Abstract
The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant τ is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.
Similar content being viewed by others
References
Abramowitz, M., & Stegun, I. R. (1972). Handbook of mathematical functions (9th ed.). New York: Dover Publications Inc.
Arnold, L. (1974). Stochastic differential equations: Theory and applications. New York: John Wiley and Sons.
Bender, C. M., & Orzag, S. A. (1978). Advanced mathematical methods for scientists and engineers. New York: McGraw Hill.
Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematical Computing, 19, 577–593.
Burkitt, A. N., & Clark, G. M. (2000). Calculation of interspike intervals for integrate-and-fire neurons with poisson distribution of synaptic inputs. Neural Computation, 12, 1789–1820.
Burkitt, A. N. (2001). Balanced neurons: Analysis of leaky integrate-and-fire neurons with reversal potentials. Biological Cybernetics, 85, 247–255.
Burkitt, A. N. (2006a). A review of the integrate-and-fire neuron model: I. Homogenous synaptic input. Biological Cybernetics, 95, 1–19.
Burkitt, A. N. (2006b). A review of the integrate-and-fire neuron model: II. Inhomogenous synaptic input and network properties. Biological Cybernetics, 95, 97–112.
Capocelli, R. M., & Ricciardi, L. M. (1972). On the inverse of the first passage time probability problem. Journal of Applied Probability, 9, 270–287.
Churchill, R. V. (1981). Operational mathematics. New York: McGraw Hill.
D’Amore, L., Laccetti, G., & Murli, A. (1999). An implementation of a fourier series method for the numerical inversion of the laplace transform. ACM Transactions on Mathematical Software, 25, 279–305.
Darling, D., & Siegert, A. (1953). The first passage problem for a continuous markov process. Annals of Mathematical Statistics, 24, 624–639.
De Hoog, F. R., Knight, J. H., & Stokes, A. N. (1982). An improved method for numerical inversion of laplace transforms. SIAM Journal of Scientific and Statistical Computing, 3, 357–366.
Ditlevsen, S., & Ditlevsen, O. (2006). Parameter estimation from observations of first-passage times of the Ornstein–Uhlenbeck process and the feller process. Presented at the fifth computational stochastic mechanics conference.
Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerves. Journal of Physiology, 117, 500–544.
Inoue, J., Sato, S., & Ricciardi, L. (1995). On the parameter estimation for diffusion models of single neuron’s activities. Biological Cybernetics, 73, 209–221.
Jolivet, R., & Gerstner, W. (2004). Predicting spike times of a detailed conductance-based neuron model driven by stochastic spike arrival. Journal of Physiology (Paris), 98, 442–451.
Jolivet, R., Lewis, T. J., & Gerstner, W. (2004). Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. Journal of Neurophysiology, 92, 959–976.
Jolivet, R., Rauch, A., Lüscher, H. R., & Gerstner, W. (2006). Predicting spike timing of neocortical pyramidal neurons by simple threshold models. Journal of Computational Neuroscience, 21, 35–49.
Kano, P. O., Moysey, B., & Moloney, J. V. (2005). Application of weeks method for the numerical inversion of the laplace transform of the matrix exponential. Computational Mathematical Sciences, 3, 335–372.
Karlin, S., & Taylor, H. (1981). A second course in stochastic processes. New York: Academic Press.
Keat, J., Reinagel, P., Reid, R. K., & Meister, M. (2001). Predicting every spike: A model for the responses of visual neurons. Neuron, 30, 803–817.
Lánský, P., Sacerdote, L., & Tomassetti, F. (1995). On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biological Cybernetics, 73, 457–465.
Lánský, P., Sanda, P., & He, J. (2006). The parameters of the stochastic leaky integrate-and-fire neuronal model. Journal of Computational Neuroscience, 21, 211–223.
Lebedev, N. N. (1972). Special functions and their applications. New York: Dover Publications.
Lehmann, E. L. (1983). Theory of point estimation. New York: John Wiley and Sons.
Miller, J. C. P. (1955). National Physical Laboratory, tables of Weber parabolic cylinder functions. London: Her Majesty’s Stationary Office.
Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213.
Paninski, L., Pillow, J. W., & Simoncelli, E. P. (2004). Maximum likelihood estimation of a stochastic integrate-and-fire neural encoding model. Neural Computation, 16, 2533–2561.
Pillow, J. W., Paninski, L., Uzzel, V. J., Simoncelli, E. P., & Chichilnisky, E. J. (2005). Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. The Journal of Neuroscience, 25(47), 11003–11013.
Plesser, H. E., & Tanaka, S. (1997). Stochastic resonance in a model neuron with reset. Physics Letters A, 225, 228–234.
Rabinovich, M. I., Varona, P., Selverston, A. I., & Abarbanel, H. D. I. (2006). Dynamical principles in neuroscience. Reviews of Modern Physics, 70, 1213–1265.
Ricciardi, L., & Sacerdote, L. (1977). The Ornstein–Uhlenbeck process as a model for neuronal activity. Biological Cybernetics, 35, 1–9.
Ricciardi, L., & Sato, S. (1988). First passage time density and moments of the Ornstein–Uhlenbeck process. Journal of Applied Probability, 25, 43–57.
Sharp, A. A., O’Neil, M. B., Abbott, L. F., & Marder, E. (1993a). The dynamic clamp: Artificial conductances in biological neurons. Trends in Neuroscience, 16, 389–394.
Sharp, A. A., O’Neil, M. B., Abbott, L. F., & Marder, E. (1993b). The dynamic clamp: Computer-generated conductances in real neurons. Journal of Neurophysiology, 69, 992–995.
Shimokawa, T., Pakdaman, K., & Sato, S. (1999). Time-scale matching in the response of a leaky integrate-and-fire neuron model to periodic stimulus with additive noise. Physical Review E, 59, 3427–3443.
Siegert, A. J. F. (1951). On the first passage time probablity functioin. Physical Review, 81, 617–623.
Stein, R. B. (1965). A theoretical analysis of neuronal variability. Biophysical Journal, 5, 173–194.
Stephens, M. A. (1974). Edf statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730–737.
Tuckwell, H. (1988). Introduction to theoretical neurobiology. Volume 2: Nonlinear and stochastic theories. Cambridge: Cambridge University Press.
Uhlenbeck, G., & Ornstein, L. (1954). On the theory of brownian motion (1930). In: N. Wax (Ed.), Selected papers in noise and stochastic processes. New York: Dover Publications.
Weeks, W. T. (1966). Numerical inversion of the laplace transform using laguerre functions. Journal of Association of Computational Mathematics, 13, 419–429.
Author information
Authors and Affiliations
Corresponding author
Additional information
Action Editor: Wulfram Gerstner
Rights and permissions
About this article
Cite this article
Mullowney, P., Iyengar, S. Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data. J Comput Neurosci 24, 179–194 (2008). https://doi.org/10.1007/s10827-007-0047-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-007-0047-5