  We investigate the modes of oscillation of heterogeneous ring-networks of
quadratic integrate-and-fire neurons with non-local, space-dependent coupling.
Perturbations of the equilibrium state with a particular wave number produce
transient standing waves with a specific frequency, analogous to those in a
tense string. In the neuronal network, the equilibrium corresponds to a
spatially homogeneous, asynchronous state. Perturbations of this state excite
the network's oscillatory modes, which reflect the interplay of episodes of
synchronous spiking with the excitatory-inhibitory spatial interactions. In the
thermodynamic limit, an exact low-dimensional neural field model describing the
macroscopic dynamics of the network is derived. This allows us to obtain
formulas for the Turing eigenvalues of the spatially-homogeneous state, and
hence to obtain its stability boundary. We find that the frequency of each
Turing mode depends on the corresponding Fourier coefficient of the synaptic
pattern of connectivity. The decay rate instead, is identical for all
oscillation modes as a consequence of the heterogeneity-induced
desynchronization of the neurons. Finally, we numerically compute the spectrum
of spatially-inhomogeneous solutions branching from the Turing bifurcation,
showing that similar oscillatory modes operate in neural bump states, and are
maintained away from onset.
