Equivalency of spike-frequency and h-current-based adaptation in a Wilson-Cowan field model

During slow-wave sleep, the brain produces traveling waves of slow oscillations (SOs; ≤ 2 Hz ), characterized by the propagation of alternating high- and low-activity states. They play a crucial role in memory consolidation and are frequently investigated, empirically and with the support of studies. The question of internal mechanisms that modulate traveling waves of SOs is still unanswered even though it is established that it is an adaptation mechanism that mediates them. One of the main mechanisms investigated is spike-frequency adaptation, a hyperpolarizing feedback current that is activated during periods of high activity. An alternative mechanism that has recently been suggested is based on hyperpolarization-activated (h-)currents, which are positive feedback currents that are activated in low-activity states. Both adaptation mechanisms were shown to feature SO-like dynamics in neuronal populations, and the inclusion of a spatial domain seems to enhance observable differences in their effects. To investigate the effects of both adaptation mechanisms on the neural dynamics in detail, we examine a spatially extended, adaptive Wilson-Cowan model of interacting excitatory and inhibitory populations of neurons with local spatial coupling. The excitatory populations are equipped with either one or the other adaptation mechanism. We describe both mechanisms using the same dynamical equation and include the inverse mode of action by changing the signs of the adaptation strength and the gain of activation function. We then show that the dynamical systems including the two feedback currents are mathematically equivalent under a compensatory external input, which depends on the adaptation strength and which in turn leads to a shift in state space of the otherwise equivalent bifurcation structure. A detailed analysis of state space shows that strong enough adaptation is required to induce traveling waves of SO-like dynamics, while only stationary or homogeneous activity patterns emerge if adaptation is weak. Additionally, adaptation modulates the properties of the spatiotemporal activity patterns, such as temporal frequencies, spatial frequencies, and the speed of the traveling waves, all of which increase with increasing strength. Though being dynamically equivalent, our results also explain why location-dependent variations in feedback strength cause differences in the propagation of traveling waves between both adaptation mechanisms.