I plan to investigate the question of metastability in stochastic models of biological neural networks through the rigorous characterization of statistical physics.During my PhD I have proven two results related to the asymptotic memorylessness of the time of extinction in a continuous-time model inspired by the original Galves-Löcherbach model. This memorylessness is the signature of metastable dynamics. However, these two results were proven under the restrictive assumption that the graph of interaction is either a one-dimensional lattice or a complete graph. It is natural to think that, while most biological neural networks actually encountered in the real world exhibit a structure of interaction which is neither a complete graph nor a one-dimensional lattice, the said structure of interaction has to lie somewhere between these two graphs, which are both at the two extremes of the spectrum with respect to connectivity. Therefore the first line of investigation, in continuation with the previously established results, is to try to prove the memorylessness of the time of extinction when the interaction is given by intermediate and possibly more complicated graphs. The most natural candidates would first be the regular and irregular trees and then random graphs such as the Erdös-Renyi graph, which are technically more complicated but also more interesting from a neurobiological point of view.Moreover all the results previously evoked are proven in the specific case in which the activation function-which gives the spiking rate of a neuron at any time, depending on its membrane potential-is an indicator function of the form Æ(x) = 1x>0. This assumption is mathematically convenient, but it is not the most realistic activation function from a biological point of view. A second Interesting line of investigation would then be to consider linear or sigmoid functions.Finally, a third possible line of investigation would be to study the temporal means of the process before extinction. The second characteristic property of metastable dynamics is indeed that these temporal means should be close in some sense to the asymptotic distribution of the infinite process restricted to the finite portion of the state space on which the finite version is defined.
News published in Agência FAPESP Newsletter about the scholarship: