Abstract
Diffusion processes have been extensively used to describe membrane potential behavior. In this approach the interspike interval has a theoretical counterpart in the first-passage-time of the diffusion model employed. Since the mathematical complexity of the first-passage-time problem increases with attempts to make the models more realistic it seems useful to compare the features of different models in order to highlight their relative performance. In this paper we compare the Feller and Ornstein-Uhlenbeck models under three different criteria derived from the level of information available about their parameters. We conclude that the Feller model is preferable when complete knowledge of the characterizing parameters is assumed. On the other hand, when only limited information about the parameters is available, such as the mean firing time and the histogram shape, no advantage arises from using this more complex model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aertsen A (ed) (1993) Brain theory, Elsevier, Amsterdam
Buonocore A, Nobile AG, Ricciardi LM (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv Appl Prob 19:784–800
Capocelli RM, Ricciardi LM (1971) Diffusion approximation and the first passage time for a model neuron. Kybernetik 8:214–223
Cox DR, Miller HD (1965) The theory of stochastic processes. Chapman and Hall, London
Durbin J (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J Appl Prob 8:431–453
Feller W (1951) Diffusion processes in genetics. In: Neyman J (ed) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp 227–246
Gibbons JD (1971) Nonparametric statistical inference. McGraw-Hill, New York
Giorno V, Lánský P, Nobile AG, Ricciardi LM (1988) Diffusion approximation and first-passage-time problem for a model neuron. III. A birth-and-death process approach. Biol Cybern 58:387–404
Giorno V, Nobile AG, Ricciardi LM, Sato S (1989) On the evaluation of first-passage-time probability densities via nonsingular integral equations. Adv Appl Prob 21:20–36
Giorno V, Nobile AG, Ricciardi LM (1990) On the asymptotic behavior of first-passage-time densities for one-dimensional diffusion processes and varying boundaries. Adv Appl Prob 22:883–914
Hanson FB, Tuckwell HC (1983) Diffusion approximation for neuronal activity including synaptic reversal potentials. J Theor Neurobiol 2:127–153
Inoue J, Sato S, Ricciardi LM (1994) On the parameter estimation for diffusion models of single neurons' activity. Biol Cybern 73:209–221
Kallianpur G (1983) On the diffusion approximation to a discontinuous model for a single neuron. In: Sen PK (ed) Contributions to statistics. North-Holland, Amsterdam
Kallianpur G, Wolpert RL (1987) Weak convergence of stochastic neuronal models. In: Kimura M, Kallianpur G, Hida T (eds) Stochastic methods in biology. Springer, Berlin Heidelberg New York, pp 116–145
Karlin S, Taylor HM (1981) A second course in stochastic processes. Academic Press, New York
Keilson J, Ross HF (1975) Passage time distributions for Gaussian Markov (Ornstein-Uhlenbeck) statistical processes. Selected Tables in Mathematical Statistics 3:233–327
Lánská V, Lánsky P, Smith CA (1994) Synaptic transmission in a diffusion model for neural activity. J Theor Biol 166:393–406
Lánský P (1984) On approximations of Stein's neuronal model. J Theor Biol 107:631–647
Lánský P, Lánská V (1987) Diffusion approximations of the neuronal model with synaptic reversal potentials. Biol Cybern 56:19–26
Lánsky P, Lánská V (1994) First-passage-time problem for simulated stochastic diffusion processes. Comput Biol Med 24:91–101
Lánský P, Musila M (1991) Variable initial depolarization in Stein's neuronal model with synaptic reversal potentials. Biol Cybern 64:285–291
McKenna T, Davis J, Zornetzer SF (eds) (1992) Single neuron computation. Academic Press, Boston, Mass.
Musila M, Lánský P (1994) On the interspike intervals calculated from diffusion approximations of Stein's neuronal model with reversal potential. J Theor Biol 171:225–232
Nobile AG, Ricciardi LM, 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
Pribram KH (eds) (1994) Origins: Brain and self organization. Erlbaum, Hillsdale, NJ
Ricciardi LM (1977) Diffusion processes and related topics in biology, Springer, Berlin
Ricciardi LM, Sacerdote L (1979) The Ornstein-Uhlenbeck process as a model of neuronal activity. Biol Cybern 35:1–9
Ricciardi LM, Sato S (1990) Diffusion processes and first-passage-time problems. In: Ricciardi LM (ed) Lectures in applied mathematics and informatics. Manchester University Press, Manchester
Ricciardi LM, Sacerdote L, Sato S (1983) Diffusion approximation and first-passage-time problem for a model neuron. II. Outline of a computation method. Math Biosci 64:29–44
Sacerdote L, Tomassetti F (1994) On evaluation and asymptotic approximations of first-passage-time probabilities. Preprint
Sato S (1978) On the moments of the firing interval of diffusion approximated neuron. Math Biosci 34:53–70
Schmidt RF (ed) (1984) Fundamentals of neurophysiology. Springer, Berlin Heidelberg New York
Stein RB (1965) A theoretical analysis of neuronal variability. Biophys J 5:173–195
Tuckwell HC (1979) Synaptic transmission in a model for stochastic neural activity. J Theor Biol 77:65–81
Tuckwell HC (1988) Introduction to theoretical neurobiology. Cambridge University Press, Cambridge
Tuckwell HC, Cope DK (1980) Accuracy of neuronal interspike times calculated from a diffusion approximation. J Theor Biol 83:377–387
Tuckwell HC, Richter W (1978) Neuronal interspike time distributions and the estimation of neurophysiological and neuroanatomical arameters. J Theor Biol 71:167–180
Ventriglia F (eds) (1994) Neural modeling and neural networks. Pergamon, Oxford
Wilbur AJ, Rinzel J (1983) A theoretical basis for large coefficient of variation and bimodality in neuronal interspike interval distribution. J Theor Biol 105:345–368
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lánský, P., Sacerdote, L. & Tomassetti, F. On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity. Biol. Cybern. 73, 457–465 (1995). https://doi.org/10.1007/BF00201480
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00201480