# American Institute of Mathematical Sciences

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

## Stochastic modeling of the firing activity of coupled neurons periodically driven

 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.
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.
 [1] Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011 [2] Marie Levakova. Effect of spontaneous activity on stimulus detection in a simple neuronal model. Mathematical Biosciences & Engineering, 2016, 13 (3) : 551-568. doi: 10.3934/mbe.2016007 [3] Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1689-1707. doi: 10.3934/jimo.2021039 [4] Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $MAP/M/s+G$ queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021078 [5] Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 [6] Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks and Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019 [7] Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065 [8] Vladimir Kazakov. Sampling - reconstruction procedure with jitter of markov continuous processes formed by stochastic differential equations of the first order. Conference Publications, 2009, 2009 (Special) : 433-441. doi: 10.3934/proc.2009.2009.433 [9] Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132 [10] Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks and Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233 [11] Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079 [12] Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17 [13] Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics and Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639 [14] Artur Stephan, Holger Stephan. Memory equations as reduced Markov processes. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2133-2155. doi: 10.3934/dcds.2019089 [15] Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 [16] Zhenzhong Zhang, Enhua Zhang, Jinying Tong. Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2433-2455. doi: 10.3934/dcdsb.2018053 [17] Linyi Qian, Wei Wang, Rongming Wang. Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model. Journal of Industrial and Management Optimization, 2013, 9 (2) : 411-429. doi: 10.3934/jimo.2013.9.411 [18] Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116 [19] Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons. Mathematical Biosciences & Engineering, 2014, 11 (2) : 189-201. doi: 10.3934/mbe.2014.11.189 [20] Jan Haskovec, Ioannis Markou. Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime. Kinetic and Related Models, 2020, 13 (4) : 795-813. doi: 10.3934/krm.2020027

Impact Factor: