Mathematical Foundations of Neuroscience pp 331-367 | Cite as

# Firing Rate Models

## Abstract

One of the most common ways to model large networks of neurons is to use a simplification called a firing rate model. Rather than track the spiking of every neuron, instead one tracks the averaged behavior of the spike rates of groups of neurons within the circuit. These models are also called population models since they can represent whole populations of neurons rather than single cells. In this book, we will call them rate models although their physical meaning may not be the actual firing rate of a neuron. In general, there will be some invertible relationship between the firing rate of the neuron and the variable at hand. We derive the individual model equation in several different ways, some of the derivations are rigorous and are directly related to some biophysical model and other derivations are ad hoc. After deriving the rate models, we apply them to a number of interesting phenomena, including working memory, hallucinations, binocular rivalry, optical illusions, and traveling waves. We also describe a number of theorems about asymptotic states as well as some of the now classical work on attractor networks.

## Keywords

Firing Rate Hopf Bifurcation Bifurcation Diagram Spike Train Stable Manifold## References

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