Dynamics of neural networks with elapsed time model and learning processes

Delphine Salort (1) , Nicolas Torres (2, 3)

1 LCQB - Biologie Computationnelle et Quantitative = Laboratory of Computational and Quantitative Biology
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 MAMBA - Modelling and Analysis for Medical and Biological Applications

We introduce and study a new model of interacting neural networks, incorporating the spatial dimension (e.g. position of neurons across the cortex) and some learning processes. The dynamic of each neural network is described via the elapsed time model, that is, the neurons are described by the elapsed time since their last discharge and the chosen learning processes are essentially inspired from the Hebbian rule. We then obtain a system of integro-differential equations, from which we analyze the convergence to stationary states by the means of entropy method and Doeblin's theory in the case of weak interconnections. We also consider the situation where neural activity is faster than the learning process and give conditions where one can approximate the dynamics by a solution with a similar profile of a steady state. For stronger interconnections, we present some numerical simulations to observe how the parameters of the system can give different behaviors and pattern formations.

Mots clés: Doeblin Theory; Weak interconnections; Neural network; Mathematical Biology; Elapsed time; Renewal equation; Connectivity kernel; Learning rule; Convergence to equilibrium; Entropy method

Domaines: Equations aux dérivées partielles [math.AP]; Neurosciences [q-bio.NC]

Delphine Salort, Nicolas Torres. Dynamics of neural networks with elapsed time model and learning processes. Acta Applicandae Mathematicae, 2020, ⟨10.1007/s10440-020-00367-1⟩. ⟨hal-02521979v2⟩