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Time to first spike in stochastic Hodgkin-Huxley systems
Henry Tuckwell and Frederic Y. M. Wan
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 $\mu$ and whose variance parameter is $\sigma$. 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 $\sigma$, including the deterministic case $\sigma=0$, 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 $\sigma$ when $\mu$ is above threshold for action potentials in the absence of noise and over a considerable range of values of $\sigma$ when $\mu$ 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.