Before revision:  We study the classic model of linear activation cascades and find that in special but important cases the output of an entire cascade can be represented analytically as a function of the input and a lower incomplete gamma function.
After revision:  We study the classic model of linear activation cascades and find that in special but important case of optimal gain cascades (i.e., when all the deactivation rates are identical) the downstream output of an entire cascade can be represented analytically as a function of the input and a lower incomplete gamma function.