Limits and Dynamics of Randomly Connected Neuronal Networks
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Limits and Dynamics of Randomly Connected Neuronal Networks Cristobal Quiñinao · Jonathan Touboul
Received: 21 June 2013 / Accepted: 21 May 2014 © Springer Science+Business Media Dordrecht 2014
Abstract Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we analyze the mesoscopic and macroscopic limits of networks with random correlated connectivity weights and delays. We address both averaged and quenched limits, and show propagation of chaos and convergence to a complex integral McKean-Vlasov equations with distributed delays. We then instantiate a completely solvable model illustrating the role of such random architectures in the emerging macroscopic activity. We particularly focus on the role of connectivity levels in the emergence of periodic solutions. Keywords Heterogeneous neuronal networks · Mean-field limits · Delay differential equations · Bifurcations Mathematics Subject Classification 82C22 · 82C44 · 37N25 1 Introduction Neuronal networks in the cortex are composed of large structures, called cortical columns, that are in charge of collective information processing. Neurons are characterized by a non-
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C. Quiñinao ( ) · J. Touboul Mathematical Neuroscience Team, CIRB-Collège de France, 11, place Marcelin Berthelot, 75005 Paris, France e-mail: [email protected] J. Touboul e-mail: [email protected] J. Touboul INRIA Mycenae Team, Domaine de Voluceau, B.P. 105, Rocquencourt, France C. Quiñinao Laboratoire Jacques-Louis Lions, Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, 75005 Paris, France
C. Quiñinao, J. Touboul
linear activity subject to an intense noise. They interact by sending action potentials (spikes) to those neurons they are connected to. The transmission of the information takes a specific time, related to the characteristic time of the synaptic chemical machinery and to the transport of signals at finite speed through the axons (and therefore function of the anatomical distance between the cells). The macroscopic behaviors emerging from such large-scale systems provide relevant signals that are recorded by usual imaging techniques and from which physicians can infer hallmarks of function and dysfunction. Large-scale networks are therefore adequate scales to uncover the function of the cells, and as such have attracted much work in the past few years. Indeed, while properties of single cells have been well known since the seminal works of Hodgkin and Huxley [16, 17], models of macroscopic behaviors are less understood and computational studies have mainly relied on heuristic descriptions of macroscopic behaviors through firing-rate models, following the important work of Wilson and Cowan (WC) [32, 33]. In this class of models, we will make a distinction between macroscopic models in which the activi
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