E8 Network dynamics (firing rate models)

Topic

The cortical network is composed of excitatory and inhibitory neuronal populations; at this population scale, it is useful to describe cortical network activity not in terms of individual neuronal spikes, but rather in terms of the average firing rate of each excitatory or inhibitory population within the network.

The Wilson-Cowan model describes the temporal evolution of a cortical population's firing rate as a relaxation toward a value determined by a sigmoid gain function applied to the net input received by that population. This net input comprises recurrent connections within the population itself, recurrent connections from other cortical populations, and external input to the network. The higher the net input, the higher the value toward which the population's firing rate relaxes; however, due to the sigmoid shape of the gain function, this relationship ceases to be proportional and flattens out when the net input is very high or very low.

When generalizing the Wilson-Cowan model to multiple coupled populations or units within the cortical network, the connections between populations are summarized in a connectivity matrix. The entire cortical network system can then be analyzed via its fixed points—states where the firing rates of all populations cease to change—and through the linear stability of these fixed points. This stability analysis examines how the network reacts to small perturbations around each fixed point: if the perturbations are damped, the fixed point is stable; if they grow, the network moves away from that fixed point toward a different activity regime. This analysis of fixed points and stability makes it possible to predict transitions in the cortical network between different activity regimes: a low-activity regime, a sustained-activity regime, or an oscillatory regime—the latter being analyzable using the same type of synchronization indicator previously employed to describe pathological synchronization in the cortical network in another model.

When the number of populations in the cortical network becomes very large and is distributed continuously across space, the network's discrete connectivity matrix can be replaced by a continuous spatial connectivity kernel—a mean-field extension useful for modeling entire cortical maps. Ultimately, it is this cortical network—described in terms of the average firing rates of its excitatory and inhibitory populations—that the Wilson-Cowan model allows us to analyze through its fixed points, stability, and transitions between activity regimes, whether in its discrete version involving a few populations or in its continuous extension for entire cortical maps.

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