Effects of the local dynamics in the synchronization of neural models

The cooperative behavior of neurons and neuronal areas associated with the synchronization behavior proves to be a fundamental neural mechanism. In addition, abnormal levels of synchronization have been related to unhealthy neural behaviors. Throughout this thesis, it is explored different synchronization phenomena which emerge through the collective dynamics of models of neurons coupled in a network. Firstly, it is shown a strong correlation between the individual dynamics of the neuron with the global behavior of the synchronization, in which the periodicity seen in the isolated neuron is reflected in a phase synchronization in the weak coupling region. Secondly, it is studied the role of bistability in the synchronization of a network of identical neurons coupled through a mean-field scheme. It is shown that the simple existence of two distinct stable states can lead the network to different states of synchronization, depending on the initialization of the system. Lastly, it is investigated the mechanism for explosive synchronization of a complex neural network composed of non-identical neurons. The presence of this regime is accompanied by a hysteresis loop on the network dynamics as the coupling parameter is adiabatically increased and decreased. It is shown that the abrupt synchronization transitions are associated with routes to chaos. The dynamical mechanisms for the bistability region, are given in terms of a saddle-node bifurcation and a boundary crisis. Therefore, the results of this thesis show a richness of synchronization behaviors associated with small changes of the neuronal dynamics bringing new insights to the theoretical study of neural networks.Comment: Thesis presented as a requirement for the degree of Ph.D. in Science in the Graduate Program in Physics, Exact Sciences Sector, of the Universidade Federal do Paran\'a. Advisor: Prof. Dr. Sergio Roberto Lope