Linear and nonlinear information flow in spatially extended systems

Infinitesimal and finite amplitude error propagation in spatially extended systems are numerically and theoretically investigated. The information transport in these systems can be characterized in terms of the propagation velocity of perturbations $V_p$. A linear stability analysis is sufficient to capture all the relevant aspects associated to propagation of infinitesimal disturbances. In particular, this analysis gives the propagation velocity $V_L$ of infinitesimal errors. If linear mechanisms prevail on the nonlinear ones $V_p = V_L$. On the contrary, if nonlinear effects are predominant finite amplitude disturbances can eventually propagate faster than infinitesimal ones (i.e. $V_p > V_L$). The finite size Lyapunov exponent can be successfully employed to discriminate the linear or nonlinear origin of information flow. A generalization of finite size Lyapunov exponent to a comoving reference frame allows to state a marginal stability criterion able to provide $V_p$ both in the linear and in the nonlinear case. Strong analogies are found between information spreading and propagation of fronts connecting steady states in reaction-diffusion systems. The analysis of the common characteristics of these two phenomena leads to a better understanding of the role played by linear and nonlinear mechanisms for the flow of information in spatially extended systems.Comment: 14 RevTeX pages with 13 eps figures, title/abstract changed minor changes in the text accepted for publication on PR

Chimera states in pulse coupled neural networks: the influence of dilution and noise

We analyse the possible dynamical states emerging for two symmetrically pulse coupled populations of leaky integrate-and-fire neurons. In particular, we observe broken symmetry states in this set-up: namely, breathing chimeras, where one population is fully synchronized and the other is in a state of partial synchronization (PS) as well as generalized chimera states, where both populations are in PS, but with different levels of synchronization. Symmetric macroscopic states are also present, ranging from quasi-periodic motions, to collective chaos, from splay states to population anti-phase partial synchronization. We then investigate the influence disorder, random link removal or noise, on the dynamics of collective solutions in this model. As a result, we observe that broken symmetry chimera-like states, with both populations partially synchronized, persist up to 80 \% of broken links and up to noise amplitudes 8 \% of threshold-reset distance. Furthermore, the introduction of disorder on symmetric chaotic state has a constructive effect, namely to induce the emergence of chimera-like states at intermediate dilution or noise level.Comment: 15 pages, 7 figure, contribution for the Workshop "Nonlinear Dynamics in Computational Neuroscience: from Physics and Biology to ICT" held in Turin (Italy) in September 201

Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to unstable growth with mound formation whose tipical linear size L increases in time. In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models for which the largest surface slope remains constant in time (model B): coarsening is known to be logarithmic in the absence of noise (L(t)=log t) and to follow a power law (L(t)=t^{1/3}) when noise is present. If surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation: here we study the late stage of coarsening through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree with regard to the conclusion that steepening of mounds makes deterministic coarsening faster: if alpha is the exponent describing the steepening of the maximal slope M of mounds (M^alpha = L) we find that L(t)=t^n: n is equal to 1/4 for 1<alpha<2 and it decreases from 1/4 to 1/5 for alpha>2, according to n=alpha/(5*alpha -2). On the other side, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t)=t^{1/4}, irrespectively of alpha. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model alpha=1, both in the absence and in the presence of noise.Comment: One figure and relative discussion changed. To be published in Eur. Phys. J.

Stable chaos in fluctuation driven neural circuits

We study the dynamical stability of pulse coupled networks of leaky integrate-and-fire neurons against infinitesimal and finite perturbations. In particular, we compare current versus fluctuations driven networks, the former (latter) is realized by considering purely excitatory (inhibitory) sparse neural circuits. In the excitatory case the instabilities of the system can be completely captured by an usual linear stability (Lyapunov) analysis, on the other hand the inhibitory networks can display the coexistence of linear and nonlinear instabilities. The nonlinear effects are associated to finite amplitude instabilities, which have been characterized in terms of suitable indicators. For inhibitory coupling one observes a transition from chaotic to non chaotic dynamics by decreasing the pulse width. For sufficiently fast synapses the system, despite showing an erratic evolution, is linearly stable, thus representing a prototypical example of Stable Chaos.Comment: 32 pages with 19 figures, submitted to Chaos, Solitons and Fractal

Sisyphus Effect in Pulse Coupled Excitatory Neural Networks with Spike-Timing Dependent Plasticity

The collective dynamics of excitatory pulse coupled neural networks with spike timing dependent plasticity (STDP) is studied. Depending on the model parameters stationary states characterized by High or Low Synchronization can be observed. In particular, at the transition between these two regimes, persistent irregular low frequency oscillations between strongly and weakly synchronized states are observable, which can be identified as infraslow oscillations with frequencies 0.02 - 0.03 Hz. Their emergence can be explained in terms of the Sisyphus Effect, a mechanism caused by a continuous feedback between the evolution of the coherent population activity and of the average synaptic weight. Due to this effect, the synaptic weights have oscillating equilibrium values, which prevents the neuronal population from relaxing into a stationary macroscopic state.Comment: 18 pages, 24 figures, submitted to Physical Review

Analytical Estimation of the Maximal lyapunov Exponent in Oscillator Chains

An analytical expression for the maximal Lyapunov exponent $\lambda_1$ in generalized Fermi-Pasta-Ulam oscillator chains is obtained. The derivation is based on the calculation of modulational instability growth rates for some unstable periodic orbits. The result is compared with numerical simulations and the agreement is good over a wide range of energy densities $\epsilon$. At very high energy density the power law scaling of $\lambda_1$ with $\epsilon$ can be also obtained by simple dimensional arguments, assuming that the system is ruled by a single time scale. Finally, we argue that for repulsive and hard core potentials in one dimension $\lambda_1 \sim \sqrt{\epsilon}$ at large $\epsilon$.Comment: Latex, 10 pages, 5 Figs - Contribution to the Conference "Disorder and Chaos" held in memory of Giovanni Paladin (Sept. 1997 - Rome) - submitted to J. de Physiqu

Order Parameter for the Transition from Phase to Amplitude Turbulence

The maximal conserved phase gradient is introduced as an order parameter to characterize the transition from phase- to defect-turbulence in the complex Ginzburg-Landau equation. It has a finite value in the phase-turbulent regime and decreases to zero when the transition to defect-turbulence is approached. Solutions with a non-zero phase gradient are studied via a Lyapunov analysis. The degree of "chaoticity" decreases for increasing values of the phase gradient and finally leads to stable travelling wave solutions. A modified Kuramoto-Sivashinsky equation for the phase-dynamics is able to reproduce the main features of the stable waves and to explain their origin.Comment: 11 pages letter (PS file) and 3 figures (PS files

Linear stability in networks of pulse-coupled neurons

In a first step towards the comprehension of neural activity, one should focus on the stability of the various dynamical states. Even the characterization of idealized regimes, such as a perfectly periodic spiking activity, reveals unexpected difficulties. In this paper we discuss a general approach to linear stability of pulse-coupled neural networks for generic phase-response curves and post-synaptic response functions. In particular, we present: (i) a mean-field approach developed under the hypothesis of an infinite network and small synaptic conductances; (ii) a "microscopic" approach which applies to finite but large networks. As a result, we find that no matter how large is a neural network, its response to most of the perturbations depends on the system size. There exists, however, also a second class of perturbations, whose evolution typically covers an increasingly wide range of time scales. The analysis of perfectly regular, asynchronous, states reveals that their stability depends crucially on the smoothness of both the phase-response curve and the transmitted post-synaptic pulse. The general validity of this scenarion is confirmed by numerical simulations of systems that are not amenable to a perturbative approach.Comment: 13 pages, 7 figures, submitted to Frontiers in Computational Neuroscienc