Synchronization properties of heterogeneous neuronal networks with mixed excitability type.

We study the synchronization of neuronal networks with dynamical heterogeneity, showing that network structures with the same propensity for synchronization (as quantified by master stability function analysis) may develop dramatically different synchronization properties when heterogeneity is introduced with respect to neuronal excitability type. Specifically, we investigate networks composed of neurons with different types of phase response curves (PRCs), which characterize how oscillating neurons respond to excitatory perturbations. Neurons exhibiting type 1 PRC respond exclusively with phase advances, while neurons exhibiting type 2 PRC respond with either phase delays or phase advances, depending on when the perturbation occurs. We find that Watts-Strogatz small world networks transition to synchronization gradually as the proportion of type 2 neurons increases, whereas scale-free networks may transition gradually or rapidly, depending upon local correlations between node degree and excitability type. Random placement of type 2 neurons results in gradual transition to synchronization, whereas placement of type 2 neurons as hubs leads to a much more rapid transition, showing that type 2 hub cells easily "hijack" neuronal networks to synchronization. These results underscore the fact that the degree of synchronization observed in neuronal networks is determined by a complex interplay between network structure and the dynamical properties of individual neurons, indicating that efforts to recover structural connectivity from dynamical correlations must in general take both factors into account.