Nonlinear and Stochastic Methods in Neurosciences

The brain is a very complex system in the strong sense. It features a huge amount of individual cells, in particular the neurons presenting a highly nonlinear dynamics, interconnected in a very intricate fashion, and which receive noisy complex informations. The problem of understanding the function of the brain, the neurons' behavior in response to different kinds of stimuli and the global behavior of macroscopic or mesoscopic populations of neurons has received a lot of attention during the last decades, and a critical amount of biological and computational data is now available and makes the field of mathematical neurosciences very active and exciting. In this manuscript we will be interested in bringing together advanced mathematical tools and biological problems arising in neuroscience. We will be particularly interested in understanding the role of nonlinearities and stochasticity in the brain, at the level of individual cells and of populations. The study of biological problems will bring into focus new and unsolved mathematical problems we will try to address, and mathematical studies will in turn shed a new light on biological processes in play. After a quick and selective description of the basic principles of neural science and of the different models of neuronal activity, we will introduce and study a general class of nonlinear bidimensional neuron models described from a mathematical point of view by an hybrid dynamical system. In these systems the membrane potential of a neuron together with an additional variable called the adaptation, has free behavior governed by an ordinary differential equation, and this dynamics is coupled with a spike mechanism described by a discrete dynamical system. An extensive study of these models will be provided in the manuscript, which will lead us to define electrophysiological classes of neurons, i.e. sets of parameters for which the neuron has similar behaviors for different types of stimulations. We will then deal with the statistics of spike trains for neurons driven by noisy currents. We will show that the problem of characterizing the probability distribution of spike timings can be reduced to the problem of first hitting times of certain stochastic process, and we shall review and develop methods in order to solve this problem. We will eventually turn to popoulation modelling. The first level of modelization is the network level. At this level, we will propose an event-based description of the network activity for noisy neurons. The network-level description is in general not suitable in order to understand the function of cortical areas or cortical columns, and in general at the level of the cell, the properties of the neurons and of the connectivities are unknown. That is why we will then turn to more mesoscopic models. We first present the derivation of mesoscopic description from first principles, and prove that the equation obtained, called the mean-field equation, is well posed in the mathematical sense. We will then simplify this equation by neglecting the noise, and study the dynamics of periodic solutions for cortical columns models, which can be related to electroencephalogram signals, with a special focus on the apparition of epileptic activity.