Bursting without slow kinetics: a role for a small world?

Bursting, a dynamical phenomenon whereby episodes of neural action potentials are punctuated by periodic episodes of inactivity, is ubiquitous in neural systems. Examples include components of the respiratory rhythm generating circuitry in the brain stem, spontaneous activity in the neonatal rat spinal cord, and developing neural networks in the retina of the immature ferret. Bursting can also manifest itself in single neurons. Bursting dynamics require one or more kinetic processes slower than the timescale of the action potentials. Such processes usually manifest themselves in intrinsic ion channel properties, such as slow voltage-dependent gating or calcium-dependent processes, or synaptic mechanisms, such as synaptic depression. In this note, we show rhythmic bursting in a simulated neural network where no such slow processes exist at the cellular or synaptic level. Rather, the existence of rhythmic bursting is critically dependent on the connectivity of the network and manifests itself only when connectivity is characterized as small world. The slow process underlying the timescale of bursting manifests itself as a progressive synchronization of the network within each burst.