2014, 11(1): 1-10. doi: 10.3934/mbe.2014.11.1

A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

1. 

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

2. 

Istituto per le Appplicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, Via Pietro Castellino, Napoli, Italy

Received  December 2012 Revised  May 2013 Published  September 2013

A method to generate first passage times for a class of stochastic processes is proposed. It does not require construction of the trajectories as usually needed in simulation studies, but is based on an integral equation whose unknown quantity is the probability density function of the studied first passage times and on the application of the hazard rate method. The proposed procedure is particularly efficient in the case of the Ornstein-Uhlenbeck process, which is important for modeling spiking neuronal activity.
Citation: Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Maria Francesca Carfora. A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (1) : 1-10. doi: 10.3934/mbe.2014.11.1
References:
[1]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Advances in Applied Probability, 19 (1987), 784-800. doi: 10.2307/1427102.

[2]

A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.

[3]

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, Advances in Applied Probability, 33 (2001), 453-482. doi: 10.1239/aap/999188324.

[4]

Y. Dong, S. 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.

[5]

V. Giorno, A. G. Nobile, L. M. Ricciardi and S. Sato, On the evaluation of first-passage-time probability densities via non-singular integral equations, Advances in Applied Probability, 21 (1989), 20-36. doi: 10.2307/1427196.

[6]

M. T. Giraudo and L. Sacerdote, Simulation methods in neuronal modelling, Biosystems, 48 (1998), 77-83. doi: 10.1016/S0303-2647(98)00052-5.

[7]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Communications in Statistics-Simulation and Computation, 28 (1999), 1135-1163. doi: 10.1080/03610919908813596.

[8]

M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodology and Computing in Applied Probability, 3 (2001), 215-231. doi: 10.1023/A:1012261328124.

[9]

R. Gutiérrez Jáimez, P. Román Román and F. Torres Ruiz, A note on the Volterra integral equation for the first passage time probability density, Journal of Applied Probability, 32 (1995), 635-648. doi: 10.2307/3215118.

[10]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992.

[11]

P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes, Computers in Biology and Medicine, 24 (1994), 91-101. doi: 10.1016/0010-4825(94)90068-X.

[12]

P. Lánský, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model, Journal of Computational Neuroscience, 21 (2006), 211-223. doi: 10.1007/s10827-006-8527-6.

[13]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, Journal of Applied Probability, 22 (1985), 360-369. doi: 10.2307/3213779.

[14]

L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process, Journal of Mathematical Analysis and Applications, 54 (1976), 185-199. doi: 10.1016/0022-247X(76)90244-4.

[15]

S. M. Ross, "Introduction to the Probability Models,'' Academic Press, Elsevier, 2007.

[16]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Computation, 11 (1999), 935-951. doi: 10.1162/089976699300016511.

[17]

T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries, Journal of Statistical Physics, 140 (2010), 1130-1156. doi: 10.1007/s10955-010-0033-6.

[18]

H. C. Tuckwell and D. K. Cope, Accuracy of neuronal interspike times calculated from a diffusion approximation, Journal of Theoretical Biology, 83 (1980), 377-387. doi: 10.1016/0022-5193(80)90045-4.

[19]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 2. Nonlinear and Stochastic Theories," Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988.

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, Advances in Applied Probability, 19 (1987), 784-800. doi: 10.2307/1427102.

[2]

A. N. Burkitt, A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input, Biological Cybernetics, 95 (2006), 1-19. doi: 10.1007/s00422-006-0068-6.

[3]

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, Advances in Applied Probability, 33 (2001), 453-482. doi: 10.1239/aap/999188324.

[4]

Y. Dong, S. 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.

[5]

V. Giorno, A. G. Nobile, L. M. Ricciardi and S. Sato, On the evaluation of first-passage-time probability densities via non-singular integral equations, Advances in Applied Probability, 21 (1989), 20-36. doi: 10.2307/1427196.

[6]

M. T. Giraudo and L. Sacerdote, Simulation methods in neuronal modelling, Biosystems, 48 (1998), 77-83. doi: 10.1016/S0303-2647(98)00052-5.

[7]

M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Communications in Statistics-Simulation and Computation, 28 (1999), 1135-1163. doi: 10.1080/03610919908813596.

[8]

M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodology and Computing in Applied Probability, 3 (2001), 215-231. doi: 10.1023/A:1012261328124.

[9]

R. Gutiérrez Jáimez, P. Román Román and F. Torres Ruiz, A note on the Volterra integral equation for the first passage time probability density, Journal of Applied Probability, 32 (1995), 635-648. doi: 10.2307/3215118.

[10]

P. E. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992.

[11]

P. Lánský and V. Lánská, First-passage-time problem for simulated stochastic diffusion processes, Computers in Biology and Medicine, 24 (1994), 91-101. doi: 10.1016/0010-4825(94)90068-X.

[12]

P. Lánský, P. Sanda and J. He, The parameters of the stochastic leaky integrate-and-fire neuronal model, Journal of Computational Neuroscience, 21 (2006), 211-223. doi: 10.1007/s10827-006-8527-6.

[13]

A. G. Nobile, L. M. Ricciardi and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, Journal of Applied Probability, 22 (1985), 360-369. doi: 10.2307/3213779.

[14]

L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process, Journal of Mathematical Analysis and Applications, 54 (1976), 185-199. doi: 10.1016/0022-247X(76)90244-4.

[15]

S. M. Ross, "Introduction to the Probability Models,'' Academic Press, Elsevier, 2007.

[16]

S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Computation, 11 (1999), 935-951. doi: 10.1162/089976699300016511.

[17]

T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries, Journal of Statistical Physics, 140 (2010), 1130-1156. doi: 10.1007/s10955-010-0033-6.

[18]

H. C. Tuckwell and D. K. Cope, Accuracy of neuronal interspike times calculated from a diffusion approximation, Journal of Theoretical Biology, 83 (1980), 377-387. doi: 10.1016/0022-5193(80)90045-4.

[19]

H. C. Tuckwell, "Introduction to Theoretical Neurobiology. Vol. 2. Nonlinear and Stochastic Theories," Cambridge Studies in Mathematical Biology, 8, Cambridge University Press, Cambridge, 1988.

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