Partial differential equations
Dynamics of time elapsed inhomogeneous neuron network model
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1111-1115.

Models for neural networks have been proposed, which describe the probability to find a neuron for which time s has elapsed since the last discharge. These are written under the form of a nonlinear age-structured equation where the total network activity modulates the firing rate. Here, we consider an inhomogeneous network with variability on the refractory period. We give conditions on the connectivity, leading to total desynchronization of the network.

Pour décrire l'activité de réseaux de neurones, des modèles qui représentent la probabilité qu'un neurone ait passé le temps s depuis sa dernière décharge ont été proposés. Ce sont des équations structurées en âge, non linéaires, où l'activité totale du réseau contrôle le taux de décharge. Ici, nous considérons un réseau inhomogène prenant en compte la variabilité des périodes réfractaires. Nous donnons une condition sur la connectivité qui conduit à la désynchronisation totale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.029
Kang, Moon-Jin 1; Perthame, Benoît 2; Salort, Delphine 3

1 Department of Mathematics, Texas University, Austin, United States
2 Sorbonne Universités, UPMC (Université Paris-6), UMR 7598, CNRS, INRIA, Laboratoire Jacques-Louis-Lions, France
3 Sorbone Universités, UPMC (Université Paris-6), UMR 7238, CNRS, Laboratoire de biologie computationnelle et quantitative, France
@article{CRMATH_2015__353_12_1111_0,
     author = {Kang, Moon-Jin and Perthame, Beno{\^\i}t and Salort, Delphine},
     title = {Dynamics of time elapsed inhomogeneous neuron network model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1111--1115},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.029},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.029/}
}
TY  - JOUR
AU  - Kang, Moon-Jin
AU  - Perthame, Benoît
AU  - Salort, Delphine
TI  - Dynamics of time elapsed inhomogeneous neuron network model
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1111
EP  - 1115
VL  - 353
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.09.029/
DO  - 10.1016/j.crma.2015.09.029
LA  - en
ID  - CRMATH_2015__353_12_1111_0
ER  - 
%0 Journal Article
%A Kang, Moon-Jin
%A Perthame, Benoît
%A Salort, Delphine
%T Dynamics of time elapsed inhomogeneous neuron network model
%J Comptes Rendus. Mathématique
%D 2015
%P 1111-1115
%V 353
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.09.029/
%R 10.1016/j.crma.2015.09.029
%G en
%F CRMATH_2015__353_12_1111_0
Kang, Moon-Jin; Perthame, Benoît; Salort, Delphine. Dynamics of time elapsed inhomogeneous neuron network model. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1111-1115. doi : 10.1016/j.crma.2015.09.029. http://www.numdam.org/articles/10.1016/j.crma.2015.09.029/

[1] Bertini, L.; Giacomin, G.; Poquet, C. Synchronization and random long time dynamics for mean-field plane rotators, Probab. Theory Relat. Fields, Volume 160 (2014), pp. 593-653

[2] Cáceres, M.J.; Carrillo, J.A.; Perthame, B. Analysis of nonlinear noisy integrate&fire neuron models: blow-up and steady states, J. Math. Neurosci., Volume 1 (2011) no. 7 (33 pp.)

[3] Cáceres, M.J.; Carrillo, J.A.; Tao, L. A numerical solver for a nonlinear Fokker–Planck equation representation of network dynamics, J. Comput. Phys., Volume 230 (2011) no. 4, pp. 1084-1099

[4] Cai, D.; Tao, L.; Shelley, M.; McLaughlin, D.W. An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex, Proc. Natl. Acad. Sci., Volume 101 (2004) no. 20, pp. 7757-7762

[5] Carrillo, J.A.; Choi, Y.-P.; Ha, S.-Y.; Kang, M.-J.; Kim, Y. Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., Volume 156 (2014), pp. 395-415

[6] Chevallier, J.; Cáceres, M.-J.; Doumic, M.; Reynaud-Bouret, P. Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 14, pp. 2669-2719

[7] Dumont, G. Analyse de modèles de population de neurones: cas des neurones à réponse postsynaptique par saut de potentiel, Université de Bordeaux, France, 2012 (PhD thesis)

[8] Ly, C.; Tranchina, D. Spike train statistics and dynamics with synaptic input from any renewal process: a population density approach, Neural Comput., Volume 21 (2009), pp. 360-396

[9] Mischler, S.; Quiñinao, C.; Touboul, J. On a kinetic Fitzhugh–Nagumo model of neuronal network, Commun. Math. Phys. (2015) (in press)

[10] S. Mischler, Q. Weng, Relaxation in time elapsed neuron network models in the weak connectivity regime, Preprint, hal-01148645, 2015.

[11] Pakdaman, K.; Perthame, B.; Salort, D. Dynamics of a structured neuron population, Nonlinearity, Volume 23 (2010), pp. 55-75

[12] Pakdaman, K.; Perthame, B.; Salort, D. Relaxation and self-sustained oscillations in the time elapsed neuron network model, SIAM J. Appl. Math., Volume 73 (2013) no. 3, pp. 1260-1279

[13] Pakdaman, K.; Perthame, B.; Salort, D. Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., Volume 4 (2014)

[14] Perthame, B.; Salort, D. On a voltage-conductance kinetic system for integrate and fire neural networks, Kinet. Relat. Models, Volume 6 (2013) no. 4, pp. 841-864

[15] C. Quiñinao, A microscopic spiking neuronal network for the age structured model, Preprint, hal-01121061v3, 2015.

Cited by Sources: