Fluctuations for mean field limits of interacting systems of spiking neurons

Multiple Phase Transitions for an Infinite System of Spiking Neurons

We consider a stochastic model describing the spiking activity of a countable set of neurons spatially organized into a homogeneous tree of degree $d$, $d \geq 2$; the degree of a neuron is just the number of connections it has. Roughly, the model is as follows. Each neuron is represented by its membrane potential, which takes non-negative integer values. Neurons spike at Poisson rate 1, provided they have strictly positive membrane potential. When a spike occurs, the potential of the spiking neuron changes to 0, and all neurons connected to it receive a positive amount of potential. Moreover,…

Phase transition for infinite systems of spiking neurons

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…

Metastability in a Stochastic System of Spiking Neurons with Leakage

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…

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 mea…

具有超越一致可和性权重的无限脉冲神经元交互系统

Interacting systems of infinite spiking neurons with weights beyond uniform summability

本文研究了具有漂移及兴奋和抑制性连接的无限脉冲神经元系统，探讨了非爆炸性和不变测度唯一性的条件，并将 Galves-Löcherbach 模型扩展至超越一致可和性权重的限制。

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.