State-dependent fire models and related renewal processes

We introduce a general class of stochastic processes forced by instantaneous random fires (i.e., jumps) that reset the state variable x to a given value. Since in many physical systems the fire activity is often dependent on the actual value of the state variable, as in the case of natural fires in ecosystems and firing dynamics in neuronal activity, the frequency of fire occurrence is assumed to be state dependent. Such dynamics leads to independent interfire statistics-i.e., to renewal point processes. Various functions relating the frequency of fire occurrence to x(t) are analyzed and compared. The relation between the probabilistic dynamics of x(t) and the interfire statistics is derived and some exact probability distribution of both x(t) and the interfire times are obtained for systems with different degrees of complexity. After studying processes in which the fire activity is coupled only to a deterministic drift, we also analyze processes forced by either additive or multiplicative Gaussian white noise.