Analysis of inverse stochastic resonance and the long-term firing of Hodgkin-Huxley neurons with Gaussian white noise

Abstract

In previous articles we have investigated the firing properties of the standard Hodgkin-Huxley (HH) systems of ordinary and partial differential equations in response to input currents composed of a drift (mean) and additive Gaussian white noise. For certain values of the mean current, as the noise amplitude increased from zero, the firing rate exhibited a minimum and this phenomenon was called inverse stochastic resonance (ISR). Here we analyse the underlying transitions from a stable equilibrium point to the limit cycle and vice-versa. Focusing on the case of a mean input current density $\mu =6.8$ at which repetitive firing occurs and ISR had been found to be pronounced, some of the 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 based on the correlations between the pairs of HH variables and the small magnitudes of the fluctuations in two of them. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically and also the nature of distribution of spikes at very small noise corresponding to trajectories which do not ever enter the basin of attraction of the equilibrium point. Long term trials of duration 500000 ms are carried out for values of the noise parameter $\sigma$ from 0 to 2.0, with results appearing in Section 3. 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.

27 pages, 16 figures

References in corpus (3)

• The what and where of adding channel noise to the Hodgkin-Huxley equations
• On stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons
• The emergence of the deterministic Hodgkin--Huxley equations as a limit from the underlying stochastic ion-channel mechanism

Cited by in corpus (3)

• Channel noise induced stochastic facilitation in an auditory brainstem neuron model
• Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation
• Firing statistics in the bistable regime of neurons with homoclinic spike generation