Section: New Results

Neurosciences

Participants : Jonathan Touboul, Gilles Wainrib, Tanguy Cabana, Mathieu Galtier, Luis Garcia Del Molino, Khashayar Pakdaman.

We pursued our studies of disordered networks of the brain and collective phenomena in neuroscience. We have been more interested this year in the role of disorder in the spontaneous emergence of synchronized activity. In order to study these phenomena, we have been establishing limit equations for randomly coupled networks [11] , and the analysis of this equation reveal a number of transitions due to the level of disorder in the connectivity. A universal transition observed in such randomly coupled networks is a transition to chaotic activity for large levels of noise. These transitions were investigated [24] and were shown to be related to an explosion of complexity at the edge of chaos, i.e. the number of equilibria is exponentially large with the network size at the phase transition, and the exponential factor was related to the Lyapunov exponent. These large-scale limits give rise to nonlinear reduced equations that we have been introducing in [15] . Eventually, when considering that the network is structured into different populations and that the connectivity weights satisfy a balance condition, which is postulated as a natural scaling of the synaptic input, we have shown that the network shows random transitions to periodic activity depending on the spectrum of the random connectivity matrix [25] , yielding up and down states or synchronized oscillations depending on the eigenvalue of larger real part of the connectivity matrix.