Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T11:52:33.841Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lower Bound for the First Passage Time Density of the Suprathreshold Ornstein-Uhlenbeck Process

Part of: Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  14 July 2016

Peter J. Thomas
Peter J. Thomas*
Affiliation:
Case Western Reserve University
*
∗Postal address: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA. Email address: pjthomas@case.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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[-pe6βt] 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.

Keywords

Leaky integrate and fire neuronOrnstein-Uhlenbeck processreliabilitysynchronizationneural modellower bound

MSC classification

Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92C20: Neural biology
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 420 - 434
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Capocelli, R. M. and Ricciardi, L. M. (1971). Diffusion approximation and first passage time problem for a model neuron. Kybernetik 8, 214–223.Google Scholar
[2] Coombes, S. and Bressloff, P. C. (1999). Mode locking and Arnold tongues in integrate-and-fire neural oscillators. Phys. Rev. E 60, 2086–2096.Google Scholar
[3] Cox, D. R. and Miller, H. D. (1965). The Theory of Stochastic Processes. John Wiley, New York.Google Scholar
[4] 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.Google Scholar
[5] 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.Google Scholar
[6] 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.Google Scholar
[7] 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.CrossRefGoogle ScholarPubMed
[8] 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.Google Scholar
[9] Lasota, A. and Mackey, M. C. (1994). Chaos, Fractals, and Noise (Appl. Math. Sci. 97). Springer, New York.Google Scholar
[10] 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.Google Scholar
[11] Lindner, B. (2004). Moments of the first passage time under external driving. J. Statist. Phys. 117, 703–737.Google Scholar
[12] Loader, C. R. and Deely, J. J. (1987). Computations of boundary crossing probabilities for the Wiener process. J. Statist. Comput. Simul. 27, 95–105.CrossRefGoogle Scholar
[13] Mainen, Z. F. and Sejnowski, T. J. (1995). {Reliability of spike timing in neocortical neurons}. Science 268, 1503–1506.Google Scholar
[14] 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.CrossRefGoogle Scholar
[15] 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.Google Scholar
[16] Pakdaman, K. (2001). Periodically forced leaky integrate-and-fire model. Phys. Rev. E 63, 041907, 5 pp.CrossRefGoogle ScholarPubMed
[17] Pauwels, E. J. (1987). Smooth first-passage densities for one-dimensional diffusions. J. Appl. Prob. 24, 370–377.Google Scholar
[18] Pinsky, M. and Karlin, S. (2011). An Introduction to Stochastic Modeling, 4th edn. Academic Press, Burlington, CA.Google Scholar
[19] Reinagel, P. and Reid, R. C. (2002). Precise firing events are conserved across neurons. J. Neurosci. 22, 6837–6841.Google Scholar
[20] 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.Google Scholar
[21] Ricciardi, L. M. and Sato, S. (1988). First-passage-time density and moments of the Ornstein-Uhlenbeck process. J. Appl. Prob. 25, 43–57.Google Scholar
[22] 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.Google Scholar
[23] Shimokawa, T. et al. (2000). A first-passage-time analysis of the periodically forced noisy leaky integrate-and-fire model. Biol. Cybernet. 83, 327–340.Google Scholar
[24] 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.Google Scholar
[25] Tateno, T. (1998). Characterization of stochastic bifurcations in a simple biological oscillator. J. Statist. Phys. 92, 675–705.Google Scholar
[26] Tateno, T. (2002). Noise-induced effects on period-doubling bifurcation for integrate-and-fire oscillators. Phys. Rev. E 65, 021901, 10 pp.Google Scholar
[27] Tateno, T. and Jimbo, Y. (2000). Stochastic mode-locking for a noisy integrate-and-fire oscillator. Phys. Lett. A 271, 227–236.Google Scholar
[28] 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.Google Scholar
[29] 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.Google Scholar
[30] Tiesinga, P., Fellous, J.-M. and Sejnowski, T. J. (2008). Regulation of spike timing in visual cortical circuits. Nature Rev. Neurosci. 9, 97–107.CrossRefGoogle ScholarPubMed
[31] Toups, J. V. et al. (2011). Finding the event structure of neuronal spike trains. To appear in Neural Computation.Google Scholar
[32] Tuckwell, H. C. and Wan, F. Y. M. (1984). First-passage time of Markov processes to moving barriers. J. Appl. Prob. 21, 695–709.Google Scholar
[33] Wan, F. Y. M. and Tuckwell, H. C. (1982). Neuronal firing and input variability. J. Theoret. Neurobiol. 1, 197–218.Google Scholar