Dynamics, synchronization and inverse problem in mean field neural networks with synaptic plasticity

This thesis regards the dynamics of neural ensembles, investigated through mathematical models. When the parameters defining the dynamics of single elements are inhomogeneous, i.e. disorder is present in the system, the model taken under consideration is able to reproduce a wide range of dynamical phases, typically observed in experiments. After describing the dynamical regimes of the model, it is proposed an heterogeneous mean–field approach to neural dynamics on random networks, that explicitly preserves the disorder on the parameters of the system at growing network sizes, and leads to a set of self-consistent equations. Within this approach, an effective description of microscopic and large scale temporal signals is provided. The mean field equations provide a clear analytical picture of the dynamics and can be applied in presence of disorder on network structure or on other parameters of the model. A great advantage of the mean field model is the possibility to formulate and solve a global inverse problem of reconstructing the in-degree distribution of the network from the knowledge of the average activity field detected from a finite size sample. Furthermore, the method applies in presence of inhibitory neurons, reconstructing also the fraction of inhibitory neurons from the knowledge of the only global activity field. The method is very general and applies to a large class of dynamical models on dense random networks