  Transient or partial synchronization can be used to do computations, although
a fully synchronized network is frequently related to epileptic seizures. Here,
we propose a homeostatic mechanism that is capable of maintaining a neuronal
network at the edge of a synchronization transition, thereby avoiding the
harmful consequences of a fully synchronized network. We model neurons by maps
since they are dynamically richer than integrate-and-fire models and more
computationally efficient than conductance-based approaches. We first describe
the synchronization phase transition of a dense network of neurons with
different tonic spiking frequencies coupled by gap junctions. We show that at
the transition critical point, inputs optimally reverberate through the network
activity through transient synchronization. Then, we introduce a local
homeostatic dynamic in the synaptic coupling and show that it produces a robust
self-organization toward the edge of this phase transition. We discuss the
potential biological consequences of this self-organization process, such as
its relation to the Brain Criticality hypothesis, its input processing
capacity, and how its malfunction could lead to pathological synchronization.
