Asymptotic Phase for Stochastic Oscillators

Abstract

Oscillations and noise are ubiquitous in physical and biological systems. When oscillations arise from a deterministic limit cycle, entrainment and synchronization may be analyzed in terms of the asymptotic 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 the Kolmogorov 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 the limit of vanishing noise. The phase can be obtained either by solving an eigenvalue problem, or by empirical observation of an oscillating density's approach to its steady state.Introduction. Limit cycles (LC) appear in deterministic models of nonlinear oscillators such as spiking nerve cells [1], central pattern generators [2], and nonlinear circuits [3]. The reduction of LC systems to onedimensional "phase" variables is an indispensable tool for understanding entrainment and synchronization of weakly coupled oscillators [4,5]. Within the deterministic framework, all initial points converge to the LC, on which we can define a phase that progresses at a constant rate (θ = ω LC = 2π/T LC ). The phase θ(x 0 ) of any point x 0 is then defined by the asymptotic convergence of the trajectory to that phase on the LC. However, stochastic oscillations are ubiquitous, for example in biological systems [6], and in this setting the classical definition of the phase breaks down. For a noisy dynamics, all initial densities will converge to the same stationary density. Thus the large-t asymptotic behavior no longer disambiguates initial conditions, and the classical asymptotic phase is not well defined.Schwabedal and Pikovsky attacked this problem by defining the phase for a stochastic oscillator in terms of the mean first passage times (MFPT) between surfaces analogous to the isochrons (level curves of the phase function θ(x)) of deterministic LC [7][8][9]. Here we formulate an alternative definition that is tied directly to the asymptotic behavior of the density, rather than the first passage time, and is grounded in the analysis of the forward and backward operators governing the evolution of system densities. Our operator approach leads to two distinct notions of "phase" for stochastic systems. As we argue below, the phase associated with the backward or adjoint operator is closely related to the classical asymptotic phase.General framework. Consider the conditional density ρ(y, t|x, s), for times t > s, evolving according to the forward and backward equations

Weakly nonlinear response of noisy neurons

“…We also perform numerical simulations of a model of excitable membrane dynamics that was proposed by Izhikevich [25] and is similar to the two-dimensional Morris-Lecar model [26]. Following reference [27] the model is endowed with channel noise, which is due to the finite number of potassium selective channels. The voltage dynamics of the Izhikevich model are given by…”

Long-time analytic approximation of large stochastic oscillators: simulation, analysis and inference

“…1 limit cycle [10] and/or the amplitude variability in the transversal direction of the limit cycle [11][12][13]. Other papers derive related descriptions of the asymptotic phase of stochastic oscillators [14,15].…”

Thomas and Lindner Reply:

“…For instance, in [4] we showed that the persistent sodium-potassium model driven by channel noise [1,2] can exhibit sustained subthreshold oscillations alternating with large amplitude limit cycle oscillations (action potentials). Under these conditions the eigenvalue spectrum of the adjoint equation [2] has two complex eigenvalue pairs with similar real parts. The system violates our criterion iii, and we do not expect it to have a single well defined phase.…”

“…In [2] we considered, for instance, a heteroclinic 2d system that does not possess a limit cycle in the deterministic limit. Pikovsky's first example, a spiral focus with additive noisė…”

“…(1) Our definition of the asymptotic phase [2] applies to a broad class of stochastic oscillations, including, but not limited to, the important example of limit cycles perturbed by noise. In [2] we considered, for instance, a heteroclinic 2d system that does not possess a limit cycle in the deterministic limit.…”