Network of interacting neurons with random synaptic weights
ESAIM. Proceedings, Tome 65 (2019), pp. 445-475
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Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.
Affiliations des auteurs :
Paolo Grazieschi 1 ; Marta Leocata 2 ; Cyrille Mascart 3 ; Julien Chevallier 4 ; François Delarue 5 ; Etienne Tanré 6
@article{EP_2019_65_a19,
author = {Paolo Grazieschi and Marta Leocata and Cyrille Mascart and Julien Chevallier and Fran\c{c}ois Delarue and Etienne Tanr\'e},
title = {Network of interacting neurons with random synaptic weights},
journal = {ESAIM. Proceedings},
pages = {445--475},
year = {2019},
volume = {65},
doi = {10.1051/proc/201965445},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201965445/}
}
TY - JOUR AU - Paolo Grazieschi AU - Marta Leocata AU - Cyrille Mascart AU - Julien Chevallier AU - François Delarue AU - Etienne Tanré TI - Network of interacting neurons with random synaptic weights JO - ESAIM. Proceedings PY - 2019 SP - 445 EP - 475 VL - 65 UR - http://geodesic.mathdoc.fr/articles/10.1051/proc/201965445/ DO - 10.1051/proc/201965445 LA - en ID - EP_2019_65_a19 ER -
%0 Journal Article %A Paolo Grazieschi %A Marta Leocata %A Cyrille Mascart %A Julien Chevallier %A François Delarue %A Etienne Tanré %T Network of interacting neurons with random synaptic weights %J ESAIM. Proceedings %D 2019 %P 445-475 %V 65 %U http://geodesic.mathdoc.fr/articles/10.1051/proc/201965445/ %R 10.1051/proc/201965445 %G en %F EP_2019_65_a19
Paolo Grazieschi; Marta Leocata; Cyrille Mascart; Julien Chevallier; François Delarue; Etienne Tanré. Network of interacting neurons with random synaptic weights. ESAIM. Proceedings, Tome 65 (2019), pp. 445-475. doi: 10.1051/proc/201965445
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