Asymptotic Phase for Stochastic Oscillators

Peter J. ThomasBernstein Center for Computational Neuroscience. Humboldt University, 10115 Berlin, Germany.

Department of Mathematics, Applied Mathematics,and Statistics. Case Western Reserve University, Cleveland, Ohio, 44106, USA.

Benjamin LindnerBernstein Center for Computational Neuroscience and Department of Physics. Humboldt University, 10115 Berlin, Germany.

(Dated: January 18, 2015)

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arisefrom a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of theasymptotic phase function. In the presence of noise, the asymptotic phase is no longer well defined.We introduce a new definition of asymptotic phase in terms of the slowest decaying modes of theKolmogorov backward operator. Our stochastic asymptotic phase is well defined for noisy oscillators,even when the oscillations are noise dependent. It reduces to the classical asymptotic phase in thelimit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or byempirical observation of an oscillating density’s approach to its steady state.

Introduction. Limit cycles (LC) appear in deter-ministic models of nonlinear oscillators such as spikingnerve cells [1], central pattern generators [2], and non-linear circuits [3]. The reduction of LC systems to one-dimensional “phase” variables is an indispensable toolfor understanding entrainment and synchronization ofweakly coupled oscillators [4, 5]. Within the determinis-tic framework, all initial points converge to the LC, onwhich we can define a phase that progresses at a constantrate (θ = ωLC = 2π/TLC). The phase θ(x0) of any pointx0 is then defined by the asymptotic convergence of thetrajectory to that phase on the LC. However, stochasticoscillations are ubiquitous, for example in biological sys-tems [6], and in this setting the classical definition of thephase breaks down. For a noisy dynamics, all initial den-sities will converge to the same stationary density. Thusthe large-t asymptotic behavior no longer disambiguatesinitial conditions, and the classical asymptotic phase isnot well defined.

Schwabedal and Pikovsky attacked this problem bydefining the phase for a stochastic oscillator in termsof the mean first passage times (MFPT) between sur-faces analogous to the isochrons (level curves of the phasefunction θ(x)) of deterministic LC [7–9]. Here we formu-late an alternative definition that is tied directly to theasymptotic behavior of the density, rather than the firstpassage time, and is grounded in the analysis of the for-ward and backward operators governing the evolution ofsystem densities. Our operator approach leads to twodistinct notions of “phase” for stochastic systems. As weargue below, the phase associated with the backward oradjoint operator is closely related to the classical asymp-totic phase.

General framework. Consider the conditional densityρ(y, t|x, s), for times t > s, evolving according to theforward and backward equations

∂

∂tρ(y, t|x, s) = Ly[ρ],

∂

∂sρ(y, t|x, s) = −L†x[ρ], (1)

where L and L† are adjoint with respect to the usualinner product on the space of densities. We assume thatthe conditional density can be written as a sum

ρ(y, t|x, s) = P0(y) +∑λ

eλ(t−s)Pλ(y)Q∗λ(x), (2)

where the eigentriples (λ, P,Q∗) satisfy

L[Pλ] = λPλ, L†[Q∗λ] = λQ∗λ, (3)

〈Qλ|Pλ′〉 =

∫dxQ∗λ(x)Pλ′(x) = δλ,λ′ . (4)

Here P0 is the unique stationary distribution correspond-ing to eigenvalue 0, Q0 ≡ 1, and for all other eigenval-ues λ, we assume <[λ] < 0. Thus, as (t − s) → ∞,ρ(y, t|x, s) → P0(y). We refer to the system as robustlyoscillatory if (i) the nontrivial eigenvalue with least neg-ative real part λ1 = µ + iω is complex (with ω > 0),(ii) |ω/µ| � 1 and (iii) for all other eigenvalues λ′,<[λ′] ≤ 2µ. These conditions guarantee that the slow-est decaying mode, as the density approaches its steadystate, will oscillate with period 2π/ω, and decay withtime constant 1/|µ|. Writing the eigenfunctions of λ1,the slowest decaying eigenvalue of the forward and back-ward operators, in polar form, we have Pλ1

= ve−iφ andQ∗λ1

= ueiψ, where u, v ≥ 0 and ψ, φ ∈ [0, 2π). Asymp-totically, we obtain with this notation from eq. (1)

ρ(y, t|x, s)− P0(y)

2u(x)v(y)' eµ(t−s) cos (ω(t−s) + ψ(x)−φ(y))

(5)

As we now argue, ψ(x), the polar angle associated withthe backward eigenfunction, is the natural generalizationof the deterministic asymptotic phase.

For a deterministic LC system, a given asymptoticphase is assigned to points off the LC by identifyingthose points which at an earlier time were positionedso that their subsequent paths would converge. Suppose

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we observe a density of points ρ(y, t) concentrated neara position on the LC corresponding to a certain phaseθ(y) ≈ θ0. Fixing a point x away from the LC, the den-sity ρ(x, s) at earlier times s < t will show transient oscil-lations with period TLC as the density propagates awayfrom the stable LC in reverse time. The oscillations ob-served at two distinct points x and x′ will be offset bythe difference in their asymptotic phase. Looking for-ward in time, all trajectories will continue converging tothe LC, so the density for a point away from the LC willnot oscillate – it will remain zero.

Figure 1 illustrates the analogous measurement of thephase at a point x from the conditional density at ear-lier times, ρ(x, s|y, t), for a stochastic oscillator. For astationary stochastic time series this density is related tothe conditional density ρ(y, t|x, s) appearing in eq. (5)by ρ0(x, s;y, t) = ρ(y, t|x, s)P0(x) = ρ(x, s|y, t)P0(y)(not to be confused with the detailed balance condition),which can be used to rewrite eq. (5) as follows

ρ(x, t− τ |y, t)− P0(x)

2u(x)v(y)P0(x)' eµτ

P0(y)cos (ωτ + ψ(x)−φ(y)) ,

(6)where we have switched to s = t − τ with τ > 0. If weselect from a stationary ensemble the trajectories thatend up at time t in y, we can estimate the conditionaldensity ρ(x, t−τ |y, t) and the steady state P0(x). Fittingthen the left-hand-side of eq. (6) to a damped cosine inτ (see Fig. 1), we can by virtue of eq. (6) infer the phaseψ(x) at any point x.

We may also obtain the backward-looking phase bysolving the eigenvalue problem eq. (3) for Q∗. Com-parison with the deterministic case again points to thecomplex angle of Q∗ as the analog of the classical phase.For a deterministic system, dx/dt = A(x), the condi-tional density ρ(y, t|x, s) obeys eq. (1) with L†x[Q] =∑iAi(x)∂Q(x)/∂xi. The function Q1 = eiθ(x) with

u ≡ 1 and ψ(x) ≡ θ(x) is an eigenfunction of L†x witheigenvalue λ = iωLC . The analogous eigenfunction ofthe forward operator, Ly[P ] = −

∑i ∂(Ai(y)P (y))/∂yi,

is identically zero except on the LC, at which it has adelta-mass radial distribution. Thus P1 is unsuitable fordefining a “phase” anywhere except on the limit cycleitself.

Noisy Heteroclinic Oscillator. Consider the system

Y1 = cos(Y1) sin(Y2) + α sin(2Y1) +√

2Dξ1(t)

Y2 = − sin(Y1) cos(Y2) + α sin(2Y2) +√

2Dξ2(t), (7)

with α = 0.1, reflecting boundary conditions on the do-main −π/2 ≤ {Y1, Y2} ≤ π/2, and independent whitenoise sources 〈ξi(t)ξj(t′)〉 = δ(t − t′)δi,j . Without noise(D = 0) the system has an attracting heteroclinic cycle,but does not possess a finite-period limit cycle. There-fore, in the noiseless case, there is no classical asymptoticphase [10].

For weak noise, the system displays pronounced oscil-lations (Fig. 1, B), manifest as irregular clockwise rota-tions in the (y1, y2) plane (Fig. 1, A). We can use large

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0 50 τ

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Δ/ω=8.3

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reference point (end point)

(A)

(B)

(C)

(D)

2

1

1

2

,Y ) 2(Y 1

ρ(x , x ,t-τ|y , y ,t) - P (x , x )1 2 1 2 1 20

FIG. 1: (color online) Trajectory of the heteroclinic os-cillator and the histogram method to estimate theasymptotic phase. Trajectories in the (Y1, Y2) plane likethe one shown in (A) that all end up in the neighborhoodof the reference point (Y1, Y2) (red box) are used to estimatethe time-dependent probability in the past in other points(X1, X2) in the plane (blue boxes). This probability displaysasymptotically damped oscillations (C, D), characterized bythe smallest non-vanishing eigenvalue and a space-dependentphase-shift ∆(x1, x2, y1, y2) = ψ(x1, x2) − φ(y1, y2), fromwhich the asymptotic phase ψ(x1, x2) can be extracted [theconstant off-set still depends on the reference point (y1, y2)].Stochastic oscillations of the variables are shown in (B).

trajectories and condition them on their end point (redbox in Fig. 1, A). As argued above, looking back into thepast of such an ensemble of trajectories, we see for largetimes a damped oscillation (Fig. 1, C and D), the damp-ing constant and frequency of which should be related tothe real and imaginary parts of the first non-vanishingeigenvalue. Indeed, we have checked by fitting a dampedcosine according to eq. (6) to the counting histogramsof the backward probability at different positions, thatthe estimate of µ and ω is largely independent of loca-tion (not shown). More importantly, fitting a dampedcosine function also provides an estimate of the asymp-totic phase ψ(x1, x2) in eq. (6). We verified that (upto a fixed phase shift at every point (x1, x2)) the result-ing phase does not depend on the choice of the referencepoint (y1, y2).

As outlined above, the asymptotic phase is also givenby the complex phase of the eigenfunction for the slowesteigenvalue of the system. For the process eq. (7), thebackward operator reads explicitly

L† = [cos(x1) sin(x2) + α sin(2x1)]∂x1 +D∂2x1

+ [− sin(x1) cos(x2) + α sin(2x2)]∂x2 +D∂2x2. (8)

We solve the eigenvalue problem eq. (3) for the sys-tem by expanding the eigenfunctions in a Fourier basis

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FIG. 2: (color online) Asymptotic phase of the stochasticheteroclinic oscillator for two different noise levels.The complex phase of the backward eigenfunction (solid lines)is compared to the results of the histogram method [11] forD = 0.1 (A) and D = 0.01125 (B). Eigenfunctions used in(A) and (B) correspond to the slowest eigenvalues, markedby dashed boxes in (C). Isochrons at lower noise level [blackin (D)] are more curled than for stronger noise [red in (D)].Thick lines in (D) denote 2π-jump in phase.

Q∗λ =∑cm,n,λe

i(mx1+nx2) and computing the eigenval-ues and eigenvectors of the corresponding matrix equa-tion numerically. The leading eigenvalues are shown inFig. 2C for two different noise values. Under both noiseconditions, the first nonvanishing eigenvalues form a com-plex conjugate pair (framed) that is well separated fromthe remaining eigenvalues. As we would expect, for alower noise level (D = 0.01125, black filled circles) thisseparation is more pronounced than for a higher level(D = 0.1, red empty circles).

The complex phase of the eigenfunction for the two dis-tinct noise levels is shown in Fig. 2A and B. The phaseincreases in the same direction as the local mean velocity(clockwise) in both cases. For weaker noise, the phase

winds inward more steeply, i.e. the inward radial compo-nent of ∇ψ is larger.

In Fig. 2A and B we also superimpose data (bluepoints) generated by the histogram method, subject to auniform constant vertical offset. The agreement of thesetwo surfaces demonstrates that the asymptotic phase canbe obtained by the solution of the partial differentialeq. (3) for model systems, for which this equation isknown, but also from trajectories of the system obtainedeither by stochastic simulations (for a model) or mea-surements (experimental data).Neural Oscillator with Ion Channel Noise. Izhikevich

introduced a planar conductance-based model for ex-citable membrane dynamics [12] that is similar to the wellknown two-dimensional Morris-Lecar model [13, 14]. Weconsider a jump Markov process version of Izhikevich’smodel, in which noise arises from the random gating ofa small, discrete population of Ntot potassium (K) chan-nels, which switch between an open and a closed state.Conditional on N(t), the number of open channels attime t, the voltage V evolves deterministically:

CdV

dt

∣∣∣∣N

= I0 − IL(V )− INaP(V )− IK(V,N)

= Cf(V,N) (9)

where I0 is an applied current, IL is a passive leak cur-rent, INaP is a deterministic “persistent sodium” currentand IK is a potassium current gated by the number ofopen potassium channels, 0 ≤ N ≤ Ntot. We used stan-dard parameters [15].

The number of open channels N(t) comprises a contin-uous time Markov jump process with voltage dependentper capita transition rates α(v) for channel opening andβ(v) for channel closing [12]. We generated trajectoriesof the joint (V,N) process using an exact stochastic simu-lation algorithm that takes into account the time-varyingtransition rates α and β [16, 17]. Fig. 3A shows a tra-jectory in the (v, n) plane for Ntot = 100 channels andapplied current I0 = 60. The light and dark gray dashedlines show the v-nullcline and n-nullcline, respectively. Incontrast to the noisy heteroclinic oscillator, this systemhas a stable limit cycle in the limit of vanishing noise(Ntot →∞) with finite period TLC ≈ 5.9825.

The forward and backward equations for this systemare given in terms of f(v, n) (eq. (9)), α(v) and β(v) [15]:

∂

∂tρ(v′, n′, t|v, n, s) = Lv′ [ρ] = − ∂

∂v′[f(v′, n′)ρ]− (α(v′) (Ntot − n′) + β(v′)n′) ρ

+α(v′) (Ntot − (n− 1)) ρ(v′, n′ − 1, t|v, n, s) + β(v′)(n′ + 1)ρ(v′, n′ + 1, t|v, n, s) (10)

− ∂

∂sρ(v′, n′, t|v, n, s) = L†v[ρ] = f(v, n)

∂ρ

∂v+ α(v) (Ntot − n) {ρ(v′, n′, t|v, n+ 1, s)− ρ(v′, n′, t|v, n, s)}

+β(v)n {ρ(v′, n′, t|v, n− 1, s)− ρ(v′, n′, t|v, n, s)} (11)

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tot

FIG. 3: (color online) Trajectory, nullclines, eigenvalues of the backward operator, and asymptotic phase lines forthe persistent-sodium–potassium model. (A) Sample trajectory (thin black line) for the (V,N) process for Ntot = 100channels, and nullclines for the deterministic v (thick grey line) and n (thick black line) dynamics. (B) Low-lying spectrum forL† for two different channel numbers, Ntot = 100 (black dots) and Ntot = 25 (red crosses). Dashed boxes indicate the leadingcomplex conjugate eigenvalue pairs. (C) Level curves (isochrons) of the asymptotic phase for Ntot = 25 (red), Ntot = 100(black), and Ntot = ∞ (blue; deterministic case). The thick lines indicate the locations of the phase jump by 2π, which havebeen adjusted to coincide for the three cases. Isochrons are marked in equal increments of 2π/20. Nullclines as in (A).

We approximate the operator L† with a finite differencescheme by discretizing the voltage axis −80 ≤ v ≤ 20 into200 bins of equal width. We obtain the eigenvalues andeigenvectors of the matrices approximating L and L† us-ing standard methods (MATLAB, The Mathworks). Fig. 3Bshows the dominant (slowest decaying) part of the eigen-value spectrum. Note the occurrence of a family of eigen-values of the form λk ≈ ±iωk− µk2, k = 0, 1, 2, . . . . Thequadratic relationship between the real and imaginaryparts of the eigenvalues of this form is consistent withthe existence of a change of coordinates under which the

evolution takes the approximate form of diffusion on aring with constant drift, ϕ = ω +

√2µξ(t). Here the

eigensystem is exactly solvable, and the spectrum lies onthe same paraboloa.

In Figure 3B, the first nonzero pair (framed) for Ntot =100 is λ1 ≈ −0.031± 1.0475i, corresponding to a periodfor the decaying oscillation of T ≈ 5.9985 (cf. TLC above)and ω/|µ| ≈ 33.7 � 1. All other eigenvalues have realpart less than or equal to 4µ, so the system is “robustlyoscillatory” according to our criteria (i-iii).

Fig. 3C shows level curves of the asymptotic phasefunction ψ(v, n) in three cases, along with the nullclinesfrom panel A. For Ntot → ∞ the process converges tothe solution of a system of nonlinear ordinary differentialequations for v and n [18]. This system possesses a stablelimit cycle for which the phase θ and isochrons are ob-tained in the standard way [12] (blue curves). Near theunstable spiral fixed point at the intersection of the null-clines, the deterministic isochrons exhibit a pronouncedtwisting. For Ntot = 100, with moderately noisy dynam-ics, the level curves of the asymptotic phase ψ for thestochastic system (black curves) lie close to the deter-ministic isochrons. The greatest differences appear in ararely visited region, in the neighborhood of the unstablefixed point. As in the heteroclinic system (Fig. 2D), theless noisy system has more tightly wound isochrons. ForNtot = 25, corresponding to an even larger noise level,the stochastic isochrons (red curves) show even less twist-ing. At both noise levels, the stochastic isochrons showgreatest similarity to the deterministic isochrons in theregion corresponding to the upstroke of the action poten-tial, and show the greatest discrepancy at subthresholdvoltages.

Discussion. Most investigations have approached noisyoscillators by studying the effects of weak noise on a de-terministically defined phase [19–23]. We generalize theclassical asymptotic phase to the stochastic case in termsof the eigenfunctions of the backward operator describ-ing the evolution of densities with respect to the initialtime. As with the stochastic phase defined via the MFPT[7–9], the backward-looking asymptotic phase is well de-fined whether or not the underlying deterministic systemhas a well defined phase. However, if the classical phaseexists, in the absence of noise, our asymptotic phase isconsistent with the classical definition.

The MFPT approach has been applied to non-Markovian systems [9]. Our operator approach would notapply to a non-Markovian process unless it can be em-bedded in a higher-dimensional Markovian system [24].Moreover, for a Markovian system, the MFPT from apoint x to a given surface obeys an inhomogeneous par-tial differential equation involving the same adjoint op-erator L†x, an eigenfunction of which defines our asymp-totic phase. Thus, the relationship between Schwabedaland Pikovsky’s phase description of stochastic oscillatorsand our asymptotic phase remains an appealing topic for

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future research.Acknowledgments. PJT was supported by grant

#259837 from the Simons Foundation, by the Coun-cil for the International Exchange of Scholars, and byNational Science Foundation grant DMS-1413770. BL

was supported by the Bundesministerium fur Bildungund Forschung (FKZ: 01GQ1001A). The authors thankH. Chiel, M. Gyllenberg, L. van Hemmen, A. Pikovsky,M. Rosenblum, L. Schimansky-Geier, J. Schwabedal, andF. Wolf for helpful discussions.

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[15] Applied current I0 = 60µA/cm2, passive leak currentIL = gL(V − VL) with gL = 1mS/cm2 and VL = −78mV,“persistent sodium” current INaP = gNaPm∞(V )(V −VNaP) with gNaP = 4mS/cm2, VNaP = 60mV and volt-age dependent activation m∞(v) = 1/(1 + exp((−30 −v)/7))); potassium current IK = (gKN/Ntot)(V − VK)with gK = 4mS/cm2, VK = −90mV, and open chan-nel number 0 ≤ N ≤ Ntot. Membrane capacitance isC = 1µF/cm2. Per capita transition rate for channelopening is α(v) = 1/(1+exp((−45−v)/5)) and for closingis β(v) = 1− α(v).

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