Synchronization transitions in a system of superdiffusively coupled neurons: Interplay of chimeras, solitary states, and phase waves

In this paper, a network of interacting neurons based on a two-component system of reaction-superdiffusion equations with fractional Laplace operator responsible for the coupling configuration and nonlinear functions of the Hindmarsh-Rose model is considered. The process of synchronization transition in the space of the fractional Laplace operator exponents is studied. This parametric space contains information about both the local interaction strength and the asymptotics of the long-range couplings for both components of the system under consideration. It is shown that in addition to the homogeneous transition, there are regions of inhomogeneous synchronization transition in the space of the fractional Laplace operator exponents. Weak changes of the corresponding exponents in inhomogeneous zones are associated with the significant restructuring of the dynamic modes in the system. The parametric regions of chimera states, solitary states, phase waves, as well as dynamical modes combining them, are determined. The development of filamentary structures associated with the manifestation of different partial synchronization modes has been detected. In view of the demonstrated link between changes in network topology and internal dynamics, the data obtained in this study may be useful for neuroscience tasks. The approaches used in this study can be applied to a wide range of natural science disciplines.