Geodesic
Network of interacting neurons with random synaptic weights
Paolo Grazieschi1 ; Marta Leocata2 ; Cyrille Mascart3 ; Julien Chevallier4 ; François Delarue5 ; Etienne Tanré6
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.
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},
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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. http://geodesic.mathdoc.fr/articles/10.1051/proc/201965445/

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