Geodesic
Network of interacting neurons with random synaptic weights
Paolo Grazieschi1 ; Marta Leocata2 ; Cyrille Mascart3 ; Julien Chevallier4 ; François Delarue5 ; Etienne Tanré6
1Mathematics Institute, University of Warwick, Coventry, United Kingdom
2Department of Mathematics, University of Pisa, Pisa, Italy
3Université Côte d’Azur, CNRS, I3S, France
4Laboratoire Jean Kuntzmann, Université Grenoble Alpes (UFR IM2AG), Grenoble, France
5Université Côte d’Azur, CNRS, LJAD, France
6Université Côte d’Azur, Inria, France.
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.
DOI : 10.1051/proc/201965445

Paolo Grazieschi  1   ; Marta Leocata  2   ; Cyrille Mascart  3   ; Julien Chevallier  4   ; François Delarue  5   ; Etienne Tanré  6

1 Mathematics Institute, University of Warwick, Coventry, United Kingdom
2 Department of Mathematics, University of Pisa, Pisa, Italy
3 Université Côte d’Azur, CNRS, I3S, France
4 Laboratoire Jean Kuntzmann, Université Grenoble Alpes (UFR IM2AG), Grenoble, France
5 Université Côte d’Azur, CNRS, LJAD, France
6 Université Côte d’Azur, Inria, France. 
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     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},
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     doi = {10.1051/proc/201965445},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201965445/}
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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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