Time to first spike in stochastic Hodgkin-Huxley systems

Henry Tuckwell and Frederic Y. M. Wan

Contact the author: Please use for correspondence this email.
Submission date: 02. Dec. 2004
published in: Physica / A, 351 (2005) 2/4, p. 427-438 
DOI number (of the published article): 10.1016/j.physa.2004.11.059
Bibtex
with the following different title: Time to first spike in Hodgkin-Huxley stochastic systems
Keywords and phrases: spiking neuron, stochastic model

Abstract:
The time to first spike is an experimentally observed quantity in laboratory experiments. In the auditory, somatic and visual sensory modalities, the times of first spikes in the corresponding cortical neurons have been implicated as coding much of the information about stimulus properties. We describe an analytical approach for determining the time to first spike from a given initial state which may be applied to a general nonlinear stochastic model neuron. We illustrate with a standard Hodgkin-Huxley model with a Gaussian white noise input current whose drift parameter is and whose variance parameter is . Partial differential equations (PDE's) of second order are obtained for the first two moments of the time taken for the depolarization to reach a threshold value from rest state, as functions of the initial values. Simulation confirms that for small noise amplitudes a 2-component model is reasonably accurate. For small values of the noise parameter , including the deterministic case , perturbation methods are used to find the moments of the firing time and the results compare favorably with those from simulation. The approach is accurate for almost all when is above threshold for action potentials in the absence of noise and over a considerable range of values of when is as small as 2. The same methods may be applied to models similar to Hodgkin-Huxley which involve channels for additional or different ionic currents.