First Passage Time Densities in Models of Resonate-and-Fire Neurons

Neurons can be considered as excitable dynamical systems, which respond upon changes of the membrane potential due to incoming current and intrinsic noise of ion channels. One believes, that information is coded in the spike times of neurons, and, thus, is contained in the statistics of interspike intervals (ISI), in particular, in the ISI probability density function (PDF). Mathematically formulated, the problem of obtaining the ISI density of a neuron is the problem of finding the probability density F(T) for the first passage time T (FPT). In neuronal modelling the membrane potential represents the stochastic process, which should exceed the excitation threshold for the first time in order to spike. Methods of finding FPT PDF F(T) are well worked out for one dimensional Markovian processes [1]. The FPT densities in this case have a very habitual form: The density exhibits a single maximum and than it decays monotonically either exponentially or as a power low. The ISI histograms obtained experimentally from output of some neurons have this form and can be reproduced by FPT densities of one dimensional diffusion process [2]. Nevertheless, the ISI histograms obtained from the output of resonant neurons (neuronal cells showing the subthreshold oscillations of the membrane potential) exhibit a sequence of decaying peaks shifted one from each other by the inverse frequency of the subthreshold oscillations. The multimodal FPT density is obtained only for stochastic dynamical systems, which have at least two dynamical variables, exhibit weakly or moderately damped or self-amplifying oscillations and after spiking put the reset to initial values which are not a fixed point (so called "Resonate-and-Fire" neurons [3]). In the present work we model excitable behavior with damped subthreshold oscillations.