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 Lognormal Á Inverse Gaussian and BirnbaumSaunders distributions Á Integrate-and-fire model Á Interspike distribution In a recent communication, Kish et al. (2015) highlight lognormal-like features in the inter-spike distribution of a single neuron under a simple integrate-and-fire scheme with an additive noise with no long-tail but exponential cutoff. To be more precise, let V(k) be the membrane potential of a typical neuron at discrete time k 2 N. Its dynamics is given by:with initial condition Vð0Þ ¼ V 0 . The injected current is described through the process fnðkÞg k2N , which represents a white noise process (typically normally or uniformly distributed), and the parameters d and D, which represent the drift velocity and the diffusion coefficient, respectively. Here, the condition V 0 VðkÞ is enforced during the whole motion (i.e., V 0 is a reflecting boundary). The process, then, stops at time j ¼ inffk : VðkÞ [ V th g, where V th represents a constant (membrane potential) threshold at which the neuron spikes. Once it occurs, the process resets at V 0 . Although this model is too simple to capture a great part of reality, it is still a reasonable approximation when the input rate is high in comparison with the membrane capacitance. A lognormal-like feature in the inter-spike distribution appears whenActually, this is not a theoretical result but a graphical similarity. Indeed, if we assume that fnðkÞg k2N in (1) is a standard normal process, when no condition on the motion is imposed, it is well-known that the resulting inter-spike times are inverse Gaussian distributed (which is already mentioned in Kish et al. 2015). The fact that the inverse Gaussian and the lognormal distributions are graphically similar under some specific values of its parameters is just an akin of the pictures of its probability density functions.In Takagi et al. (1997)

Fatigue-life distributions for reaction time data

“…(1) and (2) are also used to describe simple stochastic versions of the Lapicque's 1907 integrate-and-fire model (Abbott 1999), although h could be included in the modeling by playing the role of the ''refractory period''. and Tejo and Niklitschek-Soto (2016) proposed a fatigue-life family of distributions for inter-spike interval (ISI) data under this kind of processes, where such ISIs are mathematically defined in a similar way to the RTs. This fatiguelife family contains the inverse Gaussian (or Wald) and the Birnbaum-Saunders (BS) distributions.…”