Dynamics of Neural Networks with Elapsed Time Model and Learning Processes
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Dynamics of Neural Networks with Elapsed Time Model and Learning Processes Nicolas Torres1,2
· Delphine Salort3,2
Received: 13 April 2020 / Accepted: 27 September 2020 © Springer Nature B.V. 2020
Abstract 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. Keywords Mathematical biology · Neural network · Elapsed time · Learning rule · Connectivity kernel · Weak interconnections Mathematics Subject Classification (2010) 35B40 · 35F20 · 35R09 · 92B20
1 Introduction The study and modeling of neural networks have been expanded significantly in the past years and still lead to several stimulating open problems. In the case of homogeneous networks, evolution equations describing neural assemblies derived from stochastic processes
B N. Torres
[email protected]
1
Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France
2
Present address: 4 Place Jussieu, 75005, Paris, France
3
Laboratoire de Biologie Computationnelle et Quantitative, Sorbonne Université, Paris, France
N. Torres, D. Salort
and microscopic models have become a very active area. Among them, the elapsed time model, has known a growth interest and has been studied by several authors such as Cañizo et al. in [4], Chevalier et al. in [6], Ly et al. in [16], Mischler et al. in [18] and Pakdaman et al. in [19–21]. In particular, the work of Chevalier et al. in [6] establishes a bridge between Poisson point processes that model spike trains and the time elapsed model. However, the incorporation of spatial dimension, using those homogeneous models for each unit has not been investigated much yet. Recent works of J. Crevat et al. in [7–9] consider the case with spatial dimension, where each neuron is described via a kinetic PDE derived from FitzHugh-Nagumo model. Else, the main models used for the incorporation of space variable via integro-differential equations are inspired from the Wilson-Cowan [25] and Amari [2] models, where several theoretical and numerical results has been obtained, see Faye et al. in [10–12] and Pham et al. in [24]. Here, we consider the evolution of interacting neu
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