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International Conference on Computer Aided Systems Theory

EUROCAST 2019: Computer Aided Systems Theory – EUROCAST 2019 pp 211–219Cite as

On the Integration of Fractional Neuronal Dynamics Driven by Correlated Processes

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12013))

Abstract

The stochastic Leaky Integrate-and-Fire (LIF) model is revisited adopting a fractional derivative instead of the classical one and a correlated input in place of the usual white noise. The aim is to include in the neuronal model some physiological evidences such as correlated inputs, codified input currents and different time-scales. Fractional integrals of Gauss-Markov processes are considered to investigate the proposed model. Two specific examples are given. Simulations of paths and histograms of first passage times are provided for a specific case.

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References

  1. Buonocore, A., Caputo, L., Pirozzi, E., Carfora, M.F.: A leaky integrate-and-fire model with adaptation for the generation of a spike train. Math. Biosci. Eng. 13(3), 483–493 (2016)

    Article  MathSciNet  Google Scholar 

  2. Kim, H., Shinomoto, S.: Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation. Math. Bios. Eng. 11, 49–62 (2014)

    Article  MathSciNet  Google Scholar 

  3. Sakai, Y., Funahashi, S., Shinomoto, S.: Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons. Neural Netw. 12, 1181–1190 (1999)

    Article  Google Scholar 

  4. Kobayashi, R., Kitano, K.: Impact of slow K\(^+\) currents on spike generation can be described by an adaptive threshold model. J. Comput. Neurosci. 40, 347–362 (2016)

    Article  MathSciNet  Google Scholar 

  5. Shinomoto, S., Sakai, Y., Funahashi, S.: The Ornstein-Uhlenbeck process does not reproduce spiking statistics of cortical neurons. Neural Comput. 11, 935–951 (1997)

    Article  Google Scholar 

  6. Pirozzi, E.: Colored noise and a stochastic fractional model for correlated inputs and adaptation in neuronal firing. Biol. Cybern. 112(1–2), 25–39 (2018)

    Article  MathSciNet  Google Scholar 

  7. Ascione, G., Pirozzi, E.: On fractional stochastic modeling of neuronal activity including memory effects. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2017. LNCS, vol. 10672, pp. 3–11. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74727-9_1

    Chapter  Google Scholar 

  8. Ascione, G., Pirozzi, E.: On a stochastic neuronal model integrating correlated inputs. Math. Biosci. Eng. 16(5), 5206–5225 (2019)

    Article  MathSciNet  Google Scholar 

  9. Abundo, M., Pirozzi, E.: On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications. arXiv preprint arXiv:1905.08167

  10. Bazzani, A., Bassi, G., Turchetti, G.: Diffusion and memory effects for stochastic processes and fractional Langevin equations. Phys. A Stat. Mech. Appl. 324, 530–550 (2003)

    Article  MathSciNet  Google Scholar 

  11. Teka, W., Marinov, T.M., Santamaria, F.: Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol. 10, e1003526 (2014)

    Article  Google Scholar 

  12. Abundo, M., Pirozzi, E.: Integrated stationary Ornstein-Uhlenbeck process, and double integral processes. Phys. A: Stat. Mech. Appl. 494, 265–275 (2018). https://doi.org/10.1016/j.physa.2017.12.043

    Article  MathSciNet  Google Scholar 

  13. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  14. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993)

    MATH  Google Scholar 

  15. Baldi, P.: Stochastic Calculus, Universitext. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62226-2

    Book  Google Scholar 

  16. Haugh, M.: Generating Random Variables and Stochastic Processes, Monte Carlo Simulation. Columbia University, New York (2017)

    Google Scholar 

  17. Garrappa, R., Kaslik, E., Popolizio, M.: Evaluation of fractional integrals and derivatives of elementary functions: overview and tutorial. Mathematics 7, 407 (2019). https://doi.org/10.3390/math7050407

    Article  Google Scholar 

  18. Kazem, S.: Exact solution of some linear fractional differential equation by Laplace transform. Int. J. Nonlinear Sci. 16(1), 3–11 (2013)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli FEDERICO II, Via Cintia, Monte S.Angelo, 80126, Napoli, Italy

    Enrica Pirozzi

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  1. Enrica Pirozzi
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Correspondence to Enrica Pirozzi .

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Editors and Affiliations

  1. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Roberto Moreno-Díaz

  2. Johannes Kepler University Linz, Linz, Austria

    Franz Pichler

  3. University of Las Palmas de Gran Canaria, Las Palmas de Gran Canaria, Spain

    Alexis Quesada-Arencibia

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Pirozzi, E. (2020). On the Integration of Fractional Neuronal Dynamics Driven by Correlated Processes. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2019. EUROCAST 2019. Lecture Notes in Computer Science(), vol 12013. Springer, Cham. https://doi.org/10.1007/978-3-030-45093-9_26

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  • DOI: https://doi.org/10.1007/978-3-030-45093-9_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-45092-2

  • Online ISBN: 978-3-030-45093-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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