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A new firing paradigm for integrate and fire stochastic neuronal models
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
1. | Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria |
References:
[1] |
M. Abundo, Some results about boundary crossing for Brownian motion, Ric. Mat., 50 (2001), 283-301. |
[2] |
L. Alili, P. Patie and J. Pedersen, Representation of the first hitting time density of an Ornstein-Uhlenbeck process, Stoch. Models, 21 (2005), 967-980.
doi: 10.1080/15326340500294702. |
[3] |
K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process, J. Appl. Probab., 42 (2005), 82-92.
doi: 10.1239/jap/1110381372. |
[4] |
A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model, Math. Biosci. Eng., 11 (2014), 1-10. |
[5] |
A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19 (1987), 784-800.
doi: 10.2307/1427102. |
[6] |
R. M. Capocelli and L. M. Ricciardi, On the transformation of diffusion process into the Feller process, Math. Biosci., 29 (1976), 219-234.
doi: 10.1016/0025-5564(76)90104-8. |
[7] |
M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343. |
[8] |
M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Comput., 15 (2003), 253-278.
doi: 10.1162/089976603762552915. |
[9] |
M. J. Chacron, A. Longtin, M. St-Hilaire and L. Maler, Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors, Phys. Rev. Lett., 85 (2000), 1576-1579.
doi: 10.1103/PhysRevLett.85.1576. |
[10] |
R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications, Marcel Dekker, New York, 1989. |
[11] |
D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, CRC Press, 1977. |
[12] |
G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68.
doi: 10.1016/S0006-3495(64)86768-0. |
[13] |
M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Commun. Stat. Simulat., 28 (1999), 1135-1163.
doi: 10.1080/03610919908813596. |
[14] |
M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodol. Comput. App. Probab., 3 (2001), 215-231.
doi: 10.1023/A:1012261328124. |
[15] |
J. Honerkamp, Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis, Wiley/VCH, Weinheim, 1993. |
[16] |
R. Jolivet, A. Roth, F. Schurmann, W. Gerstner and W. Senn, Special issue on quantitative neuron modeling, Biol. Cybern., 99 (2008), 237-239.
doi: 10.1007/s00422-008-0274-5. |
[17] |
R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci., 3 (2009), 1-11.
doi: 10.3389/neuro.10.009.2009. |
[18] |
B. Lindner, Moments of the first passage time under weak external driving, J. Stat. Phys., 117 (2004), 703-737.
doi: 10.1007/s10955-004-2269-5. |
[19] |
B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521.
doi: 10.1016/j.jtbi.2004.08.030. |
[20] |
A. Metzler, On the first passage problem for correlated Brownian motion, Stat. Probabil. Lett., 80 (2010), 277-284.
doi: 10.1016/j.spl.2009.11.001. |
[21] |
A. Molini, P. Talkner, G. G. Katul and A. Porporato, First passage time statistics of Brownian motion with purely time dependent drift and diffusion, Physica A, 390 (2011), 1841-1852.
doi: 10.1016/j.physa.2011.01.024. |
[22] |
A. Novikov, V. Frishling and N. Kordzakhia, Approximations of boundary crossing probabilities for a Brownian motion, J. Appl. Probab., 36 (1999), 1019-1030.
doi: 10.1239/jap/1032374752. |
[23] |
K. Pötzelberger and L. Wang, Boundary crossing probability for Brownian motion, J. Appl. Probab., 38 (2001), 152-164.
doi: 10.1239/jap/996986650. |
[24] |
R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2014. |
[25] |
L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process, J. Math. Anal. Appl., 54 (1976), 185-199.
doi: 10.1016/0022-247X(76)90244-4. |
[26] |
L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Lecture notes in Biomathematics, 14, Springer Verlag, Berlin, 1977. |
[27] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japonica, 50 (1999), 247-322. |
[28] |
L. Sacerdote and M. T. Giraudo, Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications, in Stochastic Biomathematical Models, Lecture Notes in Mathematics, 2058, Springer Berlin Heidelberg, 2013, 99-148.
doi: 10.1007/978-3-642-32157-3_5. |
[29] |
L. Sacerdote, M. Tamborrino and C. Zucca, First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes, J. Comput. Appl. Math., 296 (2016), 275-292.
doi: 10.1016/j.cam.2015.09.033. |
[30] |
L. Sacerdote, O. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time dependent boundaries, Adv. Appl. Probab., 46 (2014), 186-202.
doi: 10.1239/aap/1396360109. |
[31] |
T. H. Scheike, A boundary-crossing results for Brownian motion, J. Appl. Probab., 29 (1992), 448-453.
doi: 10.2307/3214581. |
[32] |
S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Comput., 11 (1999), 935-951.
doi: 10.1162/089976699300016511. |
[33] |
T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries, J. Stat. Phys., 140 (2010), 1130-1156.
doi: 10.1007/s10955-010-0033-6. |
[34] |
M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion, Lifetime Data Anal., 21 (2015), 331-352.
doi: 10.1007/s10985-014-9307-7. |
[35] |
H. C. Tuckwell, Recurrent inhibition and afterhyperpolarization: Effects on neuronal discharge, Biol. Cybernet., 30 (1978), 115-123.
doi: 10.1007/BF00337325. |
[36] |
H. C. Tuckwell, Introduction to Theoretical Neurobiology, Volume 2. Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1988. |
[37] |
H. C. Tuckwell and F. Y. M. Wan, First passage time of Markov processes to moving barriers, J. Appl. Probab., 21 (1984), 695-709.
doi: 10.2307/3213688. |
[38] |
E. Urdapilleta, Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift, Phys. Rev. E, 83 (2011), 021102.
doi: 10.1103/PhysRevE.83.021102. |
[39] |
L. Wang and K. Pötzelberger, Boundary crossing probability for Brownian motion and general boundaries, J. App. Probab., 34 (1997), 54-65.
doi: 10.2307/3215174. |
[40] |
L. Wang and K. Pötzelberger, Crossing probabilities for diffusion processes with piecewise continuous boundaries, Methodol. Comput. Appl. Probab., 9 (2007), 21-40.
doi: 10.1007/s11009-006-9002-6. |
[41] |
C. Zucca and L. Sacerdote, On the inverse first-passage-time problem for a Wiener process, Ann. Appl. Probab., 19 (2009), 1319-1346.
doi: 10.1214/08-AAP571. |
show all references
References:
[1] |
M. Abundo, Some results about boundary crossing for Brownian motion, Ric. Mat., 50 (2001), 283-301. |
[2] |
L. Alili, P. Patie and J. Pedersen, Representation of the first hitting time density of an Ornstein-Uhlenbeck process, Stoch. Models, 21 (2005), 967-980.
doi: 10.1080/15326340500294702. |
[3] |
K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process, J. Appl. Probab., 42 (2005), 82-92.
doi: 10.1239/jap/1110381372. |
[4] |
A. Buonocore, L. Caputo, E. Pirozzi and M. F. Carfora, A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model, Math. Biosci. Eng., 11 (2014), 1-10. |
[5] |
A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19 (1987), 784-800.
doi: 10.2307/1427102. |
[6] |
R. M. Capocelli and L. M. Ricciardi, On the transformation of diffusion process into the Feller process, Math. Biosci., 29 (1976), 219-234.
doi: 10.1016/0025-5564(76)90104-8. |
[7] |
M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343. |
[8] |
M. J. Chacron, K. Pakdaman and A. Longtin, Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue, Neural Comput., 15 (2003), 253-278.
doi: 10.1162/089976603762552915. |
[9] |
M. J. Chacron, A. Longtin, M. St-Hilaire and L. Maler, Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors, Phys. Rev. Lett., 85 (2000), 1576-1579.
doi: 10.1103/PhysRevLett.85.1576. |
[10] |
R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications, Marcel Dekker, New York, 1989. |
[11] |
D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, CRC Press, 1977. |
[12] |
G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68.
doi: 10.1016/S0006-3495(64)86768-0. |
[13] |
M. T. Giraudo and L. Sacerdote, An improved technique for the simulation of first passage times for diffusion processes, Commun. Stat. Simulat., 28 (1999), 1135-1163.
doi: 10.1080/03610919908813596. |
[14] |
M. T. Giraudo, L. Sacerdote and C. Zucca, A Monte Carlo method for the simulation of first passage times of diffusion processes, Methodol. Comput. App. Probab., 3 (2001), 215-231.
doi: 10.1023/A:1012261328124. |
[15] |
J. Honerkamp, Stochastic Dynamical Systems. Concepts, Numerical Methods, Data Analysis, Wiley/VCH, Weinheim, 1993. |
[16] |
R. Jolivet, A. Roth, F. Schurmann, W. Gerstner and W. Senn, Special issue on quantitative neuron modeling, Biol. Cybern., 99 (2008), 237-239.
doi: 10.1007/s00422-008-0274-5. |
[17] |
R. Kobayashi, Y. Tsubo and S. Shinomoto, Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold, Front. Comput. Neurosci., 3 (2009), 1-11.
doi: 10.3389/neuro.10.009.2009. |
[18] |
B. Lindner, Moments of the first passage time under weak external driving, J. Stat. Phys., 117 (2004), 703-737.
doi: 10.1007/s10955-004-2269-5. |
[19] |
B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521.
doi: 10.1016/j.jtbi.2004.08.030. |
[20] |
A. Metzler, On the first passage problem for correlated Brownian motion, Stat. Probabil. Lett., 80 (2010), 277-284.
doi: 10.1016/j.spl.2009.11.001. |
[21] |
A. Molini, P. Talkner, G. G. Katul and A. Porporato, First passage time statistics of Brownian motion with purely time dependent drift and diffusion, Physica A, 390 (2011), 1841-1852.
doi: 10.1016/j.physa.2011.01.024. |
[22] |
A. Novikov, V. Frishling and N. Kordzakhia, Approximations of boundary crossing probabilities for a Brownian motion, J. Appl. Probab., 36 (1999), 1019-1030.
doi: 10.1239/jap/1032374752. |
[23] |
K. Pötzelberger and L. Wang, Boundary crossing probability for Brownian motion, J. Appl. Probab., 38 (2001), 152-164.
doi: 10.1239/jap/996986650. |
[24] |
R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2014. |
[25] |
L. M. Ricciardi, On the transformation of diffusion processes into the Wiener process, J. Math. Anal. Appl., 54 (1976), 185-199.
doi: 10.1016/0022-247X(76)90244-4. |
[26] |
L. M. Ricciardi, Diffusion Processes and Related Topics in Biology, Lecture notes in Biomathematics, 14, Springer Verlag, Berlin, 1977. |
[27] |
L. M. Ricciardi, A. Di Crescenzo, V. Giorno and A. Nobile, An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling, Math. Japonica, 50 (1999), 247-322. |
[28] |
L. Sacerdote and M. T. Giraudo, Stochastic Integrate and Fire Models: A Review on Mathematical Methods and Their Applications, in Stochastic Biomathematical Models, Lecture Notes in Mathematics, 2058, Springer Berlin Heidelberg, 2013, 99-148.
doi: 10.1007/978-3-642-32157-3_5. |
[29] |
L. Sacerdote, M. Tamborrino and C. Zucca, First passage times of two-dimensional correlated processes: Analytical results for the Wiener process and a numerical method for diffusion processes, J. Comput. Appl. Math., 296 (2016), 275-292.
doi: 10.1016/j.cam.2015.09.033. |
[30] |
L. Sacerdote, O. Telve and C. Zucca, Joint densities of first hitting times of a diffusion process through two time dependent boundaries, Adv. Appl. Probab., 46 (2014), 186-202.
doi: 10.1239/aap/1396360109. |
[31] |
T. H. Scheike, A boundary-crossing results for Brownian motion, J. Appl. Probab., 29 (1992), 448-453.
doi: 10.2307/3214581. |
[32] |
S. Shinomoto, Y. Sakai and S. Funahashi, The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex, Neural Comput., 11 (1999), 935-951.
doi: 10.1162/089976699300016511. |
[33] |
T. Taillefumier and M. O. Magnasco, A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries, J. Stat. Phys., 140 (2010), 1130-1156.
doi: 10.1007/s10955-010-0033-6. |
[34] |
M. Tamborrino, S. Ditlevsen and P. Lansky, Parameter inference from hitting times for perturbed Brownian motion, Lifetime Data Anal., 21 (2015), 331-352.
doi: 10.1007/s10985-014-9307-7. |
[35] |
H. C. Tuckwell, Recurrent inhibition and afterhyperpolarization: Effects on neuronal discharge, Biol. Cybernet., 30 (1978), 115-123.
doi: 10.1007/BF00337325. |
[36] |
H. C. Tuckwell, Introduction to Theoretical Neurobiology, Volume 2. Nonlinear and Stochastic Theories, Cambridge University Press, Cambridge, 1988. |
[37] |
H. C. Tuckwell and F. Y. M. Wan, First passage time of Markov processes to moving barriers, J. Appl. Probab., 21 (1984), 695-709.
doi: 10.2307/3213688. |
[38] |
E. Urdapilleta, Survival probability and first-passage-time statistics of a Wiener process driven by an exponential time-dependent drift, Phys. Rev. E, 83 (2011), 021102.
doi: 10.1103/PhysRevE.83.021102. |
[39] |
L. Wang and K. Pötzelberger, Boundary crossing probability for Brownian motion and general boundaries, J. App. Probab., 34 (1997), 54-65.
doi: 10.2307/3215174. |
[40] |
L. Wang and K. Pötzelberger, Crossing probabilities for diffusion processes with piecewise continuous boundaries, Methodol. Comput. Appl. Probab., 9 (2007), 21-40.
doi: 10.1007/s11009-006-9002-6. |
[41] |
C. Zucca and L. Sacerdote, On the inverse first-passage-time problem for a Wiener process, Ann. Appl. Probab., 19 (2009), 1319-1346.
doi: 10.1214/08-AAP571. |
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