Spatially structured waves and oscillations in neuronal networks with synaptic depression and adaptation

We analyze the spatiotemporal dynamics of systems of nonlocal integro--differential equations, which all represent neuronal networks with synaptic depression and spike frequency adaptation. These networks support a wide range of spatially structured waves, pulses, and oscillations, which are suggestive of phenomena seen in cortical slice experiments and in vivo. In a one--dimensional network with synaptic depression and adaptation, we study traveling waves, standing bumps, and synchronous oscillations. We find that adaptation plays a minor role in determining the speed of waves; the dynamics are dominated by depression. Spatially structured oscillations arise in parameter regimes when the space--clamped version of the network supports limit cycles. Analyzing standing bumps in the network with only depression, we find the stability of bumps is determined by the spectrum of a piecewise smooth operator. We extend these results to a two--dimensional network with only depression. Here, when the space--clamped network supports limit cycles, both target wave emitting oscillating cores and spiral waves arise in the spatially extended network. When additive noise is included, the network supports oscillations for a wider range of parameters. In the high--gain limit of the firing rate function, single target waves and standing bumps exist in the network. We then proceed to study binocular rivalry in a competitive neuronal network with synaptic depression. The network consists of two separate populations each corresponding to cells receiving input from a single eye. Different regions in these populations respond preferentially to a particular stimulus orientation. In a space--clamped version of the model, we identify a binocular rivalry state with limit cycles, whose period we can compute analytically using a fast--slow analysis. In the spatially--extended model, we study rivalry as the destabilization of double bumps, using the piecewise smooth stability analysis we developed for the single population model. Finally, we study the effect of inhomogeneities in the spatial connectivity of a neuronal network with linear adaptation. Periodic modulation of synaptic connections leads to an effective reduction in the speed of traveling pulses and even wave propagation failure when inhomogeneities have sufficiently large amplitude or period.