
j=O
?)
Z E?y
j=O
Q(T)
whereT is a slow timevariabledefinedby
(2.8)
T
=
Et.
(2.7) and (2.8) into(2.1) and
The coefficients
vj(r) and Wj(T) are obtainedby inserting
E.
The
of
each
analysis of the firsttwo
of
coefficients
power
the
zero
to
equating
THE
SLOW
PASSAGE
A HOPF
THROUGH
BIFURCATION
59
problemsleads to the followingresults:
(2.9)
2
v5(T, E) V=V(I(T))
+ ?6bB
byB
Vs(T)+
()
and
(2.10)
w(T, E) =
VS(T)+
v(I(T))-- EB
O(E2)
Bys
and A, B and W0are definedby (2.4b), (2.4c), and (2.6), respectively.
wherev' = -dvs/dT
From (2.9) and (2.10), we notethatthe expansionof the slowlyvaryingsolutiondoes
notbecome singularat the Hopf bifurcation
pointI = I_. Indeed at I = I_, B =
#0?
and the functionsin (2.9) and (2.10) remain0(1) quantities.This contrastswiththe
case of a steadybifurcationor limitpointwherethe expansionof the slowlyvarying
solutionbecomes singularat and near the bifurcationor limitpoint [1], [10]. Note
howeverfrom(2.9) and using the definitions(2.4) and (2.6) that the expansion is
nonuniform
ifb = O(8) or y = 0(e). Bothcases are ofpracticalinterest
and we consider
themin ?? 4 and 5, respectively.
Numericalcomputationswereperformedon a Vax 8600 using a classical fourthorderRunge-Kuttamethodwithfixedstepsize (DT = 0.1). Resultswerealso computed
using a Gear method[18] for stiffdifferential
equations. The two methodsshowed
excellentagreement.To retainaccuracyusing Gear's methodin problemswithslow
passage throughbifurcationpoints,a tightcontrolof relativeerroris imperative(we
used TOL = 10-12). Hence we foundthe RK4 methodto be more efficient
forthese
calculations.In addition,thesimplicity
ofthemethodmakesourresultseasilyreproducible. When a controlparametervariesslowlyand/orwhen I_- Ii is large,numerical
solutionsto (2.1) are particularlysensitiveto roundofferror.Thus we were carefulto
comparecomputationsin single,double, and quadrupleprecision.The resultsforall
figures(except as noted in Figs. 1(c) and 4) were computedin double precision.In
numericalcalculationsthe onsetof oscillationswas definedas the time tj whenthe v
firstcrossedthe value v = 0.4. The bifurcationdiagramin Fig. l(b)
versust trajectory
was computedusingAUTO [19].
3. Stabilityof theslowlyvaryingsolution.In thissection,we analyzethe stability
of the slowlyvaryingsolution(v, w) = (v5,wv).Afterintroducing
the deviations
V(t, ?)
=
v(t, ?)
W(tg ?) =W(t,
-
i(T, ?),
?-) -
(TF
)
into (2.1), we obtainthe followinglinearizedequationsfor V and W:
dV -f'(D(, e))Vdt
(3.2)
W
dW= b(V- yW).
dt
Assumingnow zero initialconditionsfor V and W,we solve (3.2) by a WKB method
[24]. Specifically,we seek a solutionof (3.2) of the form
V(t, ?)
(3.3)
=
V(T, ?)
=
exp [o0(T)/e]
E
?3Vj (T),
j=O
00
i=O
60
AND
S. M. BAER, T. ERNEUX,
J. RINZEL
of each power of s,
Introducing(3.3) into (3.2) and equatingto zero the coefficients
we obtainthe followingproblemfor VOand WO:
Vo =
cr'(T)
-f(V(T))VVo0-WWo)
V- yWO).
C '(T) WO= b(V
(3.4)
equation
A nontrivialsolutionis possible onlyif A = 0''(T) satisfiesthe characteristic
of I(T). From(3.3) we conclude
A and B are now functions
(2.4a) wherethecoefficients
thatat thetime , theslowlyvaryingsolutionis stablewithrespectto the fasttimet if
Re (C) =
(3.5)
0
Re [A(s)] ds< 0.
When the quantityRe (cr) becomes positive then the solution (3.3) exhibitsrapid
unstableon the fast
exponentialgrowthand the slowlyvaryingsolutionis therefore
a
Destabilizationof
is
effect.
memory
timescale. From (3.5) we conclude thatthere
changessign
Re
[A(s)]
when
immediately
not
occur
does
solution
the slowlyvarying
Re (A) > 0
of
effect
integrated
but
after
the
only
I
through
increases
(i.e., when
I1),
of
<
0.
is
independent
Re
Moreover,
(3.5)
influence
of
(A)
accumulated
overcomesthe
slowly.
infinitesimally
tuned
if
is
the
control
parameter
so
even
thatthedelaypersists
?
The importanceof thisintegralconditionforpredictingthedelay was seen previously
forsteadybifurcationproblems[5], [10] and forburstingoscillations[8].
We remarkthat the series (3.3) representsa valid approximationon the time
of (2.4a) givenby
intervalr if the discriminant
(3.6)
D(T)=A2(I(T))
-4B(I(T))
does not vanish.PointswhereD(T) vanishesare where(vs, w,) changesfroma node
to a focus.These pointsare called turningpoints(notto be confusedwithlimitpoints).
small,D(r) maychangesignon theintervalof interestand
If IL - Ii is not sufficiently
invalidin the neighborhoodof the turningpoints.
becomes
solution
the WKB
(3.3)
a
Nevertheless, global approximationto the solutionof (3.2) can be obtainedby the
expansions.In thisstudy,we consideronlythesimplest
methodof matchedasymptotic
case wherethereare no turningpoints,i.e., D(T) < 0 duringthetimeintervalofinterest.
pointswillbe presentedelsewhere.Its analysisleads to a stability
The case withturning
conditionsimilarto (3.5).
We have obtained explicitexpressionsfor the delay and for conditionswhich
which
guaranteethat D(T) remainsnegativeby exploitingalgebraicsimplifications
arisein theparameterrange 0 < a << 1. In thelimita -> 0, we assume I(T) = 0(a), and
findfrom(2.3) and thenfrom(2.4(b), (c)) the followingexpressionsforvS,A, and B:
(3.7)
v,(I) = yI + 0(a2),
(3.8)
A = -2y(I - IO-)+ 0(a2),
(3.9)
B = b + 0(a2b)
ofthefirst
Hopfbifurcation
whereI?. = a/2y correspondsto theleadingapproximation
pointI = IL (from(2.5), v_= a/2+ 0(a2) and thenusing(2.3), IL = I? + 0(a2)). Using
the definition(3.6), we obtain an approximateexpressionforD(Tr)
(3.10)
D(T)
4y2(I()
4b
-
or, equivalently,
(3.11)
D(T)
4y2(I -1- b'12/ )(I
-
i+
I'2/y).
THE
At I= I0 D()
SLOW
PASSAGE
THROUGH
A HOPF
61
BIFURCATION
< 0 and remainsnegativeprovidedthat
(3.12)
I?
-
b1/2/y
<I(T)
< I? + b1/2/y.
Thus, if Ii> I? - b1/2/y,thenD(r) is negativeuntilI = Io + b1/2Y is reached.Because
we assume that D(T) <O duringthe timeintervalof interest,the stabilitychange of
the slowlyvaryingsolutionappears at T = Tj,whichis definedby the condition
(3.13a)
JRe[A(s)]
0
ds-=O
or, equivalently,
(1.13b)|
0
A(s) ds--2^y
o
(I(s)- IO-)ds=-YTi
(I(Ti)
-IO)
-(Io
- I)
0.
Thus, since, I-= Io + 0(a2), we conclude from(3.13b) that
(3.14)
I(j)-I-
= I -Ii
to lowest order. Using (3.14) we easily verifythat D(T) <0 for 0 T Tj. We call
I(Tj) - I_ thedelay ofthebifurcation
transition.
The expression(3.14) emphasizestwo
importantfeaturesof the slow passage throughthe Hopf bifurcation:first,it is
independentof E, the rate of change of the controlparameterI; second,the stability
change of the slowlyvaryingreferencesolution appears at a distancethat is 0(1),
withrespectto E, fromthe staticbifurcation
point I__as seen in Fig. 1. This distance
can be controlledbychangingIi, theinitialvalue of I. We thusobservea memoryeffect.
In Fig. 1(c), we illustratethe memoryeffectby integrating
(2.1) numerically.Our
calculations confirmthat increasingI_-Ii increases the delay of the bifurcation
transition.Moreover,(3.14) is in excellentagreementwiththenumericalresultswhen
I_ -Ii > 0.2. For I_ - Ii < 0.2 thenumericsapparentlydeviatefromour analyticprediction. This is due to the bifurcationbeing supercritical.The bifurcatingbranch of
periodic solutionsis locally stable,so when Ii is near the staticHopf pointthereare
severalsmall oscillationswhose amplituderemainbelow the prescribed"threshold."
For largerdelays thereis usually only one or two such oscillations.Anotherfeature
observedin Fig. 1(c) is a sawtoothjump patternthatoccursbecause thefinalsubthreshold oscillationbeforeonset shiftsin phase as I_- Ii increases.Eventuallya value is
reached thatdelays the onset forone more subthresholdoscillation.The size of the
therampspeed ? bytheperiodof theoscillation
jump Alj is estimatedby multiplying
21r/Wo,that is AIj = E(21T/to).When I_- Ii = O, six subthresholdoscillationsoccur
beforeonset.Thus thejump magnitudein thiscase is about 6E(2v1/wo).
the
4. Slow passagethrougha lowfrequency
Hopf bifurcation.We now investigate
of the slowlyvarying
dynamicsof the case b small,whichappeared as a singularity
reference
solution(2.9) and (2.10). A detailedstudyofthissingularity
(E = 0(b)) leads
to a rich discussionand will be presentedelsewhere.In this section,we considera
particularrelationbetweenE, Ii - I_, and b thatis motivatedby the parametervalues
used in our numericalstudyof theFHN equations(E = 0(b3/2) and Ii - I_= 0(b"/2)).
of
This special relationbetweentheparametersdoes not correspondto the singularity
the slowlyvaryingsolution.However,it can be shown thatthe Hopf bifurcationis
singularin thiscriticalregime[20]. This motivatesa carefulanalysisof thiscase. To
lowestorder,we findthatthe dynamicaldescriptionis givenby a nonlinearproblem.
we obtainusefulinsightshowingthatin this
Althoughwe do not solve it analytically,
case the onsetof oscillationsexhibitsan advance ratherthan a delay.
62
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
In orderto analyzetheslow passage throughthislow frequencyHopf bifurcation,
we firstintroducea new fasttimeand a slow timedefinedby
(4.1)
T= b1/2t
(4.2)
S= bt.
T and S correspondto the timescales of the oscillationsand the controlparameter,
respectively.
For mathematicalsimplicity,
we shall restrictour analysisto thevicinity
of the Hopf bifurcationpoint (v, w,I) = (v_, w_,I) definedby (2.3) and (2.5), and
assume thatthe deviationIL - Ii is small. Specifically,we seek a solutionof the FHN
equations (2.1) of the form
, S)+bV2(T, S)+.*
(4.3)
v(T, S, b"2)- v++b"2 V
(4.4)
w(T S, b"/2)- w_+ b"/2W,(1TS)+ bW2(7TS)+.**,
(4.5)
I(?t)
=
I(S) = I +
b1/21,(S)
,
+ bI2(S) + * - .
The expansion(4.5) fortheslowlyvaryingcontrolparameterI(S) and therequirement
thatI is a functionof the slow timeS imply
Ii-L- = b1/2P+ bP2+**
(4.6)
and
(4.7)
= b312Q
+ b2Q2+...
whereP and Q are prescribed0(1) quantities.Consequently,we may writethat
I,(S) = P+ QS.
(4.8)
Introducing(4.3)-(4.7) into (2.1) and equatingto zero the coefficients
of each power
of b1/2leads to a sequenceofproblemsforthecoefficients
VI, V2, - -. and W,, W2,* Applyingthe standardtechniquesof multi-scateanalysis,we obtainthat
(4.9)
W,=P+QS
and
(4.10)
aV _
w2-f"a(v
aT
(4.11)
aW2=
V=
a9T
)-+P2
2
+
Q2S
-Y(P+ QS)-Q
or, equivalently,if we eliminateW2,
(4.12)
-a2v
=-V
d9T2
a
-f"(V )VV,a+(P
dT
+PQS)+Q.
DefiningU(T, S) =f'(v)
formas
in a simpler
V,(T, S) and using(4.8), (4.12) can be rewritten
(4.13)
U+ U d U= R (S) =f'(v(YII,
2
aTU+
aT
(S) + Q).
Equation (4.13) admitsa slowlyvaryingsolutiongivenby
(4.14)
U(S) = f"(V4(yI,(S) + Q)
Using the expansions(4.3)-(4.7) it can be shownthat
(4.15)
v = iv(S, b"/2) = v + b/2 U(S)/f"(v-) + O(b)
THE
SLOW
PASSAGE
BIFURCATION
A HOPF
THROUGH
63
matches,as b - 0, the outerexpansionof the slowlyvaryingreferencesolution,given
by (2.9). To analyze the linear stabilityof (4.14) withrespectto the fasttime T, we
mustconsiderthe followinglinearizedproblem
(4.16)
a__au_
aT2 +f(V)(YI1(S)
=
+Q) a3U
T UO.
The stabilityproblemis similarto theproblemstudiedin ?3. If I1(S) is consideredas
a constantparameter,the criticalpointdefinedby
(4.17)
I(SC) =-Qy
representsa Hopf bifurcationpointof (4.16). However,since I1(S) is slowlyvarying,
the changeof stabilityof (4.14) will occur later.We analyze (4.16) by usingthe WKB
D(S) = [f"'(v_)4(yl(S) + Q)]2 -4<0 duringtheintervalof
method.If thediscriminant
we
find
that
interest,
(4.18)
I,(S)
-
I,(SC) = I(SC)
-
P
whereI,(Sj) correspondsto theonsetof therapidoscillations.If -2Q/ y - P < 0, (i.e.,
(b)
(a)
R?
R
UT
s
?
UT
~~~~~~~~u
,
(C)
R <0
UT1
<H
W
~~~~~~~u
withlowfrequency.
for slowlyvaryingsolutionin the case of Hopf bifurcation
FIG. 2. Approximation
Sequenceofphase plane portraitsof (4.13) for R > 0, R = 0, and R < 0. (a) R > 0: all initialpointsclose to
to a solutionwhichtrackstheslowly
thesingularpointare attractedtowardthesingularpoint; thiscorresponds
to thetimeat whichI reachesa criticalvalue,I (Sc). Theseparatrix
steadystate.(b) R = 0 corresponds
varying
(c) R < 0 corresponds
familyofperiodicorbitsfromunboundedtrajectories.
UT = -1 dividesa one-parameter
All initialpointslead tounbounded
neartheslowlyvarying
steadystategrowinamplitude.
towhentheoscillations
trajectories.
64
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
ifI_ - Ii <2b112Q/ y) thenI,(Sj) is negative.Consequently,thelargeamplitudeoscillationsthatappear at
Ij = I+
(4.19)
+ O(b)
b2(S)
occur before I reaches the Hopf bifurcationpoint I_. Then thereis no delay, but
ratheran advance.
The precedingconclusionis based on the linearizedequation (4.16). To analyze
the full nonlinearproblem(4.13) in which R is a slowlyvaryingfunctionof S, we
considera slowlyvaryingphase plane technique.Thus,in Fig.2 we examinea sequence
close
of phase planes forR > 0, R = 0, R < 0. In Fig. 2(a), all initialpointssufficiently
to the singularpoint lead to trajectoriesspiralingtoward the singularpoint. This
correspondsto the initiationof the slowlyvaryingsolutionwhen the responsetracks
the slowlychangingsteadystate.In Fig. 2(a) note thatpointslocated below a critical
separatrixlead to unboundedtrajectories.They correspondto the earlierstage of an
[20]. As R(S) becomes zero, or, equivalently,I1(S) = I1(SC) (Fig.
excitabletrajectory
2(b)), the separatrixis a straightline given by UT= -1 and it separates the oneparameterfamilyof periodic orbits surroundingthe center U = UT=0 and the
Finally,Fig. 2(c) showsthephase plane portraitforR < 0 that
unboundedtrajectories.
describesthe slow growthof subthresholdoscillationsjust priorto the onset. The
numericalresultsof Fig. 3 illustratethe phenomenonof an advance and confirmthe
For theseparametervalues (see figurelegend), we have
above asymptotictreatment.
I_ = 0.251,Q = 0.5,and P = -2.01, whichyieldI1(S,) = -1.25 from(4.17) and Ij 0.202,
from(4.18), (4.19).
0.3
0.0
I(sc)
I
Ij I-
0.3
FIG. 3. Advance ratherthan delayfor lowfrequencyoscillations(small b); comparisonof numerical
(4.13) (solid) for
solutionstofull problem(2.1) (dashed) and to the lowestordernonlinearapproximation
parametervaluesa = 0.2, y = 0.4, b = 0.01, and ? = 5 x 10-4. ThesolutionplottedversusI showsthattheonset
pointI- = 0.25. ForIi = 0.05,I(Sc) = I_ -0.125 =
is advancedratherthandelayedrelativetotheHopfbifurcation
an advance. The numericalidentification
0.125 and oscillationscommencenearI) = 0.225< I-, thusindicating
oscillation.
one subthreshold
of (4.19) byapproximately
fromtheprediction
of Ij differs
5. Effectof therate of changeof thecontrolparameter.In the previoussections,
we have consideredE, the rateof changeof the applied current,as a smallparameter.
Except when b is small (low frequencyHopf bifurcation),the delayed bifurcation
SLOW
THE
A HOPF
THROUGH
PASSAGE
BIFURCATION
65
transitiondoes notdepend on s in firstapproximation.However,in a real experiment,
increasesfromsmall to moderate
we mightstudythe dependenceas ? progressively
values and determineif its increasehas a stabilizingor destabilizingeffect.In other
words,we wantto knowifthe onsetof thelargeamplitudeand rapid 6scillationscan
facilitatedas a resultof changingE. Figure
be considerablydelayedor,on thecontrary,
transition
Ij - I_ as a function
4(a) showsnumericalevidenceofthedelayedbifurcation
of 1/8. For now, we disregardthe pointslabeled SP; these data will be discussedin
(a)
0.3'
3
-
~ ~
-
-
-
--
--
-
-
012
0.0
t
-0.1-1
-0.2
(b)
0
1000 2000
3000
l/e
4000
5
5000
10
5-
I-I
j-5
-10*
0
50
1/1
100
150
200
FIG. 4. Increasingtherampspeeddecreasesthemagnitude
ofthedelay.Numericalsolutionsto (2.1) show
transition
thedelayedbifurcation
Ij - I_ as a functionof I/E. (a) Theparametervalues as in Fig. 1, butwith
the asymptotic
Ii=0. Computedresultsasymptoteto Ij - IL = 0.29 as E -> 0. The dashed curverepresents
as se-*0, determined
of (3.5). Pointslabeled (SP) are discussedin
fromthedirectintegration
approximation,
in scale in Figs. 4(a) and 4(b)). Parametervalues
? 6. (b) Delayed transition
for low y case (note difference
are a = 0.2, b = 0.4, y = 0.01, and I, = 0. The dashedcurveis a WKB approximation
of thedelayedtransition,
from(5.7) and (5.8).
computed
66
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
? 6. Note thatas E -*0 (or 1/e - oox) the curveasymptotesto a value consistentwith
the memoryeffectpredictedby (3.5). Also, notethatthe magnitudeof thejumps due
to skippedoscillationsdecreasesas e ->0, since AIj-= (217-Ito)-0 in thislimit.Curves
and theoreticalstudiesof nerveaccomodasuch as thisone are usefulin experimental
tion [21].
To betterunderstandthe effectof ? we analyze the limity-e 0, withE, a, and b
fixed in the FHN equations (2. 1). At y = 0, these equations admit an exact time
dependentsolutiongivenby
(5.1)
v(t) = E/b,
w(t)
I(?t) -f(?/b)
=
where I is given by (2.1d). We consider(5.1) as our new basic referencesolution.
Because v is constant,it is possible to study the linear stabilityof (5.1) by first
reformulating
theevolutionequations(2.1) as a second-orderequation forv onlyand
then by linearizingthis equation about v = E/b. We findthat the small deviation
V(t) = v(t) - ivsatisfiesthe followingequation:
d2+f'(/b)
f(
(5.2)
dV+bV=O.
dt
dt2-
equation, which is exactly(2.4) with
From (5.2) we easily obtain the characteristic
thereference
solution(5.1) is stable
y = 0 and v.(I) replacedby v-= e/b.Conisequently,
(unstable)if v = e/b < v_or v = E/b> v+ (if v_< v = E/b< v,) wherev?(a) is defined
by (2.5) withy = 0. Thus forsmall or largevalues of E,thereferencesolutionis always
stable.On the otherhand formoderatevalues of E, thissolutionis unstableand rapid
oscillationswill develop as soon as timeincreases.
We now considerthe case of a referencesolution,which is stable when y =0
(v < v_or v > v+), and examinethelimity - 0. We firstseek a slowlyvaryingsolution
of (2.1) of the form
(5.3)
i(0, y)
ii(0, y) =
v0(0)+yv1(0)
1
wo(O) + W (0) +
where0 is a new slow timedefinedby
0= yt.
(5.4)
We obtainthecoefficients
v0,v,, wo,w, - , byinserting(5.3) intotheFHN equations
of each power of y. We thenobtainthat
(2.1) and by equatingto zero thecoefficients
v3and w are givenby
(5.5)
3(e, 'y)= E/b+ EO+ 0(y),
w(0, y) = Y'Es0 + 0(1).
We now considerthelinearizedevolutionequationsand analyzethe stabilityof (5.5).
As y - 0, we obtainthefollowingequationforthesmalldeviationV(t) = v(t) - i(0, y)
(5.6)
d2
f'(e/b + e@) d + bV = 0.
'f
dt2
~dt
of dV/dt is now
Note thatthisequation is similarto (5.2) exceptthatthe coefficient
a function0. Since Ii does not appear in (5.6), we do not expect,forfixede and as
y - 0, thatthestabilityof the slowlyvaryingreferencesolutiondepends on the initial
positionof I (recall thatthe memoryeffecthas been foundforfixedy and as e -> 0).
However,we stillhave a delayedbifurcationtransition.This delay can be foundby a
in ? 3. The analysisis tedious and we
WKB analysisof (5.6) similarto the treatment
summarizethe results.
THE
SLOW
PASSAGE
THROUGH
A HOPF
BIFURCATION
67
The Hopf bifurcationpoint I_ and the point Ij wherethejump transitionoccurs
are givenby (assumingf'(i)2 - 4b <0):
(5.7)
L_ y 1(Sc -)/b + 0(1),
(5.8)
I=
2 l{a + 1-3s/b -[(a+
1-3e/b)2 -4f(sl/b)]1/1}+ 0(1)
whereec is definedbytheconditionf'(e,/b)= 0, i.e., s?,= bv_and v_is definedby (2.5).
In conclusion,if e < ?,c the systemapproaches a slowlyvaryingsolutionwhich
remainsstableuntilI = Ij is reached(IJ> I), butife > sc, thentheonsetofoscillations
occursbeforeI = I_. We have analyzedthesepredictionsnumericallyby considering
the followingvalues of the parametersy = 10-2, a = 0.2, and b = 0.4. We determine
SC= 0.038. In Fig. 4(b) we comparethe numericalresultsfor (2.1) withthe analytic
values of s. For
approximation(dashed curve) givenby (5.7) and (5.8) fordifferent
oftheslowlyvaryingsolutionis considerablydelayed;
smallvalues of E, theinstability
however,forlargervalues of S(S > ?c) oscillationsquicklyappear.
6. Discussion. In this paper we have consideredthe effectof a slow monotonic
variationin a controlparameteron the response of a systemas it passes througha
Hopf bifurcation.We have foundthatthe onset of oscillationscan be considerably
delayed if the initialpoint is near the steadystatein its stable regime(Fig. 1). That
is, the systemcontinuesto track,forsome measurabletime,a "slowlyvaryingsteady
state" even afterit has lost stability(determinedwhen the controlvariableis treated
as a static parameter).Eventuallythe destabilizinginfluencesaccumulate and the
from
responsebecomes oscillatory.The delay is greaterif the initialpoint is further
expresses
which
condition
(3.5),
the staticbifurcationpoint (Fig. 1(c)). The integral
methodsforsmall
thenew stabilityconditionwas derivedanalyticallyby perturbation
applies over a
condition
The
variation.
E, where E is the rate of the slow parameter
(associfrequency
characteristic
of
the
scale
time
the
in
which
robustparameterrange
the
whether
on
not
depend
result
does
The
is
0(1).
bifurcation)
the
Hopf
ated with
a
to
particular
our
strategy
we
applied
Although
or
is
subsupercritical.
bifurcation
model problem,the FitzHugh-Nagumoequation,our generalresultis applicable to a
wide class of problems.For this,A in (3.5) is identifiedwiththe eigenvalueof largest
real part.
In certainparameterranges,we have gained additional insightto the model
problemby consideringlimitingparametervalues and usingasymptoticmethods.For
example,if the Hopf bifurcationleads to a slow oscillation,e.g., as in the case of a
relaxationoscillation[20], we findthattheonsetof oscillationmayexhibitan advance
insteadof a delay.
We have also consideredthe dependenceof the onseton the rateof variationof
thecontrolparameter.We have seen numerically(cf.Fig. 4) thatthedelayedinstability
occursearlier.We are uncertain
pointincreaseswith1/s. For fastramps,theinstability
behavior.
However,numericalsimulathis
exhibit
of
as to how largea class problems
the
equation is
which
FitzHugh-Nagumo
model
(for
tions of the Hodgkin-Huxley
in
curves
4(a) [21],
the
Fig.
similar
to
behavior
reveal
a
simplification)
considered
an
description
analytic
have
obtained
we
equation
FitzHugh-Nagumo
For
the
[23].
forthe rate dependencein a special case, y<< 1, and we findusing a WKB analysis
thatthe onsetpointincreasesmonotonicallywith1/s.
We re-emphasizethatnumericalsupportforsome of our analyticresultsrequires
carefulerrorcontrol.For example,when S is verysmall, we employhighprecision
numericsto revealthepredictedasymptoticvalue of the delayed Ij (Fig. 4(a)). In this
68
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
influences(i.e., roundoff
range,thenumericalmodel is subjectto sustainedperturbing
errors)fora considerabledurationas I passes intotheunstableregimebeyondI_ but
beforeIj is reached; quadruple precisionreducesthe effectof roundoff.By analogy,
of delays and advances foran experimental
we would anticipatesimilarsensitivities
These observations
or imposed fluctuations.
systemthatis exposed to environmental
(sinusoidal,as well
of sustainedperturbations
led us to explorespecificallythe effects
as stochastic)on thedelayphenomena.For this,we again considereda modelproblem
of the FHN-type:
(6.1)
d = -f(v) - w+ I(et) +3 sin (ct)
dw
(62)
d- = b(v- yw)
dt
in whichboth 3 and ? are small parameters.Note thatthe problemalso depends on
co,the frequencyof the time periodicperturbation.As co-- 0, sin (cot)- 0 if t<< 1/co
and thereis no effectof the perturbation(if 3 cos (cot) was consideredinstead of
theinitialvalue ofI as I = Ii + 3,
3 sin (cot),then3 cos (art) 3Sift<< 1/co;byredefining
perturbations).On the otherhand, as
we again findno effectof the time-dependent
Co- 00, the periodic forcingrepresentsrapid oscillationsand only its average value
will contributeto the long time behavior of the solution.This can be shown by a
analysiswhereT- cotis now consideredas thebasic fasttime.The average
multi-time
value of the periodicoscillationis zero and consequently,we do not expectan effect
studiedpreviously
We concludethatthedelay and memoryeffects
oftheperturbation.
remainunchangedin the presenceof small amplitudeperiodicforcingif the forcing
high.
small or sufficiently
frequencyis eithersufficiently
As we expect,the delay is most sensitiveto frequenciesnear co0.Figure 5(a)
illustratesthatthe delay is reducedconsiderably.The reductionis moredramaticfor
withdelay
resonanceeffects
larger3, and we also see subharmonicand superharmonic
reductionsforconear coo/3,coo/2,and 2co.
amplitude3 as an adjustableparameter,then
If we now considertheperturbation
exhibitsthreeregimesof behavior (solid curvesof Fig. 5(b)). For 3
the sensitivity
small,3<<S, thedelayis maximaland independentof 3. If 3 is sufficiently
sufficiently
3, Ij - I_ decreasesas 3 increases
large,thereis an advancewithIj - Ii. For intermediate
with a sizable range of approximatelylinear dependence on (-ln 3)1/2 for 3 just
below SC.
Some featuresof the above numericalstudy of sensitivityare supportedby
analyticresultsfromconsideringthe linearstabilityof the slowlyvarying
preliminary
solution(V(Et, e), W(?t, ?)) as 3 -> 0 when e is small but fixed.In particular,we find
that Sc = O(e-l/6), and thatIj - I- depends linearlyon (-ln 3)1/2 for3 just below SC.
A similarbehaviorhas been found by an asymptoticanalysis of a problemwhich
exhibitsa staticbifurcation[22]. Our analysisalso indicatesthepossibilityof subharmonicresonanceas seen in Fig. 5(b).
of the small 3
For our model problemwe conclude thataccurateidentification
with
requiresthatperturbations
in thepresenceof sustainedperturbations,
asymptote,
to verysmall amplitudes,say less than
frequencycomponentsnear co be restricted
with
about 10-7 10-8 relativeto I(st). Generally,a systemis subjectto fluctuations
componentsand we shouldnotassumethatselectedfrequency
frequency
manydifferent
rangeswould be absent.To emphasizethiswe have simulatedtheeffectof whitenoise
superimposedupon the controlparameter,i.e., we have replaced,in (6.1), (6.2), the
THE
SLOW
PASSAGE
A HOPF
THROUGH
69
BIFURCATION
() 0.250
0.000
(b)
0
*
1
I .
*
*
I
I
2
*
I
I
3
0.3 -I
0.0
1/
(e ln(/2
-':"
-i
n(d)1/
=
:14
-0.30.0
2.5
(
2 x 1
ln
)125.0
fluctuations(values of a, b, y as in
periodicand stochastic
FIG. 5. Delay is sensitiveto small amplitude
Fig. 1, but with? = 10-'). (a) Numericalsolutionsto (6.1) and (6.2) showthatthedelay is mostsensitiveto
at frequenciesnear w0= 0.223, givenby (2.6). Dashed linesindicatesubharmonic
smallperiodicfluctuations
w near wo/3,O0/2,(oO and 2wo. Thetwocurves
withdelayreductionsfor
resonanceeffects
and superharmonic
amplitude
totheperturbation
8 = 5 x 10-6) showthatthedelaysare sensitive
(lowerforS= 2 x 10-3 and upperfor
amplitude3 is treatedas an adjustableparameter,and numericalsolutionsto (6.1) and
3. (b) Perturbation
solutions
withnumerical
(6.2) (solid curves)fornearresonantfrequencies
o1/2,wo,and 2woare superimposed
to (2.1) for simulatedwhitenoise. The value of Ij for each of 100 diferentvalues of 3 is plottedas a discrete
transition
datapointtorepresent
thenoisedata. Thedelayedbifurcation
Ij- I, forall cases,has an approximate
lineardependenceon (-In 8)1/2.
sinusoidalforcingbythestochasticforcingterm8o(t), wherecr(t)is a randomnumber
values of
in [-0.5,0.5]. The value of Ij foreach of 100 different
distributed
uniformly
8 is plottedin Fig. 5(b) as a discretedata point.The trendis again that,evenforsmall
8, thereis a deviationfromthe predicteddelay for 8 = 0. We should also noticethe
70
S. M. BAER, T. ERNEUX,
AND
J. RINZEL
sudden dropoffat a critical8 whenthe observedIj approximatelyequals I_. If this
featureis robust,thenit could be used to estimateI_.
The above calculationsfortheeffects
ofperiodicperturbations
and smallamplitude
whitenoise help us to understandwhynumericalcalculationsinvolvingslow passage
througha Hopf bifurcationcan be particularlysensitiveto roundofferror.Random
evenas smallas 10-8,can reducethedelaywhenE is small.This amplitude
fluctuations,
is approximatelyequal to thatof singleprecision"machine noise" due to roundoff.
To emphasizethispointwe compare,in Fig.4(a), thedependenceofIj on 1/? computed
withboth quadruple and singleprecision;the latterresultsare distinguishedby the
label SP. Note how roundofferrorseriouslyaffectsthe single precisionresultas decreases.The value of - below whichthe roundofferrorfirstappears is dependent
on thespecificnumericalalgorithm.However,deviationfromthedeterministic
prediction due to roundofferroris unavoidableif s is small.
Everybiological or physicalexperimentis subjectto noise. Noise can influence
the outcomeof an experimentif the systemis particularly
sensitive.In thispaper,the
parameterrange forwhich most of our analyticresultsare applicable, - << 1, is also
the parameterrangeforwhichthe FHN systemseems to be quite sensitiveto noise.
We have shownthatin generalwe may expectdelays in the onsetof oscillationsbut
that small amplitudefluctuationmay decrease the delay and diminishthe memory
effect.
We suggestthatbothdeterministic
and stochasticapproacheswill be important
for comparingtheoreticaland experimentalresultsin systemswhere slow passage
througha Hopf bifurcationis theunderlyingmechanismforthe onsetof oscillations.
We thankShihab Shamma forhelpfuldiscussionsand suggesAcknowledgment.
tions.
Noteaddedinproof.It has recently
come to our attention
thata different
approach
can be used to analyzethedelay due to theslow passage throughtheHopf bifurcation
point[A. I. Neishtadt,Persistence
ofstability
lossfordynamicalbifurcations
I, Differential Equations,23 (1987), pp. 1385-1391].
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