Nonlinear analysis of periodic waves in a neural field model

Various types of brain activity, including motor, visual, and language, are accompanied by the propagation of periodic waves of electric potential in the cortex, possibly providing the synchronization of the epicenters involved in these activities. One example is cortical electrical activity propagating during sleep and described as traveling waves [Massimini et al., J. Neurosci. 24, 6862-6870 (2004)]. These waves modulate cortical excitability as they progress. Clinically related examples include cortical spreading depression in which a wave of depolarization propagates not only in migraine but also in stroke, hemorrhage, or traumatic brain injury [Whalen et al., Sci. Rep. 8, 1-9 (2018)]. Here, we consider the possible role of epicenters and explore a neural field model with two nonlinear integrodifferential equations for the distributions of activating and inhibiting signals. It is studied with symmetric connectivity functions characterizing signal exchange between two populations of neurons, excitatory and inhibitory. Bifurcation analysis is used to investigate the emergence of periodic traveling waves and of standing oscillations from the stationary, spatially homogeneous solutions, and the stability of these solutions. Both types of solutions can be started by local oscillations indicating a possible role of epicenters in the initiation of wave propagation. Published under license by AIP Publishing. The concept of periodic traveling waves appeared in the early 1970s and since that time investigators have actively explored themechanisms, computational principles, and functional role of these waveforms.3,4 This paper focuses on howmodeling information can be used to guide electrical stimulation in the targeted manipulation of brain networks in human patients affected by neurological disorders. We explore a neural field model with two nonlinear integrodifferential equations and perform a bifurcation analysis. Results suggest that it is possible to determine the direction of propagation and that epicenters may play a role in the initiation of wave propagating in the cortex. These results will guide the determination of the optimal localization to apply external brain stimulation (forcing term) to restore wave propagation in damaged cortical tissue.