A mean-field model with discontinuous coefficients for neurons with spatial interaction

Franco Flandoli; Enrico Priola; Giovanni Zanco

doi:10.3934/dcds.2019126

Article Contents

2019, Volume 39, Issue 6: 3037-3067. Doi: 10.3934/dcds.2019126

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A mean-field model with discontinuous coefficients for neurons with spatial interaction

1.
Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy
2.
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Torino, Italy
3.
Institute of Science and Technology Austria (IST Austria), Am Campus 1, Klosterneuburg, Austria
4.
LUISS Università Guido Carli, Viale Romania 32, Roma, Italy

^* Corresponding author: Giovanni Zanco
^* Corresponding author: Giovanni Zanco

Received: January 31, 2018

Published: February 2019

The second author has been partially supported by INdAM through the GNAMPA Research Project (2017) "Sistemi stocastici singolari: buona posizione e problemi di controllo". The third author was partly funded by the Austrian Science Fund (FWF) project F 65

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Abstract

Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.

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Mathematics Subject Classification: Primary: 60H10, 35Q84; Secondary: 92C20, 62M45.

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Figure 1. Simulation of the system (2) for $ 3 $ neurons with no spatial interaction

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Figure 2. Solution of system (2) for 10 neurons with strong interaction uniform in space. A spike from one neuron propagates to all other neurons in the network

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Figure 3. A network made of two subnetworks with localized strong interaction. Each blue line corresponds to the presence of interaction; only one neuron of the first subnetwork interacts with a single neuron of the second one. The vertical axis represents time; the network is drawn on the $ t = 0 $ plane and to each spike of a neuron corresponds a sphere above it, with the same color as the neuron. Near time $ t = 1 $ many neurons in the first network spike and the signal is propagated to the second network, while at time $ t = 2 $ the signal does not propagate to the second network.

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