  Many networks of physical and biological interest are characterized by a
long-range coupling mediated by a chemical which diffuses through a medium in
which oscillators are embedded. We considered a one-dimensional model for this
effect for which the diffusion is fast enough so as to be implemented through a
coupling whose intensity decays exponentially with the lattice distance. In
particular, we analyzed the bursting synchronization of neurons described by
two timescales (spiking and bursting activity), and coupled through such a
long-range interaction network. One of the advantages of the model is that one
can pass from a local (Laplacian) type of coupling to a global (all-to-all) one
by varying a single parameter in the interaction term. We characterized
bursting synchronization using an order parameter which undergoes a transition
as the coupling parameters are changed through a critical value. We also
investigated the role of an external time-periodic signal on the bursting
synchronization properties of the network. We show potential applications in
the control of pathological rhythms in biological neural networks.
