No Arabic abstract
Neuromorphic networks can be described in terms of coarse-grained variables, where emergent sustained behaviours spontaneously arise if stochasticity is properly taken in account. For example it has been recently found that a directed linear chain of connected patch of neurons amplifies an input signal, also tuning its characteristic frequency. Here we study a generalization of such a simple model, introducing heterogeneity and variability in the parameter space and long-range interactions, breaking, in turn, the preferential direction of information transmission of a directed chain. On one hand, enlarging the region of parameters leads to a more complex state space that we analytically characterise; moreover, we explicitly link the strength distribution of the non-local interactions with the frequency distribution of the network oscillations. On the other hand, we found that adding long-range interactions can cause the onset of novel phenomena, as coherent and synchronous oscillations among all the interacting units, which can also coexist with the amplification of the signal.
We perform an analytical analysis of the long-range degree correlation of the giant component in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the giant component is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the giant component becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of the $l$-distant node pairs, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The ErdH{o}s-R{e}nyi random graph is employed to confirm this critical behavior.
We consider two neuronal networks coupled by long-range excitatory interactions. Oscillations in the gamma frequency band are generated within each network by local inhibition. When long-range excitation is weak, these oscillations phase-lock with a phase-shift dependent on the strength of local inhibition. Increasing the strength of long-range excitation induces a transition to chaos via period-doubling or quasi-periodic scenarios. In the chaotic regime oscillatory activity undergoes fast temporal decorrelation. The generality of these dynamical properties is assessed in firing-rate models as well as in large networks of conductance-based neurons.
We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long term prediction, which we propose as the primary signature of (Strong) Emergent Behaviour. The key feature of our definition of PMI is the omission of an interval of present time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behaviour or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalisation of the notion of ordered phases.
Statistical mechanical models with local interactions in $d>1$ dimension can be regarded as $d=1$ dimensional models with regular long range interactions. In this paper we study the critical properties of Ising models having $V$ sites, each having $z$ randomly chosen neighbors. For $z=2$ the model reduces to the $d=1$ Ising model. For $z= infty$ we get a mean field model. We find that for finite $z > 2$ the system has a second order phase transition characterized by a length scale $L={rm ln}V$ and mean field critical exponents that are independent of $z$.
Interconnecting power systems has a number of advantages such as better electric power quality, increased reliability of power supply, economies of scales through production and reserve pooling and so forth. Simultaneously, it may jeopardize the overall system stability with the emergence of so-called inter-area oscillations, which are coherent oscillations involving groups of rotating machines separated by large distances up to thousands of kilometers. These often weakly damped modes may have harmful consequences for grid operation, yet despite decades of investigations, the mechanisms that generate them are still poorly understood, and the existing theories are based on assumptions that are not satisfied in real power grids where such modes are observed. Here we construct a matrix perturbation theory of large interconnected power systems that clarifies the origin and the conditions for the emergence of inter-area oscillations. We show that coherent inter-area oscillations emerge from the zero-modes of a multi-area network Laplacian matrix, which hybridize only weakly with other modes, even under significant capacity of the inter-area tie-lines, i.e. even when the standard assumption of area partitioning is not satisfied. The general theory is illustrated on a two-area system, and numerically applied to the well-connected PanTaGruEl model of the synchronous grid of continental Europe.