# American Institute of Mathematical Sciences

2014, 11(2): 189-201. doi: 10.3934/mbe.2014.11.189

## Gauss-diffusion processes for modeling the dynamics of a couple of interacting neurons

 1 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli 2 Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli

Received  October 2012 Revised  April 2013 Published  October 2013

With the aim to describe the interaction between a couple of neurons a stochastic model is proposed and formalized. In such a model, maintaining statements of the Leaky Integrate-and-Fire framework, we include a random component in the synaptic current, whose role is to modify the equilibrium point of the membrane potential of one of the two neurons and when a spike of the other one occurs it is turned on. The initial and after spike reset positions do not allow to identify the inter-spike intervals with the corresponding first passage times. However, we are able to apply some well-known results for the first passage time problem for the Ornstein-Uhlenbeck process in order to obtain (i) an approximation of the probability density function of the inter-spike intervals in one-way-type interaction and (ii) an approximation of the tail of the probability density function of the inter-spike intervals in the mutual interaction. Such an approximation is admissible for small instantaneous firing rates of both neurons.
Citation: 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
##### References:
 [1] K. Amemori and S. Ishii, Gaussian process approach to spiking neurons for inhomogeneous Poisson inputs, Neural Comp., 13 (2001), 2763-2797. doi: 10.1162/089976601317098529. [2] A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for evaluation of first-passage-time probability densities, Advances in Applied Probability, 19 (1987), 784-990. doi: 10.2307/1427102. [3] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Comput., 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023. [4] 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 (2011), 29-57. doi: 10.1007/s11009-009-9132-8. [5] A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [6] A. Di Crescenzo, B. Martinucci and E. Pirozzi, Feedback effects in simulated Stein's coupled neurons, in Computer Aided Systems Theory – EUROCAST 2005, Lecture Notes in Computer Science, 3643, Springer, Berlin-Heidelberg, 2005, 436-446. doi: 10.1007/11556985_57. [7] 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. doi: 10.1016/j.biosystems.2004.09.020. [8] E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, A computational approach to first-passage-time problems for Gauss-Markov processes, Adv. Appl. Prob., 33 (2001), 453-482. doi: 10.1239/aap/999188324. [9] Y. Dong, F. Mihalas and E. Niebur, Improved integral equation solution for the first passage time of leaky integrate-and-fire neurons, Neural Computation, 23 (2011), 421-434. doi: 10.1162/NECO_a_00078. [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, Adv. Appl. Prob., 22 (1990), 883-914. doi: 10.2307/1427567. [11] D. Golomb and G. B. Ermentrout, Bistability in pulse propagation in networks of excitatory and inhibitory populations, Phys. Rev. Lett., 68 (2001), 4179-4182. doi: 10.1103/PhysRevLett.86.4179. [12] A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On the estimation of first-passage time densities for a class of Gauss-Markov processes, in Computer Aided Systems Theory – EUROCAST 2007, Lecture Notes in Computer Science, 4739, Springer, Berlin-Heidelberg, 2007, 146-153. doi: 10.1007/978-3-540-75867-9_19. [13] 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 (2008), 241-266. Available from: http://www.jams.or.jp/scm/contents/e-2008-2/2008-12.pdf. [14] A. Politi and S. Luccioli, Dynamics of networks of leaky-integrate-and-fire neurons, in Network Science, Springer, London, 2010, 217-242. doi: 10.1007/978-1-84996-396-1_11. [15] L. Sacerdote, M. Tamborrino and C. Zucca, Detecting dependencies between spike trains of pairs of neurons through copulas, Brain Research, 1434 (2012), 243-256. doi: 10.1016/j.brainres.2011.08.064. [16] H. Sakaguchi, Oscillatory phase transition and pulse propagation in noisy integrate-and-fire neurons, Phys. Rev. E., 70 (2004), 1-4. doi: 10.1103/PhysRevE.70.022901. [17] H. Sakaguchi and S. Tobiishi, Synchronization and spindle oscillation in noisy integrate-and-fire-or-burst neurons with inhibitory coupling, Progress of Theoretical Physics, 114 (2005), 1-18. doi: 10.1143/PTP.114.539. [18] R. Sirovich, L. Sacerdote and A. E. P. Villa, Effect of increasing inhibitory inputs on information processing within a small network of spiking neurons, in Computational and Ambient Intelligence, Lecture Notes in Computer Science, 4507, Springer, Berlin-Heidelberg, 2007, 23-30. doi: 10.1007/978-3-540-73007-1_4. [19] H. Soula and C. C. Chow, Stochastic dynamics of a finite-size spiking neural network, Neural Comput., 19 (2007), 3262-3292. doi: 10.1162/neco.2007.19.12.3262.

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##### References:
 [1] K. Amemori and S. Ishii, Gaussian process approach to spiking neurons for inhomogeneous Poisson inputs, Neural Comp., 13 (2001), 2763-2797. doi: 10.1162/089976601317098529. [2] A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for evaluation of first-passage-time probability densities, Advances in Applied Probability, 19 (1987), 784-990. doi: 10.2307/1427102. [3] A. Buonocore, L. Caputo, E. Pirozzi and L. M. Ricciardi, On a stochastic leaky integrate-and-fire neuronal model, Neural Comput., 22 (2010), 2558-2585. doi: 10.1162/NECO_a_00023. [4] 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 (2011), 29-57. doi: 10.1007/s11009-009-9132-8. [5] A. N. Burkitt, A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input, Biol. Cybern., 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6. [6] A. Di Crescenzo, B. Martinucci and E. Pirozzi, Feedback effects in simulated Stein's coupled neurons, in Computer Aided Systems Theory – EUROCAST 2005, Lecture Notes in Computer Science, 3643, Springer, Berlin-Heidelberg, 2005, 436-446. doi: 10.1007/11556985_57. [7] 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. doi: 10.1016/j.biosystems.2004.09.020. [8] E. Di Nardo, A. G. Nobile, E. Pirozzi and L. M. Ricciardi, A computational approach to first-passage-time problems for Gauss-Markov processes, Adv. Appl. Prob., 33 (2001), 453-482. doi: 10.1239/aap/999188324. [9] Y. Dong, F. Mihalas and E. Niebur, Improved integral equation solution for the first passage time of leaky integrate-and-fire neurons, Neural Computation, 23 (2011), 421-434. doi: 10.1162/NECO_a_00078. [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, Adv. Appl. Prob., 22 (1990), 883-914. doi: 10.2307/1427567. [11] D. Golomb and G. B. Ermentrout, Bistability in pulse propagation in networks of excitatory and inhibitory populations, Phys. Rev. Lett., 68 (2001), 4179-4182. doi: 10.1103/PhysRevLett.86.4179. [12] A. G. Nobile, E. Pirozzi and L. M. Ricciardi, On the estimation of first-passage time densities for a class of Gauss-Markov processes, in Computer Aided Systems Theory – EUROCAST 2007, Lecture Notes in Computer Science, 4739, Springer, Berlin-Heidelberg, 2007, 146-153. doi: 10.1007/978-3-540-75867-9_19. [13] 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 (2008), 241-266. Available from: http://www.jams.or.jp/scm/contents/e-2008-2/2008-12.pdf. [14] A. Politi and S. Luccioli, Dynamics of networks of leaky-integrate-and-fire neurons, in Network Science, Springer, London, 2010, 217-242. doi: 10.1007/978-1-84996-396-1_11. [15] L. Sacerdote, M. Tamborrino and C. Zucca, Detecting dependencies between spike trains of pairs of neurons through copulas, Brain Research, 1434 (2012), 243-256. doi: 10.1016/j.brainres.2011.08.064. [16] H. Sakaguchi, Oscillatory phase transition and pulse propagation in noisy integrate-and-fire neurons, Phys. Rev. E., 70 (2004), 1-4. doi: 10.1103/PhysRevE.70.022901. [17] H. Sakaguchi and S. Tobiishi, Synchronization and spindle oscillation in noisy integrate-and-fire-or-burst neurons with inhibitory coupling, Progress of Theoretical Physics, 114 (2005), 1-18. doi: 10.1143/PTP.114.539. [18] R. Sirovich, L. Sacerdote and A. E. P. Villa, Effect of increasing inhibitory inputs on information processing within a small network of spiking neurons, in Computational and Ambient Intelligence, Lecture Notes in Computer Science, 4507, Springer, Berlin-Heidelberg, 2007, 23-30. doi: 10.1007/978-3-540-73007-1_4. [19] H. Soula and C. C. Chow, Stochastic dynamics of a finite-size spiking neural network, Neural Comput., 19 (2007), 3262-3292. doi: 10.1162/neco.2007.19.12.3262.
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