American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 195-203. doi: 10.3934/proc.2015.0195

Stochastic modeling of the firing activity of coupled neurons periodically driven

Maria Francesca Carfora 1, and  Enrica Pirozzi 2,

1. 

Istituto per le Applicazioni del Calcolo CNR, Napoli, Italy
2. 

Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Napoli

Received  September 2014 Revised  November 2014 Published  November 2015

A stochastic model for describing the firing activity of a couple of interacting neurons subject to time-dependent stimuli is proposed. Two stochastic differential equations suitably coupled and including periodic terms to represent stimuli imposed to one or both neurons are considered to describe the problem. We investigate the first passage time densities through specified firing thresholds for the involved time non-homogeneous Gauss-Markov processes. We provide simulation results and numerical approximations of the firing densities. Asymptotic behaviors of the first passage times are also given.
Keywords: LIF neuronal model, first passage time, asymptotic regime., Gauss--Markov processes, periodic stimulus.
Mathematics Subject Classification: Primary: 60J70; Secondary: 92--0.
Citation: Maria Francesca Carfora, Enrica Pirozzi. Stochastic modeling of the firing activity of coupled neurons periodically driven. Conference Publications, 2015, 2015 (special) : 195-203. doi: 10.3934/proc.2015.0195
References:
[1]

K. Amemori and S. Ishii, Gaussian Process Approach to Spiking Neurons for Inhomogeneous Poisson Inputs. Neural Comp., 13(12) (2001), 2763-2797.

[2]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons, Mathematical Biosciences and Engineering, 11(2) (2014), 189-201.

[3]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model. Neural Comput. 22(10) (2010), 2558-2585.

[4]

A. Buonocore, L. Caputo, M. F. Carfora and E. Pirozzi, On the Dynamics of a Couple of Mutually Interacting Neurons, in Computer Aided Systems Theory-EUROCAST 2013, LNCS, Springer, (2013), 36-44.

[5]

A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models, Mathematical Biosciences, 214(1-2) (2008), 122-133

[6]

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. Probab., 13(2) (2011), 29-57.

[7]

A. Di Crescenzo, B. Martinucci and E. Pirozzi, Feedback Effects in Simulated Stein's Coupled Neurons, Lecture Notes in Computer Science 3643 (2005), 436-446.

[8]

A. Di Crescenzo, B. Martinucci and E. Pirozzi, On the dynamics of a pair of coupled neurons subject to alternating input rates, BioSystems, 79 (2005), 109-116.

[9]

E. Di Nardo, A.G. Nobile, E. Pirozzi and L.M. Ricciardi, A computational approach to the first-passage-time problems for Gauss-Markov processes, Adv Appl Prob, 33 (2001), 453-482.

[10]

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, Advances in applied probability, (1990), 883-914.

[11]

P. Lánský, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Phys. Rev. E 55(2) (1997), 2040-2043.

[12]

A.G. Nobile, E. Pirozzi and L.M. Ricciardi, Asymptotics and evaluations of FPT densities through varying boundaries for Gauss-Markov processes, Scientiae Mathematicae Japonicae, 67(2) (2008), 241-266.

[13]

A. Politi and S. Luccioli, Dynamics of Networks of Leaky-Integrate-and-Fire Neurons, Network Science, Springer (2010), 217-242.

[14]

L. Sacerdote, M. Tamborrino and C. Zucca, Detecting dependencies between spike trains of pairs of neurons through copulas, Brain Research 1434 (2011), 243-256.

[15]

M. Schindler, P. Talkner and P. Hanggi, Escape rates in periodically driven Markov processes, Physica A 351 (2005), 40-50.

[16]

R. Sirovich, L. Sacerdote and A.E.P. Villa, Effect of increasing inhibitory inputs on information processing within a small network of spiking neuron, Computational and Ambient Intelligence, Lecture Notes in Computer Science, Volume 4507/2007 (2007), 23-30.

[17]

H. Soula and CC. Chow, Stochastic dynamics of a finite-size spiking neural network. Neural Comput. 19(12) (2007), 3262-3292.

show all references

References:
[1]

K. Amemori and S. Ishii, Gaussian Process Approach to Spiking Neurons for Inhomogeneous Poisson Inputs. Neural Comp., 13(12) (2001), 2763-2797.

[2]

A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons, Mathematical Biosciences and Engineering, 11(2) (2014), 189-201.

[3]

A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model. Neural Comput. 22(10) (2010), 2558-2585.

[4]

A. Buonocore, L. Caputo, M. F. Carfora and E. Pirozzi, On the Dynamics of a Couple of Mutually Interacting Neurons, in Computer Aided Systems Theory-EUROCAST 2013, LNCS, Springer, (2013), 36-44.

[5]

A. Buonocore, L. Caputo and E. Pirozzi, On the evaluation of firing densities for periodically driven neuron models, Mathematical Biosciences, 214(1-2) (2008), 122-133

[6]

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. Probab., 13(2) (2011), 29-57.

[7]

A. Di Crescenzo, B. Martinucci and E. Pirozzi, Feedback Effects in Simulated Stein's Coupled Neurons, Lecture Notes in Computer Science 3643 (2005), 436-446.

[8]

A. Di Crescenzo, B. Martinucci and E. Pirozzi, On the dynamics of a pair of coupled neurons subject to alternating input rates, BioSystems, 79 (2005), 109-116.

[9]

E. Di Nardo, A.G. Nobile, E. Pirozzi and L.M. Ricciardi, A computational approach to the first-passage-time problems for Gauss-Markov processes, Adv Appl Prob, 33 (2001), 453-482.

[10]

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, Advances in applied probability, (1990), 883-914.

[11]

P. Lánský, Sources of periodical force in noisy integrate-and-fire models of neuronal dynamics, Phys. Rev. E 55(2) (1997), 2040-2043.

[12]

A.G. Nobile, E. Pirozzi and L.M. Ricciardi, Asymptotics and evaluations of FPT densities through varying boundaries for Gauss-Markov processes, Scientiae Mathematicae Japonicae, 67(2) (2008), 241-266.

[13]

A. Politi and S. Luccioli, Dynamics of Networks of Leaky-Integrate-and-Fire Neurons, Network Science, Springer (2010), 217-242.

[14]

L. Sacerdote, M. Tamborrino and C. Zucca, Detecting dependencies between spike trains of pairs of neurons through copulas, Brain Research 1434 (2011), 243-256.

[15]

M. Schindler, P. Talkner and P. Hanggi, Escape rates in periodically driven Markov processes, Physica A 351 (2005), 40-50.

[16]

R. Sirovich, L. Sacerdote and A.E.P. Villa, Effect of increasing inhibitory inputs on information processing within a small network of spiking neuron, Computational and Ambient Intelligence, Lecture Notes in Computer Science, Volume 4507/2007 (2007), 23-30.

[17]

H. Soula and CC. Chow, Stochastic dynamics of a finite-size spiking neural network. Neural Comput. 19(12) (2007), 3262-3292.

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