A variational method for analyzing limit cycle oscillations in stochastic hybrid systems

A universal description of stochastic oscillators

Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = μ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width μ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

Determinants of Brain Rhythm Burst Statistics

Brain rhythms recorded in vivo, such as gamma oscillations, are notoriously variable both in amplitude and frequency. They are characterized by transient epochs of higher amplitude known as bursts. It has been suggested that, despite their short-life and random occurrence, bursts in gamma and other rhythms can efficiently contribute to working memory or communication tasks. Abnormalities in bursts have also been associated with e.g. motor and psychiatric disorders. It is thus crucial to understand how single cell and connectivity parameters influence burst statistics and the corresponding brain states. To address this problem, we consider a generic stochastic recurrent network of Pyramidal Interneuron Network Gamma (PING) type. Using the stochastic averaging method, we derive dynamics for the phase and envelope of the amplitude process, and find that they depend on only two meta-parameters that combine all the model parameters. This allows us to identify an optimal parameter regime of healthy variability with similar statistics to those seen in vivo; in this regime, oscillations and bursts are supported by synaptic noise. The probability density for the rhythm’s envelope as well as the mean burst duration are then derived using first passage time analysis. Our analysis enables us to link burst attributes, such as duration and frequency content, to system parameters. Our general approach can be extended to different frequency bands, network topologies and extra populations. It provides the much needed insight into the biophysical determinants of rhythm burst statistics, and into what needs to be changed to correct rhythms with pathological statistics.

Synchronization of stochastic hybrid oscillators driven by a common switching environment

Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate.

Stochastic Hybrid Systems in Cellular Neuroscience

We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.