# American Institute of Mathematical Sciences

2016, 13(3): 495-507. doi: 10.3934/mbe.2016003

## Successive spike times predicted by a stochastic neuronal model with a variable input signal

 1 Dipartimento di Matematica e Applicazioni, Università degli studi di Napoli, FEDERICO II, Via Cinthia, Monte S.Angelo, Napoli, 80126, Italy 2 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli

Received  April 2015 Revised  November 2015 Published  January 2016

Two different stochastic processes are used to model the evolution of the membrane voltage of a neuron exposed to a time-varying input signal. The first process is an inhomogeneous Ornstein-Uhlenbeck process and its first passage time through a constant threshold is used to model the first spike time after the signal onset. The second process is a Gauss-Markov process identified by a particular mean function dependent on the first passage time of the first process. It is shown that the second process is also of a diffusion type. The probability density function of the maximum between the first passage time of the first and the second process is considered to approximate the distribution of the second spike time. Results obtained by simulations are compared with those following the numerical and asymptotic approximations. A general equation to model successive spike times is given. Finally, examples with specific input signals are provided.
Citation: Giuseppe D'Onofrio, Enrica Pirozzi. Successive spike times predicted by a stochastic neuronal model with a variable input signal. Mathematical Biosciences & Engineering, 2016, 13 (3) : 495-507. doi: 10.3934/mbe.2016003
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##### References:
 [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] 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 & Management Optimization, 2021  doi: 10.3934/jimo.2021039 [3] 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 & Management Optimization, 2021  doi: 10.3934/jimo.2021078 [4] Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170 [5] 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 & Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019 [6] 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 [7] 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 [8] Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks & Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233 [9] Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17 [10] Mathias Staudigl. A limit theorem for Markov decision processes. Journal of Dynamics & Games, 2014, 1 (4) : 639-659. doi: 10.3934/jdg.2014.1.639 [11] Artur Stephan, Holger Stephan. Memory equations as reduced Markov processes. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2133-2155. doi: 10.3934/dcds.2019089 [12] Zhenzhong Zhang, Enhua Zhang, Jinying Tong. Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2433-2455. doi: 10.3934/dcdsb.2018053 [13] Linyi Qian, Wei Wang, Rongming Wang. Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 411-429. doi: 10.3934/jimo.2013.9.411 [14] 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 [15] H.Thomas Banks, Shuhua Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. Mathematical Biosciences & Engineering, 2012, 9 (1) : 1-25. doi: 10.3934/mbe.2012.9.1 [16] Thomas Kruse, Mikhail Urusov. Approximating exit times of continuous Markov processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3631-3650. doi: 10.3934/dcdsb.2020076 [17] Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061 [18] A. M. Vershik. Polymorphisms, Markov processes, quasi-similarity. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1305-1324. doi: 10.3934/dcds.2005.13.1305 [19] S. Mohamad, K. Gopalsamy. Neuronal dynamics in time varying enviroments: Continuous and discrete time models. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 841-860. doi: 10.3934/dcds.2000.6.841 [20] Lakhdar Aggoun, Lakdere Benkherouf. A Markov modulated continuous-time capture-recapture population estimation model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1057-1075. doi: 10.3934/dcdsb.2005.5.1057

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