Stochastic phase dynamics: multi-scale behaviour and coherence measures
Na Yu, R. Kuske, and Y.X. Li
Dept. of Math, U. British Columbia, Vancouver, BC
Corresponding author: rachel@math.ubc.ca
A novel multi-scale approach is used to derive stochastic amplitude and phase dynamics for a
canonical noise-sensitive model exhibiting coherence resonance. Explicit expressions for the dependence on noise levels and model type are compared with computational coherence measures.
PACS numbers: numbers
I.
INTRODUCTION
Coherence resonance (CR) or autonomous stochastic
resonance refers to coherent behaviors such as rhythmic
oscillations at an optimal level of noise in a system that is
quiescent without noise. It was first reported in a numerical study of a two-dimensional autonomous system [1],
whose deterministic behavior is characterized by two stable equibria separated in its circular phase space by two
unstable equilibria. If the system is perturbed beyond
the threshold distance between neighboring stable equilibria, the system evolves from one to the other. Similar
noisy behavior has been observed in excitable systems,
such as the FitzHugh-Nagumo (FHN) [2] and HodgkinHuxley (HH) [3] models. There the deterministic systems
have a single stable equilibrium, but large “excited” excursions occur for perturbations beyond a threshold. In
all of these settings higher noise levels increase the frequency of transitions or excursions, providing the intuition behind a phase plane analysis [4] and the relative
first passage time [2] used to explain the increased frequency of the noise-induced coherent oscillations.
These studies have two important characteristics of CR
in common. Firstly, the coherence of the dynamics, defined as [1]
β = hp (∆ω/ωp )−1 ,
(1)
reaches a maximum at an optimal level of noise. Here
hp and ∆ω are the height and the width of the averaged
spectrum peak at frequency ωp . Secondly, the frequency
of the coherent oscillations depends on the noise level.
A large number of studies of noise-induced synchrony in
networks of coupled excitable systems, including coupled
integrate-and-fire models [5], FHN models [6], HH models
[3], and bursting models [7] also illustrate these characteristics of CR. Optimal coherence at a finite noise level
was explained by Wiesenfeld [8] who revealed how the
noise controls the structure of the power spectrum. Similarly, [9] used logistic maps to explain the peak values in
the coherence measure.
These earlier results examined coherence resonance for
oscillations composed of transitions between steady equilibria, where the frequency naturally increases with noise
level. In other contexts, the relationship between frequency and noise may vary. For parameter values that
are close to a saddle-node point in the periodic branch
in the HH model [10], noise variation gives little or no
change in the frequency of the coherent oscillations. Instead noise apparently “shifts” the bifurcation structure,
yielding stable large amplitude oscillations associated
with solution branches far from the Hopf point. CR is
also observed in transitions between steady state and oscillatory modes, [1], [11]-[16]. There the relationship between resonance frequency and noise level depends on the
model type.
In this paper we focus on CR via the canonical model
for a normal form near a Hopf bifurcation,
dx = [(λ + αr2 + βr4 )x − (ω0 + ω1 r2 )y]dt + δ1 dη1
dy = [(ω0 + ω1 r2 )x + (λ + αr2 + βr4 )y]dt + δ2 dη2
r 2 = x2 + y 2 ,
(2)
where η1 and η2 are independent standard Brownian motions (SBMs). This model captures the generic behaviour
of excitable systems near a critical Hopf point λ = 0. We
present explicit analytical results in the context where
system parameters approach the bifurcation point so that
λ 1. This is a noise-sensitive regime, well-known from
computations to exhibit CR [15]-[16]. We restrict our attention to λ < 0, corresponding to a quiescent regime in
the absence of noise. Through CR the system exhibits
oscillations in this regime, and the analysis reveals the dependence of the phase and amplitude on both the noise
level and the model parameters, providing critical scaling
relationships related to resonance.
The key to our results is a stochastic multiple scales
expansion which exploits the resonance phenomenon. We
derive a reduced system of stochastic equations for the
amplitude R(T ) and phase correction φ(T ) of the oscillations. Here T is a slow time, T = 2 t, compared
to the original time scale t. The parameter 1
measures proximity to the Hopf bifurcation point via
λ = 2 λ2 , and −1 appears as an amplification factor
in the reduced equations. We derive analytical quantities, including moments for R and φ, which illustrate
the model dependency of the stochastic behaviour and
the noise-amplification factor related to the Hopf bifurcation. In addition these quantities compare well with
numerical simulations and provide quantitative insight
into the stochastic resonance-type maximum in the coherence measure β defined in (1) and shown in Figure
2. Computed coherence measures are commonly used
as indicators of CR, while the more desirable analytical
2
Power Spectral Density
0.6
0.3
x(t)
expressions for the stochastic phase dynamics, derived
directly from the model, are rare.
0
-0.3
-0.6
1850
RESULTS AND ANALYSIS
1950
2000
0.5
0
0
0.1
0.2
t
Power Spectral Density
0.6
x(t)
0.3
0
-0.3
-0.6
1850
1900
1950
1.5
(b)
1
0.5
0
2000
t
0
0.1 0.2
Normalized Frequency
FIG. 1: Left column: time series for x(t) for δ1 = δ2 = δ.
Right column: corresponding PSD. The parameters are α =
−0.2, β = −0.2, ω0 = 0.9, ω1 = 0, (a) δ = .01, λ = −0.03
(solid line) and λ = −0.003 (dashed line). Recall λ = 2 λ2 ,
measures distance from the Hopf point. (b) λ = −0.03,
δ = 0.07 (solid line) and δ = 0.1 (dashed line). Oscillations
increase as λ → 0 and as δ increases.
(a)
1.2
2.5
(c)
1
(e)
0.75
2
1
ω
We begin with (2) for α < 0 and β < 0, so that λ = 0
is a super-critical Hopf bifurcation point in the absence
of noise. For δj = 0, j = 1, 2, the oscillations decay over
time when λ < 0, which we consider here. Oscillations
with amplitude r0 and phase Φ = (ω0 + ω1 r02 )t are stable
when λ > 0. For parameter values near the critical point,
that is, λ = 2 λ2 with 1 and λ2 = −1 for λ < 0, the
system is particularly sensitive even to small noise. The
parameter ω1 is an important link between the amplitude
and phase; different values of ω1 represent different model
types with different phase dynamics.
Figure 1 shows time series with small additive white
noise, 0 < δj = δ 1 for j = 1, 2. The variance of
the amplitude of the slowly modulated oscillations increases with δ and as |λ| → 0, that is, approaching the
Hopf point at λ = 0. A strong peak in the power spectral density (PSD) indicates a dominant frequency due
to CR. This response is induced by the noise since, without noise, the oscillations decay for λ < 0. There is
also a slow phase variation (not apparent on the graphs).
Figure 2 (bottom row) gives the numerically computed
coherence measure β, based on the shape of the PSD.
It is obtained from a large number (1200) of realizations
for (2), averaging the PSD’s and using the shape of the
frequency peak to compute β in (1). The graphs show
roughly the same behaviour for β for vanishing, positive,
and negative values of ω1 which characterizes the model
type. For δ1 = δ2 = δ near 0, β is small, followed by a
sharp increase to its maximum for values 0 < δ < .1 and
decreasing β for δ > .1.
While β gives a measure of the coherence, it does not
provide any details about the phase. In Figure 2 (top
row) we show the numerically and analytically (11) obtained expected value for the frequency associated with
the peak of the PSD; that is, it is the coherence frequency
induced by resonance with the noise. For ω1 = 0, the expected value of the phase does not depend on the noise,
while for ω1 > (<) 0, the coherence frequency increases
(decreases) with the noise. Figure 2 shows good agreement between simulations and analysis for δ/ < 1. The
results are shown for symmetric noise (δ1 = δ2 = δ), and
similar behaviour is observed for δ1 6= δ2 .
The analytical results below give a complete description of this stochastic behaviour both in terms of the
noise level and the model parameters, providing a view
beyond the numerical results of β. For example, we get
the quantitative relationships for the dependence of the
phase on ω1 and δ, the amplification factors near the
Hopf point, and the behaviour of β shown in Figure 2.
We obtain expressions for the stochastic amplitude and
phase through the method of multiple-scales [17], combined with a projection onto the resonant modes. The
1900
(a)
1
0.5
1.5
0.8
0.25
1
0.6
0.01
δ
0.1
5
0.01
(b)
δ
0
0.1
5
(d)
0.01
δ
0.1
5
4
4
4
3
3
3
2
2
2
(f)
β
II.
1.5
1
1
1
0
0
0
0.01
δ
0.1
0.01
δ
0.1
0.01
δ
0.1
FIG. 2: Upper: peak frequency ω of the PSD peak vs. the
noise intensity δ1 = δ2 = δ; solid line: asymptotic results (11);
dotted line: numerical results. Lower: coherence measure β
vs. δ. The parameters are α = −0.2, β = −0.2, λ = −0.03,
with ω0 = 0.9, and ω1 = 0 in (a) and (b); ω1 = 1.2 in (c) and
(d); ω1 = −0.5 in (e) and (f).
approach exploits the fact that the system is near critical, using a slow time scale T = 2 t, related to proximity
to the critical Hopf point through λ = 2 λ2 . We allow
x(t, T ) and y(t, T ) to be functions of both time scales t
and T , replacing the time derivative xt with xt + 2 xT ,
and similarly for yt . Writing
x = R(T ) cos(ω0 t − φ(T )), y = R(T ) sin(ω0 t − φ(T )),(3)
we look for stochastic amplitude and phase equations in
terms of A and B, where
R2 = A2 + B 2 , φ = tan−1 B/A
dA = ψA dT + σA1 dξ11 + σA2 dξ12
dB = ψB dT + σB1 dξ21 + σB2 dξ22 ,
(4)
(5)
3
The properties of white noise are used to rewrite dηj on
the T scale and furthermore
dη1 (t) = −1 dη1 (T ) = −1 (cos ω0 tdηA1 + sin ω0 tdηB1 )(7)
and similarly for η2 , with ηAj and ηBj independent
SBMs. Using (7) in (6), we project onto the primary
modes (cos ω0 t, sin ω0 t) and (− sin ω0 t, cos ω0 t). That is,
the dot product of these modes with (6) is integrated
over one period of the oscillation on the t time scale.
The multi-scale ansatz is also applied, treating functions
of the slow time T as independent of the fast time t. This
projection yields
σA1 dξ11 + σA2 dξ12 = dζA /(22 )
σB1 dξ21 + σB2 dξ22 = dζB /(22 ),
(8)
where dζm = δ1 dηm1 + δ2 dηm2 , m = A, B. We let ξ1j =
ηAj and ξ2j = ηBj to get the diffusion coefficients in (5)
σAj = σBj = δj −2 /2. A similar procedure is used to get
the drift coefficients ψA and ψB in (5).
This derivation of stochastic amplitude equations (5)
has been used in applications where autonomous stochastic resonance appears due to noise sensitivity (see [18]
and references therein), and it is shown in detail for this
system in [19]. A key component is the projection onto
the primary modes combined with the multi-scale ansatz.
While it is true that the noise is white with all frequency
components, this projection selects out the components
corresponding to the resonant mode. Note that on the
long time scale, the noise has an additional factor of −2 .
The primary mode dominates the dynamics through CR
over a long time scale (5), while the other modes decay
at a sufficiently fast rate to be of higher order for δ ,
as discussed in the next section.
From (4), (5), (8) we then obtain the equations for the
stochastic amplitude and phase, and the approximation
1.2
x(t)
where ξij are independent SBMs. The coefficients ψA ,
ψB , σAj , σBj are derived through the analysis by equating expressions for dx and dy obtained by two methods: firstly, using Ito’s formula, which relates (2) to (5)
through (3) and (4), and secondly, by direct substitution
of (3) and (4) into the original equations (2). After equating these expressions, we project them onto the primary
mode of oscillations with frequency ω0 on the time scale
t. Combined with the multi-scale representation, this
projection leaves terms which depend on T only, thus
yielding equations for ψA , ψB , and σAj , σBj , j = 1, 2 in
(5). We give some details here for the derivation of the
diffusion (noise) coefficients in (5).
The noise terms from (2) are set equal to the noise
terms obtained using Ito’s formula and (5),
σA1 dξ11 + σA2 dξ12
cos ω0 t sin ω0 t
σB1 dξ21 + σB2 dξ22
− sin ω0 t cos ω0 t
1 δ1 dη1 (T )
.
(6)
=
δ2 dη2 (T )
0
-1.2
250
300
350
400
450
t
FIG. 3: Time series for the subcritical case when α = 0.2, β =
−0.2, λ = −0.03, ω0 = 0.9, ω1 = 1.2. (a)δ = 0.02 (bold
line),(b)δ = 0.04 (thin line).
to their first moments,
dR2 = [2R2 (λ2 + αR2 ) + ∆]dT
+R[cos φ dζA + sin φ dζB ]/2
(9)
dφ = −R2 ω1 dT + [cos φ dζB − sin φ dζA ]/(22 R),
α∆2
∆
−
(10)
E[R2 ] ∼ −
2λ2
4λ32
∆
α∆2
E[φ] ∼ (
+
)ω1 T + ψ0 ,
(11)
2λ2
4λ32
where ∆ = (δ12 + δ22 )/(24 ). In (10)-(11) we omit higher
order corrections of the form ∆2 multiplying exponentially decaying terms. Here ψ0 is a constant phase shift,
which can be set to zero. The expected phase E[φ] depends on the noise through the term −R2 ω1 dT in (9),
giving the phase behaviour shown in Figure 2.
III.
SUMMARY AND EXTENSIONS
A stochastic multiple scales method yields analytical
quantities for the stochastic amplitude and phase for an
oscillator exhibiting CR near a Hopf bifurcation. Explicit
expressions show how the phase and amplitude dynamics depend on the model, providing additional information beyond numerically computed coherence measures.
Phase variations due to noise level and model type are
characterized by a parameter ω1 which captures the coupling between the phase and amplitude. The phase behaviour differs from the increased coherence frequency
typically observed in systems with a threshold. Amplification of the oscillations is related to proximity to criticality measured by .
The multi-scale analysis is valid for noise levels that do
not overwhelm the resonant oscillations. The stochastic
equations (9), transformed for the amplitude R in the
single mode approximation (3), yield validity criteria in
terms of the amplified noise level δj /. Thus the quantitative results give the asymptotic range for the coherence
effect in terms of the noise level and proximity to criticality. The amplitude R is related to the numerator for
the coherence measure β as
hp = O(|R|) = O(δj /).
(12)
For very small δj /, j = 1, 2, (9) and (12) show that the
resonant oscillation is present with small power, so that
4
x1(t), x2(t)
0.4
0
-0.2
x1(t), x2(t)
-0.4
500
0.4
550
600
(b)
0.2
0
-0.2
-0.4
500
0.4
x1(t), x2(t)
(a)
0.2
550
600
(c)
0.2
0
-0.2
-0.4
500
550
600
t
FIG. 4: Time series for diffusively coupled system when α =
−0.2, β = −0.2, λ = −0.03, ω0 = 2, ω1 = 1 and small initial
condition, solid and dashed line for x(t) in first and second
oscillator, respectively. (a) coupling coefficient: d = 0.05;
noise levels: δ1 = 0.05, δ2 = 0.05 (b)d = 0.05, δ1 = 0.01,
δ2 = 0.05,(c) d = 0.001, δ1 = 0.01, δ2 = 0.05.
cate or expensive. Stochastic phase and amplitude equations were derived for the relaxation oscillations of an
elliptic burster in [20], and we mention two important
problems related to (2) for which we have preliminary
results.
1)Transitions from steady state to large amplitude oscillations in the context of subcritical bifurcations
This phenomenon occurs when α > 0 and β < 0 in (2).
Small perturbations or oscillations decay to zero in the
absence of noise since small amplitude oscillations are
unstable. However, larger perturbations trigger jumps
to stable large amplitude oscillations. The probability
or expected time of this transition is directly related to
the amplitude E[R], derived from (9)-(11). In Figure 3
the transition to large amplitude is highly improbable
for small enough noise (bold line), but it can occur for
slightly larger noise (thinner line). A similar phenomenon
is observed if λ approaches the critical Hopf point.
β is small using (1) and (12). For increasing δj / < 1 this
amplitude is increased so that β increases, with (3) still a
valid approximation. For δj / = O(1) or larger, (3) is less
accurate since additional modes can not be neglected in
the approximation [18]. Due to the contributions of these
other modes, |x(t)−R cos(ω0 t+φ)| increases as does the
width of the PSD peak. That is, the variation from (3)
for δj / = O(1) or larger is mirrored in the denominator
∆ω/ωp of β but not in the numerator (12). Therefore
the loss of coherence is reflected simultaneously in the
decrease of β and the breakdown of the approximation
(3) for δj / = O(1) or larger.
Extensions of the analysis are particularly valuable in
noise-sensitive systems where computations can be deli-
2)Interaction and competition between coupling and
noise in systems near critical.
We consider two diffusively coupled oscillators of the form
(2), where a similar multiscale analysis can be used to
derive the coupled slowly varying stochastic phase and
amplitude equations. Through CR the amplification of
the noise depends on −1 , and contributions from the
coupling are similarly amplified. Then there is an interplay between noise and coupling in synchronization. In
Figure 4 (top) we show intermittency between periods
of phase locking and phase drift for noise and coupling
at the same strength. If the noise is decreased (middle)
the synchronization is strengthened, even when the noise
levels are asymmetric. For decreased coupling and asymmetric noise levels, the signals are dissimilar in amplitude
and phase (bottom). These dynamics appear directly
in the stochastic phase and amplitude equations derived
with the stochastic multi-scale analysis.
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