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We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with…

Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural…

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To…

In a recent work, we introduced a rigorous framework to describe the mean field limit of the gradient-based learning dynamics of multilayer neural networks, based on the idea of a neuronal embedding. There we also proved a global…

Large nonlinear recurrent neural networks with random couplings generate high-dimensional, potentially chaotic activity whose structure is of interest in neuroscience and other fields. A fundamental object encoding the collective structure…

If the behavior of a system with many degrees of freedom can be captured by a small number of collective variables, then plausibly there is an underlying mean-field theory. We show that simple versions of this idea fail to describe the…

Dedicated analog neurocomputing circuits are promising for high-throughput, low power consumption applications of machine learning (ML) and for applications where implementing a digital computer is unwieldy (remote locations; small, mobile,…

Understanding neural networks is becoming increasingly important. Over the last few years different types of visualisation and explanation methods have been proposed. However, none of them explicitly considered the behaviour in the presence…

A stochastic model of excitatory and inhibitory interactions which bears universality traits is introduced and studied. The endogenous component of noise, stemming from finite size corrections, drives robust inter-nodes correlations, that…

We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the…

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then…

Complex networks, from neuronal assemblies to social systems, can exhibit abrupt, system-wide transitions without external forcing. These endogenously generated ``noise-induced transitions'' emerge from the intricate interplay between…

We study Turing bifurcations on one-dimensional random ring networks where the probability of a connection between two nodes depends on the distance between the two nodes. Our approach uses the theory of graphons to approximate the graph…

Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic…

Describing the collective dynamics of large neural populations using low-dimensional models for averaged variables has long been an attractive task in theoretical neuroscience. Recently developed reduction methods make it possible to derive…

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a highly symmetric geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical…

We consider pulse-coupled Leaky Integrate-and-Fire neural networks with randomly distributed synaptic couplings. This random dilution induces fluctuations in the evolution of the macroscopic variables and deterministic chaos at the…

Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from…

Existing analyses of neural network training often operate under the unrealistic assumption of an extremely small learning rate. This lies in stark contrast to practical wisdom and empirical studies, such as the work of J. Cohen et al.…

A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we…