Related papers: Mean field limits for nonlinear spatially extended…
Random fields are useful mathematical tools for representing natural phenomena with complex dependence structures in space and/or time. In particular, the Gaussian random field is commonly used due to its attractive properties and…
We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the…
Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the…
Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other…
We use mean field theory to study the response properties of a simple randomly-connected model cortical network of leaky integrate-and-fire neurons with balanced excitation and inhibition. The formulation permits arbitrary temporal…
We consider optimizing two-layer neural networks in the mean-field regime where the learning dynamics of network weights can be approximated by the evolution in the space of probability measures over the weight parameters associated with…
In this article, we fill a gap in the literature on Hawkes processes. In particular, we derive a CLT for a non linear compound marked Hawkes process. We also provide an upper bound on the convergence rate using the functional 1-Wasserstein…
In this paper, we develop an efficient nonparametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their…
Large-scale systems with inherent heterogeneity often exhibit complex dynamics that are crucial for their functional properties. However, understanding how such heterogeneity shapes these dynamics remains a significant challenge,…
Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit time-delays, modelled by $N\gg 1$ stochastic delay-differential equations is derived. The resulting…
The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes n goes to +$\infty$) being granted by the study performed in (Chevallier, 2015), the aim of…
We present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn. The theory is complemented by a description of a numerical procedure for solving the mean-field equations quantitatively. With our…
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…
Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent…
This paper is devoted to establishing the full scaling limit theorems for multivariate Hawkes processes. Under some mild conditions on the exciting kernels, we develop a new way to prove that after a suitable time-spatial scaling, the…
Neural field models are commonly used to describe wave propagation and bump attractors at a tissue level in the brain. Although motivated by biology, these models are phenomenological in nature. They are built on the assumption that the…
Many self-exciting systems change because endogenous amplification, as opposed to exogenous forcing, varies. We study a Hawkes process with fixed background rate and kernel, but piecewise time-varying productivity. For exponential kernels…
Existing spatio-temporal Hawkes process models typically rely on either parametric or semiparametric assumptions, limiting the model's ability to capture complex endogenous and exogenous event dynamics. We propose a fully Bayesian…
In this paper we consider multivariate Hawkes processes with baseline hazard and kernel functions that depend on time. This defines a class of locally stationary processes. We discuss estimation of the time-dependent baseline hazard and…
We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit…