  Cortical networks exhibit synchronized activity which often occurs in
spontaneous events in the form of spike avalanches. Since synchronization has
been causally linked to central aspects of brain function such as selective
signal processing and integration of stimulus information, participating in an
avalanche is a form of a transient synchrony which temporarily creates neural
assemblies and hence might especially be useful for implementing flexible
information processing. For understanding how assembly formation supports
neural computation, it is therefore essential to establish a comprehensive
theory of how network structure and dynamics interact to generate specific
avalanche patterns and sequences. Here we derive exact avalanche distributions
for a finite network of recurrently coupled spiking neurons with arbitrary
non-negative interaction weights, which is made possible by formally mapping
the model dynamics to a linear, random dynamical system on the $N$-torus and by
exploiting self-similarities inherent in the phase space. We introduce the
notion of relative unique ergodicity and show that this property is guaranteed
if the system is driven by a time-invariant Bernoulli process. This approach
allows us not only to provide closed-form analytical expressions for avalanche
size, but also to determine the detailed set(s) of units firing in an avalanche
(i.e., the avalanche assembly). The underlying dependence between network
structure and dynamics is made transparent by expressing the distribution of
avalanche assemblies in terms of the induced graph Laplacian. We explore
analytical consequences of this dependence and provide illustrating examples.
