# Threshold setting and mean firing rates in networks of integrate-and-fire neurons

## Author

**Hamish Meffin** and **William G. Gibson**

## Status

Research Report 98-16

Date: 24 July 1998

## Abstract

The time-course and stability of the firing rates in a network of
interconnected spiking integrate-and-fire neurons is analyzed for the
simplified case where the conduction of the neuron's membrane is taken to be
zero. The treatment is valid for networks of arbitrary size, no symmetry
requirements are placed on the inter-neuronal connections and constant
external inputs are allowed. An equation is given that explicitly relates the
mean firing rates to the neuronal threshold, the synaptic response functions
and the external inputs.
For purely excitatory networks the Perron-Frobenius theorem for non-negative
matrices allows the existence and stability of the steady states to be related
to the eigenvalues and eigenvectors of the matrix of synaptic connections.

The addition of inhibition allows more complex behaviour which is studied
using a mixture of theory and computer simulations. In contrast to the
excitatory case, the final state now depends on the initial state, thus
allowing the possibility of memory storage. In addition, such a network can
exhibit multistable and oscillatory behaviours.

## Key phrases

neural network. integrate-and-fire neurons.

## AMS Subject Classification (1991)

Primary: 92B20

Secondary:

## Content

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Sydney Mathematics and Statistics