  We introduce a mathematically rigorous framework based on rough path theory
to model stochastic spiking neural networks (SSNNs) as stochastic differential
equations with event discontinuities (Event SDEs) and driven by c\`adl\`ag
rough paths. Our formalism is general enough to allow for potential jumps to be
present both in the solution trajectories as well as in the driving noise. We
then identify a set of sufficient conditions ensuring the existence of pathwise
gradients of solution trajectories and event times with respect to the
network's parameters and show how these gradients satisfy a recursive relation.
Furthermore, we introduce a general-purpose loss function defined by means of a
new class of signature kernels indexed on c\`adl\`ag rough paths and use it to
train SSNNs as generative models. We provide an end-to-end autodifferentiable
solver for Event SDEs and make its implementation available as part of the
$\texttt{diffrax}$ library. Our framework is, to our knowledge, the first
enabling gradient-based training of SSNNs with noise affecting both the spike
timing and the network's dynamics.
