no. 1
An Interacting Neuronal Network with Inhibition: theoretical analysis and perfect simulation
Goncalves, Branda P. I.1
1 Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS UMR-8089, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France
MathematicS In Action, Maths Bio, Tome 12 (2023) no. 1, pp. 3-22

We study a purely inhibitory neural network model where neurons are represented by their state of inhibition. The study we present here is partially based on the work of Cottrell [6] and Fricker et al. [8]. The spiking rate of a neuron depends only on its state of inhibition. When a neuron spikes, its state is replaced by a random new state, independently of anything else, and the inhibition states of the other neurons increase by a positive value. Using the Perron–Frobenius theorem, we show the existence of a Lyapunov function for the process. Furthermore, we prove a local Doeblin condition which implies the existence of an invariant probability measure for the process. Finally, we extend our model to the case where the neurons are indexed by ℤ. We construct a perfect simulation algorithm to show the recurrence of the process under certain conditions. To do this, we rely on the classical contour technique used in the study of contact processes, and assuming that the spiking rate takes values in the interval [β * ,β * ], we show that there is a critical threshold for the ratio δ=β * β * -β * over which the process is ergodic.

Publié le :
DOI : 10.5802/msia.29
Classification : 00X99
Keywords: Spiking rate, Interacting neurons, Perfect simulation algorithm, Classical contour technique

Goncalves, Branda P. I. 1

1 Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS UMR-8089, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Goncalves, Branda P. I. An Interacting Neuronal Network with Inhibition: theoretical analysis and perfect simulation. MathematicS In Action, Maths Bio, Tome 12 (2023) no. 1, pp. 3-22. doi: 10.5802/msia.29

[1] André, Morgan A result of metastability for an infinite system of spiking neurons, J. Stat. Phys., Volume 177 (2019) no. 5, pp. 984-1008 | Zbl | MR | DOI

[2] Athreya, Krishna B.; Ney, Peter E. Branching processes, Courier Corporation, 1972 | DOI

[3] Benaïm, Michel; Le Borgne, Stéphane; Malrieu, Florent; Zitt, Pierre-André Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 3, pp. 1040-1075 | Numdam | Zbl | DOI | MR

[4] Comets, Francis; Fernández, Roberto; Ferrari, Pablo A. Processes with long memory: regenerative construction and perfect simulation, Ann. Appl. Probab., Volume 12 (2002) no. 3, pp. 921-943 | Zbl | MR

[5] Cormier, Quentin; Tanré, Etienne; Veltz, Romain Long time behavior of a mean-field model of interacting neurons, Stochastic Processes Appl., Volume 130 (2020) no. 5, pp. 2553-2595 | Zbl | MR | DOI

[6] Cottrell, Marie Mathematical analysis of a neural network with inhibitory coupling, Stochastic Processes Appl., Volume 40 (1992) no. 1, pp. 103-126 | Zbl | MR | DOI

[7] Ferrari, Pablo A.; Galves, Antonio; Grigorescu, Ilie; Löcherbach, Eva Phase transition for infinite systems of spiking neurons, J. Stat. Phys., Volume 172 (2018) no. 6, pp. 1564-1575 | Zbl | MR | DOI

[8] Fricker, Christine; Robert, Philippe; Saada, Ellen; Tibi, Danielle Analysis of a network model, Cellular Automata and Cooperative Systems, Springer, 1993, pp. 159-172 | Zbl | DOI

[9] Galves, Antonio; Garcia, Nancy Lopes; Löcherbach, Eva; Orlandi, Enza Kalikow-type decomposition for multicolor infinite range particle systems, Ann. Appl. Probab., Volume 23 (2013) no. 4, pp. 1629-1659 | Zbl | MR

[10] Galves, Antonio; Löcherbach, Eva Infinite systems of interacting chains with memory of variable length—a stochastic model for biological neural nets, J. Stat. Phys., Volume 151 (2013) no. 5, pp. 896-921 | Zbl | MR | DOI

[11] Galves, Antonio; Löcherbach, Eva; Orlandi, Enza Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations, J. Stat. Phys., Volume 138 (2010) no. 1, pp. 476-495 | Zbl | MR | DOI

[12] Goncalves, Branda; Huillet, Thierry; Löcherbach, Eva On decay-surge population models (2020) (https://arxiv.org/abs/2012.00716)

[13] Griffeath, David The basic contact processes, Stochastic Processes Appl., Volume 11 (1981) no. 2, pp. 151-185 | Zbl | MR | DOI

[14] Meyn, Sean P; Tweedie, Richard L Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes, Adv. Appl. Probab., Volume 25 (1993) no. 3, pp. 518-548 | Zbl | MR | DOI

[15] Robert, Philippe; Touboul, Jonathan On the dynamics of random neuronal networks, J. Stat. Phys., Volume 165 (2016) no. 3, pp. 545-584 | Zbl | MR | DOI

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