arXiv subjects

Morgan André

Publications and source records attributed to Morgan André.

7 recordsLinked to original sources

arXiv2026

The Pathwise Approach to Metastability and its Applications to Galves--Löcherbach Models

Metastability is the tendency of a system to dwell for a very long time near an apparently stable equilibrium before a rare fluctuation drives it, on a comparatively short time scale, towards another. Among the rigorous frameworks developed to capture this phenomenon, the pathwise approach proceeds by identifying the ``typical'' trajectories of the stochastic dynamics at hand and estimating their probabilities. In this article we review the pathwise approach and its application to the Galves--Löcherbach (GL) class of stochastic models of spiking neural networks. After recalling the conceptual and historical roots of the theory -- which goes from chemistry to rigorous probability theory, with fundamental ideas coming mainly from statistical physics -- and illustrating them on two classical examples, we give a general definition encompassing the known variants of the GL model and survey the metastability results already established for some of these variants. As far as we can, we do so in a self-contained fashion, and we sketch the proofs when possible, highlighting their common structure. We close with a discussion on open problems and point to possible further directions.

math.PR↗

arXiv2026

Estimates on Escape Times for the Elephant Random Walk

We study the gambler's ruin problem for the Elephant Random Walk, focusing on escape time from a symmetric interval of the form $\{-N, \ldots, N\}$. As our main result, we derive tight exponential bounds for the tail of this escape time. We then illustrate the usefulness of such bounds by proving that, in the diffusive regime, the Elephant's average behavior mirrors that of the traditional symmetric random walk: the expected escape time grows quadratically with $N$.

math.PR↗

arXiv2024

A Quasi-Stationary Approach to Metastability in a System of Spiking Neurons with Synaptic Plasticity

After reviewing the behavioral studies of working memory and of the cellular substrate of the latter, we argue that metastable states constitute candidates for the type of transient information storage required by working memory. We then present a simple neural network model made of stochastic units whose synapses exhibit short-term facilitation. The Markov process dynamics of this model was specifically designed to be analytically tractable, simple to simulate numerically and to exhibit a quasi-stationary distribution (QSD). Since the state space is finite this QSD is also a Yaglom limit, which allows us to bridge the gap between quasi-stationarity and metastability by considering the relative orders of magnitude of the relaxation and absorption times. We present first analytical results: characterization of the absorbing region of the Markov process, irreducibility outside this absorbing region and consequently existence and uniqueness of a QSD. We then apply Perron-Frobenius spectral analysis to obtain any specific QSD, and design an approximate method for the first moments of this QSD when the exact method is intractable. Finally we use these methods to study the relaxation time toward the QSD and establish numerically the memorylessness of the time of extinction.

q-bio.NC↗

arXiv2022

Convergence of the Temporal Averages of a Metastable System of Spiking Neurons

We consider a stochastic system of spiking neurons which was previously proven to present a metastable behavior for a suitable choice of the parameter, in the sense that the time of extinction is asymptotically memory-less when the number of components in the system goes to $\infty$. In the present article we complete this work by showing that, previous to extinction, the system tends to stabilize in the sense that temporal means taken on an appropriate time scale converge in probability to some fixed value. This property is sometime called thermalization.

math.PR↗

arXiv2020

The Effect of Graph Connecitivity on Metastability on a Stochastic System of Spiking Neurons

We consider a continuous-time stochastic model of spiking neurons. In this model, we have a finite or countable number of neurons which are vertices in some graph $G$ where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction $G$ that is chosen. This model depends on a leakage parameter $γ$, and it was previously proven that when the graph $G$ is the infinite one-dimensional lattice, this model presents a phase transition with respect to $γ$. It was also proven that, when $γ$ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result doesn't hold if $γ$ is not small anymore, in fact we prove that for $γ> 1$ the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if $G$ is the complete graph, the result of metastability holds for any positive $γ$.

math.PR↗

arXiv2020

A Result of Metastability for an Infinite System of Spiking Neurons

In 2018, Ferrari et al. wrote a paper called "Phase Transition for Infinite Systems of Spiking Neurons" in which they introduced a continuous time stochastic model of interacting neurons. This model has a parameter $γ$, corresponding to the rate of the leaking times of the neurons and, as the title says, it was proven there to present a phase transition phenomenon with respect to this $γ$. Here we prove that this model also exhibit a metastable behavior. By this we mean that if $γ$ is small enough, then the re-normalized time of extinction converges toward an exponential random variable of mean 1 as the number of neurons goes to infinity.

math.PR↗

arXiv2019

A Numerical Study of the Time of Extinction in a Class of Systems of Spiking Neurons

In this paper we present a numerical study of a mathematical model of spiking neurons introduced by Ferrari et al. in an article entitled Phase transition forinfinite systems of spiking neurons. In this model we have a countable number of neurons linked together in a network, each of them having a membrane potential taking value in the integers, and each of them spiking over time at a rate which depends on the membrane potential through some rate function $ϕ$. Beside being affected by a spike each neuron can also be affected by leaking. At each of these leak times, which occurs for a given neuron at a fixed rate $γ$, the membrane potential of the neuron concerned is spontaneously reset to $0$. A wide variety of versions of this model can be considered by choosing different graph structures for the network and different activation functions. It was rigorously shown that when the graph structure of the network is the one-dimensional lattice with a hard threshold for the activation function, this model presents a phase transition with respect to $γ$, and that it also presents a metastable behavior. By the latter we mean that in the sub-critical regime the re-normalized time of extinction converges to an exponential random variable of mean 1. It has also been proven that in the super-critical regime the renormalized time of extinction converges in probability to 1. Here, we investigate numerically a richer class of graph structures and activation functions. Namely we investigate the case of the two dimensional and the three dimensional lattices, as well as the case of a linear function and a sigmoid function for the activation function. We present numerical evidence that the result of metastability in the sub-critical regime holds for these graphs and activation functions as well as the convergence in probability to $1$ in the super-critical regime.

cs.NE↗