...$\mu(t-t_0)$

Mathematically, a very similar model has been employed to describe the propagation of diseases through a population (Daley & Vere-Jones 2003). In this context, the function $\mu (t)$ is called ``infectivity'', for obvious reasons. We will adopt the same name for it, although the medical connotation is irrelevant here.

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...$I(t)\propto\exp(-t/\tau)$

A single exponentially-decaying pulse is described by function $I(t,\xi)=I_0~{\rm e}^{-t/\tau}\theta(t)$. Notice that in this simple case the only parameter, $\tau\equiv\xi$, has a meaning of the decay timescale.

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