The role of stochastic diffusion processes for modeling purposes is discussed. Special emphasis is put on neuronal firing problems and on the description of population dynamics, for which the first passage time distribution and its statistics carry a fundamental relevance, being representative of the firing mechanism and of extinction or survival, respectively. Methods to solve the first passage time problems, both of a theoretical and of a computational nature, are outlined and a number of accessory results are also presented. These include a quantitative treatment of the so-called return process, which describes the sequence of spikes released by a model neuron within Wiener, Ornstein-Uhlenbeck and Feller approximations. A thorough investigation is also performed to shade some light on the question of the role of environmental fluctuations on population growth processes. Following a simple procedure, it is argued that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of asymptotic limits of first passage time density and its moments is given, which is for instance useful to discuss neuronal firing under certain slow activity conditions.
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