  We investigate front propagation and synchronization transitions in
dependence on the information transmission delay and coupling strength over
scale-free neuronal networks with different average degrees and scaling
exponents. As the underlying model of neuronal dynamics, we use the efficient
Rulkov map with additive noise. We show that increasing the coupling strength
enhances synchronization monotonously, whereas delay plays a more subtle role.
In particular, we found that depending on the inherent oscillation frequency of
individual neurons, regions of irregular and regular propagating excitatory
fronts appear intermittently as the delay increases. These delay-induced
synchronization transitions manifest as well-expressed minima in the measure
for spatial synchrony, appearing at every multiple of the oscillation
frequency. Larger coupling strengths or average degrees can broaden the region
of regular propagating fronts by a given information transmission delay and
further improve synchronization. These results are robust against variations in
system size, intensity of additive noise and the scaling exponent of the
underlying scale-free topology. We argue that fine-tuned information
transmission delays are vital for assuring optimally synchronized excitatory
fronts on complex neuronal networks, and indeed, they should be seen as
important as the coupling strength or the overall density of interneuronal
connections. We finally discuss some biological implications of the presented
results.
