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On a stochastic neuronal model integrating correlated inputs

Dipartimento di Matematica e Applicazioni ”Renato Caccioppoli”, Università degli studi di Napoli Federico II, Via Cintia, Monte S.Angelo Napoli, 80126, Italy

Special Issues: Neural Coding 2018

A modified LIF-type stochastic model is considered with a non-delta correlated stochastic process in place of the traditional white noise. Two different mechanisms of reset are specified with the aim to model endogenous and exogenous correlated input stimuli. Ornstein-Uhlenbeck processes are used to model the two different cases. An equivalence between different ways to include currents in the model is also shown. The theory of integrated stochastic processes is evoked and the main features of involved processes are obtained, such as mean and covariance functions. Finally, a simulation algorithm of the proposed model is described; simulations are performed to provide estimations of firing densities and related comparisons are given.
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1. Y. Sakai, S. Funahashi and S. Shinomoto, Temporally correlated inputs to leaky integrate-and-fire models can reproduce spiking statistics of cortical neurons, Neural Networks, 12 (1999), 1181– 1190.

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

3. L. Sacerdote and M. T. Giraudo, Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications, in Stochastic Biomathematical Models, Volume 2058 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg (2012), 99–148.

4. A. Buonocore, L. Caputo, E. Pirozzi, et al., The first passage time problem for gauss-diffusion processes: Algorithmic approaches and applications to LIF neuronal model, Methodol. Comput. Appl. Probab., 13 (2011), 29–57.

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

6. C. F. Stevens and A. M. Zador, Input synchrony and the irregular firing of cortical neurons, Nat. Neurosci., 1 (1998), 210–217.

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

8. A. Buonocore, L. Caputo, M. F. Carfora, et al., A Leaky Integrate-And-Fire Model With Adapta-tion For The Generation Of A Spike Train. Math. Bios. Eng., 13 (2016), 483–493.

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

10. W. Teka, T. M. Marinov and F. Santamaria, Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model, PLoS Comput Biol., 10 (2014).

11. G. Ascione and E. Pirozzi, On fractional stochastic modeling of neuronal activity including mem-ory effects, in Computer Aided Systems Theory EUROCAST 2017, LNCS, 10672, (2018), 3–11.

12. E. Salinas and T. J. Sejnowski, Impact of correlated synaptic input on output firing rate and vari-ability in simple neuronal models, J. Neurosci., 20 (2000), 6193–6209.

13. J. Feng and P. Zhang, Behavior of integrate-and-fire and Hodgkin-Huxley models with correlated inputs, Phys. Rev. E, 63 (2001), 051902.

14. N. Brunel and S. Sergi, Firing frequency of leaky intergrate-and-fire neurons with synaptic current dynamics, J. Theor. Biol., 195 (1998), 87–95.

15. E. Salinas and T. J. Sejnowski, Integrate-and-fire neurons driven by correlated stochastic input, Neural. Comput., 14 (2002), 2111–2155.

16. H. C. Tuckwell, F. Y. M. Wan and J.-P. Rospars, A spatial stochastic neuronal model with Orn-steinUhlenbeck input current, Biol. Cybern., 86 (2002), 137–145.

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

18. M. Abundo, On the first passage time of an integrated Gauss-Markov process, SCMJ, 28 (2015), 1–14.

19. M. Abundo and E. Pirozzi, Integrated stationary Ornstein-Uhlenbeck process, and double integral processes , Phys. A Stat. Mech. Appl., 494 (2018), 265–275

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

21. M. Pospischil, M. Toledo-Rodriguez, C. Monier, et al., Minimal HodgkinHuxley type models for different classes of cortical and thalamic neurons, Biol. Cybern., 99 (2008), 427–441.

22. R. F. Fox, Stochastic versions of the Hodgkin-Huxley equations, Biophys. J., 72 (1997), 2068–2074.

23. R. Höpfner, E. Löcherbach and M. Thieullen, Ergodicity for a stochastic HodgkinHuxley model driven by OrnsteinUhlenbeck type input, Ann. I. H. Poincare-Pr., 52, Institut Henri Poincaré, 2016.

24. A. Buonocore, L. Caputo, E. Pirozzi,et al., Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons. Math. Biosci. Eng., 11 (2014), 189–201.

25. M.F.Carfora and E.Pirozzi, Linked Gauss-Diffusion processes for modeling a finite-size neuronal network, Biosystems, 161 (2017), 15–23.

26. A. Jentzen and A. Neuenkirch, A random Euler scheme for Carathèodory differential equations, J. Comput. Appl. Math., 224 (2009), 346–359.

27. P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8, (2003).

28. A. Tonnelier, H. Belmabrouk and D. Martinez, Event-driven simulations of nonlinear integrate-and-fire neurons, Neural. Comput., 19 (2007), 3226–3238.

29. V. J. Barranca, D. C. Johnson, J. L. Moyher, et al., Dynamics of the exponential integrate-and-fire model with slow currents and adaptation, J. Comput. Neurosci., 37 (2014), 161–180.

30. P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT press (2001).

31. P. Lánský, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Phys. Rev. E, 55 (1997), 2040–2043.

32. Y. Gai, B. Doiron, V. Kotak, et al, Noise-gated encoding of slow inputs by auditory brainstem neurons with a low-threshold K+ current, J. Neurophysiol., 102 (2009), 3447–3460.

33. R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosc., 3 (2009), 9.

34. E. Di Nardo, A. G. Nobile, E. Pirozzi, et al., A computational approach to first-passage-time problems for GaussMarkov processes, Adv. Appl. Probab., 33 (2001), 453–482.

35. J. L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Stat., 20 (1949), 393–403.

36. S. Asmussen and P. W. Glynn, Stochastic simulation: algorithms and analysis, Springer Science & Business Media (2007).

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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