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Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications

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Stochastic Biomathematical Models

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 2058))

Abstract

Mathematical models are an important tool for neuroscientists. During the last 30 years many papers have appeared on single neuron description and specifically on stochastic Integrate and Fire models. Analytical results have been proved and numerical and simulation methods have been developed for their study. Recent reviews collect the main features of these models but do not focus on the methodologies employed to obtain them. The aim of this paper is to fill this gap by upgrading old reviews. The idea is to collect the existing methods and the available analytical results for the most common one dimensional stochastic Integrate and Fire models to make them available for studies on networks. An effort to unify the mathematical notation is also made. The review is divided in two parts:

  1. 1.

    Derivation of the models with the list of the available closed forms expressions for their characterization.

  2. 2.

    Presentation of the existing mathematical and statistical methods for the study of these models.

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Authors and Affiliations

  1. Department of Mathematics, University of Torino, Via Carlo Alberto 10, Torino, Italy

    Laura Sacerdote & Maria Teresa Giraudo

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  1. Laura Sacerdote
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  2. Maria Teresa Giraudo
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Correspondence to Laura Sacerdote .

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  1. College of Sciences, Department of Mathematics, King Saud University, King Khalid Road, Diriyah Campus, Riyadh, 11451, Saudi Arabia

    Mostafa Bachar

  2. Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, Graz, 8010, Austria

    Jerry Batzel

  3. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen, 2100, Denmark

    Susanne Ditlevsen

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Sacerdote, L., Giraudo, M.T. (2013). Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_5

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