  For spiking neural networks we consider the stability problem of global
synchrony, arguably the simplest non-trivial collective dynamics in such
networks. We find that even this simplest dynamical problem -- local stability
of synchrony -- is non-trivial to solve and requires novel methods for its
solution. The dynamics in the vicinity of the synchronous state is determined
by a multitude of linear operators, in contrast to a single stability matrix in
conventional linear stability theory. This unusual property qualitatively
depends on network topology and may be neglected for globally coupled
homogeneous networks. For generic networks, however, the number of operators
increases exponentially with the size of the network. We present methods to
treat this multi-operator problem exactly. First, based on the Gershgorin and
Perron-Frobenius theorems, we derive bounds on the eigenvalues that provide
important information about the synchronization process but are not sufficient
to establish the asymptotic stability or instability of the synchronous state.
We then present a complete analysis of asymptotic stability for topologically
strongly connected networks using simple graph-theoretical considerations. For
inhibitory interactions between dissipative (leaky) oscillatory neurons the
synchronous state is stable, independent of the parameters and the network
connectivity. These results indicate that pulse-like interactions play a
profound role in network dynamical systems, and in particular in the dynamics
of biological synchronization, unless the coupling is homogeneous and
all-to-all.
