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

Previous Article
Stability and optimal control for some classes of tritrophic systems
MBE Home
This Issue
Next Article
Local versus nonlocal barycentric interactions in 1D agent dynamics

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

Abstract
Related Papers

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. doi: 10.2307/1427102.
[2]	A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models,, Math. Biosci., 214 (2008), 122. doi: 10.1016/j.mbs.2008.02.003.
[3]	R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron,, Kybernetik (Berlin), 8 (1971), 214. doi: 10.1007/BF00288750.
[4]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Phys. Rev. E (3), 71 (2005). doi: 10.1103/PhysRevE.71.011907.
[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*, (1028), 171. doi: 10.1063/1.2965085.
[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.
[7]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.
[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. doi: 10.2307/1427567.
[9]	M. T. Giraudo and L. Sacerdote, Jump-diffusion processes as models for neuronal activity,, BioSystems, 40 (1997), 75. doi: 10.1016/0303-2647(96)01632-2.
[10]	P. Lansky and L. Sacerdote, The Ornstein-Uhlenbeck neuronal model with signal-dependent noise,, Physics Letters A, 285 (2001), 132. doi: 10.1016/S0375-9601(01)00340-1.
[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. doi: 10.1007/s10827-006-8527-6.
[12]	L. M. Ricciardi and F. Esposito, On some distribution functions for non-linear switching elements with finite dead time,, Kybernetik, 3 (1966), 148. doi: 10.1007/BF00288925.
[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.
[14]	L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity,, Biol. Cyb., 35 (1979), 1. doi: 10.1007/BF01845839.
[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. doi: 10.1364/JOSA.68.000386.

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. doi: 10.2307/1427102.
[2]	A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models,, Math. Biosci., 214 (2008), 122. doi: 10.1016/j.mbs.2008.02.003.
[3]	R. M. Capocelli and L. M. Ricciardi, Diffusion approximation and first passage time problem for a model neuron,, Kybernetik (Berlin), 8 (1971), 214. doi: 10.1007/BF00288750.
[4]	S. Ditlevsen and P. Lansky, Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model,, Phys. Rev. E (3), 71 (2005). doi: 10.1103/PhysRevE.71.011907.
[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*, (1028), 171. doi: 10.1063/1.2965085.
[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.
[7]	G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron,, Biophys. J., 4 (1964), 41. doi: 10.1016/S0006-3495(64)86768-0.
[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. doi: 10.2307/1427567.
[9]	M. T. Giraudo and L. Sacerdote, Jump-diffusion processes as models for neuronal activity,, BioSystems, 40 (1997), 75. doi: 10.1016/0303-2647(96)01632-2.
[10]	P. Lansky and L. Sacerdote, The Ornstein-Uhlenbeck neuronal model with signal-dependent noise,, Physics Letters A, 285 (2001), 132. doi: 10.1016/S0375-9601(01)00340-1.
[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. doi: 10.1007/s10827-006-8527-6.
[12]	L. M. Ricciardi and F. Esposito, On some distribution functions for non-linear switching elements with finite dead time,, Kybernetik, 3 (1966), 148. doi: 10.1007/BF00288925.
[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.
[14]	L. M. Ricciardi and L. Sacerdote, The Ornstein-Uhlenbeck process as a model for neuronal activity,, Biol. Cyb., 35 (1979), 1. doi: 10.1007/BF01845839.
[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. doi: 10.1364/JOSA.68.000386.

[1]	Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451
[2]	Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235
[14]	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
[15]	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
[16]	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
[17]	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
[18]	Simona Fornaro, Abdelaziz Rhandi. On the Ornstein Uhlenbeck operator perturbed by singular potentials in $L^p$--spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5049-5058. doi: 10.3934/dcds.2013.33.5049
[19]	Giuseppe Da Prato. Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 637-647. doi: 10.3934/dcdss.2013.6.637
[20]	Ummugul Bulut, Edward J. Allen. Derivation of SDES for a macroevolutionary process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777

2017 Impact Factor: 1.23

American Institute of Mathematical Sciences