A Variational Method for Analyzing Stochastic Limit Cycle Oscillators

Abstract

We introduce a variational method for analyzing limit cycle oscillators in R d driven by Gaussian noise. This allows us to derive exact stochastic differential equations (SDEs) for the amplitude and phase of the solution, which are accurate over times over order exp Cbǫ −1 , where ǫ is the amplitude of the noise and b the magnitude of decay of transverse fluctuations. Within the variational framework, different choices of the amplitude-phase decomposition correspond to different choices of the inner product space R d . For concreteness, we take a weighted Euclidean norm, so that the minimization scheme determines the phase by projecting the full solution on to the limit cycle using Floquet vectors. Since there is coupling between the amplitude and phase equations, even in the weak noise limit, there is a small but non-zero probability of a rare event in which the stochastic trajectory makes a large excursion away from a neighborhood of the limit cycle. We use the amplitude and phase equations to bound the probability of it doing this: finding that the typical time the system takes to leave a neighborhood of the oscillator scales as exp Cbǫ −1 .

Phase Reduction of Stochastic Biochemical Oscillators

“…An alternative approach is to introduce a weighted inner product on R K and to specify v(t) by projecting the full solution on to the limit cycle using Floquet vectors. In a sufficiently small neighborhood of the limit cycle, the resulting phase variable coincides with the isochronal phase [8,9]. This has the advantage that the amplitude and phase decouple to linear order in 1/Ω.…”

“…The method of isochrons determines the phase θ(t) by tracing where the isochron through X(t) intersects the limit cycle.The response to perturbations depends on the phase response curve R(θ), which is normal to the isochron at the point of intersection with the limit cycle. and transverse (amplitude) fluctuations of the limit cycle [24,7,28,8,9]. The basic intuition is that Gaussian-like transverse fluctuations are distributed in a tube of radius 1/…”

“…The second, and more significant issue, is that even for large Ω there is a small but non-zero probability that a stochastic trajectory generated by the SDE (2.18) leaves any neighborhood of the limit cycle, in which case the notion of phase breaks down. In previous work, we have developed a variational principle for SDEs that provides strong exponential bounds on the escape probability [9]. (We have also extended the variational formulation to the case of piecewise-deterministic Markov processes [10].)…”

“…However, as we have previously mentioned, the amplitude-phase decomposition is not unique, since it depends on the particular definition of the phase θ(t). Recently, we introduced a variational principle for SDEs that uniquely specifies the amplitude-phase decomposition by minimizing X(t) − Φ(θ(t)) ρ for a specific choice of ρ, which has two significant features: (i) sufficiently close to the limit cycle, the phase corresponds to the isochronal phase, and (ii) one can obtain rigorous exponential bounds on the probability of escape [9,10]. We will apply the variational approach to the stochastic CRN model, and show that both of these results carry over; (ii) will be established in §5.…”

“…In §3 we show how to derive a corresponding phase equation for the exact discrete Markov process using Kurtz's random time change representation [31,32,33,3]. This analysis is placed on a more rigorous footing in §4 by extending a variational principle that we recently developed for stochastic differential equations (SDEs) [9]. The latter generates a unique amplitude-phase decomposition of a stochastic oscillator by minimizing the distance of the stochastic trajectory from the limit cycle with respect to an appropriately weighted norm.…”