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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 an excitatory network with synaptic depression and adaptation, we study traveling waves, spiral waves, standing bumps, and synchronous oscillations. 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. When the space clamped network supports limit cycles, target wave emitting oscillating cores and spiral waves arise in the spatially-extended network. We then proceed to study binocular rivalry in a competitive neuronal network with synaptic depression. Rivalry arises as limit cycles in the space-clamped system and double bump instabilities in the spatially-extended system. Finally, we find inhomogeneities in the spatial connectivity of a neuronal network with adaptation can cause wave propagation failure.

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Zachary Kilpatrick is an NSF postdoctoral research fellow at the University of Pittsburgh in the Department of Mathematics and a member of the Complex Biological Systems group there. He received his bachelor´s degree in applied mathematics and history from Rice University in 2005 and doctorate in mathematics at the University of Utah in 2010.

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