Statistical Analysis of a Randomly Stimulated Neuron

We consider the dynamics of a one-dimensional neuron model under a constant stimulation at random time intervals. The timings of the stimulation are a Markov process, depending on the timing of the previous stimulation. We present a new method of calculating the natural stationary distribution of the internal states, and use this directly to calculate the Lyapunov exponent of the neuron model. Our method completely avoids having to produce long random orbits for the calculations, with the Lyapunov exponents calculated directly using an integral (a space average). We argue that random iteration is an inefficient and inaccurate way of performing statistical analyses. KEYWORDS: stationary distribution, Lyapunov exponent, invariant probability measure, space average 1. Introduction and Motivation We begin with a specific example that will be the main focus of this paper, and in the next section show how this example naturally fits into the general theory of a new method to assist statis..