We study a stochastic system of interacting neurons and its metastable properties. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential. In between successive spike times, each neuron looses potential at exponential speed. We study this system in the supercritical regime, that is, for sufficiently high values of the synaptic weight $h.$ Under very mild conditions on the spiking rate function, is has been shown in Duarte and Ost \cite{do} that the only invariant distribution of the finite system is the trivial measure $ δ_{\bf 0}$ corresponding to extinction of the process. Under minimal conditions on the behavior of the spiking rate function in the vicinity of $0$, we prove that the extinction time arrives at exponentially late times in $ N$, and discuss the stability of the equilibrium $δ_{\bf 0}$ for the non-linear mean-field limit process depending on the parameters of the dynamics. We then specify our study to the case of saturating spiking rates and show that, under suitable conditions on the parameters of the model, 1) the non-linear mean-field limit admits a unique and globally attracting equilibrium and 2) the rescaled exit times for the mean spiking rate of a finite system from a neighbourhood of the non-linear equilibrium rate converge in law to an exponential distribution, as the system size diverges. In other words, the system exhibits a metastable behavior.
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 mea…
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 …
We consider a finite system of interacting point processes with memory of variable length modeling a finite but large network of spiking neurons with two different leakage mechanisms. Associated to each neuron there are two point processes, describing its successive spiking and leakage times. For each neuron, the rate of the spiking point process is an exponential function of its membrane potential, with the restriction that the rate takes the value 0 when the membrane potential is 0. At each spiking time, the membrane potential of the neuron resets to 0, and simultaneously, the membrane poten…
We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations.
This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of local interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear meanfield process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical …
In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting neurons indexed by a lattice $\mathbb{Z}^d$. The neurons are subject to noise, which is modelled as a correlated martingale. The probability law governing the noise is strictly stationary, and we are therefore able to find a LDP for the probability laws $Π^n$ governing the stationary empirical measure $\hatμ^n$ generated by the neurons in a cube of length $(2n+1)$. We use this LDP to determine an LDP for the neural network model. The connection weights between the neurons evolve according to a …
In this paper we present a simple microscopic stochastic model describing short term plasticity within a large homogeneous network of interacting neurons. Each neuron is represented by its membrane potential and by the residual calcium concentration within the cell at a given time. Neurons spike at a rate depending on their membrane potential. When spiking, the residual calcium concentration of the spiking neuron increases by one unit. Moreover, an additional amount of potential is given to all other neurons in the system. This amount depends linearly on the current residual calcium concentrat…
We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-Löcherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation. We consequentl…
On the long time behaviour of single stochastic Hodgkin-Huxley neurons with constant signal, and a construction of circuits of interacting neurons showing self-organized rhythmic oscillations
We consider the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) and then further studied in Erny, Löcherbach and Loukianova (2021) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic f…
We study a purely inhibitory neural network model where neurons are represented by their state of inhibition. The study we present here is partially based on the work of Cottrell \cite{Cot} and Fricker et al. \cite{FRST}. The spiking rate of a neuron depends only on its state of inhibition. When a neuron spikes, its state is replaced by a random new state, independently of anything else and the inhibition state of the other neurons increase by a positive value. Using the Perron-Frobenius theorem, we show the existence of a Lyapunov function for the process. Furthermore, we prove a local Doebli…
This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential 1/N . This mimics the effect of chemical syna…
We study the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and Löcherbach (2016) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as $N \to \infty$, to…
We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. We show that as the coupling strength increases, the number of equilibrium points of the system changes from 3^N to 3. While for weak coupling, the system behaves like an Ising model with spin-flip dynamics, for strong coupling (of the order N^2), it synchronises, in the sense that all oscillators assume almost the same position in thei…
We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the segment $[0,1]$, and the connectivity between neurons is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices through a spatial kernel. The main result of the paper concerns the longtime stability of the synaptic current of the population, as $N\to\infty$, in the subcritical regime in case the synaptic memory kernel is exponential, up to time horizons that a…
We study the long time behavior of the solution to some McKean-Vlasov stochastic differential equation (SDE) driven by a Poisson process. In neuroscience, this SDE models the asymptotic dynamic of the membrane potential of a spiking neuron in a large network. We prove that for a small enough interaction parameter, any solution converges to the unique (in this case) invariant measure. To this aim, we first obtain global bounds on the jump rate and derive a Volterra type integral equation satisfied by this rate. We then replace temporary the interaction part of the equation by a deterministic ex…
In this paper we study the hydrodynamic limit for a stochastic process describing the time evolution of the membrane potentials of a system of neurons with spatial dependency. We do not impose on the neurons mean-field type interactions. The values of the membrane potentials evolve under the effect of chemical and electrical synapses and leak currents. The system consists of $ε^{-2}$ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to a resti…
We consider a system of $N$ neurons, each spiking randomly with rate depending on its membrane potential. When a neuron spikes, its potential is reset to $0$ and all other neurons receive an additional amount $h/N$ of potential, where $ h > 0$ is some fixed parameter. In between successive spikes, each neuron's potential undergoes some leakage at constant rate $ α. $ While the propagation of chaos of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation has already been established in a series of papers, see De Masi et al. (2015) and Fournier and Löcherba…
We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N^2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurca…
In this article we study the convergence of a stochastic particle system that interacts through threshold hitting times towards a novel equation of McKean-Vlasov type. The particle system is motivated by an original model for the behavior of a network of neurons, in which a classical noisy integrate-and-fire model is coupled with a cable equation to describe the dendritic structure of each neuron.
We consider a system of $ N \in \mathbb{N} $ mean-field interacting stochastic differential equations that are driven by a single-site potential of double-well form and by Brownian noise. The strength of the noise is measured by a small parameter $ \varepsilon >0$ (which we interpret as the \emph{temperature}), and we suppose that the strength of the interaction is given by $ J>0 $. Choosing the \emph{empirical mean} ($ P:\mathbb{R}^N \rightarrow \mathbb{R} $, $ Px =1/N \sum_i x_i $) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two g…
This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations, and compares them to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models were investigated, and mechanisms leading to these discrepancies were identified. Chief among them is correlations in spiking, or partial synchronization, working in concert with "nonlinearities" in the time evolution of membrane potentials. Additionally, simple random walk mo…
Consider a large number $n$ of neurons, each being connected to approximately $N$ other ones, chosen at random. When a neuron spikes, which occurs randomly at some rate depending on its electric potential, its potential is set to a minimum value $v_{min}$, and this initiates, after a small delay, two fronts on the (linear) dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_n$. Between jumps, the …
We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate $1 $ whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all {\it presynaptic neurons} since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to $0$ and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effe…
We address the questions of identifying pairs of interacting neurons from the observation of their spiking activity. The neuronal network is modeled by a system of interacting point processes with memory of variable length. The influence of a neuron on another can be either excitatory or inhibitory. To identify the existence and the nature of an interaction we propose an algorithm based only on the observation of joint activity of the two neurons in successive time slots. This reduces the amount of computation and storage required to run the algorithm, thereby making the algorithm suitable for…
We study a system of interacting renewal processes which is a model for neuronal activity. We show that the system possesses an exponentially large number (with respect to the number of neurons in the network) of limiting configurations of the "firing neurons". These we call patterns. Furthermore, under certain conditions of symmetry we find an algorithm to control limiting patterns by means of the connection parameters.
We consider a population of $N$ interacting neurons, represented by a multivariate Hawkes process: the firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and Law of Large Numbers results as $N\to\infty$. A crucial issue will be to understand how sp…
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 neu…
We introduce a nonlinear modification of the classical Hawkes process, which allows inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons with exponential transfer functions. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally,…
We consider, as proposed and studied in Hofstad et.\ al.\ \cite{HHKR}, a class of graph-based "interacting urn"-type Polya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency' of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved/reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeatin…
We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of firing rate neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of …
