Several heuristic methods can be used to approximate the distributions and moments of first passage time for neural diffusion models [1]. Deterministic crossings, i.e. when the mean voltage trajectory crosses the threshold in a finite time, and nondeterministic crossings due to fluctuations from the mean trajectory ,which itself is below the voltage threshold, partition approximations into perturbation type methods and Poisson-like approximations. While these methods provide some intuition into the behaviour of diffusion neural models and which parameters the mean and variance of the first passage times are sensitive to, they lack the ability to determine which distributions are ordered with regard to type of model or parameter variations.
The present work uses the notion of stochastic ordering to determine sufficient conditions for the ordering of first passage time distributions given the infinitesimal mean and variance of the corresponding diffusion process. Comparisons among and between several generalizations of Steinユs model for the sub-treshold behaviour of the membrane potential of a neuron are used to illustrate the method as in [2],[3]. The Ornstein-Uhlenbeck , Feller, and Lanska-Lansky-Smith [4] process models serve to represent the presence or absence of a neural reversal potential. For example, for a suitable matching of mean membrance voltage trajectories which of the two processes tend to produce faster firing in various scenarios such as high excitatory inputs, roughly equal excitatory and inhibitory inputs, and a dominant inhibitory input. Compared to earlier heuristic methods the whole range of behaviour can be examined in a common framework, namely stochastic ordering.
While the parameter values appropriate to a neural context are considered here, the methodology used appears to have promise in other biological applications where diffusion models are used, such as population growth and tumor growth models.
REFERENCES
[1] Smith C.E., A heuristic approach to stochastic models of single neurons. In Single Neuron Computation: Neural Nets foundations and applications. (T. McKenna, J. Davis and S. Zornetzer, eds.), Academic Press, New York,, pp. 561-328588, 1992.
[2] Sacerdote L., Smith C.E., New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural model. Biosystems 58:59-65, 2000.
[3] Sacerdote L., Smith C.E., A qualitative comparison of some diffusion models for neural activity via stochastic ordering. Biol. Cybern 83:543-551, 2001.
[4] Lanska V., Lansky P., Smith C.E. Synaptic transmission in a diffusion model for neural activity. J. Theor. Biol. 166:393-406, 1994.