The Cauchy problem for one-dimensional spiking neuron models

by Romain Brette
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 [Formula: see text] 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.
Reference:
Romain Brette, 2008. The Cauchy problem for one-dimensional spiking neuron models, Cognitive neurodynamics, volume 2.
Bibtex Entry:
@article{Brette2008,
 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 [Formula: see text] 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.},
 author = {Brette, Romain},
 day = {15},
 doi = {10.1007/s11571-007-9032-y},
 issn = {1871-4080},
 journal = {Cognitive neurodynamics},
 language = {eng},
 month = {Mar},
 number = {1},
 pages = {21--27},
 title = {The Cauchy problem for one-dimensional spiking neuron models.},
 url = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2289251/pdf/11571_2007_Article_9032.pdf},
 volume = {2},
 year = {2008}
}