Synchronization of unidirectional time delay chaotic networks and the greatest common divisor

We present the interplay between synchronization of unidirectional coupled chaotic nodes with heterogeneous delays and the greatest common divisor (GCD) of loops composing the oriented graph. In the weak chaos region and for GCD=1 the network is in chaotic zero-lag synchronization, whereas for GCD=m>1 synchronization of m-sublattices emerges. Complete synchronization can be achieved when all chaotic nodes are influenced by an identical set of delays and in particular for the limiting case of homogeneous delays. Results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps.Comment: 7 pages, 5 figure

Nonlocal mechanism for cluster synchronization in neural circuits

The interplay between the topology of cortical circuits and synchronized activity modes in distinct cortical areas is a key enigma in neuroscience. We present a new nonlocal mechanism governing the periodic activity mode: the greatest common divisor (GCD) of network loops. For a stimulus to one node, the network splits into GCD-clusters in which cluster neurons are in zero-lag synchronization. For complex external stimuli, the number of clusters can be any common divisor. The synchronized mode and the transients to synchronization pinpoint the type of external stimuli. The findings, supported by an information mixing argument and simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity and synaptic noise, call for reexamining sources of correlated activity in cortex and shorter information processing time scales.Comment: 8 pges, 6 figure