A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process

A lower bound for the first passage time density of the suprathreshold Ornstein-Uhlenbeck process

Peter J. Thomas

Source: J. Appl. Probab. Volume 48, Number 2 (2011), 420-434.

Abstract

We prove that the first passage time density ρ(t) for an Ornstein-Uhlenbeck process X(t) obeying dX = -β Xdt + σdW to reach a fixed threshold θ from a suprathreshold initial condition x₀ > θ > 0 has a lower bound of the form ρ(t) > kexp[-pe^6βt] for positive constants k and p for times t exceeding some positive value u. We obtain explicit expressions for k, p, and u in terms of β, σ, x₀, and θ, and discuss the application of the results to the synchronization of periodically forced stochastic leaky integrate-and-fire model neurons.

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Primary Subjects: 60J70

Secondary Subjects: 92C20

Keywords: Leaky integrate and fire neuron; Ornstein-Uhlenbeck process; reliability; synchronization; neural model; lower bound

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Permanent link to this document: http://projecteuclid.org/euclid.jap/1308662636
Digital Object Identifier: doi:10.1239/jap/1308662636
Zentralblatt MATH identifier: 05918592
Mathematical Reviews number (MathSciNet): MR2840308

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References

Capocelli, R. M. and Ricciardi, L. M. (1971). Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8, 214–223.

Mathematical Reviews (MathSciNet): MR444097

Digital Object Identifier: doi:10.1007/BF00288750

Zentralblatt MATH: 0224.92008

Coombes, S. and Bressloff, P. C. (1999). Mode locking and Arnold tongues in integrate-and-fire neural oscillators. Phys. Rev. E 60, 2086–2096.

Mathematical Reviews (MathSciNet): MR1707378

Digital Object Identifier: doi:10.1103/PhysRevE.60.2086

Cox, D. R. and Miller, H. D. (1965). The Theory of Stochastic Processes. John Wiley, New York.

Mathematical Reviews (MathSciNet): MR192521

Doi, S., Inoue, J. and Kumagai, S. (1998). Spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der Pol oscillator with additive noise. J. Statist. Phys. 90, 1107–1127.

Mathematical Reviews (MathSciNet): MR1628320

Zentralblatt MATH: 0923.60067

Digital Object Identifier: doi:10.1023/A:1023271109747

Fellous, J.-M., Tiesinga, P. H. E., Thomas, P. J. and Sejnowski, T. J. (2004). Discovering spike patterns in neuronal responses. J. Neurosci. 24, 2989–3001.

Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1990). On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries. Adv. Appl. Prob. 22, 883–914.

Mathematical Reviews (MathSciNet): MR1082198

Zentralblatt MATH: 0717.60088

Digital Object Identifier: doi:10.2307/1427567

Hunter, J. D., Milton, J. G., Thomas, P. J. and Cowan, J. D. (1998). Resonance effect for neural spike time reliability. J. Neurophysiology 80, 1427–1438.

Keener, J. P., Hoppensteadt, F. C. and Rinzel, J. (1981). Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math. 41, 503–517.

Mathematical Reviews (MathSciNet): MR639130

Digital Object Identifier: doi:10.1137/0141042

Lasota, A. and Mackey, M. C. (1994). Chaos, Fractals, and Noise (Appl. Math. Sci. 97). Springer, New York.

Mathematical Reviews (MathSciNet): MR1244104

Zentralblatt MATH: 0784.58005

Lehmann, A. (2002). Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes. Adv. Appl. Prob. 34, 869–887.

Mathematical Reviews (MathSciNet): MR1938946

Zentralblatt MATH: 1026.60046

Digital Object Identifier: doi:10.1239/aap/1037990957

Project Euclid: euclid.aap/1037990957

Lindner, B. (2004). Moments of the first passage time under external driving. J. Statist. Phys. 117, 703–737.

Mathematical Reviews (MathSciNet): MR2099734

Zentralblatt MATH: 1105.60054

Digital Object Identifier: doi:10.1007/s10955-004-2269-5

Loader, C. R. and Deely, J. J. (1987). Computations of boundary crossing probabilities for the Wiener process. J. Statist. Comput. Simul. 27, 95–105.

Mathematical Reviews (MathSciNet): MR966812

Zentralblatt MATH: 0622.60090

Digital Object Identifier: doi:10.1080/00949658708810984

Mainen, Z. F. and Sejnowski, T. J. (1995). Reliability of spike timing in neocortical neurons. Science 268, 1503–1506.

Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of Ornstein-Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.

Mathematical Reviews (MathSciNet): MR789359

Zentralblatt MATH: 0564.60076

Digital Object Identifier: doi:10.2307/3213779

Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution. J. Appl. Prob. 22, 611–618.

Mathematical Reviews (MathSciNet): MR799284

Zentralblatt MATH: 0575.60076

Digital Object Identifier: doi:10.2307/3213864

Pakdaman, K. (2001). Periodically forced leaky integrate-and-fire model. Phys. Rev. E 63, 041907, 5pp.

Pauwels, E. J. (1987). Smooth first-passage densities for one-dimensional diffusions. J. Appl. Prob. 24, 370–377.

Mathematical Reviews (MathSciNet): MR889801

Zentralblatt MATH: 0622.60063

Digital Object Identifier: doi:10.2307/3214261

Pinsky, M. and Karlin, S. (2011). An Introduction to Stochastic Modeling, 4th edn. Academic Press, Burlington, CA.

Reinagel, P. and Reid, R. C. (2002). Precise firing events are conserved across neurons. J. Neurosci. 22, 6837–6841.

Rescigno, A., Stein, R. B., Purple, R. L. and Poppele, R. E. (1970). A neuronal model for the discharge patterns produced by cyclic inputs. Bull. Math. Biophys. 32, 337–353.

Ricciardi, L. M. and Sato, S. (1988). First-passage-time density and moments of the Ornstein-Uhlenbeck process. J. Appl. Prob. 25, 43–57.

Mathematical Reviews (MathSciNet): MR929503

Zentralblatt MATH: 0651.60080

Digital Object Identifier: doi:10.2307/3214232

Sato, S. (1977). Evaluation of the first-passage time probability to a square root boundary for the Wiener process. J. Appl. Prob. 14, 850–856.

Mathematical Reviews (MathSciNet): MR471056

Zentralblatt MATH: 0408.60077

Digital Object Identifier: doi:10.2307/3213358

Shimokawa, T. \et (2000). A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model. Biol. Cybernet. 83, 327–340.

Stiefel, K. M., Fellous, J. M., Thomas, P. J. and Sejnowski, T. J. (2010). Intrinsic subthreshold oscillations extend the influence of inhibitory synaptic inputs on cortical pyramidal neurons. Eur. J. Neurosci. 31, 1019–1026.

Tateno, T. (1998). Characterization of stochastic bifurcations in a simple biological oscillator. J. Statist. Phys. 92, 675–705.

Mathematical Reviews (MathSciNet): MR1649019

Zentralblatt MATH: 0932.92001

Digital Object Identifier: doi:10.1023/A:1023048923644

Tateno, T. (2002). Noise-induced effects on period-doubling bifurcation for integrate-and-fire oscillators. Phys. Rev. E 65, 021901, 10pp.

Mathematical Reviews (MathSciNet): MR1908292

Tateno, T. and Jimbo, Y. (2000). Stochastic mode-locking for a noisy integrate-and-fire oscillator. Phys. Lett. A 271, 227–236.

Mathematical Reviews (MathSciNet): MR1768652

Zentralblatt MATH: 1115.34335

Digital Object Identifier: doi:10.1016/S0375-9601(00)00370-4

Tateno, T., Doi, S., Sato, S. and Ricciardi, L. M. (1995). Stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise: a first-passage-time approach. J. Statist. Phys. 78, 917–935.

Thomas, P. J., Tiesinga, P. H., Fellous, J. M. and Sejnowski, T. J. (2003). Reliability and bifurcation in neurons driven by multiple sinusoids. Neurocomput. 52-54, 955–961.

Tiesinga, P., Fellous, J.-M. and Sejnowski, T. J. (2008). Regulation of spike timing in visual cortical circuits. Nature Rev. Neurosci. 9, 97–107.

Toups, J. V. et al. (2011). Finding the event structure of neuronal spike trains. To appear in Neural Computation.

Tuckwell, H. C. and Wan, F. Y. M. (1984). First-passage time of Markov processes to moving barriers. J. Appl. Prob. 21, 695–709.

Mathematical Reviews (MathSciNet): MR766808

Zentralblatt MATH: 0551.60087

Digital Object Identifier: doi:10.2307/3213688

Wan, F. Y. M. and Tuckwell, H. C. (1982). Neuronal firing and input variability. J. Theoret. Neurobiol. 1, 197–218.

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