The membrane potential process of a single neuron seen as a cumulative damage process

Abstract

A simple integrate-and-fire mechanism of a single neuron can be compared with a cumulative damage process, where the spiking process is analogous to rupture sequences of a material under cycles of stress. Although in some cases lognormal-like patterns can be recognized in the inter-spike times under a simple integrate-and-fire mechanism, fatigue life models as the inverse Gaussian distribution and the Birnbaum-Saunders distribution (which was recently introduced in the neural activity framework) provide theoretical arguments that make them more suitable for the modeling of the resulting inter-spike times.

Keywords: Integrate-and-fire model; Inter-spike distribution; Inverse Gaussian and Birnbaum–Saunders distributions; Lognormal.

Fig. 1

This histogram was generated by using (1) with ${\{\mathit{\xi }(k)\}}_{k\in \mathbb{N}}$ a sequence of independent and identically standard normal distributed random variables, $D=0,25$, $\mathit{\delta }=1$, $V_0=0$ and $V_{\text{th}}=20$, so that (2) holds and $V(k)>0$ occurs with high probability, for all $k\in \mathbb{N}$ ($P(\mathit{\delta }+\sqrt{D}\mathit{\xi }(1)<0)\approx 0,02$). 2000 spikes was simulated, and the log-likelihoods of the corresponding inter-spike times for each distribution were: −4484,396 for the inverse Gaussian distribution (the exact distribution), −4484.424 for the Birnbaum–Saunders distribution and −4484.738 for the lognormal distribution. As expected, they are close, although Birnbaum–Saunders distribution is closer to the exact distribution than the lognormal distribution, as was anticipated and theoretically explained in Leiva et al. (2015)