# American Institute of Mathematical Sciences

2016, 13(3): 495-507. doi: 10.3934/mbe.2016003

## Successive spike times predicted by a stochastic neuronal model with a variable input signal

 1 Dipartimento di Matematica e Applicazioni, Università degli studi di Napoli, FEDERICO II, Via Cinthia, Monte S.Angelo, Napoli, 80126, Italy 2 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli

Received  April 2015 Revised  November 2015 Published  January 2016

Two different stochastic processes are used to model the evolution of the membrane voltage of a neuron exposed to a time-varying input signal. The first process is an inhomogeneous Ornstein-Uhlenbeck process and its first passage time through a constant threshold is used to model the first spike time after the signal onset. The second process is a Gauss-Markov process identified by a particular mean function dependent on the first passage time of the first process. It is shown that the second process is also of a diffusion type. The probability density function of the maximum between the first passage time of the first and the second process is considered to approximate the distribution of the second spike time. Results obtained by simulations are compared with those following the numerical and asymptotic approximations. A general equation to model successive spike times is given. Finally, examples with specific input signals are provided.
Citation: Giuseppe D'Onofrio, Enrica Pirozzi. Successive spike times predicted by a stochastic neuronal model with a variable input signal. Mathematical Biosciences & Engineering, 2016, 13 (3) : 495-507. doi: 10.3934/mbe.2016003
##### References:
 [1] A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [2] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, The first passage time problem for Gauss-diffusion processes: Algorithmic approaches and applications to LIF neuronal model, Methodol. Comput. Appl. Prob., 13 (2011), 29-57. doi: 10.1007/s11009-009-9132-8. [3] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Computation, 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023. [4] A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons, Math. Biosci. Eng., 11 (2014), 189-201. [5] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Gauss-Markov processes in the presence of a reflecting boundary and applications in neuronal models, Applied Mathematics and Computation, 232 (2014), 799-809. doi: 10.1016/j.amc.2014.01.143. [6] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals, Journal of Computational and Applied Mathematics, 285 (2015), 59-71. doi: 10.1016/j.cam.2015.01.042. [7] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Gauss-markov processes for neuronal models including reversal potentials, Advances in Cognitive Neurodynamics (IV), 11 (2015), 299-305. doi: 10.1007/978-94-017-9548-7_42. [8] M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire neuron with threshold fatigue. Neural Computation, 15 (2003), 253-276. [9] E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, A computational approach to first passage-time problems for Gauss-Markov processes, Adv. Appl. Prob., 33 (2001), 453-482. doi: 10.1239/aap/999188324. [10] J. M. Fellous, P. H. Tiesinga, P. J. Thomas and T. J. Sejnowski, Discovering spike patterns in neuronal responses, The Journal of Neuroscience, 24 (2004), 2989-3001. doi: 10.1523/JNEUROSCI.4649-03.2004. [11] V. Giorno and S. Spina, On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model, Math. Bios. Eng., 11 (2014), 285-302. [12] 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. [13] P. Lánský and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models, Biol. Cybern., 99 (2008), 253-262. doi: 10.1007/s00422-008-0237-x. [14] P. Lánský, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Physical Review E, 55 (1997), 2040-2043. [15] B. Lindner, Interspike interval statistics for neurons driven by colored noise, Physical Review E, 69 (2004), 022901-1-022901-4. doi: 10.1103/PhysRevE.69.022901. [16] L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biological Cybernetics, 35 (1979), 1-9. doi: 10.1007/BF01845839. [17] L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Mathematica Japonica, 50 (1999), 247-322. [18] T. Schwalger, F. Droste and B. Lindner, Statistical structure of neural spiking under non-Poissonian or other non-white stimulation, Journal of Computational Neuroscience, 39 (2015), 29-51. doi: 10.1007/s10827-015-0560-x. [19] M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Boston (USA), 1994. [20] S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of cortical neurons, Neural Computation, 11 (1997), 935-951. [21] T. Taillefumier and M. 0. Magnasco, A phase transition in the first passage of a Brownian process through a fluctuating boundary: Implications for neural coding, PNAS, 110 (2013), E1438-E1443. doi: 10.1073/pnas.1212479110. [22] T. Taillefumier and M. Magnasco, A transition to sharp timing in stochastic leaky integrate-and-fire neurons driven by frozen noisy input, Neural Computation, 26 (2014), 819-859. doi: 10.1162/NECO_a_00577. [23] T. Taillefumier and M. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Holder continuous boundaries, J. Stat. Phys., 140 (2010), 1130-1156. doi: 10.1007/s10955-010-0033-6. [24] P. J. Thomas, A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process, J. Appl. Probab., 48 (2011), 420-434. doi: 10.1239/jap/1308662636. [25] J. V. Toups, J. M. Fellous, P. J. Thomas, T. J. Sejnowski and P. H. Tiesinga, Multiple spike time patterns occur at bifurcation points of membrane potential dynamics, PLoS Comput. Biol., 8 (2012), e1002615, 1-19. doi: 10.1371/journal.pcbi.1002615. [26] H. C. Tuckwell, Stochastic Processes in the Neurosciences, SIAM, 1989. doi: 10.1137/1.9781611970159. [27] E. Urdapilleta, Series solution to the first-passage-time problem of a Brownian motion with an exponential time-dependent drift, J. Stat. Phys., 140 (2010), 1130-1156. doi: 10.1088/1751-8113/45/18/185001.

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##### References:
 [1] A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [2] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, The first passage time problem for Gauss-diffusion processes: Algorithmic approaches and applications to LIF neuronal model, Methodol. Comput. Appl. Prob., 13 (2011), 29-57. doi: 10.1007/s11009-009-9132-8. [3] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Computation, 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023. [4] A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons, Math. Biosci. Eng., 11 (2014), 189-201. [5] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Gauss-Markov processes in the presence of a reflecting boundary and applications in neuronal models, Applied Mathematics and Computation, 232 (2014), 799-809. doi: 10.1016/j.amc.2014.01.143. [6] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Restricted Ornstein-Uhlenbeck process and applications in neuronal models with periodic input signals, Journal of Computational and Applied Mathematics, 285 (2015), 59-71. doi: 10.1016/j.cam.2015.01.042. [7] A. Buonocore, L. Caputo, A. G. Nobile and E. Pirozzi, Gauss-markov processes for neuronal models including reversal potentials, Advances in Cognitive Neurodynamics (IV), 11 (2015), 299-305. doi: 10.1007/978-94-017-9548-7_42. [8] M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire neuron with threshold fatigue. Neural Computation, 15 (2003), 253-276. [9] E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, A computational approach to first passage-time problems for Gauss-Markov processes, Adv. Appl. Prob., 33 (2001), 453-482. doi: 10.1239/aap/999188324. [10] J. M. Fellous, P. H. Tiesinga, P. J. Thomas and T. J. Sejnowski, Discovering spike patterns in neuronal responses, The Journal of Neuroscience, 24 (2004), 2989-3001. doi: 10.1523/JNEUROSCI.4649-03.2004. [11] V. Giorno and S. Spina, On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model, Math. Bios. Eng., 11 (2014), 285-302. [12] 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. [13] P. Lánský and S. Ditlevsen, A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models, Biol. Cybern., 99 (2008), 253-262. doi: 10.1007/s00422-008-0237-x. [14] P. Lánský, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Physical Review E, 55 (1997), 2040-2043. [15] B. Lindner, Interspike interval statistics for neurons driven by colored noise, Physical Review E, 69 (2004), 022901-1-022901-4. doi: 10.1103/PhysRevE.69.022901. [16] L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biological Cybernetics, 35 (1979), 1-9. doi: 10.1007/BF01845839. [17] L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. G. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Mathematica Japonica, 50 (1999), 247-322. [18] T. Schwalger, F. Droste and B. Lindner, Statistical structure of neural spiking under non-Poissonian or other non-white stimulation, Journal of Computational Neuroscience, 39 (2015), 29-51. doi: 10.1007/s10827-015-0560-x. [19] M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Boston (USA), 1994. [20] S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of cortical neurons, Neural Computation, 11 (1997), 935-951. [21] T. Taillefumier and M. 0. Magnasco, A phase transition in the first passage of a Brownian process through a fluctuating boundary: Implications for neural coding, PNAS, 110 (2013), E1438-E1443. doi: 10.1073/pnas.1212479110. [22] T. Taillefumier and M. Magnasco, A transition to sharp timing in stochastic leaky integrate-and-fire neurons driven by frozen noisy input, Neural Computation, 26 (2014), 819-859. doi: 10.1162/NECO_a_00577. [23] T. Taillefumier and M. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Holder continuous boundaries, J. Stat. Phys., 140 (2010), 1130-1156. doi: 10.1007/s10955-010-0033-6. [24] P. J. Thomas, A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process, J. Appl. Probab., 48 (2011), 420-434. doi: 10.1239/jap/1308662636. [25] J. V. Toups, J. M. Fellous, P. J. Thomas, T. J. Sejnowski and P. H. Tiesinga, Multiple spike time patterns occur at bifurcation points of membrane potential dynamics, PLoS Comput. Biol., 8 (2012), e1002615, 1-19. doi: 10.1371/journal.pcbi.1002615. [26] H. C. Tuckwell, Stochastic Processes in the Neurosciences, SIAM, 1989. doi: 10.1137/1.9781611970159. [27] E. Urdapilleta, Series solution to the first-passage-time problem of a Brownian motion with an exponential time-dependent drift, J. Stat. Phys., 140 (2010), 1130-1156. doi: 10.1088/1751-8113/45/18/185001.
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