The dance of neurons: Exploring nonlinear dynamics in brain networks

The brain is a complex, nonlinear system, exhibiting ever-evolving patterns of activities, whether in the presence or absence of external stimuli or task demands. Nonlinearity can notably obscure the link between structural constraints enforced on the interaction and its dynamical consequences. Suitable nonlinear dynamical models and their analysis serve as essential tools not only for bridging structural and functional understanding of the brain but also for predictably altering the complex dynamical organization of the brain. Here, starting from a large-scale network of threshold Hodgkin–Huxley style neurons, we formulate the average nonlinear dynamics implicitly following from the Wilson–Cowan assumptions. We investigate the influence of biophysical and structural properties on the complexity of neural dynamics at the microscale level and its relationship with the macroscopic Wilson–Cowan model. Incorporating the elements in the model can help identify more realistic regimes of activity and connect the mathematical prediction of increasing nonlinearity to physical manipulations. Our simulations of the temporal profiles reveal dependency on the binary state of interacting subpopulations and the random property of structural network at the transition points, when different synaptic weights are considered. For substantial configurations of stimulus intensity, our model provides further estimates of the neural population's dynamics, ranging from simple-periodic to aperiodic patterns and phase transition regimes. This reflects the potential contribution of the microscopic nonlinear scheme to the mean-field approximation in studying the collective behaviour of individual neurons with particularly concentrating on the occurrence of critical phenomena. We show that finite-size effects kick the system in a state of irregular modes to evolve differently from predictions of the original Wilson–Cowan reference. Additionally, we report that the complexity and temporal diversity of neural dynamics, especially in terms of limit cycle trajectory, and synchronization can be induced by either small heterogeneity in the degree of various types of local excitatory connectivity or considerable diversity in the external drive to the excitatory pool.