Table Of ContentJournalofMathematicalNeuroscience(2013)3:2
DOI10.1186/2190-8567-3-2
RESEARCH OpenAccess
Phase-Amplitude Descriptions of Neural Oscillator
Models
K.C.A.Wedgwood·K.K.Lin·R.Thul·
S.Coombes
Received:31July2012/Accepted:18January2013/Publishedonline:24January2013
©2013K.C.A.Wedgwoodetal.;licenseeSpringer.ThisisanOpenAccessarticledistributedunderthe
termsoftheCreativeCommonsAttributionLicense(http://creativecommons.org/licenses/by/2.0),which
permitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkis
properlycited.
Abstract Phaseoscillatorsareacommonstartingpointforthereduceddescription
ofmanysingleneuronmodelsthatexhibitastronglyattractinglimitcycle.Theframe-
workforanalysingsuchmodelsinresponsetoweakperturbationsisnowparticularly
welladvanced,andhasallowedforthedevelopmentofatheoryofweaklyconnected
neuralnetworks.However,thestrong-attractionassumptionmaywellnotbethenat-
ural one for many neural oscillator models. For example, the popular conductance
basedMorris–Lecarmodelisknowntorespondtoperiodicpulsatilestimulationina
chaoticfashionthatcannotbeadequatelydescribedwithaphasereduction.Inthispa-
per,wegeneralisethephasedescriptionthatallowsonetotracktheevolutionofdis-
tancefromthecycleaswellasphaseoncycle.Weuseaclassicaltechniquefromthe
theoryofordinarydifferentialequationsthatmakesuseofamovingcoordinatesys-
temtoanalyseperiodicorbits.Thesubsequentphase-amplitudedescriptionisshown
tobeverywellsuitedtounderstandingtheresponseoftheoscillatortoexternalstim-
uli (which are not necessarily weak). We consider a number of examples of neural
oscillatormodels,rangingfromplanarthroughtohighdimensionalmodels,toillus-
tratetheeffectivenessofthisapproachinprovidinganimprovementoverthestandard
phase-reductiontechnique.Asanexplicitapplicationofthisphase-amplitudeframe-
work, we considerin some detailthe response of a genericplanar modelwherethe
strong-attractionassumptiondoesnothold,andexaminetheresponseofthesystem
(cid:2)
K.C.A.Wedgwood( )·R.Thul·S.Coombes
SchoolofMathematicalSciences,UniversityofNottingham,Nottingham,NG72RD,UK
e-mail:[email protected]
R.Thul
e-mail:[email protected]
S.Coombes
e-mail:[email protected]
K.K.Lin
DepartmentofAppliedMathematics,UniversityofArizona,Tucson,AZ,USA
e-mail:[email protected]
Page2of22 K.C.A.Wedgwoodetal.
toperiodicpulsatileforcing.Inaddition,weexplorehowthepresenceofdynamical
shearcanleadtoachaoticresponse.
Keywords Phase-amplitude·Oscillator·Chaos·Non-weakcoupling
ListofAbbreviations
ML Morris–Lecar
FHN FitzHugh–Nagumo
CS Connor–Stevens
LE Lyapunovexponent
1 Introduction
Oneonlyhastolookattheplethoraofpapersandbooksonthetopicofphaseoscil-
lators in mathematical neuroscience to see the enormous impact that this tool from
dynamicalsystemstheoryhashadonthewaywethinkaboutdescribingneuronsand
neuralnetworks.Muchof thisworkhas itsroots inthetheoryof ordinarydifferen-
tialequations(ODEs)andhasbeenpromotedformanyyearsintheworkofWinfree
[1],Guckenheimer[2],Holmes[3],Kopell[4],Ermentrout[5]andIzhikevich[6]to
name but a few. For a recent survey, we refer the reader to the book by Ermentrout
and Terman [7]. At heart, the classic phase reduction approach recognises that if a
high dimensional non-linear dynamical system (as a model of a neuron) exhibits a
stablelimitcycleattractorthentrajectoriesnearthecyclecanbeprojectedontothe
cycle.
Anaturalphasevariableissimplythetimealongthecycle(fromsomearbitrary
origin)relativetotheperiodofoscillation.Thenotionofphasecanevenbeextended
off the cycle using the concept of isochrons [1]. They provide global information
aboutthe‘latentphase’,namelythephasethatwillbeasymptoticallyreturnedtofor
atrajectorywithinitialdatawithinthebasinofattractionofanexponentiallystable
periodicorbit.Moretechnically,isochronscanbeviewedastheleavesoftheinvariant
foliationofthestablemanifoldofaperiodicorbit[8].Inrotatingframecoordinates
givenbyphaseandtheleafoftheisochronfoliation,thesystemhasaskew-product
structure, i.e. the equationof the phase decouples.However,it is a major challenge
tofindtheisochronfoliation,andsinceitreliesontheknowledgeofthelimitcycleit
canonlybefoundinspecialcasesornumerically.Therearenowanumberofcom-
plementarytechniquesthattacklethislatterchallenge,andinparticularwereferthe
readertoworkofGuillamonandHuguet[9](usingLiesymmetries)andOsingaand
Moehlis[10](exploitingnumericalcontinuation).MorerecentworkbyMauroyand
Mezic[11]isespeciallyappealingasitusesasimpleforwardintegrationalgorithm,
as illustrated in Fig. 1 for a Stuart–Landau oscillator. However, it is more common
to side-step the need for constructing global isochrons by restricting attention to a
smallneighbourhoodofthelimitcycle,wheredynamicscansimplyberecastinthe
reducedformθ˙=1,whereθ isthephasearoundacycle.Thisreductiontoaphase
descriptiongivesanicesimpledynamicalsystem,albeitonethatcannotdescribeevo-
lutionoftrajectoriesinphase-spacethatarefarawayfromthelimitcycle.However,
JournalofMathematicalNeuroscience(2013)3:2 Page3of22
Fig.1 Isochronsofa
Stuart–Landauoscillatormodel:
x˙=λx/2−(λc/2+ω)y−
λ(x2+y2)(x−cy)/2,
y˙=(λc/2+ω)x+λy/2−
λ(x2+y2)(cx+y)/2.The
blackcurverepresentsthe
periodicorbitofthesystem,
whichissimplytheunitcircle
forthismodel.Thebluecurves
aretheisochronsobtainedusing
the(forward)approachof
MauroyandMezic[11].The
greendotsareanalytically
obtainedisochronalpoints[15].
Parametervaluesareλ=2.0,
c=1.0,andω=1.0
the phase reduction formalism is useful in quantifying how a system (on or close
toacycle)respondstoweakforcing,viatheconstructionoftheinfinitesimalphase
responsecurve(iPRC).Foragivenhighdimensionalconductancebasedmodelthis
canbesolvedfornumerically,thoughforsomenormalformdescriptionsclosedform
solutionsarealsoknown[12].TheiPRCatapointoncycleisequaltothegradient
of the (isochronal) phase at that point.This approach forms the basis for construct-
ing models of weakly interacting oscillators, where the external forcing is pictured
as a function of the phase of a firing neuron. This has led to a great deal of work
onphase-lockingandcentralpatterngenerationinneuralcircuitry(see,forexample
[13]). Note that the work in [9] goes beyond the notion of iPRC and introduces in-
finitesimalphaseresponsesurfaces(allowingevaluationofphaseadvancementeven
when the stimulus is off cycle), and see also the work in [14] on non-infinitesimal
PRCs.
Theassumptionthatphasealoneisenoughtocapturetheessentialsofneuralre-
sponse is one made more for mathematical convenience than being physiologically
motivated.Indeed,forthepopulartypeIMorris–Lecar(ML)firingmodelwithstan-
dardparameters,directnumericalsimulationswithpulsatileforcingshowresponses
thatcannotbeexplainedsolelywith aphase model[16]. The failure of a phasede-
scription is in itself no surprise and underlies why the community emphasises the
use of the word weakly in the phrase “weakly connected neural networks”. Indeed,
there are a number of potential pitfalls when applying phase reduction techniques
to a system that is not in a weakly forced regime. The typical construction of the
phaseresponsecurveusesonlylinearinformationabouttheisochronsandnon-linear
effectswillcomeintoplaythefurtherwemoveawayfromthelimitcycle.Thisprob-
lem can be diminished by taking higher order approximations to the isochrons and
using this information in the construction of a higher order PRC [17]. Even using
perfectinformationaboutisochrons,thephasereductionstillassumespersistenceof
thelimit-cycleandinstantaneousrelaxationbacktocycle.However,thepresenceof
nearby invariant phase-space structures such as (unstable) fixed points and invari-
ant manifolds may result in trajectories spending long periods of time away from
the limit cycle. Moreover, strong forcing will necessarily take one away from the
Page4of22 K.C.A.Wedgwoodetal.
neighbourhood of a cycle where a phase description is expected to hold. Thus, de-
veloping a reduced description, which captures some notion of distance from cycle
isakeycomponentofanytheoryofforcedlimitcycleoscillators.Thedevelopment
of phase-amplitude models that better characterise the response of popular high di-
mensional single neuron models is precisely the topic of this paper. Given that it is
amajorchallengetoconstructanisochronalfoliationweusenon-isochronalphase-
amplitudecoordinatesasapracticalmethodforobtainingamoreaccuratedescription
ofneuralsystems.Recently,Medvedev[18]hasusedthisapproachtounderstandin
moredetailthesynchronisationoflinearlycoupledstochasticlimitcycleoscillators.
InSect.2,weconsiderageneralcoordinatetransformation,whichrecaststhedy-
namicsofasystemintermsofphase-amplitudecoordinates.Thisapproachisdirectly
takenfromtheclassicaltheoryforanalysingperiodicorbitsofODEs,originallycon-
sidered for planar systems in [19], and for general systems in [20]. We advocate it
hereasonewaytomovebeyondapurelyphase-centricperspective.Weillustratethe
transformation by applying it to a range of popular neuron models. In Sect. 3, we
consider how inputs to the neuron are transformed under these coordinate transfor-
mations and derive the evolution equations for the forced phase-amplitude system.
Thisreducestothestandardphasedescriptionintheappropriatelimit.Importantly,
weshowthatthebehaviourofthephase-amplitudesystemismuchmoreabletocap-
ture that of the original single neuron model from which it is derived. Focusing on
pulsatile forcing, we explore the conditions for neural oscillator models to exhibit
shearinducedchaos[16].FinallyinSect.4,wediscusstherelevanceofthisworkto
developingatheoryofnetworkdynamicsthatcanimproveuponthestandardweak
couplingapproach.
2 Phase-AmplitudeCoordinates
Throughout this paper, we study the dynamics prescribed by the system x˙ =f(x),
x ∈Rn, with solutions x =x(t) that satisfy x(0)=x ∈Rn. We will assume that
0
the system admits an attracting hyperbolic periodic orbit (namely one zero Floquet
exponentand the others havingnegativereal part), withperiod (cid:5),such that u(t)=
u(t +(cid:5)). A phase θ ∈[0,(cid:5)) is naturally defined from θ(u(t))=tmod(cid:5). It has
longbeenknowninthedynamicalsystemscommunityhowtoconstructacoordinate
system based on this notion of phase as well as a distance from cycle; see [20] for
adiscussion.Infact,ErmentroutandKopell[21]madegooduseofthisapproachto
derive the phase-interaction function for networks of weakly connected limit-cycle
oscillatorsinthelimitofinfinitelyfastattractiontocycle.However,thisassumption
isparticularlyextremeandunlikelytoholdforabroadclassofsingleneuronmodels.
Thus,itisinterestingtoreturntothefullphase-amplitudedescription.Inessence,the
transformationtothesecoordinatesinvolvessettingupamovingorthonormalsystem
aroundthelimitcycle.Oneaxisofthissystemischosentobeinthedirectionofthe
tangent vector along the orbit, and the remaining are chosen to be orthogonal. We
introducethenormalisedtangentvectorξ as
(cid:3) (cid:3)
(cid:2)(cid:3) (cid:3)
ξ(θ)= du (cid:3)(cid:3)du(cid:3)(cid:3). (1)
dθ dθ
JournalofMathematicalNeuroscience(2013)3:2 Page5of22
Fig.2 Demonstrationofthe
movingorthonormalcoordinate
systemalonganorbitsegment.
Astevolvesfromt0tot1,the
coordinatesvarysmoothly.In
thisplanarexample,ζ always
pointstotheoutsideoftheorbit
Theremainingcoordinateaxesareconvenientlygroupedtogetherasthecolumnsof
ann×(n−1)matrixζ.Inthiscasewecanwriteanarbitrarypointx as
x(θ,ρ)=u(θ)+ζ(θ)ρ. (2)
Here,|ρ|representstheEuclideandistancefromthelimitcycle.Acaricature,inR2,
ofthecoordinatesystemalonganorbitsegmentisshowninFig.2.Throughtheuse
of the variable ρ,we canconsider pointsawayfrom theperiodicorbit. Rather than
beingisochronal,linesofconstantθ aresimplystraightlinesthatemanatingfroma
pointontheorbitinthedirectionofthenormal.Thetechnicaldetailsofspecifying
theorthonormalcoordinatesformingζ arediscussedinAppendixA.
Uponprojectingthedynamicsontothemovingorthonormalsystem,weobtainthe
dynamicsofthetransformedsystem:
θ˙=1+f (θ,ρ), ρ˙=A(θ)ρ+f (θ,ρ), (3)
1 2
where
(cid:4) (cid:5)
dζ
f (θ,ρ)=−hT(θ,ρ) ρ+hT(θ,ρ) f(u+ζρ)−f(u) , (4)
1
dθ
(cid:4) (cid:5)
dζ
f (θ,ρ)=−ζT ρf +ζT f(u+ζρ)−f(u)−Dfζρ , (5)
2 1
dθ
(cid:6)(cid:3) (cid:3) (cid:7) (cid:6) (cid:7)
h(θ,ρ)= (cid:3)(cid:3)(cid:3)du(cid:3)(cid:3)(cid:3)+ξT dζρ −1ξ, A(θ)=ζT −dζ +Dfζ , (6)
dθ dθ dθ
and Df is the Jacobian of the vector field f evaluated along the periodic orbit u.
The derivation of this system may be found in Appendix A. It is straightforward to
show that f (θ,ρ)→0 as |ρ|→0, f (θ,0)=0 and that ∂f (θ,0)/∂ρ =0. In the
1 2 2
above, f captures the shear present in the system, that is, whether the speed of θ
1
increasesordecreasesdependentonthedistancefromcycle.Aprecisedefinitionfor
shearisgivenin[22].Additionally,A(θ)describestheθ-dependentrateofattraction
orrepulsionfromcycle.
It is pertinent to consider where this coordinate transformation breaks down,
thatis,wherethedeterminantoftheJacobianofthetransformationK =det[∂x/∂θ
∂x/∂ρ] vanishes. This never vanishes on-cycle (where ρ =0), but may do so for
some|ρ|=k>0.Thissetsanupperboundonhowfarawayfromthelimitcyclewe
candescribethesystemusingthesephase-amplitudecoordinates.InFig.3,weplot
thecurvealongwhichthetransformationbreaksdownfortheMLmodel.Weobserve
that,forsomevaluesofθ,k isrelativelysmaller.Thebreakdownoccurswherelines
Page6of22 K.C.A.Wedgwoodetal.
Fig.3 Thisfigureshowsthe
determinantKofthe
phase-amplitudetransformation
fortheMLmodel.Colours
indicatethevalueofK.Thered
contourindicateswhereK=0,
andthuswherethecoordinate
transformationbreaksdown.
Notehowallofthevaluesfor
whichthisoccurshaveρ<0.
Parametervaluesasin
AppendixB.1
ofconstantθ cross,andthusthetransformationceasestobeinvertible,andtheseval-
uesofθ correspondtopointsalongwhichtheorbithashighcurvature.Wenotethat
thisissueislessproblematicalinhigherdimensionalmodels.
Ifwenowconsiderthedrivensystem,
x˙=f(x)+εg(x,t), x∈Rn, (7)
whereε isnotnecessarilysmall,wemayapplythesametransformationasaboveto
obtainthedynamicsin(θ,ρ)coordinates,whereθ ∈[0,(cid:5))andρ∈Rn−1,as
(cid:8) (cid:9)
θ˙=1+f (θ,ρ)+εhT(θ,ρ)g u(θ)+ζ(θ)ρ,t , (8)
1
(cid:8) (cid:9)
ρ˙=A(θ)ρ+f (θ,ρ)+εζTB(θ,ρ)g u(θ)+ζ(θ)ρ,t , (9)
2
with
dζ
B(θ,ρ)=I − ρhT(θ,ρ), (10)
n
dθ
and I is the n×n identity matrix. Here, h and B describe the effect in terms of θ
n
andρ thattheperturbationshave.DetailsofthederivationaregiveninAppendixA.
For planar models, B =I . To demonstrate the application of the above coordinate
2
transformation,wenowconsidersomepopularsingleneuronmodels.
2.1 A2DConductanceBasedModel
TheMLmodelwasoriginallydevelopedtodescribethevoltagedynamicsofbarnacle
giant muscle fibre [23], and is now a popular modelling choice in computational
neuroscience[7].Itiswrittenasapairofcouplednon-linearODEsoftheform
Cv˙=I(t)−gl(v−vl)−gKw(v−vK)−gCam∞(v)(v−vCa),
(cid:8) (cid:9) (11)
w˙ =φ w∞(v)−w /τw(v).
Here,v isthemembranevoltage,whilstw isagatingvariable,describingthefrac-
tionofmembraneionchannelsthatareopenatanytime.Thefirstequationexpresses
JournalofMathematicalNeuroscience(2013)3:2 Page7of22
Fig.4 TypicaltrajectoryoftheMLmodelofthetransformedsystem.Left:Timeevolutionofθ andρ.
Right:Trajectoryplottedinthe(v,w)phaseplane.Weseethatwhenρhasalocalmaximum,theevolution
ofθslowsdown,correspondingtowheretrajectoriespassneartothesaddlenode.Parametervaluesasin
AppendixB.1
Kirchoff’scurrentlawacrossthecellmembrane,withI(t)representingastimulusin
theformofaninjectedcurrent.ThedetailedformofthemodeliscompletedinAp-
pendixB.1.TheMLmodelhasaveryrichbifurcationstructure.Roughlyspeaking,
byvaryingaconstantcurrentI(t)≡I ,oneobserves,indifferentparameterregions,
0
dynamicalregimescorrespondingto sinks, limitcycles,and Hopf,saddle-nodeand
homoclinic bifurcations, as well as combinations of the above. These scenarios are
discussedindetailin[7]and[24].
As the ML model is planar, ρ is a scalar, as are the functions A and f . This
1,2
allows us to use the moving coordinate system to clearly visualise parts of phase
spacewheretrajectoriesareattractedtowardsthelimitcycle,andpartsinwhichthey
move away from it, as illustrated in Fig. 4. The functions f and A, evaluated at
1,2
ρ =−0.1 are shown in Fig. 5. The evolution of θ is mostly constant, however we
clearlyobserveportionsofthetrajectorieswherethisisslowed,alongwhichρ˙≈0.
In fact, this corresponds to where trajectories pass near to the saddle node, and the
dynamicsstall.Thisoccursaroundθ =17,andinFig.5weseethatbothA(θ)and
f (θ,ρ)areindeedcloseto0.Thereducedvelocitiesoftrajectoriesherehighlights
1
the importance of considering other phase space structures in forced systems, the
details of which are missed in standard phase only models. Forcing in the presence
ofsuchstructuresmaygiverisetocomplexandevenchaoticbehaviours,asweshall
seeinSect.3.
Inthenextexample,weshowhowthesameideasgoacrosstohigherdimensional
models.
2.2 A4DConductanceBasedModel
TheConnor–Stevens(CS)model[25]isbuiltupontheHodgkin–Huxleyformalism
+ +
andcomprisesafastNa current,adelayedK current,aleakcurrentandatransient
+
K current, termed the A-current. The full CS model consists of 6 equations: the
membranepotential,theoriginalHodgkin–Huxleygatingvariables,andanactivating
and inactivating gating variable for the A-current. Using the method of equivalent
Page8of22 K.C.A.Wedgwoodetal.
Fig.5 f1,f2,andAfortheMLmodel,evaluatedatρ=−0.1.Weclearlyseethedifferenceintheorder
ofmagnitudebetweenf1 andf2 forsmallρ.Notethat,althoughtheaverageofAoveroneperiodis
negative,itispositiveforanon-trivialintervalofθ.Thiscorrespondstomovementclosetothestable
manifoldofthesaddlenode.ParametervaluesasinAppendixB.1
potentials [26], we may reduce the dimensionality of the system, to include only 4
variables.Thereducedsystemis
Cv˙=−F(v,u,a,b)+I, u˙=G(v,u),
X˙ = X∞(v)−X, X∈{a,b}, (12)
τ (v)
X
where
F(v,u,a,b)=gKn4∞(u)(v−vK)+gNah∞(u)m3∞(v)(v−vNa)
+gl(v−v)+g a3b(v−v ). (13)
l l a a
ThedetailsofthereducedCSmodelarecompletedinAppendixB.2.Thesolutions
tothereducedCSmodelunderthecoordinatetransformationmaybeseeninFig.6,
whilst,inFig.7,weshowhowthissolutionlooksintheoriginalcoordinates.Asfor
theMLmodel,θ evolvesapproximatelyconstantlythroughout,thoughthisevolution
is sped up close to θ =(cid:5). The trajectories of the vector ρ are more complicated,
but note that there is regularity in the pattern exhibited, and that this occurs with
approximately the same period as the underlying limit cycle. The damping of the
JournalofMathematicalNeuroscience(2013)3:2 Page9of22
Fig.6 SolutionofthetransformedCSsystem.Top:Timeevolutionofθ.Bottom:Timeevolutionofρ
coordinates.Upontransformingthesecoordinatesbacktotheoriginalones,wearriveatFig.7.Parameter
valuesgiveninAppendixB.2.Inthisparameterregime,themodelexhibitstypeIfiringdynamics
amplitudeofoscillationsinρoversuccessiveperiodsrepresentstheoverallattraction
tothelimitcycle,whilsttheregularbehaviourofρ representsthespecificrelaxation
to cycle as shown in Fig. 7. Additional file 1 shows a movie of the trajectory in
(v,u,b) coordinateswiththemovingorthonormalsystemsuperimposed,aswellas
thesolutioninρ forcomparison.
3 PulsatileForcingofPhase-AmplitudeOscillators
Wenowconsiderasystemwithtime-dependentforcing,givenby(7)with
(cid:10)(cid:8) (cid:9)
g(x,t)= δ(t−nT),0,...,0 T, (14)
n∈Z
whereδ istheDiracδ-function.ThisdescribesT-periodickickstothevoltagevari-
able. Even such a simple forcing paradigm can give rise to rich dynamics [16]. For
theperiodicallykickedMLmodel,shearforcescanleadtochaoticdynamicsasfolds
and horseshoes accumulate under the forcing. This means that the response of the
neuron is extremely sensitive to the initial phase when the kicks occur. In terms of
neuralresponse,thismeansthattheneuronisunreliable[27].
Thebehaviourofoscillatorsundersuchperiodicpulsatileforcingisthesubjectof
anumberofstudies;see,e.g.[27–30].Ofparticularrelevancehereis[27],inwhich
a qualitative reasoning of the mechanisms that bring about shear in such models is
Page10of22 K.C.A.Wedgwoodetal.
Fig.7 Transformedtrajectoryin(v,u,b)spaceofthephase-amplitudedescriptionofthereducedCS
model.Thedottedblacklineisthephaseamplitudesolutiontransformedintheoriginalcoordinates,whilst
thecolouredorbitistheunderlyingperiodicorbit,wherethecolourcorrespondstothephasealongthe
orbit.Thesolutionofthephase-amplitudedescriptionofthemodelin(θ,ρ)coordinatesisshowninFig.6
supplementedbydirectnumericalsimulationstodetectthepresenceofchaoticsolu-
tions.FortheMLmodelinaparameterregionclosetothehomoclinicregime,kicks
can cause trajectories to pass near the saddle-node, and folds may occur as a result
[16].
We here would like to compare full planar neural models to the simple model,
studiedin[27]:
(cid:10)
θ˙=1+σρ, ρ˙=−λρ+εP(θ) δ(t−nT). (15)
n∈Z
This system exhibits dynamical shear, which under certain conditions, can lead to
chaotic dynamics. The shear parameter σ dictates how much trajectories are ‘sped
up’ or ‘slowed down’ dependent on their distance from the limit cycle, whilst λ is
the rate of attraction back to the limit cycle, which is independent of θ. Supposing
thatthefunction P is smoothbutnon-constant,trajectorieswillbetakena variable
distance from the cycle upon the application of the kick. When kicks are repeated,
thisgeometricmechanismcanleadtorepeatedstretchingandfoldingofphasespace.
Itisclearthatthelarger σ isin(15),themoreshearispresent,andthemorelikely
wearetoobservethefoldingeffect.Inasimilarway,smallervaluesofλmeanthat
theshearhaslongertoactupontrajectoriesandagainresultinagreaterlikelihoodof
chaos.Finally,toobservechaoticresponse,wemustensurethattheshearforceshave
sufficienttimetoact,meaningthatT,thetimebetweenkicksmustnotbetoosmall.
