American Institute of Mathematical Sciences

Advanced
2014, 11(2): 285-302. doi: 10.3934/mbe.2014.11.285

On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model

Virginia Giorno 1, and  Serena Spina 2,

1. 

Dipartimento di Studi e Ricerche Aziendali (Management &Information Technology), Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy
2. 

Dipartimento di Matematica, Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy

Received  October 2012 Revised  January 2013 Published  October 2013

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.
Keywords: jump diffusion process, constant refractoriness, exponential refractoriness., Ornstein-Uhlenbeck process.
Mathematics Subject Classification: Primary: 60J60, 92C20; Secondary: 60J7.
Citation: Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285
References:
[1]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. Appl. Prob., 19 (1987), 784-800. doi: 10.2307/1427102.  Google Scholar

[2]

A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models, Math. Biosci., 214 (2008), 122-133. doi: 10.1016/j.mbs.2008.02.003.  Google Scholar

[3]

R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybernetik (Berlin), 8 (1971), 214-223. doi: 10.1007/BF00288750.  Google Scholar

[4]

S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Phys. Rev. E (3), 71 (2005), 011907, 9 pp. doi: 10.1103/PhysRevE.71.011907.  Google Scholar

[5]

S. Ditlevsen and P. Lansky, Comparison of statistical methods for estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model from first-passage times data, in Collective Dynamics: Topics on Competition and Cooperation in the Biosciences (eds. L. M. Ricciardi, A. Buonocore and E. Pirozzi), AIP Conf. Proc., 1028, Amer. Inst. Phys., Melville, NY, 2008, 171-185. doi: 10.1063/1.2965085.  Google Scholar

[6]

G, Esposito, V. Giorno, A. G. Nobile, L. M. Ricciardi and C. Valerio, Neuronal modeling in the presence of random refractoriness, Sci. Math. Jpn., 64 (2006), 1-36.  Google Scholar

[7]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[8]

V. Giorno, A. G. Nobile and L. M. Ricciardi, On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries, Adv. in Appl. Probab., 22 (1990), 883-914. doi: 10.2307/1427567.  Google Scholar

[9]

M. T. Giraudo and L. Sacerdote, Jump-diffusion processes as models for neuronal activity, BioSystems, 40 (1997), 75-82. doi: 10.1016/0303-2647(96)01632-2.  Google Scholar

[10]

P. Lansky and L. Sacerdote, The Ornstein-Uhlenbeck neuronal model with signal-dependent noise, Physics Letters A, 285 (2001), 132-140. doi: 10.1016/S0375-9601(01)00340-1.  Google Scholar

[11]

P. Lansky, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model, J. Comput. Neurosci., 21 (2006), 211-223. doi: 10.1007/s10827-006-8527-6.  Google Scholar

[12]

L. M. Ricciardi and F. Esposito, On some distribution functions for non-linear switching elements with finite dead time, Kybernetik, 3 (1966), 148-152. doi: 10.1007/BF00288925.  Google Scholar

[13]

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, Math. Japon., 50 (1999), 247-322.  Google Scholar

[14]

L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biol. Cyb., 35 (1979), 1-9. doi: 10.1007/BF01845839.  Google Scholar

[15]

M. C. Teich, L. Matin and B. I. Cantor, Refractoriness in the maintained discharge of the cat's retinal ganglion cell, J. Opt. Soc. Am., 68 (1978), 386-402. doi: 10.1364/JOSA.68.000386.  Google Scholar

show all references

References:
[1]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. Appl. Prob., 19 (1987), 784-800. doi: 10.2307/1427102.  Google Scholar

[2]

A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models, Math. Biosci., 214 (2008), 122-133. doi: 10.1016/j.mbs.2008.02.003.  Google Scholar

[3]

R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron, Kybernetik (Berlin), 8 (1971), 214-223. doi: 10.1007/BF00288750.  Google Scholar

[4]

S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model, Phys. Rev. E (3), 71 (2005), 011907, 9 pp. doi: 10.1103/PhysRevE.71.011907.  Google Scholar

[5]

S. Ditlevsen and P. Lansky, Comparison of statistical methods for estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model from first-passage times data, in Collective Dynamics: Topics on Competition and Cooperation in the Biosciences (eds. L. M. Ricciardi, A. Buonocore and E. Pirozzi), AIP Conf. Proc., 1028, Amer. Inst. Phys., Melville, NY, 2008, 171-185. doi: 10.1063/1.2965085.  Google Scholar

[6]

G, Esposito, V. Giorno, A. G. Nobile, L. M. Ricciardi and C. Valerio, Neuronal modeling in the presence of random refractoriness, Sci. Math. Jpn., 64 (2006), 1-36.  Google Scholar

[7]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68. doi: 10.1016/S0006-3495(64)86768-0.  Google Scholar

[8]

V. Giorno, A. G. Nobile and L. M. Ricciardi, On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries, Adv. in Appl. Probab., 22 (1990), 883-914. doi: 10.2307/1427567.  Google Scholar

[9]

M. T. Giraudo and L. Sacerdote, Jump-diffusion processes as models for neuronal activity, BioSystems, 40 (1997), 75-82. doi: 10.1016/0303-2647(96)01632-2.  Google Scholar

[10]

P. Lansky and L. Sacerdote, The Ornstein-Uhlenbeck neuronal model with signal-dependent noise, Physics Letters A, 285 (2001), 132-140. doi: 10.1016/S0375-9601(01)00340-1.  Google Scholar

[11]

P. Lansky, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model, J. Comput. Neurosci., 21 (2006), 211-223. doi: 10.1007/s10827-006-8527-6.  Google Scholar

[12]

L. M. Ricciardi and F. Esposito, On some distribution functions for non-linear switching elements with finite dead time, Kybernetik, 3 (1966), 148-152. doi: 10.1007/BF00288925.  Google Scholar

[13]

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, Math. Japon., 50 (1999), 247-322.  Google Scholar

[14]

L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity, Biol. Cyb., 35 (1979), 1-9. doi: 10.1007/BF01845839.  Google Scholar

[15]

M. C. Teich, L. Matin and B. I. Cantor, Refractoriness in the maintained discharge of the cat's retinal ganglion cell, J. Opt. Soc. Am., 68 (1978), 386-402. doi: 10.1364/JOSA.68.000386.  Google Scholar

[1]

Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020143
[2]

Mondher Damak, Brice Franke, Nejib Yaakoubi. Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4093-4112. doi: 10.3934/dcds.2020173
[3]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
[4]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
[5]

Samuel Herrmann, Nicolas Massin. Exit problem for Ornstein-Uhlenbeck processes: A random walk approach. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3199-3215. doi: 10.3934/dcdsb.2020058
[6]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871
[7]

Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651
[8]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[9]

Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525
[10]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
[11]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649
[12]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092
[13]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298
[14]

Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019
[15]

Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339
[16]

Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
[17]

Kseniia Kravchuk, Alexander Vidybida. Non-Markovian spiking statistics of a neuron with delayed feedback in presence of refractoriness. Mathematical Biosciences & Engineering, 2014, 11 (1) : 81-104. doi: 10.3934/mbe.2014.11.81
[18]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735
[19]

Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[20]

Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences