Chris G. Antonopoulos

Chimera-like States in Modular Neural Networks

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the C. elegans soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter ρ, we also employ other measures of coherence, such as the chimera-like χ and metastability λ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the C. elegans neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.

CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrodinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents Li, i = 1,…, N - 1, exhibit a transition between two power laws, Li ∝ EBk, Bk > 0, k = 1, 2, occurring at the same value of E. The destabilization energy Ec per particle goes to zero as N → ∞ following a simple power-law, Ec/N ∝ N-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle hKS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

Evidence of q-exponential statistics in Greek seismicity

We study the seismicity (global seismic activity) that occurred in Greece between 1976 and 2009 based on the dataset reported in Makropoulos et al. (2012), using concepts of non-extensive Statistical Physics. By considering the entire and declustered datasets, for which the aftershocks have been removed, we initially investigate the frequency–magnitude distribution and find that both datasets are well approximated by a physical model derived in the framework of non-extensive Statistical Physics. We then carry out a study of the distribution of interevent times of seismic events for different magnitude thresholds and discover that the data are well approximated by a statistical distribution of the q-exponential type that allows us to compute analytically the risk function of earthquake production. Our analysis thus reveals further evidence that the underlying dynamical process of earthquake birth reflects a kind of nonlinear memory due to long-term persistence of seismic events.

Do Brain Networks Evolve by Maximizing Their Information Flow Capacity

We propose a working hypothesis supported by numerical simulations that brain networks evolve based on the principle of the maximization of their internal information flow capacity. We find that synchronous behavior and capacity of information flow of the evolved networks reproduce well the same behaviors observed in the brain dynamical networks of Caenorhabditis elegans and humans, networks of Hindmarsh-Rose neurons with graphs given by these brain networks. We make a strong case to verify our hypothesis by showing that the neural networks with the closest graph distance to the brain networks of Caenorhabditis elegans and humans are the Hindmarsh-Rose neural networks evolved with coupling strengths that maximize information flow capacity. Surprisingly, we find that global neural synchronization levels decrease during brain evolution, reflecting on an underlying global no Hebbian-like evolution process, which is driven by no Hebbian-like learning behaviors for some of the clusters during evolution, and Hebbian-like learning rules for clusters where neurons increase their synchronization.

PROBING THE LOCAL DYNAMICS OF PERIODIC ORBITS BY THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

Successful network inference from time-series data using mutual information rate

This work uses an information-based methodology to infer the connectivity of complex systems from observed time-series data. We first derive analytically an expression for the Mutual Information Rate (MIR), namely, the amount of information exchanged per unit of time, that can be used to estimate the MIR between two finite-length low-resolution noisy time-series, and then apply it after a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems. In particular, we show that our methodology successfully infers the connectivity for heterogeneous networks, different time-series lengths or coupling strengths, and even in the presence of additive noise. Finally, we show that our methodology based on MIR successfully infers the connectivity of networks composed of nodes with different time-scale dynamics, where inference based on Mutual Information fails.

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈106) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.

Complex statistics and diffusion in nonlinear disordered particle chains

We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion, in both cases, in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10(9), our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic Kolmogorov-Arnold-Moser torus and that diffusion continues to spread chaotically for arbitrarily long times.

Production and transfer of energy and information in Hamiltonian systems.

We present novel results that relate energy and information transfer with sensitivity to initial conditions in chaotic multi-dimensional Hamiltonian systems. We show the relation among Kolmogorov-Sinai entropy, Lyapunov exponents, and upper bounds for the Mutual Information Rate calculated in the Hamiltonian phase space and on bi-dimensional subspaces. Our main result is that the net amount of transfer from kinetic to potential energy per unit of time is a power-law of the upper bound for the Mutual Information Rate between kinetic and potential energies, and also a power-law of the Kolmogorov-Sinai entropy. Therefore, transfer of energy is related with both transfer and production of information. However, the power-law nature of this relation means that a small increment of energy transferred leads to a relatively much larger increase of the information exchanged. Then, we propose an “experimental” implementation of a 1-dimensional communication channel based on a Hamiltonian system, and calculate the actual rate with which information is exchanged between the first and last particle of the channel. Finally, a relation between our results and important quantities of thermodynamics is presented.

Spike timing-dependent plasticity induces non-trivial topology in the brain

We study the capacity of Hodgkin-Huxley neuron in a network to change temporarily or permanently their connections and behavior, the so called spike timing-dependent plasticity (STDP), as a function of their synchronous behavior. We consider STDP of excitatory and inhibitory synapses driven by Hebbian rules. We show that the final state of networks evolved by a STDP depend on the initial network configuration. Specifically, an initial all-to-all topology evolves to a complex topology. Moreover, external perturbations can induce co-existence of clusters, those whose neurons are synchronous and those whose neurons are desynchronous. This work reveals that STDP based on Hebbian rules leads to a change in the direction of the synapses between high and low frequency neurons, and therefore, Hebbian learning can be explained in terms of preferential attachment between these two diverse communities of neurons, those with low-frequency spiking neurons, and those with higher-frequency spiking neurons.