# Waves and Oscillations in Networks of Coupled Neurons

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## **Abstract**

Neural systems are characterized by the interactions of thousands of individual cells called neurons. Individual neurons vary in their properties with some of them spontaneously active and others active only when given a sufficient perturbation. In this note, I will describe work that has been done on the mathematical analysis of waves and synchronous oscillations in spatially distributed networks of neurons. These classes of behavior are observed both in vivo (that is, in the living brain) and in vitro (isolated networks, such as slices of brain tissue.) We focus on these simple behaviors rather than on the possible computations that networks of neurons can do (such as filtering sensory inputs and producing precise motor output) mainly because they are mathematically tractable. The chapter is organized as follows. First, I will introduce the kinds of equations that are of interest and from these abstract some simplified models. I will consider several different types of connectivity – from “all-to-all” to spatially organized. Typically (although not in every case), each individual neuron is represented by a scalar equation for its dynamics. These individuals can be coupled together directly or indirectly and in spatially discrete or continuous arrays.

## Keywords

Hopf Bifurcation Coupling Function Spiral Wave Couple Neuron Gastric Mill## Preview

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