Discrete vs. continuous dynamics of certain neuronal networks

The dynamics of neuronal networks can be modeled either by systems of ODE's or by discrete dynamical systems. The former models seem closer to biological reality, while the latter ones can be easier to study. But under which conditions does a discrete model accurately reflect the dynamics of the underlying ODE system? In this talk we will present a class of neuronal networks with both excitatory and inhibitory synapses for which a strict correspondence between ODE dynamics and discrete dynamics can be rigorously proved. We will describe the network architectures and both the ODE and discrete models, state the theorem about the correspondence, sketch the main idea of the proof, and review some results how the discrete dynamics depends on the connectivity of the network. The talk is based on joint work with David Terman, Sungwoo Ahn, and Xueying Wang of the Mathematical Biosciences Institute and Ohio State University.