
Mathematical Biosciences and Engineering, 2019, 16(5): 52065225. doi: 10.3934/mbe.2019260
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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
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Special Issues: Neural Coding 2018
References
1. Y. Sakai, S. Funahashi and S. Shinomoto, Temporally correlated inputs to leaky integrateandfire 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 gaussdiffusion 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 IntegrateAndFire Model With Adaptation 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 IntegrateandFire Model, PLoS Comput Biol., 10 (2014).
11. G. Ascione and E. Pirozzi, On fractional stochastic modeling of neuronal activity including memory 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 variability in simple neuronal models, J. Neurosci., 20 (2000), 6193–6209.
13. J. Feng and P. Zhang, Behavior of integrateandfire and HodgkinHuxley models with correlated inputs, Phys. Rev. E, 63 (2001), 051902.
14. N. Brunel and S. Sergi, Firing frequency of leaky intergrateandfire neurons with synaptic current dynamics, J. Theor. Biol., 195 (1998), 87–95.
15. E. Salinas and T. J. Sejnowski, Integrateandfire 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 OrnsteinUhlenbeck input current, Biol. Cybern., 86 (2002), 137–145.
17. N. Fourcaud and N. Brunel, Dynamics of the firing probability of noisy integrateandfire neurons, Neural. Comput., 14 (2002), 2057–2110.
18. M. Abundo, On the first passage time of an integrated GaussMarkov process, SCMJ, 28 (2015), 1–14.
19. M. Abundo and E. Pirozzi, Integrated stationary OrnsteinUhlenbeck 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. ToledoRodriguez, 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 HodgkinHuxley 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. PoincarePr., 52, Institut Henri Poincaré, 2016.
24. A. Buonocore, L. Caputo, E. Pirozzi,et al., Gaussdiffusion 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 GaussDiffusion processes for modeling a finitesize 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 OrnsteinUhlenbeck processes, Electron. J. Probab. 8, (2003).
28. A. Tonnelier, H. Belmabrouk and D. Martinez, Eventdriven simulations of nonlinear integrateandfire neurons, Neural. Comput., 19 (2007), 3226–3238.
29. V. J. Barranca, D. C. Johnson, J. L. Moyher, et al., Dynamics of the exponential integrateandfire 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 integrateandfire models of neuronal dynamics, Phys. Rev. E, 55 (1997), 2040–2043.
32. Y. Gai, B. Doiron, V. Kotak, et al, Noisegated encoding of slow inputs by auditory brainstem neurons with a lowthreshold K^{+} current, J. Neurophysiol., 102 (2009), 3447–3460.
33. R. Kobayashi, Y. Tsubo and S. Shinomoto, Madetoorder spiking neuron model equipped with a multitimescale adaptive threshold, Front. Comput. Neurosc., 3 (2009), 9.
34. E. Di Nardo, A. G. Nobile, E. Pirozzi, et al., A computational approach to firstpassagetime problems for GaussMarkov processes, Adv. Appl. Probab., 33 (2001), 453–482.
35. J. L. Doob, Heuristic approach to the KolmogorovSmirnov 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).
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