Propagation and stability of waves of electrical activity in the cerebral cortex

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Abstract
Nonlinear equations are introduced to model the behavior of the waves of cortical electrical activity that are responsible for signals observed in electroencephalography. These equations incorporate nonlinearities, axonal and dendritic lags, excitatory and inhibitory neuronal populations, and the two-dimensional nature of the cortex, while rendering nonlinear features far more tractable than previous formulations, both analytically and numerically. The model equations are first used to calculate steady-state levels of cortical activity for various levels of stimulation. Dispersion equations for linear waves are then derived analytically and an analytic expression is found for the linear stability boundary beyond which a seizure will occur. The effects of boundary conditions in determining global eigenmodes are also studied in various geometries and the corresponding eigenfrequencies are found. Numerical results confirm the analytic ones, which are also found to reproduce existing results in the relevant limits, thereby elucidating the limits of validity of previous approximations.