Absolute Stability and Complete Synchronization in a Class of Neural Fields Models

Neural fields are an interesting option for modeling macroscopic parts of the cortex involving several populations of neurons, like cortical areas. Two classes of neural field equations are considered: voltage- and activity-based. The spatio-temporal behavior of these fields is described by nonlinear integro-differential equations. The integral term, computed over a compact subset of $\mathbb{R}^q$, $q=1,2,3$, involves space and time varying, possibly nonsymmetric, intracortical connectivity kernels. Contributions from white matter afferents are represented as external input. Sigmoidal nonlinearities arise from the relation between average membrane potentials and instantaneous firing rates. Using methods of functional analysis, we characterize the existence and uniqueness of a solution of these equations for general, homogeneous (i.e., independent of the spatial variable), and spatially locally homogeneous inputs. In all cases we give sufficient conditions on the connectivity functions for the solutions to be absolutely stable, that is to say, asymptotically independent of the initial state of the field. These conditions bear on some compact operators defined from the connectivity kernels, the maximal slope of the sigmoids, and the time constants used in describing the temporal shape of the postsynaptic potentials. Numerical experiments are presented to illustrate the theory. To our knowledge this is the first time that such a complete analysis of the problem of the existence and uniqueness of a solution of these equations has been obtained. Another important contribution is the analysis of the absolute stability of these solutions—more difficult but more general than the linear stability analysis which it implies. The reason we have been able to complete this work is our use of the functional analysis framework and the theory of compact operators in a Hilbert space which has allowed us to provide simple mathematical answers to some of the questions raised by modelers in neuroscience.