Stochastic phase dynamics of noise driven synchronization of two
conditional coherent oscillators
W. F. Thompson, R. Kuske, and Y.X. Li
Department of Mathematics
University of British Columbia
September 7, 2011
1
Introduction
Synchronization and coherence of multiple oscillators is of general interest to many scientific and engineering disciplines, notably in biology and neurophysiology where chemical and cellular processes are
commonly modeled by dynamical systems [20, 33]. These phenomena have been studied in various contexts, such as weak coupling [32], coupling with delays [7, 8] and oscillators subject to noisy forcing [34],
and are often considered in the setting of electrophysiological behaviours of individual cells using mathematical models like the Hodgkin-Huxley (HH) [15] and Morris-Lecar (ML) models [28]. More recently
there has been a great deal of work that highlights the constructive role that noise can play in dynamical
systems, for example, in the context of stochastic resonance, spike time reliability, and noise induced
synchrony (see, e.g. [11, 23, 24, 29, 31] and references therein ).
For the last of these, there have been a number of studies showing that identical uncoupled oscillators
subject to a weak common noise source exhibit primarily synchronous dynamics. At first such a result
appeared counter-intuitive, given the expectation that noise should increase disorder. However, it is not
completely unexpected from the perspective that forcing two identical oscillating systems with the same
smooth forcing function leads to entrainment of their dynamics. Noise-induced synchronization has been
studied in a wide array of experimental contexts, for example, in olfactory bulb neurons and electroreceptors [10], [31]. There has also been a great deal of interest in noise-induced oscillator synchronization
as it pertains to the phenomenon of spike time reliability (STR) of neuronal cells. Experiments involving
prepared slices of neuronal tissue have exhibited this phenomenon [5, 25], with complementary theoretical work in [4] [9]. STR is demonstrated by the reliable reproduction of a train of spikes or oscillatory
responses of an oscillator over a series of repeated trials. In each trial an identical noisy stimulus applied to a neuron or oscillator reproduces a train of spikes or oscillatory responses with reliable timing,
while a constant current fails to do so. The same phenomenon can be demonstrated by simultaneously
subjecting similar uncoupled neuronal oscillators to the same noisy forcing simultaneously, as considered
in this paper and references discussed below. Then these oscillators exhibit a response of synchronized
spikes or in-phase oscillations. Such a response, in the absence of any coupling besides the noisy forcing,
is analogous to the reliable replication over repeated trials observed in STR. Recently, the phenomenon of
STR has been studied in the context where the oscillator is a conditional oscillator, that is, in parameter
ranges where the system is quiescent in the absence of noise [40]. There the nearly regular oscillations
are driven by a noise-induced subthreshold activity, in contrast to other contexts for STR where the
underlying deterministic system exhibits spiking, large amplitude oscillations, or a limit cycle, as in, for
example, [9].
Oscillator synchronization via common noisy input has been studied using the Fokker-Planck equation
(FPE) for the probability density function for the phase difference [13]. In that setting, each of the
oscillators has an attracting limit cycle in the absence of noise. Goldobin and Pikovsky studied the case
1
of oscillators with different angular velocities with only common noise forcing and the case of identical
oscillators subject to both common and intrinsic noise stimuli. Their analysis gave a probability density
of the phase difference concentrated near zero for parameter rangers where there was a larger probability
of the oscillators exhibiting synchronized in-phase oscillations. Teramae et al. [37] derived the general
result that weak common noise synchronizes limit-cycle oscillators, provided the second derivative of
the phase response is continuous. Their analysis is based on a calculation of the Lyapunov exponent
for the noisy phase equations. More recently, Nakao et al. studied synchrony in a general class of
limit-cycle oscillators [30] via the phase difference density, comparing with a numerical study of StuartLandau oscillators. As in other studies, a peak concentrated near zero for the density of the phase
difference indicated a large probability of observing synchronous oscillations for common noise forcing. In
addition, they used correlation functions to illustrate that there are parameter ranges where noise-induced
clustering occurs: subgroups of oscillators exhibiting synchronization within the subgroup while their
response is out of phase with the other subgroups. Several other studies analyze noise-induced clustering
by a similar characterization [9, 24], with modifications in treatments in the noise terms considered in
[38]. Recent results in [14] and [36] consider the different limiting behaviors of the phase dynamics
observed for reduced stochastic phase equations for limit-cycle oscillators and provide reductions to phase
equations for general non-Gaussian noise. In [29] it was demonstrated that common noise drives a larger
probability of synchronization in large populations of globally coupled non-identical limit-cycle oscillators,
also taking advantage of phase reduction theory. A number of these studies further study indicators for
the synchronization of the oscillations via computation of the Lyapunov exponent for the phase difference
of limit-cycle oscillations.
The studies listed above concentrate on noise-induced synchrony in the context where large amplitude limit cycle oscillations are stable in the absence of noise. In-phase, synchronous behavior of the
oscillations is observed with a large probability for the phase difference density concentrated near zero
or when the Lyapunov exponents for the stochastic phase equations are negative. In contrast, here we
consider parameter combinations corresponding to conditional oscillators, where sustained nearly regular
oscillations are generated by noisy forcing in a system that would be quiescent otherwise. That is, the
deterministic system is quiescent, with parameters in the range where there are unstable subthreshold
oscillations associate with a subcritical Hopf bifurcation. The weak damping of these oscillations near a
Hopf bifurcation corresponds to a damping over a slow time scale, which plays an important role in our
analysis below. The observed subthreshold coherent oscillations are purely noise-induced, yet they have
a time-varying frequency close to that of the system’s Hopf frequency. Over a time interval that is long
compared to the period of oscillations associated with the Hopf bifurcation, the power spectral density
of the time series of the oscillations is concentrated near the Hopf frequency, indicating that the noise is
driving nearly regular oscillations associated with the underlying deterministic system.
This type of nearly regular oscillation sustained by noisy forcing of subthreshold, weakly damped
oscillations, has been referenced as coherence resonance (CR) in the context of different applications
[22, 39]. In its broadest sense, CR is observed when a range of noise levels excites coherent oscillations in
a system that is quiescent without noise. Typically a maximum coherence, defined in terms of a specific
coherence measure appropriate for the application or context, is observed at an optimal noise level. This
phenomenon is most commonly referenced in the context of transitions between steady equilibria with
large excursions, such as relaxation oscillations, which can be driven by noise in a nearly regular manner
[34]. In this paper the CR phenomenon refers to the noise-induced subthreshold oscillations near a Hopf
bifurcation. The analysis of these CR-type oscillations in [39] indicates that they occur in the range of
noise levels around the optimal values of a coherence measure. This coherence measure is given by the
ratio of the height and width of the power spectral density of the time series of the oscillations, scaled by
the Hopf frequency of the oscillations which are excited through CR. The amplification factor of the CRtype oscillations is proportional to the noise level and inversely proportional to the parametric proximity
to the Hopf bifurcation, so that they are a prominent, prolonged feature for a range of parameter values.
To date, there is no existing theory that deals with the case of noise-driven synchronization for oscillations
2
that are driven via this CR-type phenomenon.
Understanding the phase difference dynamics of noise-induced CR-type oscillations coupled through a
common noisy input requires a different analysis than the case of limit-cycle oscillators coupled through
common noisy input studied in [13, 30, 36, 37]. In the limit-cycle oscillator case, the oscillators exhibit
regular oscillations without noise, and the leading order phase dynamics include the features of the deterministic limit-cycle and its perturbations, so it is possible to analyze phase equations without variation
in the amplitude. For the phenomenon we study here, the noise plays the role of both inducing the
oscillations and coupling the oscillators. For the CR-type driven oscillations described above, the phase
dynamics are dependent on the leading order amplitude dynamics which depend strongly on the noise
levels. This result leads to coupling of the noisy phase and amplitude dynamics [39], different from the
limit-cycle oscillator dynamics. Then an analysis of the phase dynamics for the CR-type oscillations must
reflect that coupling, as demonstrated below in the calculation of the density for the phase difference.
The main results of the paper are the derivation and analysis of a reduced Fokker-Planck equation
(FPE) for the density of the phase difference for two conditional oscillators, exhibiting CR-type oscillations
due to noisy forcing but otherwise uncoupled. Each oscillator is described by a canonical deterministic
oscillator forced both by intrinsic noise that is independent of the others intrinsic noise and by a noisy
input common to both. An asymptotic analysis of the system leads to coupled phase and amplitude
equations, and a further asymptotic analysis using the moments of the amplitudes leads to a reduced
system of the amplitude-dependent phase dynamics and the corresponding FPE for this reduced system.
The asymptotic analysis is based on approximations where the intrinsic and extrinsic noise are small
relative to the parameter measuring distance from the Hopf bifurcation. The asymptotic expression for
the phase difference density provides explicit parametric expressions for the probability for observing
different phase dynamics, yielding parametric descriptions for the different concentrations of the density
corresponding to in-phase synchronization, phase shifted oscillations, and non-synchronized states. The
different dynamics are observed for different relative sizes of the intrinsic and extrinsic noises, and the
influence of other parameters can also be seen in the expressions for the density. When the common
noise forcing clearly dominates, then the phase difference density is strongly concentrated near the origin,
resulting in larger probability of observing oscillations that synchronized in phase, as in the case of limitcycle oscillators studied in [13]. However when the intrinsic noise levels are not completely dominated by
the common noise forcing, there are several types of behavior possible. If the intrinsic noise levels are of
identical strength and smaller than that of the common noise, the phase difference density is concentrated
near zero, that is, there is a large probability of synchronization of the oscillations. In contrast, if the
intrinsic noises are not of identical strength, the phase difference density is asymmetric around zero,
indicating a lag between oscillators. If the intrinsic noise dominates, the phase difference density spreads
out, approaching a uniform density for increasing intrinsic noise.
The paper is organized as follows. In Section 2 we give a canonical model for conditional oscillators
with intrinsic and common noisy input, and perform a stochastic multiple scales analysis of the model
to derive stochastic amplitude and phase equations for the main mode of oscillation. These provide the
basis for our analysis of the phase dynamics. In Section 3 we find the density of the phase difference
using linear theory, in the setting of weak noise and symmetric stochastic forcing. We also develop an
asymptotic analysis to obtain the reduced FPE for the phase difference density, using an approximation
that is analogous to a quasi-steady approximation for the dependence on the amplitude dynamics. The
asymptotic approximation for the density describes the explicit dependence on the system parameters,
yielding a straightforward calculation of the probability of observing the different types of synchronized
and non-synchronized oscillations for different parameter combinations. In Section 4 we compare the
results for the phase different density obtained from linear theory, from both the asymptotic approximation
and the numerical result for the solution of the reduced FPE, and from the simulation of the full system of
stochastically driven oscillators. These comparisons illustrate the range of parameters where the analytical
approximation is valid, as consistent with the asymptotic approach. They also show that the numerical
solution of the reduced FPE for the phase dynamics, derived in Section 3, has the same shape as the density
3
approximated by simulation of the full system, even for parameters where the analytical expressions for
the solution of the FPE breakdown.
2
The Model
We consider the canonical λ − ω oscillator with parameters such that the system is subthreshold but near
a subcritical Hopf bifurcation. This model corresponds with the normal form for a Type-II Morris Lecar
oscillator with parameters near the critical value of the applied stimulus that induces a steady spiking
behaviour [35]. The oscillators are subject to both an intrinsic and extrinsic (common) noise forcing. The
model we study is given by the following system of stochastic differential equations:
dxi
λ(ri ) −ω(ri )
xi
δi δC
dηi (t)
=
dt +
, ri2 = x2i + yi2 , i = 1, 2. (2.1)
dyi
ω(ri ) λ(ri )
yi
0 0
dηC (t)
The functions ηi , ηC are Wiener processes (standard Brownian motions) that satisfy E[ηj (t)] = 0 and
E[ηj (t)ηk (t − τ )] = 1{j=k} δ(τ ) where j, k ∈ {1, 2, C} and E[η] denotes the time average of η. The
parameters δi and δC give the noise strength for the different noise sources. In the following we consider
cases where the intrinsic noises δi ηi (t), i = 1, 2, have either identical strengths, δ1 = δ2 , or non-identical
strengths, δ1 6= δ2 . The parameter δC gives the strength of the common noise, which is a common input
to both oscillators.
We summarize a few basic features of the deterministic model (2.1) with δ1 = δ2 = δC = 0, corresponding to two uncoupled identical oscillators. Then each oscillator is a single deterministic λ − ω system,
where the function λ(ri ) is related to the behaviour of the amplitude, while ω(ri ) gives the frequency
behaviour. For our study we consider the following functions for λ(ri ) and ω(ri ):
λ(ri ) = λb + αri2 + γri4 ,
ω(ri ) = ω0 + ω1 ri2 .
The constants λb , α and γ are parameters that determine the bifurcation structure of the oscillator
amplitudes. Taking λb as the bifurcation parameter, there is a Hopf bifurcation at λb = 0, and for
α > 0, γ < 0 the Hopf bifurcation is subcritical, as shown in Fig. 1. A standard stability analysis
on the deterministic system shows that ri = 0 is locally stable for λb < 0 and unstable for λb > 0.
The middle (dashed) branches correspond to unstable nonlinear oscillations, and the upper and lower
branches corresponds to the maximum and minimum for a large amplitude limit cycle. We refer to the
smallest value of λb for which the steady state and limit-cycle oscillations are bi-stable, as the knee of
the bifurcation structure for the system. The oscillator is a conditional oscillator for values of λb near
the knee and between the knee and the Hopf bifurcation, in the context where crossing the unstable limit
cycle and transitioning to the large stable limit cycle is unlikely. From the form of ω(ri ), one can see that
the frequency of small oscillations is, to leading order, equal to ω0 . The frequency of oscillations changes
depending on the amplitude of the oscillator and this change is determined by the ω1 ri2 term.
Here we focus on the range λb < 0 with |λb | 1 and parameter values such that small perturbations
from the steady state yield oscillations in (2.1) that decay weakly in the absence of noise. For these values
of λb , both outside and inside the knee of the bifurcation structure, weak to moderate noise drives coherent
oscillations about the fixed point, via a CR-type mechanism. As discussed in [39], the range of noise levels
for this phenomenon is O(|λb |) or smaller. In the following we consider parameters and noise strengths in
certain ranges so that the possibility of jumping to the stable limit-cycle, if it exists, is negligible, and in
the absence of noise, the amplitude decays to zero. In this context we consider the influence of the relative
noise strengths on the dynamics of the phase differences for CR-type oscillations exhibited by (2.1). Note
that the behavior of these CR-type oscillations are strongly related to the oscillations of the underlying
deterministic system, so that these oscillations have some of the characteristics of the deterministic system
which we use in our analysis. Stochastic oscillations and stochastic synchronization have been studied in
more general mathematical contexts of ordinary differential equations perturbed by noise [6], [26], [27].
4
1
r
0.5
0
−0.5
−1
−6
−4
−2
λ2
0
2
p
Figure 1: Bifurcation diagram of r = x2 + y 2 vs. λ2 in the case of the deterministic system (δ1 = δ2 =
δC = 0) for a single oscillator given by (2.1). Other parameters include: = 0.1, α = 0.2, γ = −0.2. Solid
(dashed) lines indicate stability (instability). Black (Gray) lines indicate a fixed point (periodic orbit).
The data for this diagram was generated using XPPAUT.
where the stochastic behavior does not necessarily retain characteristics of the underlying deterministic
system.
The time series shown in Fig. 2 indicate qualitatively the different types of behavior that are observed
for (2.1), depending on the relative strengths of the common noise δC and the intrinsic noises δi , and
whether the intrinsic noise strengths are equal or quite different. In the following sections we provide
explicit expressions for the asymptotic approximation of the phase difference density for the two oscillators.
This density is obtained by combining a multiple scales analysis with an asymptotic analysis of the FokkerPlanck equation (FPE) for the density of the phase difference. In the top figure the common noise forcing
dominates over the intrinsic noise contributions, resulting in a larger probability of synchronized in-phase
CR-type oscillations; that is, in-phase synchronization is observed over most of the time series. In the
middle figure, the strengths of the intrinsic noise sources are closer to that of the common noise strength,
but the intrinsic noise strengths are not identical. In this case the oscillations of the two components
have different amplitudes on average, but they appear more frequently together with a phase shift. The
bottom figure displays a low common to intrinsic noise strength ratio, so that the influence of the different
intrinsic noise processes dominate in driving the CR-type oscillations. Then there is a lower probability
of observing synchronized oscillations, which occur only rarely in the time series. These behaviors are
reflected in the shape of the phase difference density, obtained analytically in Sections 3.2 and 3.4, and
compared computationally in Section 4.
3
3.1
Analysis
Stochastic multiple scales analysis
In this section, we perform a multiple scales analysis of the original system (2.1), similar to that performed
in [19] to obtain a description of the dynamics on a long time scale. This multiple scales analysis provides
asymptotic approximations of the stochastic equations for amplitude and phase of the oscillations, which
are useful in studying the dynamics of the phase difference. The asymptotic approximation is based on a
small parameter 1 defined in terms of the distance of the bifurcation parameter λb from the critical
value 0. We set λb = 2 λ2 , for 1 and λ2 = O(1). As discussed above, in this parameter range
5
y1, y2
0.1
0
−0.1
300
350
400
450
500
550
600t
y1, y2
0.2
0
−0.2
1200
1220
1240
1260
1280
1300t
y1, y2
0.1
0
−0.1
300
350
400
450
500
550
600 t
Figure 2: Time series of (2.1) with y1 , y2 displayed (solid, dashed respectively). The top figure has
a larger δC = .01 to δ1 = δ2 = .003 ratio (as compared with the other panels), which results in a
larger probability of observing synchronized oscillations as compared with results of other panels. (Other
parameters: λ2 = −3, α = 0.2, γ = −0.2, ω0 = 0.9, ω1 = 1.2, = 0.1, ). The middle figure shows stronger
asymmetric intrinsic noise δ1 = 0.02 δ2 = 0.05 as compared with the common noise strength δC = 0.07,
resulting in different amplitudes of oscillations that appear frequently
with a relative phase shift. (Other
√
parameters: λ2 = −3, α = 1.2, γ = −1, ω0 = 2, ω1 = 25, = 0.1). The bottom figure displays a smaller
ratio of common δC = 0.005 to intrinsic noise δ1 = δ2 = 0.009, resulting in rare synchronization. (Other
parameters: λ2 = −3, α = 0.2, γ = −0.2, ω0 = 0.9, ω1 = 1.2, = 0.1,)
oscillations are weakly damped in the deterministic model. Furthermore, as shown below, the multiple
scales expansion is valid only for noise levels that are O(2 ) or smaller.
Weakly damped oscillations perturbed by small noise have been studied rigorously in a variety of
contexts. For example, in [2] and references therein, amplitude and phase equations for stochastic oscillations are obtained using stochastic averaging in the limit of small damping for the stochastic van der
Pol-Duffing equation. For both additive and multiplicative noise, weak convergence of the measure for
the averaged process is demonstrated as the small parameter characterizing the damping () vanishes. In
that case appropriate scaling relationships are identified between the damping, noise and weakly nonlinear contributions. In [21] it was shown that the multiple scales approach presented here gives the same
amplitude and phase equations as in [2]. Further comparisons in more general contexts of weakly damped
oscillations with multiplicative noise show agreement of the diffusion coefficients for rigorous approaches
in [1] and [3] and the multiple scales approach, as discussed in [19]. Rigorous results in [3] are given for
the Ornstein-Uhlenbeck processes describing amplitudes of stochastic oscillations near a fixed point, as
observed in a variety of biological models. These results show convergence in distribution of the linearized
6
approximation of the stochastic oscillator transformed by a rotation corresponding to the deterministic
oscillations. The transformed process gives the same amplitude equations for the oscillations, in the
linearized case, as those obtained by the multiple scales approximation applied here and in [22].
We look for noise-driven oscillations about the fixed point in the quiescent parameter regime, λ2 < 0.
For weak noise forcing (O(2 ) or smaller) in (2.1) it is appropriate to seek small amplitude oscillations
of O(). This scaling, together with the slow time scale T = 2 t, is consistent with a standard multiple
scales analysis [17, 19] and yields a leading order solution of the form
xi (t, T ) = Ai (T ) cos(ω0 t) − Bi (T ) sin(ω0 t)
(3.1)
yi (t, T ) = Ai (T ) sin(ω0 t) + Bi (T ) cos(ω0 t) ,
where Ai , Bi are stochastic processes varying on time scale T . This approximation is appropriate since
the primary mode (cos(ω0 t), sin(ω0 t)) decays on the slow time scale T , and on this time scale the noise
interacts with this weakly damped mode. In contrast, other modes such as(cos(mω0 t)), sin(mω0 t)) for
m 6= 0, decay on the fast t scale, so that their contribution to the dynamics is negligible in the leading
approximation. The approximation (3.1) captures the behavior of CR-type oscillations as described in
the Introduction, as the oscillations have a frequency near ω0 , the oscillation frequency associated with
the Hopf bifurcation. As in deterministic multiple scales analyses, we determine the behaviour of Ai , Bi
by deriving their governing equations. We choose an ansatz for these amplitude equations of the following
form,
dAi = ψAi dT + σAi I dξAi I (T ) + σAC dξAC (T )
(3.2)
dBi = ψBi dT + σBi I dξBi I (T ) + σBC dξBC (T ) .
Then we determine expressions for {ψZ , σZI , σZC , Z = Ai , Bi } in order to determine the behaviour
for Ai , Bi . We take dξZI , dξZC to be increments of independent standard Brownian motions on the time
scale T . The subscript I is used for terms that capture the effect of intrinsic noise forcing, while C is
used for terms that capture the effect from common noise. We separate them to better distinguish their
respective contributions.
We apply Ito’s formula to give expressions for dxi and dyi using (3.1) and substituting (3.2) where
appropriate, yielding
dxi = −ω0 [Ai (T ) sin(ω0 t) − Bi (T ) cos(ω0 t)]dt + cos(ω0 t)[ψAi dT + σAi I dξAi I + σAC dξAC ]
− sin(ω0 t)[ψBi dT + σBi I dξAi I + σBC dξBC ]
dyi = ω0 [Ai (T ) cos(ω0 t) − Bi (T ) sin(ω0 t)]dt + sin(ω0 t)[ψAi dT + σAi I dξAi I + σAC dξAC ]
+ cos(ω0 t)[ψBi dT + σBi I dξAi I + σBC dξBC ] .
(3.3)
We compare (3.3) to the equation (2.1) with xi , yi given by (3.1),
3 (cos(ω0 t) (λ2 + αRi2 )Ai − ω1 Ri2 Bi − sin(ω0 t) (λ2 + αRi2 )Bi + ω1 Ri2 Ai )dt + δi dηi (t) + δC dηC (t) (3.4)
= (cos(ω0 t)ψAi − sin(ω0 t)ψBi )dT + (cos(ω0 t) [σAi I dξAi I + σAC dξAC ] − sin(ω0 t) [σBi I dξBi I + σBC dξBC ])
3 (cos(ω0 t) (λ2 + αRi2 )Bi + ω1 Ri2 Ai + sin(ω0 t) (λ2 + αRi2 )Ai − ω1 Ri2 Bi )dt
(3.5)
= (cos(ω0 t)ψBi + sin(ω0 t)ψAi )dT + (cos(ω0 t) [σBi I dξBi I + σBC dξBC ] + sin(ω0 t) [σAi I dξAi I + σAC dξAC ]) ,
where Ri2 = A2i +Bi2 . We project the above equations (3.4) and (3.5) onto the basis functions {cos(ω0 t), sin(ω0 t)}
to determine expressions for ψAi , ψBi , focusing first on the drift terms. This is consistent with the results
of Baxendale [2] and of a multiple scales analysis [17]. Then the projection for the drift terms is given by
Z 2π/ω0
Z 2π/ω0
(cos(ω0 t), sin(ω0 t)) · ((3.4), (3.5)) dt,
(− sin(ω0 t), cos(ω0 t)) · ((3.4), (3.5)) dt(3.6)
0
0
⇒
ψAi = (λ2 + αRi2 )Ai − ω1 Ri2 Bi ,
ψBi = (λ2 + αRi2 )Bi + ω1 Ri2 Ai .
7
(3.7)
We note that this result is identical to the result that is determined from the multiple scales analysis of
the deterministic system.
To determine the coefficients of the stochastic contribution we compare the time-averaged generator
for the original system (2.1) and the generator for the equations (3.2), using the properties of Brownian
motion, dηi (t) = −1 dηi (T ). We express the diffusion operator for the generator of the ith oscillator of
the original system in the projected coordinate system:
2
2
2 2 δC
δC
δi
∂2
∂2
∂2
δi
∂2
2
2
+
+
cos (ω0 t) 2 − 2 cos(ω0 t) sin(ω0 t)
+ sin (ω0 t)
=
.
22 22 ∂x2i
24 24
∂Ai ∂Bi
∂Ai
∂Bi2
(3.8)
Averaging the diffusion operator in (3.8) over one period of the fast oscillation (cos ω0 t, sin ω0 t) yields
2
2 2
δC
δi
∂
∂2
+
+
.
(3.9)
44 44
∂A2i
∂Bi2
The diffusion operator for the generator for Ai , Bi given by our ansatz (3.2) is given by
!
!
2
2
2
2
2
σA
σ
+
σ
+
σ
∂
∂2
AC
Bi I
BC
iI
+
.
2
2
∂A2i
∂Bi2
(3.10)
Comparing the coefficients of the two diffusion operators (3.9) and (3.10), we obtain equations for σZ,I ,σZ,C
(Z = Ai , Bi ), in terms of δi , δC . We write the result in a representation that separates the intrinsic and
common noise strengths, yielding,
δi
σAi I = σBi I = √ ,
22
δC
σAC = σBC = √
.
22
(3.11)
Then (3.7) and (3.11) provide the unknown drift and diffusion coefficient in the stochastic envelope
equations (3.2) for Ai and Bi .
Because of the influence of the noise forcing, the amplitudes of the oscillations do not evolve to the
steady state where Ai = Bi = 0. In effect, the noise excites the main mode of oscillation associated with
the Hopf bifurcation with an amplitude given by (3.2), evolving on the slow T time scale. As discussed
in detail in the Appendix of [19], other modes decay on the fast T scale and do not contribute to the
leading order approximation of the dynamics. Note that noise strengths must be O(2 ) (or smaller) to
ensure consistency of the multiple scales analysis, as we discuss further below. Comparison of analytical
and computational results illustrate the range of validity in Section 4.
Ultimately we are interested in the phase dynamics of these oscillations, to understand the combined
effect of intrinsic and common extrinsic noise on the synchronization of the oscillations driven through
a CR-type phenomenon. We consider two different approximations for which we obtain analytical expressions for the stationary density of the phase difference of the two oscillators. The first of these is a
linear approximation, that includes corrections due to the correlation between the two oscillations, but
neglects nonlinear effects, significant when the intrinsic noises are not of identical strength. The second approximation includes these nonlinear effects, but does not include the correlation contributions
analytically.
3.2
Density of the phase difference for the linearized system
For small amplitudes Ri 1, we first linearize (3.2) about Ri = 0
δi
δC
dAi = λ2 Ai dT + √
dξAi I + √
dξAC ,
2
2
22
δi
δC
dBi = λ2 Bi dT + √
dξBi I + √
dξBC .
22
22
8
(3.12)
This approximation is valid in the context of weak noise forcing for parameter values where the nonlinear
effects are negligible. The advantage of (3.12) is that the terms Ai , Bi evolve as Ornstein-Uhlenbeck
(OU) processes, which are relatively straightforward to analyze. Then we get an analytical result that
includes contributions from the correlations, but neglects the nonlinear effects of the amplitude dependent
phase, described by ω1 , as well as the contributions to the nonlinear terms in (3.7). Furthermore, this
approximation neglects any contributions that arise from the possibility of a transition to the limit-cycle
that is bistable with the fixed point for certain values of λ2 (see Fig. 1).
We start with the stationary probability density function P (Ai , Bi , i = 1, 2) for the OU process (3.12),
which takes a well-known Gaussian form [12]. From there we can look for the density of the phases φi
of the stochastic oscillators, since φi depends on Ai and Bi . The phase of oscillator i is expressed as
φi = arctan(Bi /Ai ), equivalent to φi = arctan(yi /xi ) through the coordinates (3.1), and the amplitude
Ri is as above. Then the joint probability density for amplitude and phase is given by
2
2
δC
π
2
2
P (R1 , φ1 , R2 , φ2 ) = CR1 R2 exp
C × ∆2 R1 + ∆1 R2 − 4 R1 R2 cos(φ2 − φ1 )
,
(3.13)
λ2
where
C=
λ22
4 /48 )) ,
π 2 (∆1 ∆2 − (δC
∆i =
2
δC
δi2
+
.
24 24
(3.14)
As P is a Gaussian in this approximation, the stationary probability density function P(Φ) for the phase
difference Φ = φ2 − φ1 can be obtained in closed form from (3.13) by integrating,
Z π Z ∞Z ∞
P(Φ) =
P (R1 , φ1 , R2 , Φ + φ1 ) dR1 dR2 dφ1
−π 0
0
hπ
i
= β1 (Φ)
+ arctan β2 (Φ) + β3 (Φ) ,
(3.15)
2
where
β1 (Φ) =
β2 (Φ) =
β3 (Φ) =
K cos(Φ) 1 − K 2
,
2π(1 − K 2 cos2 (Φ))3/2
K cos(Φ)
p
,
1 − K 2 cos2 (Φ)
1
1 − K2
,
2π 1 − K 2 cos2 (Φ)
and
K=
2
δC
√
=
24 ∆1 ∆2
−1/2
δ2
δ2
1 + 21
1 + 22
.
δC
δC
(3.16)
(3.17)
(3.18)
(3.19)
The details of the calculations are given in the Appendix A. The parameter K is important in characterizing the shape of the probability density for the phase difference, which indicates the probability of
synchronous oscillations in the dynamics. Notice that K ∈ [0, 1] for all values of noise strengths, with
K = 0 and K = 1 corresponding to the cases of no common noise and only common noise, respectively.
In Section 4 we compare this probability density to simulations of the original system (2.1). We see
that in this approximation the ratio of intrinsic to common noise strengths is important in determining
the character of the phase difference probability density function. Notice that when K = 0, the phase
difference probability density function is uniform, as we would expect given that there is no common
forcing. When K = 1, the density is zero everywhere except at Φ = 0 where there is a singularity. This
suggests the existence of a δ-function solution centred at Φ = 0 when K = 1, as discussed further in
Section 3.4.
This linearized approximation for the SDEs (3.2) captures the correlation contributions, but does not
capture the nonlinear contributions, such as the amplitude-dependent phase dynamics. The nonlinear
9
effects enter in the full model through the amplitude-phase feedback term ω1 Ri2 and, to a lesser extent, the
amplitude-amplitude feedback term αRi2 . The former of these is important when the oscillators experience
intrinsic noise of different strengths as we show below. Next we consider a different asymptotic analysis
of the amplitude and phase equations, in order to include nonlinear effects.
3.3
Approximation of the FPE for the phase difference density
We obtain an approximate FPE for the joint probability density of φ1 and the phase difference Φ = φ2 −φ1 .
This approximation is based on the observation of multiple time scales for the amplitude and phase
dynamics, which leads to approximating certain coefficients with their moments. A further reduction via
integration yields the FPE for the marginal density of the phase difference.
Approximation based on multiple time scales: From the equations (3.2) we derive equations
for the scaled amplitude and phase. Applying Ito’s formula to get SDEs for Ri = (A2i + Bi2 )1/2 , φi =
arctan(Bi /Ai ) gives the following SDEs for amplitude and phase for i = 1, 2:
1
1
−1
2
dRi = Ri (λ2 + αRi ) + ∆i Ri
dT + √
[cos(φi )(δi dξAi I + δC dξAC ) + sin(φi )·
2
22
(δi dξBi I + δC dξBC )]
(3.20)
1
dφi = ω1 Ri2 dT + √
[cos(φi )(δi dξBi I + δC dξBC ) − sin(φi )(δi dξAi I + δC dξAC )] .
(3.21)
22 Ri
While it is straightforward to give a FPE for these four SDEs (3.20,3.21), obtaining a useful analytical
expression is not, and numerical solutions are cumbersome. The method used in [30] for separating the
amplitude and phase contributions to the FPEs is not applicable here. Instead we look for a reduction
based on the observation that the amplitude Ri and phase φi evolve on different time scales. Rescaling
Ri = ρsi , where si = O(1), and substituting into (3.20) yields
∆i
1
2 2
dsi = si (λ2 + αρ si ) + 2
[cos(φi )(δi dξAi I + δC dξAC ) + sin(φi )(δi dξBi I + δC dξBC )] .
dT + √
2ρ si
2ρ2
(3.22)
√
Then for ρ = O( ∆i ) the leading order contributions for the drift and diffusion dynamics in the equation
for s, evolve on the slow T time scale. With λ2 < 0, the linear terms in (3.22)
√ indicate mean reverting
behavior on the T scale for the amplitude. For the amplitude Ri taking O( ∆i ) values, the drift terms
in (3.21) describe evolution on the even slower ∆i T time scale for φ. We see below that this scaling
assumption for Ri is consistent with the calculation of amplitude moments for CR-type oscillations in the
weak noise case. The differences between the amplitude and phase dynamics indicates that the amplitude
has a mean-reverting behavior on a faster time than the evolution of the phase, which does not have this
mean reverting behavior. This observation for Ri and φi suggests using an approximation reminiscent of a
quasi-steady approximation in chemical kinetics: the more rapid, mean-reverting variation of Ri is treated
as
√ independent of φi , while keeping the dependence of φi on Ri . This approximation is appropriate for
∆i 1 and for consideration of long time behavior of the system. Note that the same small parameter
is used for the approximation to the stationary probability density in Section 3.4. Then we look for an
approximation of the probability density function for Ri , φi , i = 1, 2 under this quasi-steady assumption,
by averaging the SDE (3.21) with respect to Ri . The end result is that the various contributions involving
Ri are replaced with their expected values. We get the following modified SDEs for the phases,
dφi = ω1 E[Ri2 ]dT + √
1
E[Ri−1 ] [cos(φi )(δi dξBi I + δC dξBC ) − sin(φi )(δi dξAi I + δC dξAC )] ,
2
2
i = 1, 2 .
(3.23)
We thus achieve a pseudo-separation of the amplitude and phase components for the oscillators, and we
consider the phase behaviour that includes the influence of the Ri through its moments. There is some
analogy in this method to the approximation used in [13, 37], where the authors undertake an analysis of
10
the stochastic phase equations via a phase reduction technique for coupled oscillators in the parameter
range corresponding to stable large-amplitude limit cycles. In those studies, the phase reduction is based
on the observation that perturbations from the limit-cycle are shown to be transient given the strong
attraction to the limit cycle. In our setting, the quasi-regular oscillations are driven by a CR-type
phenomenon, rather than close proximity to a limit-cycle. Then the approximation here is based on the
separation of time scales, indicating the independence of Ri from the phase behavior to leading order and
allowing the moment approximation.
Using (3.23), the FPE for the joint probability density function q(Φ, φ1 ) for the phase difference
Φ = φ2 − φ1 and φ1 , is given by [12],
∂q
1
∂2q
2
2 ∂q
2 ∂q
+
[∆1 E[R1−1 ]2 + ∆2 E[R2−1 ]2 ] 2
= −ω1 (E[R2 ] − E[R1 ])
− ω1 E[R1 ]
∂T
∂Φ
∂φ1 2
∂Φ
2
2
2
2
δC
δC
∂
−1
−1 ∂
−1 2
−1
−1
− 4 E[R1 ]E[R2 ] 2 (cos (Φ)q) + 2
(−∆1 E[R1 ] + 4 E[R1 ]E[R2 ] cos (Φ))q
∂Φ
∂Φ∂φ1
2
2
∂ q
,
(3.24)
+[∆1 E[R1−1 ]2 ]
∂φ1 2
which is a 2-D advection-diffusion equation with anisotropic diffusion. Given that this is a FPE for phase
behaviour, the boundary conditions for this system are periodic, and the solution must be normalized as
a probability density.
Computation of the moments: To compute the expected values of the moments for the amplitudes
in (3.24), we consider SDEs (3.20) for the amplitude Ri of the oscillator i. The steady-state probability
density function ρi (Ri ) for the amplitude of the oscillator satisfies
∂
1
∆i ∂ 2 ρi
2
−1
0=−
R(λ2 + αR ) + ∆i R
ρi +
.
(3.25)
∂R
2
2 ∂R2
For α = 0, the density function ρi is given by
−2λ2 Ri
ρ0 i (Ri ) =
exp
∆i
λ2 2
R
∆i i
.
(3.26)
For oscillations driven by a CR-type phenomenon about the steady state, (3.26) gives the leading order
approximation to the density for Ri 1, consistent for weak noise forcing. Notice that this distribution
agrees with (3.13, 3.14) in the limit where no common noise is present, and thus the oscillators are
independent. Using (3.26), the leading order contributions to the moments can be calculated, for example,
r
r
∆i
−λ2 π
π∆i
−1
2
E[Ri ] = − , E[Ri ] =
, E[Ri ] =
.
(3.27)
λ2
∆i
−4λ2
In particular, the result for the first moment confirms our assumption that Ri 1, used above to derive
the averaged FPE (3.24) for the phases. Furthermore, the result for E[Ri2 ] in (3.27) is consistent with
that obtained using the linear distribution (3.13), as shown in Appendix A.
With these leading order results in mind, we look for a solution of (3.25) of the form
!
n
X
k
ρi = ρ0 i g, where g = g(R) = exp
bk R
,
(3.28)
k=0
for n an integer. Together with this ansatz, we consider the case where Ri 1. Substituting (3.28) into
(3.25) gives the following equation:
dg
d2 g
0 = 8∆i αR3 + 4αλ2 R5 g(R) + ∆i (2αR4 − ∆i − 2λ2 R2 )
− ∆2i R 2 .
dR
dR
11
(3.29)
Substituting in the expression for g given in (3.28) allows us to solve for the coefficients bk , b1,2,3 = 0 and
α
b4 = 2∆
with b0 free. The freedom in choosing b0 allows for normalization of the corrected density. Then
i
the corrected probability density function is
λ2 2
α 4
−2λ2 R
exp
R +
R ∼
ρi (R) ∼ ci
∆i
∆i
2∆i
−2λ2 R
λ2 2
α 4
ci
exp
R
1+
R , ci = (1 + α∆i /λ22 )−1 ,
(3.30)
∆i
∆i
2∆i
where ci is the normalization constant. Then the approximate moments for the amplitudes are
s
!
r
2
∆
3α∆
−λ
π
3α
π∆
i
2
i
i
E[Ri2 ] ∼ ci −
−
E[Ri−1 ] ∼ ci
+
.
λ2
∆i
8
λ32
−λ32
(3.31)
In order to normalize, we note that the approximation (3.28) is appropriate over a range of |R| values
within the curve of the subcritical Hopf bifurcation (dashed branches in Fig. 1), away from the large
amplitude limit cycle. The density for the amplitude over a larger range of R could be normalized
if we were to include the effect of γ < 0, the highest order nonlinearity in λ(ri ). It would result in
exponential decay of the amplitude distribution for values beyond the amplitude of the stable limit-cycle.
The contributions to ρi and the corresponding moments from the terms involving γ are negligible for R
small.
FPE for the phase difference: Now that we have expressions for the moments of the distributions,
we reduce the FPE to study the phase difference, rather than the distribution of phases themselves. We
integrate (3.24) over all φ1 ∈ [−π, π] to get the following reduced FPE:
2
2
2
δC
∂p
1
−1 2 ∂ p
−1 ∂
−1 2
−1
2
2 ∂p
= −ω1 (E[R2 ] − E[R1 ])
+
[∆1 E[R1 ] + ∆2 E[R2 ] ] 2 − 4 E[R1 ]E[R2 ] 2 (cos (Φ)p) ,
∂T
∂Φ 2
∂Φ
∂Φ
(3.32)
where
p
=
p(Φ)
is
the
marginal
probability
density
distribution
for
the
phase
difference:
p(Φ)
=
Rπ
q(Φ,
φ
)
dφ
.
The
resulting
FPE
is
an
anisotropic
advection-diffusion
equation
in
one
variable.
Sub1
1
−π
stituting the moments (3.31) for Ri , i = 1, 2 in (3.32) yields:
∂p
∂p
∂2p
∂2
= m1
+ m2 2 − m3 2 (cos (Φ)p) ,
∂T
∂Φ
∂Φ
∂Φ
(3.33)
where
m1 =
m2 =
m3 =
ω1
3α
2
2
(∆2 c2 − ∆1 c1 ) + 2 ∆2 c2 − ∆1 c1 ,
λ2
λ2
3απ
9α2 π
1
2
2
2
2
2 2
2 2
−λ2 π(c1 + c2 ) +
(∆1 c1 + ∆2 c2 ) +
(∆ c + ∆2 c2 ) ,
2
4|λ2 |
−64λ32 1 1
"
#
r
r !
2c c
δC
−λ2 π
3απ
∆1
∆2
9α2 π p
1 2
√
+
+
+
∆1 ∆2 ,
24
∆2
∆1
−64λ32
∆1 ∆2 8|λ2 |
(3.34)
(3.35)
(3.36)
and {ci , i = 1, 2} are as defined as above.
3.4
Solutions to the reduced stationary Fokker-Planck equation
We now solve for the stationary distribution for the FPE (3.33) with ∂p/∂T = 0, yielding
Z
−1
p(Φ) = ([m2 − m3 cos(Φ)] exp [f (Φ)]) ×
M1 exp[f (Φ)] dΦ + M2
!
(m2 + m3 ) tan (x/2)
p
f (x) = 2m1 arctan
m22 − m23
12
(3.37)
(3.38)
where M1 , M2 are constants chosen for normalization and periodicity. In this form the analytical solution
does not yield very much information about the qualities of the distribution. Instead we look for an
asymptotic solution of the form p = p0 +∆1 p1 +O(∆2i ), based on the small noise approximation (∆i 1).
This small parameter was also used to develop the quasi-steady approximation of the phase equations, as
discussed in Section 3.4, which led to the reduced FPE (3.33). It is possible to compute the constants M1
and M2 numerically for comparison to this expansion, but instead we compare to the numerical solution
of (3.33) in the figures in Section 4.
Substituting the asymptotic expansion for p into (3.33) we get the following order equations for the
stationary density:
O(1) :
O(∆1 ) :
∂2
[(1 − K cos (Φ))p0 ] ,
∂Φ2
∂p0
∂2
∂2
0 = a1
+
[(a
+
a
cos
(Φ))p
]
+
[(a4 + a5 cos (Φ))p0 ] ,
2
3
1
∂Φ
∂Φ2
∂Φ2
0 =
(3.39)
(3.40)
where
3α
1 + 2 , a2 = −λ2 π, a3 = λ2 πK ,
λ2
1
3
3απ
a4 = πα(1 + z)
+
, a5 = −
(1 + z)K, z = ∆2 /∆1 .
λ2 8|λ2 |
8|λ2 |
ω1
a1 = (z − 1)
λ2
(3.41)
Integrating (3.39), then enforcing both normalization and periodicity yields the leading order solution,
√
1 − K2
p0 (Φ) =
for K 6= 1 .
(3.42)
2π(1 − K cos(Φ))
From (3.42) we can notice important qualities about the solution to the FPE. If K = 0, corresponding
to the case of no common noise, the probability density for Φ is uniform on [−π, π]. As K is increased, the
height of the probability density at Φ = 0 increases as well. As K gets close to 1, the probability density
becomes very peaked until a singularity occurs at Φ = 0. Note that for K = 1, the asymptotic solution
(3.42) is no longer valid, since the standard method of integration yields a solution that is singular at
Φ = 0. The case of K = 1 is considered separately below.
Solving (3.40), normalizing the density and enforcing periodicity, we obtain the correction to the
stationary density,
tan (Φ/2)
√
Φ
√
ω1 (z − 1) arctan (1+K)
−
2
1 − K 2 (a4 − a5 cos (Φ))
1−K 2
p1 (Φ) =
−
−2λ2 π 2 (1 − K cos (Φ))2
λ22 π 2 (1 − K cos (Φ))
(a5 K − a4 )
√
for K 6= 1 .
(3.43)
+
2
2λ2 π 1 − K 2 (1 − K cos (Φ))
Note that the coefficients aj for j = 1, 4, 5 include factors of the form (1 ± z), so that the terms in ∆1 p1
have factors (∆2 ± ∆1 ).
When K = 1, then the steady state solution to p solves:
0=
∂2
[(1 − cos (Φ))p] .
∂Φ2
(3.44)
Here the advection term is dropped because K = 1 implies that δ1 , δ2 = 0 (δC < ∞), and the coefficient for
the advection term vanishes. Using the theory of generalized functions [16], one can show that p(Φ) = δ(Φ)
is a solution of (3.44). This solution is valid as it satisfies both the periodicity condition in Φ and the
normalization condition.
13
4
Numerical Results and Comparison to the Analytical Approximations
In this section, we compare the analytical probability density functions for the phase difference to the
probability densities for simulations of the original system (2.1). We use a second order explicit weak
stochastic simulation scheme given in [18] (Section 15.1) for our simulations. These results indicate
that the phase difference density characterizes the different phase behaviors for different combinations
of noisy inputs, as shown in Fig. 2. When the common noise forcing dominates over the intrinsic noise
contributions, the density is peaked around the origin, indicating a stronger probability of synchronized
in-phase CR-type oscillations. For the case when the intrinsic noise sources have different strengths,
with at least one of them of the same order of magnitude as that of the common noise strength, the
phase difference density is asymmetric about the origin, indicating that the nearly regular oscillations
appear frequently together with a phase shift. For low common to intrinsic noise strength ratio, the
phase difference density spreads out, indicating only rare instances of synchronized oscillations. The
comparisons also show that the density based on the linear approximation (3.15) agrees with the density
from the simulation of (2.1) in the tails, suggesting that correlation corrections between the two oscillations
is captured by this approximation. However, the linear approximation does not compare well with the full
simulation when the intrinsic noises are not of identical strength. In that case the nonlinear effects are
significant, and the asymptotic approximation (3.42)-(3.43) agrees with the density from the simulation
of (2.1), as shown in the figures. This comparison indicates that this asymptotic approximation (3.42)(3.43) captures the nonlinear effects in the phase different density related to amplitude-dependent phase
dynamics. The comparisons also illustrate the range of validity of the asymptotic approximations, and
show that the density computed numerically from the reduced FPE for the phase dynamics (3.33) matches
that computed from the simulation of the full system for ∆i 1, even for parameter ranges outside of
the asymptotic validity of the analytical approximation (3.42)-(3.43).
Results for identical intrinsic noise coefficients, δ1 = δ2 , are shown in Fig. 3. From left to right the
common noise is increased, while the intrinsic noise strengths are held equal and constant. Given the
analytical results for the densities when the intrinsic noises have equal strength, we expect the density
of the phase difference to be symmetric about Φ = 0. For stronger common noise indicated by larger
values δC , one would expect behavior closer to perfect synchronization. Indeed the density of the phase
difference approaches a δ-function as δC increases, corresponding to K → 1. We note that the linear
approximation (3.15) provides a better approximation in this case. The asymptotic approximation to
the stationary density (3.42)-(3.43) is not close to the behavior of the phase difference density of the full
system in the regions near Φ = ±π. This is likely due to the fact that the asymptotic solution does not
include correlation effects, while the contributions from amplitude-dependent phase evolution included in
the approximation (3.43) are not prevalent since ∆2 − ∆1 = 0.
In Fig. 4, we show results for the case of symmetric intrinsic noise for δ2 = δ1 with ω1 6= 0 and
parameter values such that the bifurcation parameter λb is outside of the knee of the bifurcation branch
shown in Fig. 1. (This region corresponds to the case where λ(r) = 0 has no real solutions for r). In
this case larger noise values are taken within the range appropriate for CR-type driven oscillations (up
to δi ∼ 2 ), as there is no potential transition to large limit cycle oscillations, in contrast to the case
where λb is inside of the knee. In the first frame of Fig. 4 we see results similar to Fig. 3, with the
density of the phase difference centred at the origin, due to the symmetry of the noise. The shape of the
asymptotic approximation for the stationary density matches that obtained from the simulation of the full
system (2.1) for values of the noise up to the range δi ∼ 2 satisfying the small noise approximation. The
approximations are no longer valid for stronger noise outside of this parameter range, where the density
approaches that of a uniform distribution, as shown in the second frame of Fig. 4.
In Fig. 5 we compare the phase difference density obtained from simulations of (2.1) with the asymptotic approximations given in (3.15) and p = p0 + ∆1 p1 for the case ω1 6= 0 and δ1 6= δ2 . Then the
phase evolution depends on the amplitude, as can be seen from the asymptotic approximations (3.42,
14
0.1
0.05
0
−2
0
2
Φ
probability density
probability density
probability density
0.15
0.3
0.2
0.1
0
−2
0
2
Φ
1
0.8
0.6
0.4
0.2
0
−2
0
2
Φ
Figure 3: Histograms of simulated phase data of the original system, (2.1), plotted against the asymptotic
FPE solution (3.42)-(3.43), p (solid), and the solution derived from the linearization of the SDEs (3.15),
P(dashed) . Parameters used in this simulation are ω0 = 0.9, ω1 = 1.2, = 0.1, λ2 = −3, α = 0.2,
γ = −0.2, and δ1 = δ2 = 0.003 (∆1 = ∆2 ). We increase the common noise strength from left to right,
(δC = 0, 0.003, 0.01). The histogram of simulated data from the original equations (2.1) is given by the
grey shaded region. All noise parameters are chosen to ensure that the probability of a transition to the
stable limit-cycle is negligible.
3.43). The correction to the leading order asymptotic solution, p1 given in (3.43), has a non-zero odd
component with coefficient ω1 (∆2 − ∆1 ), so that the density of the phase differences has a peak centred at
a non-zero value. This asymmetry due to the phase shift between the oscillations is not captured by the
density based on the linear approximation, but the asymptotic approximation to the density (3.42)-(3.43)
has this asymmetry, as shown in Fig. 5 and Fig. 6. Note that the asymmetric contribution vanishes
for δ1 = δ2 . The effect of the amplitude-dependent phase evolution is to, on average, advance or delay
the phase of one oscillator relative to the other. As expected from the form of ω(ri ), the average phase
difference has the same sign as ω1 (∆2 − ∆1 ). An increase in noise tends to increase the amplitude of
the oscillator, so that differences in δi are translated into a non-zero average difference in the amplitude.
The finite difference results indicate that the numerical solution of the reduced FPE (3.33) based on the
(quasi-steady) approximation for ∆i 1 matches the phase difference density obtained from the full simulation of (2.1). For the results shown in the lower right panel of Fig. 5, the parameters are outside the
range of validity for the asymptotic approximation (3.42)-(3.43), due to the large value of ω1 as discussed
below.
We also compare results for larger values of the parameter , ω1 and the noise coefficients for the
non-symmetric case, where δ1 6= δ2 . This comparison illustrates how the approximation for the phase
difference density gives the probability for phase difference dynamics for larger CR-type oscillations driven
by larger noise levels. By comparison with the simulation of the full system, we also illustrate parameter
combination for which the asymptotic approximation holds or breaks down, as consistent with the theory.
In Fig. 6 we consider parameters corresponding to cases of λ2 being inside or outside the knee of the
bifurcation shown in Fig. 1, and we see that the density has a similar shape and shift as in the other
cases studied above, as long as the probability of the system evolving to the large amplitude limit-cycle is
negligible. For values of λ2 inside the knee, we consider values of the noise so that a sustained transition
to large limit cycle behavior is unlikely.
In Figs. 5 and 6, we take some parameter values where the quantity Z = ω1 (∆2 − ∆1 )/(λ22 π 2 (1 − K))
is outside the range of validity for the asymptotic expansion for the solution of (3.33). This parameter
is roughly the magnitude of the odd part of the correction term (3.43). If it is on the same order as
the leading order solution (3.42), then the asymptotic solution of (3.33) should effectively break down.
Indeed, in Fig. 6, middle and Fig. 5, lower-right, the parameters are such that Z is noticeably larger than
in the other figures so that the approximation is not in the asymptotic regime (Z ≈ 0.25). Even though
the analytical approximation is no longer valid there, we see that the numerical solution of the reduced
15
probability density
probability density
0.3
0.2
0.1
0
−2
0
2
0.3
0.2
0.1
0
Φ
−2
0
2
Φ
Figure 4: Histograms of simulated phase data of the original system, (2.1) (grey shaded region), plotted
against the asymptotic solution p(Φ) (3.42-3.43) (solid) and the density derived from the linearization, P
(3.15) (dashed). Parameters used in this simulation are the same as in Fig. 3 except for λ2 = −6, and
left/right: δ1 = δ2 = δC = 0.05/0.1.
FPE for the density of the phase difference (3.33) matches the density obtained by simulation of the full
system (2.1). This result is consistent with the validity of the approximation for ∆i 1, on which the
derivation of (3.33) is based.
5
Summary and Discussion
We analyzed noise-induced synchrony in the context of identical uncoupled conditional oscillators forced
by independent intrinsic noise sources and a common noise source. In the absence of noise the system
is quiescent, and for parameter values in the proximity to a subcritical Hopf bifurcation, quasi-regular
oscillations are driven via a coherence resonance (CR) phenomenon. Even though these oscillations are
noise driven, the power spectral density of their time series is concentrated near the frequency of the
Hopf bifurcation. This phenomenon occurs in the range of noise levels around the values that optimize a
coherence measure [39]. Noise-induced synchrony has been analyzed extensively in the context of limitcycle oscillators, but this is the first study undertaken of noise-induced synchronization of uncoupled
conditional oscillators, which do not have limit-cycles.
For parameters near the Hopf bifurcation of the underlying deterministic system, a stochastic multiple
scales analysis yields stochastic differential equations (SDEs) for the amplitude and phase equations for
the CR-type quasi-regular oscillations. For the bifurcation parameter near the Hopf value, a slow time
scale naturally arises due to the weak decay in the deterministic system of oscillations with the Hopf
frequency. The stochastic phase and amplitude equations on this slow time scale are derived for noise
levels small compared with the distance of the bifurcation parameter from the Hopf bifurcation. These
equations provide the basis for analyzing the phase dynamics of the noise-induced oscillations. We derive
approximations for the density of the phase difference in order to understand the combined effect of
intrinsic and common extrinsic noise on the synchronization of the oscillations driven through a CR-type
phenomenon.
Two different analytical approximations are obtained for the stationary density of the phase difference
of the two oscillators. We obtain the probability density function for the phase difference in the context of
a linearized approximation that includes correlation effects, but neglects nonlinear contributions that arise
through the amplitude-dependence of the phase dynamics. In order to include the nonlinear amplitudedependence in the phase equations, a further asymptotic analysis of the nonlinear stochastic phase and
amplitude equations leads to a reduced system of the amplitude-dependent phase dynamics and the
16
probability density
probability density
0.25
0.2
0.15
0.1
0.05
0
−2
0
2
probability density
probability density
0.2
0.15
0.1
0.05
0
−2
0
2
0.4
0.3
0.2
0.1
0
Φ
0.25
0.5
0
2
Φ
−2
0
2
Φ
0.5
0.4
0.3
0.2
0.1
0
Φ
−2
Figure 5: Histograms of simulated phase data of the original system, (2.1) (grey shaded region), plotted
against the asymptotic solution p(Φ) (3.42-3.43) (solid) and the density derived from the linearization, P
(3.15) (dashed). Parameters used in this simulation are ω0 = 0.9, = 0.1, λ2 = −3, α = 0.2, γ = −0.2.
The intrinsic noise strengths are δ1 = 0, δ2 = 0.009. Left(Right): δC = 0.003(0.009). Top(Bottom):
ω1 = 5(20). The data in the lower right has parameters that are not in the asymptotic regime. The
(red) dash-dotted line in the lower right panel is the stationary solution to (3.33), computed using a
finite-element method.
corresponding Fokker-Planck equation (FPE) for this reduced system. A critical assumption in this
reduction is that intrinsic and extrinsic noise are small relative to the parameter measuring distance from
the Hopf bifurcation (∆i 1), which leads to the observation that the phase and amplitude evolve on
different time scales. This observation provides the basis for an approximation analogous to a quasi-steady
approximation that replaces contributions from the amplitudes in the phase equations with moments of
these terms.
The asymptotic expression for the phase difference density provides the explicit dependence of the
density on different parameters. In particular we see differences in the density of the phase difference
for different relative strengths of intrinsic and extrinsic noise levels. For common noise levels an order of
magnitude larger than the intrinsic noise levels, the density of the phase difference is centred at the origin,
yielding a greater probability of synchronized in-phase CR-type oscillations. For different strengths of the
intrinsic noise sources, with one of the intrinsic noise levels the same order of magnitude as that of the
common noise strength, the phase difference density is asymmetric about the origin, indicating a larger
probability of observing oscillations appearing together with a phase shift. The asymmetry appears in
the asymptotic approximation for the phase difference density, with a magnitude proportional both to the
17
0.2
0.1
0
−2
0
0.4
0.3
0.2
0.1
0
Φ
2
0.5
probability density
probability density
probability density
0.3
−2
0
2
Φ
0.5
0.4
0.3
0.2
0.1
0
−2
0
2
Φ
Figure 6: Histograms of simulated phase data of the original system, (2.1) (grey shaded region), plotted
against the asymptotic solution p(Φ) (3.42)-(3.43) (solid) and the density derived from the linearization
(3.15). The (red) dash-dotted line corresponds to√the finite element solution of the reduced FPE (3.33).
Parameters used for these simulations are = 0.1, ω0 = 2, ω1 = 50, α = 1.2, γ = −1 and δ1 = 0.
Left/Middle/Right: (λ2 , δ2 , δC ) = (−3, 0.05, 0.03)/(−3, 0.05, 0.05)/(−5, 0.07, 0.07). The parameters for
the left and middle figures correspond to dynamics inside the knee of the bifurcation and the parameters
for the right figure correspond to dynamics outside the knee of the bifurcation (see Fig. 1 for schematic).
coefficient ω1 of the terms for amplitude-dependent phase dynamics and to the difference of the intrinsic
noise strengths. For the intrinsic noise stronger than the common noise, the phase difference density
spreads out indicating only rare instances of synchronized oscillations.
The comparisons of the densities obtained by different methods show that the density based on the
linear approximation (3.15) captures correlation corrections between the two oscillations as evidenced by
the graphical agreement of this density in the tails with the density for the full system (2.1). However,
the linear approximation neglects the nonlinear dependence of the phase on the amplitude of the CR-type
oscillations, so it can not describe phase shifts between the two oscillators. This amplitude dependence
of the phase is included in the asymptotic approximation (3.42)-(3.43). As shown in the graphical comparisons (Section 4) of the asymptotic approximation for the phase difference density (3.42)-(3.43), the
shape and symmetry/asymmetry of this approximation agrees with the phase difference density of the full
system (2.1) for the range of parameters where ∆i 1, as is consistent with the asymptotic approach.
The asymptotic approximations are no longer valid for stronger noise outside of this parameter range.
For the case where ω1 is sufficiently large, the asymptotic expansion for the solution of the reduced FPE
(3.33) breaks down, since the correction in the asymptotic expansion is no longer O(∆i ). In this case the
graphs of the phase difference density from the simulation of the full system and the numerical solution
of the (3.33) have the same shape and asymmetry for ∆i 1, indicating that for weak noise the reduced
FPE (3.33) provides an asymptotic approximation for the phase difference FPE even if the asymptotic
solution to (3.33) is not valid.
A
Probability density for the phase difference and moments in the
linearized case
Here we evaluate the integral in (3.15) to obtain a closed form expression for the probability density
function P(Φ) as given in Section 3.2. The integral is of the form
Z ∞Z ∞
I=
rs exp −ar2 − bs2 + crs dr ds, a, b ≥ 0.
(A.1)
0
0
A linear coordinate transformation, Q, diagonalizes the quadratic form in the exponential, yielding ar2 +
T
bs2 − crs = r s QDQ−1 r s . Depending on the value of c, the result changes due to a flipping
18
of the region of integration. If c > 0, then:
I=
Z
Uŝ
Lŝ
Z
Ur̂
(Q11 Q21 r̂2 + (Q11 Q22 + Q12 Q21 )r̂ŝ + Q12 Q22 ŝ2 ) exp −D11 r̂2 − D22 ŝ2 Det[Q] dr̂dŝ, (A.2)
Lr̂
where Qij and Dij denote the ith row and j th column of Q and D, respectively, and
Lr̂ = −Q22 Q−1
21 ŝ,
Ur̂ = −Q12 Q−1
11 ŝ,
Lŝ = 0,
Uŝ = ∞. for c > 0 ,
Lŝ =
Uŝ =
−Q11 Q−1
12 r̂,
Lr̂ = 0,
Ur̂ = ∞ for c < 0 .
I=α
π
−Q21 Q−1
22 r̂,
We evaluate (A.2 ) to obtain
2
+ arctan(β) + γ
(A.3)
(A.4)
where
α=
c
,
(4ab − c2 )3/2
β=√
c
,
4ab − c2
γ=
1
.
4ab − c2
(A.5)
Comparing to (3.13,A.1), we obtain the result given by (3.15).
We calculate the moments of amplitude from the linear distribution in Section 3.2 as a consistency
check on amplitude moment results from Section 3.3. Using (3.13), we integrate over all φi for i = 1, 2,
evaluating a contour integral to obtain the marginal probability density function for the amplitudes in
the linearized case, P̂ :
2
2n #
"X
∞
2R R
λ
δ
π
1
1
2
1
2
C
P̂ (R1 , R2 ) = 4π 2 CR1 R2 exp
(A.6)
C × (∆2 R12 + ∆1 R22 ) ×
4 /44 )
λ2
(n!)2 2 4 ∆1 ∆2 − (δC
n=0
To get the expected moments for R1 , we evaluate the following integral
Z ∞Z ∞
E[R1N ] =
(A.7)
R1N P̂ (R1 , R2 ) dR1 dR2
0
0
#
2 2 2n
2
Z ∞ Z ∞ "X
∞
δC π C
1
π
N +1+2n 1+2n
2
2
2
= 4π C
R2
exp
C × (∆2 R1 + ∆1 R2 ) dR1 dR2
R1
(n!)2 24 λ2
λ2
0
0
n=0
Interchanging summation and integration and rewriting factorial terms in the form of Gamma functions
yields,
2 2n ∞
X
Γ(n + 1 + N/2) δC
−λ2 2+N/2
1
N
2
(A.8)
E[R1 ] = π C
4
2
N/2
Γ(n + 1)
2
π C
(∆1 ∆2 )1+n ∆
n=0
2
For example, the second moment is
E[R12 ]
n
2
∞
4
δC
(−λ2 )3 X
(−λ2 )3 ∆1
48
∆1
= 4 2
(n + 1)
=
=−
2
4
8
2
4
8
4 ∆1 ∆2
C π
λ2
π C ∆1 ∆2 n=0
δC − 4 ∆1 ∆2
(A.9)
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