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Mathematical Biosciences and Engineering

2014, Volume 11, Issue 2: 189-201. doi: 10.3934/mbe.2014.11.189
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Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons

  • 1.
    Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli
  • 2.
    Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli
  • Received: 01 October 2012 Accepted: 29 June 2018 Published: 01 October 2013

  • MSC : Primary: 60J60, 60J70; Secondary: 92-08.

  • With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.

    Citation: Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons[J]. Mathematical Biosciences and Engineering, 2014, 11(2): 189-201. doi: 10.3934/mbe.2014.11.189

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  • With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.


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  • This article has been cited by:

    1. M.F. Carfora, E. Pirozzi, Linked Gauss-Diffusion processes for modeling a finite-size neuronal network, 2017, 161, 03032647, 15, 10.1016/j.biosystems.2017.07.009
    2. G. Ascione, M.F. Carfora, E. Pirozzi, A stochastic model for interacting neurons in the olfactory bulb, 2019, 185, 03032647, 104030, 10.1016/j.biosystems.2019.104030
    3. Giacomo Ascione, Yuliya Mishura, Enrica Pirozzi, Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications, 2021, 23, 1387-5841, 53, 10.1007/s11009-019-09748-y
    4. Enrica Pirozzi, Maria Francesca Carfora, 2015, Stochastic modeling of the firing activity of coupled neurons periodically driven, 1-60133-018-9, 195, 10.3934/proc.2015.0195
    5. Giuseppe D’Onofrio, Claudio Macci, Enrica Pirozzi, Asymptotic Results for First-Passage Times of Some Exponential Processes, 2018, 20, 1387-5841, 1453, 10.1007/s11009-018-9659-7
    6. Aniello Buonocore, Luigia Caputo, Giuseppe D’Onofrio, Enrica Pirozzi, Closed-form solutions for the first-passage-time problem and neuronal modeling, 2015, 64, 0035-5038, 421, 10.1007/s11587-015-0248-6
    7. Giuseppe D’Onofrio, Enrica Pirozzi, Marcelo O. Magnasco, 2015, Chapter 22, 978-3-319-27339-6, 166, 10.1007/978-3-319-27340-2_22
    8. Angelo Pirozzi, Enrica Pirozzi, 2019, Chapter 100665-1, 978-1-4614-7320-6, 1, 10.1007/978-1-4614-7320-6_100665-1
    9. Giuseppe D'Onofrio, Enrica Pirozzi, Successive spike times predicted by a stochastic neuronal model with a variable input signal, 2016, 13, 1551-0018, 495, 10.3934/mbe.2016003
    10. Enrica Pirozzi, A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries, 2020, 12, 2073-8994, 279, 10.3390/sym12020279
    11. Alexander Vidybida, Olha Shchur, Relation Between Firing Statistics of Spiking Neuron with Delayed Fast Inhibitory Feedback and Without Feedback, 2018, 17, 0219-4775, 1850005, 10.1142/S0219477518500050
    12. Mario Abundo, Enrica Pirozzi, On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes, 2019, 7, 2227-7390, 991, 10.3390/math7100991
    13. Angelo Pirozzi, Enrica Pirozzi, 2022, Chapter 100665, 978-1-0716-1004-6, 1674, 10.1007/978-1-0716-1006-0_100665
    14. M. F. Carfora, 2023, Chapter 8, 978-3-031-33049-0, 137, 10.1007/978-3-031-33050-6_8
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  • © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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