Microscopic approach of a time elapsed neural model

Abstract

The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present article is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort. IntroductionIn Neuroscience, the action potentials (spikes) are the main components of the realtime information processing in the brain. Indeed, thanks to the synaptic integration, the membrane voltage of a neuron depends on the action potentials emitted by some others, whereas if this membrane potential is sufficiently high, there is production of action potentials. To access those phenomena, schematically, one can proceed in two ways: extracellularly record in vivo several neurons, at a same time, and have access to simultaneous spike trains (only the list of events corresponding to action potentials) or intracellularly record the whole membrane voltage of only one neuron at a time, being blind to the nearby neurons.Many people focus on spike trains. Those data are fundamentally random and can be modelled easily by time point processes, i.e. random countable sets of points on R + . Several point processes models have been investigated in the literature, each of them reproducing different features of the neuronal reality. The easiest model is the homogeneous Poisson process, which can only reproduce a constant firing rate for the neuron, but which, in particular, fails to reproduce refractory periods a . It is commonly admitted that this model is too poor to be realistic. Indeed, in such a model, two points or spikes can be arbitrary close as soon as their overall frequency is respected in average. Another more realistic model is the renewal process 37 , where the occurrence of a point or spike depends on the previous occurrence. More precisely, the distribution of delays between spikes (also called inter-spike intervals, ISI) is given and a distribution, which provides small weights to small delays, is able to mimic refractory periods. A deeper statistical analysis has shown that Wold processes is showing good results, with respect to goodness-of-fit test on real data sets 38 . Wold processes are point processes for which the next occurrence of a spike depends on the previous occurrence but also on the previous ISI. From another point of view, the fact that spike trains are usually non stationary can be easily modelled by inhomogeneous Poisson processes 43 . All those models do not reflect one of the main features of spike trains, which is the synaptic integration and there has been various attempts to catch such phenomenon. One of the main model is the Hawkes model, which has been introduced in 13 and which has been recently shown to fit several stationary data ...

Fluctuations for spatially extended Hawkes processes

“…The literature of neuronal modelling via Hawkes processes is vast. To cite just a few articles, see for instance [6,8,10,14,18,21,24,30,33] and the references therein.…”

Long time behavior of a mean-field model of interacting neurons

“…They derived a Volterra integral equation and used it to obtain the stability criteria. More recently, [24,25,23] have re-explored these models for neuroscience applications (see [8,7] for a rigorous derivation of some of these PDEs using Hawkes processes). PDE (3) differs from theirs in the sense that we have a non-linear transport term (theirs is constant and equal to one) and our boundary condition is more complex.…”

“…Our main tool is this Volterra equation: we use it with a Picard iteration scheme to "recover" the non-linear equation (2). The McKean-Vlasov equation (2), its "linearized" non-homogeneous version (5), the Fokker-Planck PDE (3) and the Volterra equation (8) are different ways to investigate this mean-field problem, each of these interpretations having their own strength and weakness. Here, we use mainly the Volterra equation (8) and the non-homogeneous SDE (5).…”

“…The McKean-Vlasov equation (2), its "linearized" non-homogeneous version (5), the Fokker-Planck PDE (3) and the Volterra equation (8) are different ways to investigate this mean-field problem, each of these interpretations having their own strength and weakness. Here, we use mainly the Volterra equation (8) and the non-homogeneous SDE (5). To prove that equation (2) has a path-wise unique solution, we rely on the Volterra equation (8) and show that the following mapping:…”

Spiking neurons : interacting Hawkes processes, mean field limits and oscillations

“…Hawkes processes provide good models of this synaptic integration phenomenon by the structure of their intensity processes, see (1) below. We refer to Chevallier et al (2015) [7], Chornoboy et al (1988) [8], Hansen et al (2015) [24] and to Reynaud-Bouret et al (2014) [35] for the use of Hawkes processes in neuronal modeling. For an overview of point processes used as stochastic models for interacting neurons both in discrete and in continuous time and related issues, see also Galves and Löcherbach (2016) [22].…”