MiS Preprint Repository

In order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels $\mu$ as noise level $\sigma$ increases (inverse stochastic resonance, ISR) in Hodgkin-Huxley (HH) systems, we analyse the underlying transitions from a stable equilibrium point to limit cycle and vice-versa.

For a value of $\mu$ at which ISR is pronounced, properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically. Long term trials of duration 500000 ms are carried out for values of $\sigma$ from 0 to 2.0.

The graph of mean spike count versus $\sigma$ is divided into 4 regions $R_1,...,R_4,$ where $R_3$ contains the minimum associated with ISR. In $R_1$ transitions to the basin of attraction of the rest point are not observed until a small critical value of $\sigma ={\sigma }_{c_1}$ is reached, at the beginning of $R_2$. The sudden decline in firing rate when $\sigma$ is just greater than ${\sigma }_{c_1}$ implies that there is only a small range of noise levels $0<\sigma <{\sigma }_{c_1}$ where repetitive spiking is safe from annihilation by noise. The firing rate remains small throughout $R_3$. At a larger critical value $\sigma ={\sigma }_{c_2}$ which signals the beginning of $R_4$, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing $\sigma$. The quantitative scheme underlying the ISR curve is outlined in terms of the properties of exit time random variables.

In the final subsection, several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval.