Multiscale modeling, stochastic and asymptotic approaches for analyzing neural networks based on synaptic dynamics

Ecole Normale Supérieure, Applied Mathematics and Computational Biology, IBENS/CNRS/INSERM, 75005 Paris, France

Abstract

How do neurons coordinate in complex networks to achieve higher brain functions? Answering this question has relied on experimental approaches based on functional imaging, electrophysiology and microscopy imaging, but surprisingly, what is now really missing in order to make sense of large data are analytical methods, multiscale modeling, simulations and mathematical analysis. Studying neuronal responses while accounting for the underlying geometrical organization, the details of synaptic connections and their specificity remains great challenges. With more than 10¹¹ neurons, connected by thousands of synapses per neuron, it is not clear what is the right modeling, for bridging the multiple scales starting at nanometer with the release of thousands of diffusing neurotransmitters molecules around synapses, leading to a signal that ultimately integrates into a neuronal network response (few millimeters) resulting in computations that underlie higher brain functions.

We discuss here recent progress about modeling and analysis of small and large neuronal networks. We present neural network modeling based on synaptic dynamics. The models are formulated as stochastic differential equations. We focus on the time-response of the network to stimulations. These time responses can be analyzed as an exit problem of a stochastic trajectory from the basin of attraction, the dynamics of which presents novel characteristics: the attractor is located very close to the separatrices. This property leads to novel phenomena, manifested by oscillatory peaks of the survival density probability. Finally, this report illustrates how mathematical methods in neuroscience allows a better understanding of neural network dynamics.