Periodicity Manifestations in the Turbulent Regime of Globally Coupled Map Lattice

We revisit the globally coupled map lattice (GCML). We show that in the so called turbulent regime various periodic cluster attractor states are formed even though the coupling between the maps are very small relative to the non-linearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period three motion (MSCA) due to the foliation of the period three window of the element logistic maps. An analytic approach is proposed which explains successfully the systematics of various periodicity manifestations in the turbulent regime. The linear stability of the period three cluster attractors is investigated.Comment: 34 pages, 8 Postscript figures, all in GCML-MSCA.Zi

Emergence of a non trivial fluctuating phase in the XY model on regular networks

We study an XY-rotor model on regular one dimensional lattices by varying the number of neighbours. The parameter $2\ge\gamma\ge1$ is defined. $\gamma=2$ corresponds to mean field and $\gamma=1$ to nearest neighbours coupling. We find that for $\gamma<1.5$ the system does not exhibit a phase transition, while for $\gamma > 1.5$ the mean field second order transition is recovered. For the critical value $\gamma=\gamma_c=1.5$, the systems can be in a non trivial fluctuating phase for whichthe magnetisation shows important fluctuations in a given temperature range, implying an infinite susceptibility. For all values of $\gamma$ the magnetisation is computed analytically in the low temperatures range and the magnetised versus non-magnetised state which depends on the value of $\gamma$ is recovered, confirming the critical value $\gamma_{c}=1.5$

The Role of Degree Distribution in Shaping the Dynamics in Networks of Sparsely Connected Spiking Neurons

Neuronal network models often assume a fixed probability of connection between neurons. This assumption leads to random networks with binomial in-degree and out-degree distributions which are relatively narrow. Here I study the effect of broad degree distributions on network dynamics by interpolating between a binomial and a truncated power-law distribution for the in-degree and out-degree independently. This is done both for an inhibitory network (I network) as well as for the recurrent excitatory connections in a network of excitatory and inhibitory neurons (EI network). In both cases increasing the width of the in-degree distribution affects the global state of the network by driving transitions between asynchronous behavior and oscillations. This effect is reproduced in a simplified rate model which includes the heterogeneity in neuronal input due to the in-degree of cells. On the other hand, broadening the out-degree distribution is shown to increase the fraction of common inputs to pairs of neurons. This leads to increases in the amplitude of the cross-correlation (CC) of synaptic currents. In the case of the I network, despite strong oscillatory CCs in the currents, CCs of the membrane potential are low due to filtering and reset effects, leading to very weak CCs of the spike-count. In the asynchronous regime of the EI network, broadening the out-degree increases the amplitude of CCs in the recurrent excitatory currents, while CC of the total current is essentially unaffected as are pairwise spiking correlations. This is due to a dynamic balance between excitatory and inhibitory synaptic currents. In the oscillatory regime, changes in the out-degree can have a large effect on spiking correlations and even on the qualitative dynamical state of the network

Inferring network properties of cortical neurons with synaptic coupling and parameter dispersion

Computational models at different space-time scales allow us to understand the fundamental mechanisms that govern neural processes and relate uniquely these processes to neuroscience data. In this work, we propose a novel neurocomputational unit (a mesoscopic model which tell us about the interaction between local cortical nodes in a large scale neural mass model) of bursters that qualitatively captures the complex dynamics exhibited by a full network of parabolic bursting neurons. We observe that the temporal dynamics and fluctuation of mean synaptic action term exhibits a high degree of correlation with the spike/burst activity of our population. With heterogeneity in the applied drive and mean synaptic coupling derived from fast excitatory synapse approximations we observe long term behavior in our population dynamics such as partial oscillations, incoherence, and synchrony. In order to understand the origin of multistability at the population level as a function of mean synaptic coupling and heterogeneity in the firing rate threshold we employ a simple generative model for parabolic bursting recently proposed by Ghosh et al. (2009). Further, we use here a mean coupling formulated for fast spiking neurons for our analysis of generic model. Stability analysis of this mean field network allow us to identify all the relevant network states found in the detailed biophysical model. We derive here analytically several boundary solutions, a result which holds for any number of spikes per burst. These findings illustrate the role of oscillations occurring at slow time scales (bursts) on the global behavior of the network.EC/FP7/269921/EU/Brain-inspired multiscale computation in neuromorphic hybrid systems/BrainScale

Emergent complex neural dynamics

A large repertoire of spatiotemporal activity patterns in the brain is the basis for adaptive behaviour. Understanding the mechanism by which the brain's hundred billion neurons and hundred trillion synapses manage to produce such a range of cortical configurations in a flexible manner remains a fundamental problem in neuroscience. One plausible solution is the involvement of universal mechanisms of emergent complex phenomena evident in dynamical systems poised near a critical point of a second-order phase transition. We review recent theoretical and empirical results supporting the notion that the brain is naturally poised near criticality, as well as its implications for better understanding of the brain

Transition from localized to mean field behaviour of cascading failures in the fiber bundle model on complex networks

We study the failure process of fiber bundles on complex networks focusing on the effect of the degree of disorder of fibers' strength on the transition from localized to mean field behaviour. Starting from a regular square lattice we apply the Watts-Strogatz rewiring technique to introduce long range random connections in the load transmission network and analyze how the ultimate strength of the bundle and the statistics of the size of failure cascades change when the rewiring probability is gradually increased. Our calculations revealed that the degree of strength disorder of nodes of the network has a substantial effect on the localized to mean field transition. In particular, we show that the transition sets on at a finite value of the rewiring probability, which shifts to higher values as the degree of disorder is reduced. The transition is limited to a well defined range of disorder, so that there exists a threshold disorder of nodes' strength below which the randomization of the network structure does not provide any improvement neither of the overall load bearing capacity nor of the cascade tolerance of the system. At low strength disorder the fully random network is the most stable one, while at high disorder best cascade tolerance is obtained at a lower structural randomness. Based on the interplay of the network structure and strength disorder we construct an analytical argument which provides a reasonable description of the numerical findings.Comment: 30 pages, 11 figure

Discrete scale invariance and complex dimensions

We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously'' in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in several domains of the review paper with the same title published in Physics Reports 297, 239-270 (1998