A linear stability analysis based on the population equations reveals that the asynchronous state in a homogeneous network of spiking neurons is unstable for low levels of noise. The asynchronous state is particularly vulnerable to oscillatory perturbations at the averaged firing frequency of the neurons or at harmonics thereof, which is a consequence of the tendency of the neurons to synchronize spontaneously. The axonal transmission delay plays an important role for the build-up of oscillations. However, asynchronous firing can be stabilized by a suitable choice of time constants and transmission delay if the noise level is sufficiently high.
The stability of perfectly synchronized oscillation is clarified by the locking theorem: A synchronous oscillation is stable if the spikes are triggered during the rising phase of the input potential which is the summed contribution of all presynaptic neurons. Stable synchronous oscillations can occur for a wide range of parameters and both for excitatory and inhibitory couplings.
Especially for short transmission delays in an excitatory network with pronounced refractory behavior, the fully synchronized state where all neurons are firing in ``lock-step'' is unstable. This, however, does not mean that the network switches into the asynchronous state, which may be unstable as well. Instead, the population of neurons splits up in several subgroups (`cluster') of neurons that fire alternatingly in a regular manner. Neurons within the same cluster stay synchronized over long times.
A replacement of the all-to-all connectivity by sparse random couplings can result in a network that generates highly irregular spike trains even without any additional source of noise. Neurons do not fire at every oscillation cycle, but if spike firing occurs it does so in phase with the global oscillation. The irregularity in the spike trains is due to the `frozen noise' of the connectivity and therefore purely deterministic. Restarting the population with the same initial condition thus leads to the very same sequence of spike patterns. Information on the initial condition is preserved in the spike patterns over several cycles even in the presence of synaptic transmission failures which suggests interesting applications for short-term memory and timing tasks.
Synchronization phenomena in pulse-coupled units have previously been studied in a non-neuronal context, such as the synchronous flashing of tropical fireflies (Buck and Buck, 1976), which triggered a whole series of theoretical papers on synchronization of pulse-coupled oscillators. The most important one is probably the famous work of Mirollo and Strogatz (1990). Oscillations in the visual system and the role of synchrony for feature binding has been reviewed by Singer (1994) and Singer and Gray (1995). Oscillations in sensory systems have been reviewed by Ritz and Sejnowski (1997) and, specifically in the context of the olfactory system, by Laurent (1996), and the hippocampus by O'Keefe (1993).
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