Multirhythmicity generated by slow variable diffusion in a ring of relaxation oscillators and noise-induced abnormal interspike variability.

Abstract

The deterministic and noise-dependent dynamics of a ring of three Ohmically coupled electronic relaxation oscillators are considered by means of numerical simulations. Each isolated oscillator is described by a set of two ordinary differential equations with very different characteristic times. The emergence of the limit cycle via the Hopf bifurcation results from the N-shaped current-versus-voltage characteristic of the nonlinear resistor. The phase diagram is calculated for a ring of three such oscillators in the presence of small detuning. Special attention is focused on two parameter areas, one near a transition to the homogeneous and the other near the inhomogeneous stable steady state. Along with other nontrivial limit cycles, essentially asymmetrical limit cycles termed dynamic traps may arise in these two areas. A dynamic trap is a regime in which one or two oscillators do not perform full-amplitude oscillations and, correspondingly, do not generate spikes. The interspike interval (ISI) distribution in the presence of noise is calculated as a function of the coupling strength in both areas of the parameter plane. The distributions are extremely polymodal near the homogeneous steady state even if the in-phase limit cycle is dominating. The origins of this abnormal enhancement of ISI variability are discussed in detail. A similar analysis shows that nontrivial periodic attractors are observable in the vicinity of the inhomogeneous stable steady states only if the level of noise is relatively low. In this case, the dominance of the in-phase limit cycle basin results in an almost unimodal distribution of interspike intervals.