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Gaussian processes and neuronal modeling

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Abstract

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.

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Acknowledgements

This work has been performed under partial support by MIUR (PRIN 2005), by G.N.C.S – INdAM and by Campania Region.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università della Basilicata, Contrada Macchia Romana, Potenza, Italy

    Elvira Di Nardo

  2. Dipartimento di Matematica e Informatica, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), Italy

    Amelia G. Nobile

  3. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli, Italy

    Enrica Pirozzi & Luigi M. Ricciardi

Authors
  1. Elvira Di Nardo
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  2. Amelia G. Nobile
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  3. Enrica Pirozzi
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  4. Luigi M. Ricciardi
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Corresponding author

Correspondence to Elvira Di Nardo.

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Nardo, E.D., Nobile, A.G., Pirozzi, E. et al. Gaussian processes and neuronal modeling. Nat Comput 6, 283–310 (2007). https://doi.org/10.1007/s11047-006-9010-z

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