Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.AMS Subject Classification: 35K60, 82C31, 92B20.

We study the asymptotic behavior of linear evolution equations of the type ∂tg = Dg + Lg − λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg + Lg. In the case Dg = −∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg = −∂x(x g), it is known that λ = 1 and the equation is the selfsimilar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂tf = Lf .By means of entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L 2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.

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 ...

Abstract. To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13,12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.

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The Network Noisy Leaky Integrate and Fire equation is among the simplest model allowing for a self-consistent description of neural networks and gives a rule to determine the probability to find a neuron at the potential v. However, its mathematical structure is still poorly understood and, concerning its solutions, very few results are available. In the midst of them, a recent result shows blow-up in finite time for fully excitatory networks. The intuitive explanation is that each firing neuron induces a discharge of the others; thus increases the activity and consequently the discharge rate of the full network. In order to better understand the details of the phenomena and show that the equation is more complex and fruitful than expected, we analyze further the model. We extend the finite time blow-up result to the case when neurons, after firing, enter a refractory state for a given period of time. We also show that spontaneous activity may occur when, additionally, randomness is included on the firing potential VF in regimes where blow-up occurs for a fixed value of VF.

Abstract. We prove the nonlinear stability in L p , with 1 ≤ p ≤ 2, of particular steady solutions of the Vlasov-Poisson system for charged particles in the whole space R 6 . Our main tool is a functional associated to the relative entropy or Casimir-energy functional.

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