The network of noisy leaky integrate and fire (NNLIF) model is one of the simplest self-contained mean-field models considered to describe the behavior of neural networks. Even so, in studying its mathematical properties some simplifications are required [Cáceres and Perthame, J. Theor. Biol. 350 (2014) 81–89; Cáceres and Schneider, Kinet. Relat. Model. 10 (2017) 587–612; Cáceres, Carrillo and Perthame, J. Math. Neurosci. 1 (2011) 7] which disregard crucial phenomena. In this work we deal with the general NNLIF model without simplifications. It involves a network with two populations (excitatory and inhibitory), with transmission delays between the neurons and where the neurons remain in a refractory state for a certain time. In this paper we study the number of steady states in terms of the model parameters, the long time behaviour via the entropy method and Poincaré’s inequality, blow-up phenomena, and the importance of transmission delays between excitatory neurons to prevent blow-up and to give rise to synchronous solutions. Besides analytical results, we present a numerical solver, based on high order flux-splitting WENO schemes and an explicit third order TVD Runge-Kutta method, in order to describe the wide range of phenomena exhibited by the network: blow-up, asynchronous/synchronous solutions and instability/stability of the steady states. The solver also allows us to observe the time evolution of the firing rates, refractory states and the probability distributions of the excitatory and inhibitory populations.
Mots-clés : Neural networks, Leaky integrate and fire models, noise, blow-up, steady states, entropy, long time behavior, refractory states, transmission delay
@article{M2AN_2018__52_5_1733_0,
author = {C\'aceres, Mar{\'\i}a J. and Schneider, Ricarda},
title = {Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1733--1761},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018014},
zbl = {1411.35178},
mrnumber = {3878608},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2018014/}
}
TY - JOUR AU - Cáceres, María J. AU - Schneider, Ricarda TI - Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1733 EP - 1761 VL - 52 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018014/ DO - 10.1051/m2an/2018014 LA - en ID - M2AN_2018__52_5_1733_0 ER -
%0 Journal Article %A Cáceres, María J. %A Schneider, Ricarda %T Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1733-1761 %V 52 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018014/ %R 10.1051/m2an/2018014 %G en %F M2AN_2018__52_5_1733_0
Cáceres, María J.; Schneider, Ricarda. Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1733-1761. doi : 10.1051/m2an/2018014. http://archive.numdam.org/articles/10.1051/m2an/2018014/
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