Large assemblies of stochastic oscillators

Abstract

This paper studies the occurrence of stochastic synchronization and phase locking in large assemblies of coupled stochastic oscillators which are modelled as phaseangle stochastic processes. Attention will be restricted to the case of discrete phaseangle spaces. These are technically much simpler than processes over continuous phase spaces and yet show an interesting range of properties. Stochastic synchronization occurs if the oscillator assembly obeys a large deviation principle and if the deterministic-limit dynamics possesses a stable periodic orbit. All relevant concepts are reviewed in section section 1, 2 and the Appendix. In section 4, general necessary and sufficient conditions for stochastic synchronization are obtained. Weak and robust types of synchronization are distinguished. A describing function method is formulated which is very convenient for checking the general conditions in concrete examples. Separate treatments of feedback gain, feedback type (i.e. positive or negative) and feedback sources lead to investigations of how the occurrence of stochastic synchronization depends on any one of those feedback aspects. In a sequence of gradually more realistic models (section 3, 4b, 5a), the general results are applied to assemblies of (stochastic) fireflies. The first of these models (section 3) is mathematically instructive in that it provides a good example of how the Liapunov-function character of large deviation rate functions can be exploited to study global properties of the corresponding deterministic limit ODE. The analysis of firefly assemblies culminates in a proof of stochastic firefly synchronization. This result is the precise stochastic analogue of the well-known proof of deterministic firefly synchronization given by Mirollo & Strogatz (1990). In section 5, the general results of section 4 are applied to neuronal assemblies. Particular attention is paid to the dependence of neuronal synchronization on the feedback type, that is excitatory or inhibitory signalling. We find it natural and instructive to distinguish between external and internal variants of inhibition. The latter (in particular, in the form of graded refractoriness of neurons) is shown to be crucial for stochastic synchronization within one neural assembly, whereas the former is instrumental for phase locking between distinct assemblies. These findings are summarized and sharpened in a number of concrete predictions for real biological networks of neurons. Except for some topical aspects of large deviation theory (reviewed in the Appendix), the mathematics employed in this paper are straightforward.