Synchronization, heterogeneity and inhibitory hubs in neural networks with synaptic plasticity

How brain can perform highly specialized and differentiated functions at the same time, generating collective dynamics among its elements, is a very intriguing question. From the point of view of Statistical Physics, brain is the complex system by definition, since we cannot explain its macroscopic behaviour as a simple sum of its microscopic units dynamics. Within this system, a phenomenon which largely affects brain functions is synchronization. Synchronization is one of the most fundamental and surprising dynamical states, observed in oscillatory systems belonging to many different research branches, such as engineering, physics, chemistry, life sciences and social life. In particular, in neural tissues, synchronization emerges as collective oscillations, where all units of a macroscopic population evolve in a coherent way. In recent years great efforts have been performed in experimental and theoretical neurosciences to interrelate the synchronous regimes detected in different brain areas with specific cognitive functions, such as memory and learning processes, but also with neural pathologies. The onset of synchronous phases has been shown to depend on many factors, such as the connections structures and the presence of inhibitory components. Due to their high complexity, numerical simulations of neural networks are able to correctly reproduce the collective phases detected in real neuronal ensembles, providing an useful investigation method to advance predictions about the dynamical properties of neural tissues, which have to be experimentally tested. In this framework, in this thesis we investigate theoretically and numerically the role played by heterogeneity and the interplay between inhibition and connectivity structure, in defining synchronization properties and the capacity of storing information of a neural population. As a model of neural network we consider a system of integrate-and-fire neurons coupled through a synaptic mechanism characterized by a short-term plasticity. Since all neurons are identical units, the only source of heterogeneity will be encoded by connectivity pattern. In particular, we take advantage of a mean field formulation of this model, which allows us to preserve the structural heterogeneity of our systems and it turns out to be extremely efficient in reproducing the dynamics of the model in the limit of large connectivity. Furthermore, to emphasize the role of heterogeneity, we introduce a single-site degree correlation, setting the same input and output connectivities for each neuron. In the purely excitatory case, as synaptic coupling strength increases, a synchronization phase transition occurs from a quasisynchronous regime to an asynchronous one, while increasing the connections heterogeneity makes in general the network less synchronizable. However, considering broad distributions is not the only determinant factor in driving the networks synchronization, as clearly appears in scale-free topologies. Indeed, a †œGaussian synchronization condition†� emerges: in order to have a synchronizable network the degree distribution peak should be high enough, that is it should contain enough neurons with similar degrees. In an excitatory and inhibitory networks, where in particular inhibitory neurons are highly connected (hubs), we tune the fraction of inhibitory neurons present in the system and their connectivity level. What clearly emerges is that the hub character of inhibitory neurons is a fundamental ingredient to produce a highly synchronous regime in correspondence of an inhibitory fraction equal to 10-30\\%, which is the experimentally observed range of values. Furthermore, around this regime of complete synchronization we find an interesting metastable dynamical phase, which show long-time memory of external inputs applied to the network.