, Volume 145, Issue 3, pp 713-733

Nonlocal Mechanism for Synchronization of Time Delay Networks

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We present the interplay between synchronization of networks with heterogeneous delays and the greatest common divisor (GCD) of loops composing the network. We distinguish between two types of networks; (I) chaotic networks and (II) population dynamic networks with periodic activity driven by external stimuli. For type (I), in the weak chaos region, the units of a chaotic network characterized by GCD=1 are in a chaotic zero-lag synchronization, whereas for GCD>1, the network splits into GCD-clusters in which clustered units are in zero-lag synchronization. These results are supported by simulations of chaotic systems, self-consistent and mixing arguments, as well as analytical solutions of Bernoulli maps. Type (II) is exemplified by simulations of Hodgkin Huxley population dynamic networks with unidirectional connectivity, synaptic noise and distribution of delays within neurons belonging to a node and between connecting nodes. 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 network splits into clusters equal to the greatest common divisor of loops composing the network (spatial) and the periodicity of the external stimuli (temporal). The results suggest that neural information processing may take place in the transient to synchronization and imply a much shorter time scale for the inference of a perceptual entity.

It is an honor and a privilege to contribute an article to this volume dedicated to Professor Cyril Domb, FRS. I have known Professor Domb for many years in two different capacities, both as an outstanding researcher in statistical mechanics and also as a human being. I need not write anything regarding the first capacity—Professor Domb’s accomplishments as a researcher are world-renowned. But I do want to comment on the second capacity. Professor Domb has served me as a guiding light regarding what it means to be an exemplary human being. His honesty and forthrightness, his thoughtfulness toward others, and his high moral standards give special meaning to the term “humanity.'' His life throughout nine decades personifies the words of the Psalmist: “to walk honestly, to act justly, and to speak the truth'' (15:2). This stirring passage describes the essence of Professor Cyril Domb. (Ido Kanter)