# The Cauchy problem for one-dimensional spiking neuron models

Research Article

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## Abstract

I consider spiking neuron models defined by a one-dimensional differential equation and a reset—i.e., neuron models of the integrate-and-fire type. I address the question of the existence and uniqueness of a solution on \({\mathbb{R}}\) for a given initial condition. It turns out that the reset introduces a countable and ordered set of backward solutions for a given initial condition. I discuss the implications of these mathematical results in terms of neural coding and spike timing precision.

### Keywords

Integrate-and-fire Cauchy problem Spike timing precision Reliability Neuron models## Notes

### Acknowledgments

This work was partially supported by the EC IP project FP6-015879, FACETS, and the EADS Corporate Research Foundation.

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© Springer Science+Business Media B.V. 2007