American Institute of Mathematical Sciences

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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.  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.  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.  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).  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, (1028), 171.  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.   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.  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.  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.  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.  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.  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.  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.   Google Scholar

[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.  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.  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.  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.  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.  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).  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, (1028), 171.  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.   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.  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.  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.  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.  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.  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.  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.   Google Scholar

[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.  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.  doi: 10.1364/JOSA.68.000386.  Google Scholar

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