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. I. INRODUCTIONThe spiking activity of neurons within a local cortical population is typically correlated [1][2][3][4]. As a result, local cortical signals are robust to noise, which is a prerequisite for reliable signal processing in cortex. Under special conditions, coherent activity in a local cortical population is an inevitable consequence of shared presynaptic input [5-9]. Nevertheless, the mechanism for the emergence of correlation, synchronization or even nearly zero-lag synchronization (ZLS) among two or more cortical areas which do not share the same input is one of the main enigmas in neuroscience [7][8][9]. It has been argued that nonlocal synchronization is a marker of binding activities in different cortical areas into one perceptual entity [8,[10][11][12]. This prompted the hypothesis that synchronization may hold key information about higher and complex functionalities of the network. To investigate the synchronization of complex neural circuits we studied the activity modes of networks in which the properties of solitary neurons, population dynamics, delays, connectivity and background noise mimic the inter-columnar connectivity of the neocortex. II. NEURONAL CIRCUITWe start with a description of the neuronal circuits and define the properties of a neuronal cell, the structure of a node in a network representing one cortical patch, and the connection between nodes. Each neural cell was simulated using the well known Hodgkin Huxley model [13] (see Appendix for details). Each node in the network was comprised of a balanced population of 30 neurons, 80/20 percent of which were excitatory/inhibitory (Fig. 1a). The lawful reciprocal connections within each node were only between pairs of excitatory and inhibitory neurons and were selected at random with probability p in . In terms of biological properties it was assumed that distant cortico-cortical connections are (almost) exclusively excitatory whereas local connections are both excitatory and inhibitory [14,15]. In this framework, cortical areas are connected reciprocally across the two hemispheres and within a single hemisphere [16], where small functional cortical units (patches) connect to other cortical...

A network of chaotic units is investigated where the units are coupled by signals with a transmission delay. Any arbitrary finite network is considered where the chaotic trajectories of the uncoupled units are a solution of the dynamic equations of the network. It is shown that chaotic trajectories cannot be synchronized if the transmission delay is larger than the time scales of the individual units. For several models the master stability function is calculated which determines the maximal delay time for which synchronization is possible.

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.

Zero-lag synchronization (ZLS) between chaotic units, which do not have self-feedback or a relay unit connecting them, is experimentally demonstrated for two mutually coupled chaotic semiconductor lasers. The mechanism is based on two mutual coupling delay times with certain allowed integer ratios, whereas for a single mutual delay time ZLS cannot be achieved. This mechanism is also found numerically for mutually coupled chaotic maps where its stability is analyzed using the Schur-Cohn theorem for the roots of polynomials. The symmetry of the polynomials allows only specific integer ratios for ZLS. In addition, we present a general argument for ZLS when several mutual coupling delay times are present.

This Letter was published online on 19 March 2010 with an incorrect caption to Fig. 4. Figure 4's caption should read as ''(Color online) 10 ns recording of the time dependent intensity of two semiconductor lasers (thin blue and thick red).To demonstrate the slow decorrelation of the intensities with increasing time we show in panel (a) (dashed line) the intensity of the laser after a delay of 4 1 . Corresponding correlation values can be seen in Fig. 3.''