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

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2016, 13(3): 613-629. doi: 10.3934/mbe.2016011

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

Massimiliano Tamborrino ^1,

1.	Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria

Received June 2015 Revised August 2015 Published January 2016

Abstract
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The first passage time density of a diffusion process to a time varying threshold is of primary interest in different fields. Here, we consider a Brownian motion in presence of an exponentially decaying threshold to model the neuronal spiking activity. Since analytical expressions of the first passage time density are not available, we propose to approximate the curved boundary by means of a continuous two-piecewise linear threshold. Explicit expressions for the first passage time density towards the new boundary are provided. First, we introduce different approximating linear thresholds. Then, we describe how to choose the optimal one minimizing the distance to the curved boundary, and hence the error in the corresponding passage time density. Theoretical means, variances and coefficients of variation given by our method are compared with empirical quantities from simulated data. Moreover, a further comparison with firing statistics derived under the assumption of a small amplitude of the time-dependent change in the threshold, is also carried out. Finally, maximum likelihood and moment estimators of the parameters of the model are derived and applied on simulated data.

Keywords: spike time, Hitting time, time-varying threshold, Brownian motion, maximum likelihood estimator., firing statistic, adaptive-threshold model, piecewise-linear threshold, boundary crossing probability.

Mathematics Subject Classification: Primary: 60G40, 62Mxx, 60J65, 60J70; Secondary: 62F1.

Citation: 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

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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