# Continuous neural network with windowed Hebbian learning

## Abstract

We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented.

## Keywords

Neural field Continuous network Bump Traveling front Delay equation Existence Stability## Mathematics Subject Classification

35B35 35C07 45K05 92B20 92C20## Notes

### Acknowledgments

It is pleasure for the authors to thank A. Abbasian for his useful comments about the biological aspects of the problem. This research was in part supported by a grant from IPM (No. 91920410). The third author was also partially supported by National Elites Foundation.

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