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First Passage Time Problems in Biology

When the voltage at a particular place on a neuron reaches a threshold, an action potential (nerve impulse) is produced. Many point processes in biology have similar origins as ``first passage times''; that is, they occur when some underlying process first reaches a critical level or threshold. In an environmental context, the level of pollutant in a lake may be considered as a recovery process with inputs at variable times; the threshold here may be a biological one, or an artificial one such as a government standard. In a growth process context, the threshold may be the optimal time to market a particular crop or animal.

Even for simple models of the underlying process (1-dimensional stochastic differential equations), very few analytical results are available for first passage times. Through simulation and heuristic approximation methods, several different types of behavior have been identified. The main current research activities are further development of approximation methods, and new application areas for the basic methods, including reliability theory and survival analysis. For example, of the common distributions used to model time to death (of an organism) or failure (of a manufactured product), what underlying stochastic process could produce them if failure is viewed as a first passage time?

CRSC research on this topic is mainly by C.E. Smith, in collaboration with graduate students and experimental neurobiologists.
















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