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Population density models of integrate-and-fire neurons with jumps: well-posedness

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Abstract

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation.

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References

  • Brunel N, van Rossum M (2007) Lapicque’s 1907 paper: from frogs to integrate-and-fire. Biol Cybern 97: 341–349

    Article  MATH  Google Scholar 

  • Burkitt AN (2006) A review of the integrate-and-fire neuron model: I. homogeneous synaptic input. Biol Cybern 95: 1–19

    Article  MathSciNet  MATH  Google Scholar 

  • Cáceres MJ, Carrillo JA, Perthame B (2011) Analysis of nonlinear noisy integrate and fire neuron models: blow-up and steady states. J Math Neurosci 1:7

    Google Scholar 

  • Cai D, Tao L, Rangan AV, McLaughlin DW (2006) Kinetic theory for neuronal network dynamics. Commun Math Sci 4: 97–127

    MathSciNet  MATH  Google Scholar 

  • de Kamps M (2003) A simple and stable numerical solution for the population density equation. Neural Comput 15: 2129–2146

    Article  MATH  Google Scholar 

  • DeVille REL, Peskin CS (2008) Synchrony and asynchrony in a fully stochastic neural network. Bull Math Biol 70: 1608–1633

    Article  MathSciNet  MATH  Google Scholar 

  • DeVille REL, Peskin CS, Spencer JH (2010) Dynamics of stochastic neuronal networks and the connections to random graph theory. Math Model Nat Phenom 5: 26–66

    Article  MathSciNet  MATH  Google Scholar 

  • Ermentrout GB, Terman D (2010) Mathematical foundations of neuroscience. Springer, New York

    Book  MATH  Google Scholar 

  • Fourcaud N, Brunel N (2002) Dynamics of the firing probability of noisy integrate-and-fire neurons. Neural Comput 14: 2057–2110

    Article  MATH  Google Scholar 

  • Garenne A, Henry J, Tarniceriu O (2010) Analysis of synchronization in a neural population by a population density approach. Math Model Nat Phenom 15: 5–25

    Article  MathSciNet  Google Scholar 

  • Gerstner W, Kistler W (2002) Spiking neuron models. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, Cambridge

    Google Scholar 

  • Knight B, Manim D, Sirovich L (1996) Dynamical models of interacting neuron populations in visual cortex. Robot Cybern 54: 4–8

    Google Scholar 

  • Knight BW, Omurtag A, Sirovich L (2000) The approach of a neuron population firing rate to a new equilibrium: an exact theoretical result. Neural Comput 12: 1045–1055

    Article  Google Scholar 

  • Ly C, Tranchina D (2007) Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Comput 19: 2032–2092

    Article  MathSciNet  MATH  Google Scholar 

  • Mirollo R, Strogatz S (1990) Synchronization of pulse-coupled biological oscillators. SIAM J Appl Math 50(4): 1645–1662

    Article  MathSciNet  MATH  Google Scholar 

  • Newhall KA, Kovacic G, Kramer PR, Cai D (2010) Cascade-induced synchrony in stochastically-driven neuronal networks. Phys Rev 82

  • Newhall KA, Kovacic G, Kramer PR, Zhou D, Rangan AV, Cai D (2010) Dynamics of current-based, poisson driven, integrate-and-fire neuronal networks. Commun Math Sci 8: 541–600

    MathSciNet  MATH  Google Scholar 

  • Nykamp DQ, Tranchina D (2000) A population density appraoch that facilitates large-scale modeling of neural networks : analysis and an application to orientation tuning. J Comput Neurosci 8: 19–50

    Article  MATH  Google Scholar 

  • Omurtag A, Knight B, Sirovich L (2000) On the simulation of large population of neurons. J Comput 8: 51–63

    MATH  Google Scholar 

  • Ostojic S, Brunel N, Hakim V (2009) Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities. J Comput Neurosci 26: 369–392

    Article  MathSciNet  Google Scholar 

  • Pakdaman K, Perthame B, Salort D (2009) Dynamics of a structured neuron population. Nonlinearity 23: 23–55

    MathSciNet  Google Scholar 

  • Perthame B (2007) Transport equation in biology. Birkhauser, Basel

    Google Scholar 

  • Perthame B, Tumuluri SM (2008) Nonlinear renewal equations. Math Stat 1: 1–32

    Google Scholar 

  • Sirovich L, Omurtag A, Knight B (2000) Dynamics of neuronal populations: the equilibrium solution. J Appl Math 60: 2009–2028

    MathSciNet  MATH  Google Scholar 

  • Sirovich L, Omurtag A, Lubliner K (2006) Dynamics of neural populations: stability synchrony. Comput Neural Syst 17: 3–29

    Article  Google Scholar 

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Authors and Affiliations

  1. CNRS, IMB Bordeaux, Bordeaux, France

    Grégory Dumont

  2. INRIA Bordeaux Sud Ouest, IMB Bordeaux, Bordeaux, France

    Jacques Henry

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  1. Grégory Dumont
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  2. Jacques Henry
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Correspondence to Grégory Dumont.

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Dumont, G., Henry, J. Population density models of integrate-and-fire neurons with jumps: well-posedness. J. Math. Biol. 67, 453–481 (2013). https://doi.org/10.1007/s00285-012-0554-5

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  • DOI: https://doi.org/10.1007/s00285-012-0554-5

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