Coherent neural oscillations induced by weak synaptic noise

Jürgen Jost,Marius E. Yamakou

doi:10.1007/s11071-018-4311-1

Abstract

We analyze the effect of synaptic noise on the dynamics of the FitzHugh–Nagumo (FHN) neuron model. In our deterministic parameter regime, a limit cycle solution cannot emerge through a singular Hopf bifurcation, but such a limit cycle can nevertheless arise as a stochastic effect, as a consequence of weak synaptic noise in a regime of strong timescale separation (varepsilon rightarrow 0) between the slow and fast variables of the model. We investigate the mechanism behind this phenomenon, known as self-induced stochastic resonance (SISR) (Muratov et al. in Physica D 210:227–240, 2005), by using multiple-time perturbation techniques and by analyzing the escape mechanism of the random trajectories from the stable manifolds of the model equation. Even though SISR occurs only in the limit as the singular parameter varepsilon rightarrow 0, decreasing varepsilon does not increase the coherence of the oscillations in the FHN model, but rather increases the interval of the noise amplitude sigma for which coherence occurs. This is in contrast to the dynamical system studied in Muratov et al. (2005). Moreover, the phenomenon is robust under parameter tuning. Numerical simulations exhibit the results predicted by the theoretical analysis. In fact, our analysis together with that in Yamakou and Jost (Weak noise-induced transitions with inhibition and modulation of neural oscillations, 2017) reveals that the FHN model can support different stochastic resonance phenomena. While it had been known (Pikovsky and Kurths in Phys Rev Lett 78:775–778, 1997) that coherence resonance can occur when the slow variable is subjected to noise, we show that, when noise is added to the fast variable, two other types, inverse stochastic resonance and SISR, may emerge in the same weak noise limit and that the transition between these phenomena can be induced by varying a simple parameter.

Highlights

Since electrical information in the nervous system is encoded, processed, and transmitted by trains of neuronal action potentials [4,5], a major goal in neuroscience is to understand how neurons generate action potentials both spontaneously and in response to possibly random synaptic and ion channel inputs
Our analysis together with that in Yamakou and Jost (Weak noise-induced transitions with inhibition and modulation of neural oscillations, 2017) reveals that the FHN model can support different stochastic resonance phenomena. While it had been known (Pikovsky and Kurths in Phys Rev Lett 78:775–778, 1997) that coherence resonance can occur when the slow variable is subjected to noise, we show that, when noise is added to the fast variable, two other types, inverse stochastic resonance and self-induced stochastic resonance (SISR), may emerge in the same weak noise limit and that the transition between these phenomena can be induced by varying a simple parameter
In the FHN neuron model, coherent spike trains emerge through very different mechanisms depending on whether the noise term is added to the fast variable (SISR) or to the slow variable (CR); the former is analyzed in the present paper, and for the latter, see [3,24,25,45]

Summary

Introduction

Since electrical information in the nervous system is encoded, processed, and transmitted by trains of neuronal action potentials [4,5], a major goal in neuroscience is to understand how neurons generate action potentials both spontaneously and in response to possibly random synaptic and ion channel inputs. In 1997, Pikovsky and Kurths [3] reported on another noise-induced phenomenon in an excitable system They considered the FHN neuron model with the noise term added to the slow recovery variable equation containing the Hopf bifurcation parameter. In the FHN neuron model, coherent spike trains emerge through very different mechanisms depending on whether the noise term is added to the fast variable (SISR) or to the slow variable (CR); the former is analyzed in the present paper, and for the latter, see [3,24,25,45]. We thereby produce a unified mathematical setting to analyze both ISR and SISR This allows us to understand how a neuron could effectively switch from one phenomenon to the other ( encoding different information) without changing the weak synaptic noise limit required for both phenomena to occur.

Model equation

Deterministic dynamics and its timescale on stable manifolds

Asymptotic matching of timescales and resonance

Characterization of the limit cycle

Numerical simulations and discussion

Concluding remarks