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Ensembles of phase-oscillators are known to exhibit a variety of collective regimes. Here, we show that a simple mean-field model involving two heterogenous populations of pulse-coupled oscillators, exhibits, in the strong-coupling limit, a robust irregular macroscopic dynamics. The resulting, strongly synchronized, regime is sustained by a homeostatic mechanism induced by the shape of the phase-response curve combined with adaptive coupling strength, included to account for energy dissipated by the pulse emission. The proposed setup mimicks a neural network composed of excitatory and inhibitory neurons., Comment: 4 pages, 3 figures; supplemental material 3 opages, 3 figures

The observation of negative-temperature states in the localized phase of the the Discrete Nonlinear Schr\"odinger (DNLS) equation has challenged statistical mechanics for a long time. For isolated systems, they can emerge as stationary extended states through a large-deviation mechanism occurring for finite sizes, while they are formally unstable in grand-canonical setups, being associated to an unlimited growth of the condensed fraction. Here, we show that negative-temperature states in open setups are metastable but their lifetime $\tau$ is exponentially long with the temperature, $\tau \approx \exp(\lambda |T|)$ (for $T<0$ and $\lambda>0$). More precisely, we find that condensation on a given site (i.e. emergence of a tall discrete breather) can advance only once a critical mass has been accumulated therein. This result has been obtained by combining the development of an effective grand-canonical formalism with the implementation of suitable heat baths. A general expression for $\lambda$ is obtained in the case of a simplified stochastic model of non-interacting particles. In the DNLS model, the presence of an adiabatic invariant, makes $\lambda$ even larger because of the resulting freezing of the breather dynamics. This mechanism, based on the existence of two conservation laws, provides a new perspective over the statistical description of condensation processes., Comment: Supplemental material is included

Dynamical balance of excitation and inhibition is usually invoked to explain the irregular low firing activity observed in the cortex. We propose a robust nonlinear balancing mechanism for a random network of spiking neurons, which works also in absence of strong external currents. Biologically, the mechanism exploits the plasticity of excitatory-excitatory synapses induced by short-term depression. Mathematically, the nonlinear response of the synaptic activity is the key ingredient responsible for the emergence of a stable balanced regime. Our claim is supported by a simple self-consistent analysis accompanied by extensive simulations performed for increasing network sizes. The observed regime is essentially fluctuation driven and characterized by highly irregular spiking dynamics of all neurons., Comment: 9 pages, 4 figures

The dynamics of a resonant, free-running ring laser, in the common case of a fast relaxation of the atomic polarization, is unexpectedly highly singular. As shown in [Phys. Rev. Research, {\bf 5}, 023059 (2023)], this is due to the closeness to a pure Hamiltonian dynamics ruled by a nonlinear wave equation, herein named Klein-Gordon-Toda model. In this paper, we derive a quasi-Hamiltonian model which allows describing realistic systems. In particular, we identify two nearly conserved, energy-like quantities, which ``naturally" exhibit an ultra-slow dynamics confirmed and highlighted by numerical simulations. A minimal version of the quasi-Hamiltonian model is finally derived, which does not only reproduce the laser thresholds, but also helps understanding the origin of the nearly integrable character of the laser dynamics.

We study nonequilibrium steady states of a one-dimensional stochastic model, originally introduced as an approximation of the Discrete Nonlinear Schr\"odinger equation. This model is characterized by two conserved quantities, namely mass and energy; it displays a ``normal", homogeneous phase, separated by a condensed (negative-temperature) phase, where a macroscopic fraction of energy is localized on a single lattice site. When steadily maintained out of equilibrium by external reservoirs, the system exhibits coupled transport herein studied within the framework of linear response theory. We find that the Onsager coefficients satisfy an exact scaling relationship, which allows reducing their dependence on the thermodynamic variables to that on the energy density for unitary mass density. We also determine the structure of the nonequilibrium steady states in proximity of the critical line, proving the existence of paths which partially enter the condensed region. This phenomenon is a consequence of the Joule effect: the temperature increase induced by the mass current is so strong as to drive the system to negative temperatures. Finally, since the model attains a diverging temperature at finite energy, in such a limit the energy-mass conversion efficiency reaches the ideal Carnot value., Comment: 25 pages, 13 figures

The Arecchi-Bonifacio (or Maxwell-Bloch) model is the benchmark for the description of active optical media. However, in the presence of a fast relaxation of the atomic polarization, its implementation is a challenging task even in the simple ring-laser configuration, due to the presence of multiple time scales. In this Article we show that the dynamics is nearly Hamiltonian over time scales much longer than those of the cavity losses. More precisely, we prove that it can be represented as a pseudo spatio-temporal pattern generated by a nonlinear wave equation equipped with a Toda potential. The existence of two constants of motion (identified as pseudo energies), thereby, elucidates the reason why it is so hard to simplify the original model: the adiabatic elimination of the polarization must be accurate enough to describe the dynamics correctly over unexpectedly long time scales. Finally, since the nonlinear wave equation with Toda potential can be simulated on much longer times than the previous models, this opens up the route to the numerical (and theoretical) investigation of realistic setups.

The Discrete Nonlinear Schr\"odinger (DNLS) equation is a Hamiltonian model displaying an extremely slow relaxation process when discrete breathers appear in the system. In [Iubini S, Chirondojan L, Oppo G L, Politi A and Politi P 2019 Physical Review Letters 122 084102], it was conjectured that the frozen dynamics of tall breathers is due to the existence of an adiabatic invariant (AI). Here, we prove the conjecture in the simplified context of a unidirectional DNLS equation, where the breather is "forced" by a background unaffected by the breather itself. We first clarify that the nonlinearity of the breather dynamics and the deterministic nature of the forcing term are both necessary ingredients for the existence of a frozen dynamics. We then derive perturbative expressions of the AI by implementing a canonical perturbation theory and via a more phenomenological approach based on the estimate of the energy flux. The resulting accurate identification of the AI allows revealing the presence and role of sudden jumps as the main breather destabilization mechanism, with an unexpected similarity with L\'evy processes., Comment: 23 pages. Abstract and Introduction fully rewritten, also other minor changes

The acclaimed Maxwell-Bloch (or Arecchi-Bonifacio) equations are a valid dynamical model, effectively describing wave propagation in nonlinear optical media: from the amplification in input-output devices to multimode instabilities arising in laser systems. However, the inherent spatial variability of the physical observables represents an obstacle to fast simulations and analysis, especially whenever networks of active elements have to be considered. In this paper, we propose an approach which, stripping the spatial dependence of its role as a generator of dynamical richness, allows for a compelling simple portrait. It leads to (a few) ordinary differential equations in input-output configurations, complemented by a time-delayed feedback in closed-loop setups. Such scheme reproduces accurately the dynamics, paving the way to a plain treatment of the wealth of phenomena described by the Maxwell-Bloch equations., Comment: 3 figures

We present a detailed analysis of the dynamical regimes observed in a balanced network of identical Quadratic Integrate-and-Fire (QIF) neurons with a sparse connectivity for homogeneous and heterogeneous in-degree distribution. Depending on the parameter values, either an asynchronous regime or periodic oscillations spontaneously emerge. Numerical simulations are compared with a mean field model based on a self-consistent Fokker-Planck equation (FPE). The FPE reproduces quite well the asynchronous dynamics in the homogeneous case by either assuming a Poissonian or renewal distribution for the incoming spike trains. An exact self consistent solution for the mean firing rate obtained in the limit of infinite in-degree allows identifying balanced regimes that can be either mean- or fluctuation-driven. A low-dimensional reduction of the FPE in terms of circular cumulants is also considered. Two cumulants suffice to reproduce the transition scenario observed in the network. The emergence of periodic collective oscillations is well captured both in the homogeneous and heterogeneous setups by the mean field models upon tuning either the connectivity, or the input DC current. In the heterogeneous situation we analyze also the role of structural heterogeneity.

Scientific research changed profoundly over the last 30 years, in all its aspects. Scientific publishing has changed as well, mainly because of the strong increased number of submitted papers and because of the appearance of Open Access journals and publishers. We propose some reflections on these issues., Comment: One new contribution has been added, now 22 pages

We analyze the chaotic dynamics of a one-dimensional discrete nonlinear Schr\"odinger equation. This nonintegrable model, ubiquitous in several fields of physics, describes the behavior of an array of coupled complex oscillators with a local nonlinear potential. We explore the Lyapunov spectrum for different values of the energy density, finding that the maximal value of the Kolmogorov-Sinai entropy is attained at infinite temperatures. Moreover, we revisit the dynamical freezing of relaxation to equilibrium, occurring when large localized states (discrete breathers) are superposed to a generic finite-temperature background. We show that the localized excitations induce a number of very small, yet not vanishing, Lyapunov exponents, which signal the presence of extremely long characteristic time-scales. We widen our analysis by computing the related Lyapunov covariant vectors, to investigate the interaction of a single breather with the various degrees of freedom., Comment: 7 pages, 7 figures

The idealisation of neuronal pulses as $\delta$-spikes is a convenient approach in neuroscience but can sometimes lead to erroneous conclusions. We investigate the effect of a finite pulse-width on the dynamics of balanced neuronal networks. In particular, we study two populations of identical excitatory and inhibitory neurons in a random network of phase oscillators coupled through exponential pulses with different widths. We consider three coupling functions, inspired by leaky integrate-and-fire neurons with delay and type-I phase-response curves. By exploring the role of the pulse-widths for different coupling strengths we find a robust collective irregular dynamics, which collapses onto a fully synchronous regime if the inhibitory pulses are sufficiently wider than the excitatory ones. The transition to synchrony is accompanied by hysteretic phenomena (i.e. the co-existence of collective irregular and synchronous dynamics). Our numerical results are supported by a detailed scaling and stability analysis of the fully synchronous solution. A conjectured first-order phase transition emerging for $\delta$-spikes is smoothed out for finite-width pulses., Comment: 12 pages, 12 figures

We analyse the collective behavior of a mean-field model of phase-oscillators of Kuramoto-Daido type coupled through pairwise interactions which depend on phase differences: the coupling function is composed of three harmonics. We provide convincing evidence of a transient but long-lasting chaotic collective chaos, which persists in the thermodynamic limit. The regime is analysed with the help of clever direct numerical simulations, by determining the maximum Lyapunov exponent and assessing the transversal stability to the self-consistent mean field. The structure of the invariant measure is finally described in terms of a resolution-dependent entropy.

Energy transport in one-dimensional chains of particles with three conservation laws is generically anomalous and belongs to the Kardar-Parisi-Zhang dynamical universality class. Surprisingly, some examples where an apparent normal heat diffusion is found over a large range of length scales were reported. We propose a novel physical explanation of these intriguing observations. We develop a scaling analysis which explains how this may happen in the vicinity of an integrable limit, such as, but not only, the famous Toda model. In this limit, heat transport is mostly supplied by quasi-particles with a very large mean free path $\ell$. Upon increasing the system size $L$, three different regimes can be observed: a ballistic one, an intermediate diffusive range, and, eventually, the crossover to the anomalous (hydrodynamic) regime. Our theoretical considerations are supported by numerical simulations of a gas of diatomic hard-point particles for almost equal masses and of a weakly perturbed Toda chain. Finally, we discuss the case of the perturbed harmonic chain, which exhibits a yet different scenario., Comment: Published version

The recently introduced complex active optical network (LANER) generalizes the concept of laser system to a collection of links, building a bridge with random-laser physics and quantum-graphs theory. So far, LANERs have been studied with a linear approach. Here, we develop a nonlinear formalism in the perspective of describing realistic experimental devices. The propagation along active links is treated via suitable rate equations, which require the inclusion of an auxiliary variable: the population inversion. Altogether, the resulting mathematical model can be viewed as an abstract network, its nodes corresponding to the (directed) fields in the physical links. The dynamical equations differ from standard network models in that, they are a mixture of differential delay (for the active links) and algebraic equations (for the passive links). The stationary states of a generic setup with a single active medium are thoroughly discussed, showing that the role of the passive components can be combined into a single transfer function that takes into account the corresponding resonances., Comment: 11 pages, 7 figures

The stability of synchronous states is analysed in the context of two populations of inhibitory and excitatory neurons, characterized by different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the stability depends crucially on the relative pulse-width. This has implications for the overall property of the asynchronous (balanced) regime., Comment: 9 pages, 7 figures

The activity of a sparse network of leaky integrate-and-fire neurons is carefully revisited with reference to a regime of a bona-fide asynchronous dynamics. The study is preceded by a finite-size scaling analysis, carried out to identify a setup where collective synchronization is negligible. The comparison between quenched and annealed networks reveals the emergence of substantial differences when the coupling strength is increased, via a scenario somehow reminiscent of a phase transition. For sufficiently strong synaptic coupling, quenched networks exhibit a highly bursting neural activity, well reproduced by a self-consistent approach, based on the assumption that the input synaptic current is the superposition of independent renewal processes. The distribution of interspike intervals turns out to be relatively long-tailed; a crucial feature required for the self-sustainment of the bursting activity in a regime where neurons operate on average (much) below threshold. A semi-quantitative analogy with Ornstein-Uhlenbeck processes helps validating this interpretation. Finally, an alternative explanation in terms of Poisson processes is offered under the additional assumption of mutual correlations among excitatory and inhibitory spikes., Comment: 12 pages, 12 figures

This Chapter contains an overview of the effects of nonlinear interactions in selected problems of non-equilibrium statistical mechanics. Most of the emphasis is put on open setups, where energy is exchanged with the environment. With reference to a few models of classical coupled anharmonic oscillators, we review anomalous but general properties such as extremely slow relaxation processes, or non-Fourier heat transport., Comment: Review paper, to appear in the volume "Nonlinear Science: a 20/20 vision", Springer Frontiers Collection (J. Cuevas, P. Kevrekidis and A. Saxena Editors)

In globally coupled ensembles of identical oscillators so-called chimera states can be observed. The chimera state is a symmetry-broken regime, where a subset of oscillators forms a cluster, a synchronized population, while the rest of the system remains a collection of non-synchronized, scattered units. We describe here a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It is characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new, reshuffled configuration. We identify three different kinds of rare blinking events and give a quantitative characterization by applying stability analysis to the long-lived chaotic state and to the short-lived regular regimes which arise when the cluster dissolves., Comment: 18 pages; 6 figueres

Understanding heat transport in one-dimensional systems remains a major challenge in theoretical physics, both from the quantum as well as from the classical point of view. In fact, steady states of one-dimensional systems are commonly characterized by macroscopic inhomogeneities, and by long range correlations, as well as large fluctuations that are typically absent in standard three-dimensional thermodynamic systems. These effects violate locality --material properties in the bulk may be strongly affected by the boundaries, leading to anomalous energy transport-- and they make more problematic the interpretation of mechanical microscopic quantities in terms of thermodynamic observables. Here, we revisit the problem of heat conduction in chains of classical nonlinear oscillators, following a Lagrangian and an Eulerian approach. The Eulerian definition of the flux is composed of a convective and a conductive component. The former component tends to prevail at large temperatures where the system behavior is increasingly gas-like. Finally, we find that the convective component tends to be negative in the presence of a negative pressure., Comment: 8 pages, 7 figures

Time-series analysis in terms of ordinal patterns is revisited by introducing a generalized permutation entropy $H_p(w,L)$, which depends on two different window lengths: $w$, implicitly defining the resolution of the underlying partition; $L$, playing the role of an embedding dimension, analogously to standard nonlinear time-series analysis. The $w$-dependence provides information on the structure of the corresponding invariant measure, while the $L$-dependence helps determining the Kolmogorov-Sinai entropy. We finally investigate the structure of the partition with the help of principal component analysis, finding that, upon increasing $w$, the single atoms become increasingly elongated., Comment: 6 pages; 6 figures

We provide evidence of an extremely slow thermalization occurring in the Discrete NonLinear Schr\"odinger (DNLS) model. At variance with many similar processes encountered in statistical mechanics - typically ascribed to the presence of (free) energy barriers - here the slowness has a purely dynamical origin: it is due to the presence of an adiabatic invariant, which freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events, where energy is suddenly released towards the background. We conjecture that this exponentially slow relaxation is a key ingredient contributing to the non-ergodic behavior recently observed in the negative temperature region of the DNLS equation., Comment: Corrected some misprints. Version accepted for publication in Physical Review Letters. Supplementary material is included

We analyze an intermediate collective regime where amplitude oscillators distribute themselves along a closed, smooth, time-dependent curve $\mathcal{C}$, thereby maintaining the typical ordering of (identical) phase oscillators. This is achieved by developing a general formalism based on two partial differential equations, which describe the evolution of the probability density along $\mathcal{C}$ and of the shape of $\mathcal{C}$ itself. The formalism is specifically developed for Stuart-Landau oscillators, but it is general enough to be applied to other classes of amplitude oscillators. The main achievements consist in: (i) identification and characterization of a new transition to self-consistent partial synchrony (SCPS), which confirms the crucial role played by higher Fourier hamonics in the coupling function; (ii) an analytical treatment of SCPS, including a detailed stability analysis; (iii) the discovery of a new form of collective chaos, which can be seen as a generalization of SCPS and characterized by a multifractal probability density. Finally, we are able to describe given dynamical regimes both at the macroscopic as well as the microscopic level, thereby shedding further light on the relationship between the two different levels of description.

We investigate the geometrical structure of instabilities in the two-scales Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer time scales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom, and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a non-trivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, "mixing" their evolution into a set of vectors which simultaneously carry information on both scales. We suggest these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.

We extensively explore networks of weakly unbalanced, leaky integrate-and-fire (LIF) neurons for different coupling strength, connectivity, and by varying the degree of refractoriness, as well as the delay in the spike transmission. We find that the neural network does not only exhibit a microscopic (single-neuron) stochastic-like evolution, but also a collective irregular dynamics (CID). Our analysis is based on the computation of a suitable order parameter, typically used to characterize synchronization phenomena and on a detailed scaling analysis (i.e. simulations of different network sizes). As a result, we can conclude that CID is a true thermodynamic phase, intrinsically different from the standard asynchronous regime., Comment: 12 pages, 13 figures

We revisit the dynamics of a prototypical model of balanced activity in networks of spiking neutrons. A detailed investigation of the thermodynamic limit for fixed density of connections (massive coupling) shows that, when inhibition prevails, the asymptotic regime is not asynchronous but rather characterized by a self-sustained irregular, macroscopic (collective) dynamics. So long as the connectivity is massive, this regime is found in many different setups: leaky as well as quadratic integrate-and-fire neurons; large and small coupling strength; weak and strong external currents., Comment: 5 pages, 4 figures

We discuss the behavior of the largest Lyapunov exponent $\lambda$ in the incoherent phase of large ensembles of heterogeneous, globally-coupled, phase oscillators. We show that the scaling with the system size $N$ depends on the details of the spacing distribution of the oscillator frequencies. For sufficiently regular distributions $\lambda \sim 1/N$, while for strong fluctuations of the frequency spacing, $\lambda \sim \ln N/N$ (the standard setup of independent identically distributed variables belongs to the latter class). In spite of the coupling being small for large $N$, the development of a rigorous perturbative theory is not obvious. In fact, our analysis relies on a combination of various types of numerical simulations together with approximate analytical arguments, based on a suitable stochastic approximation for the tangent space evolution. In fact, the very reason for $\lambda$ being strictly larger than zero is the presence of finite size fluctuations. We trace back the origin of the logarithmic correction to a weak synchronization between tangent and phase space dynamics.

We investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schroedinger chain in contact with a heat reservoir (a bath) at temperature $T_L$ and a pure dissipator (a sink) acting on opposite edges. We observe two different regimes. For small heat-bath temperatures $T_L$ and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures $T_L$, the transport of energy is strongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death events of the breathers. The corresponding statistics exhibits the typical signature of rare events of processes with large deviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law., Comment: Submitted to Entropy Special Issue "Thermodynamics and Statistical Mechanics of Small Systems"

We illustrate a counter-intuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show numerically and semi-analytically that a very small white noise is able to stabilize an otherwise linearly unstable collective periodic regime. Microscopic simulations reveal two noise-induced bifurcations of different nature towards self-consistent partial synchrony. We develop a macroscopic treatment solving the corresponding nonlinear Fokker-Planck equation by means of a perturbative approach. The associated linear stability analysis confirms the results anticipated by the numerics. We also argue about the generality of the phenomenon.

The Discrete NonLinear Schr\"odinger (DNLS) equation displays a parameter region characterized by the presence of localized excitations (breathers). While their formation is well understood and it is expected that the asymptotic configuration comprises a single breather on top of a background, it is not clear why the dynamics of a multi-breather configuration is essentially frozen. In order to investigate this question, we introduce simple stochastic models, characterized by suitable conservation laws. We focus on the role of the coupling strength between localized excitations and background. In the DNLS model, higher breathers interact more weakly, as a result of their faster rotation. In our stochastic models, the strength of the coupling is controlled directly by an amplitude-dependent parameter. In the case of a power-law decrease, the associated coarsening process undergoes a slowing down if the decay rate is larger than a critical value. In the case of an exponential decrease, a freezing effect is observed that is reminiscent of the scenario observed in the DNLS. This last regime arises spontaneously when direct energy diffusion between breathers and background is blocked below a certain threshold.

A powerful tool is developed for the characterization of chaotic signals. The approach is based on the symbolic encoding of time series (according to their ordinal patterns) combined with the ensuing characterization of the corresponding cylinder sets. Quantitative estimates of the Kolmogorov-Sinai entropy are obtained by introducing a modified permutation entropy which takes into account the average width of the cylinder sets. The method works also in hyperchaotic systems and allows estimating the fractal dimension of the underlying attractors., Comment: 5 pages - 4 figures

We show that self-consistent partial synchrony in globally coupled oscillatory ensembles is a general phenomenon. We analyze in detail appearance and stability properties of this state in possibly the simplest setup of a biharmonic Kuramoto-Daido phase model as well as demonstrate the effect in limit-cycle relaxational Rayleigh oscillators. Such a regime extends the notion of splay state from a uniform to distribution of phases to an oscillating one. Suitable collective observables such as the Kuramoto order parameter allow detecting the presence of a inhomogeneous distribution. The characteristic and most peculiar property of partial synchrony is the difference between the frequencies of single units and that of the macroscopic field.

We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of a suitably correlated background noise., Comment: 5 pages, 3 figures; submitted to PRL

A new method (`explosive immunization' (EI)) is proposed for immunization and targeted destruction of networks. It combines the explosive percolation (EP) paradigm with the idea of maintaining a fragmented distribution of clusters. The ability of each node to block the spread of an infection (or to prevent the existence of a large cluster of connected nodes) is estimated by a score. The algorithm proceeds by first identifying low score nodes that should not be vaccinated/destroyed, analogously to the links selected in EP if they do not lead to large clusters. As in EP, this is done by selecting the worst node (weakest blocker) from a finite set of randomly chosen `candidates'. Tests on several real-world and model networks suggest that the method is more efficient and faster than any existing immunization strategy. Due to the latter property it can deal with very large networks., Comment: 9 pages, 9 figures. Supplemental material included as appendices

In this chapter we introduce some of the basic models and concepts that will be discussed throughout the volume. In particular we describe systems of nonlinear oscillators arranged on low-dimensional lattices and summarize the phenomenology of their transport properties., Comment: Review paper, to appear in the Springer Lecture Notes in Physics volume "Thermal transport in low dimensions: from statistical physics to nanoscale heat transfer" (S. Lepri ed.)

An ensemble of pulse-coupled phase-oscillators is thoroughly analysed in the presence of a mean-field coupling and a dispersion of their natural frequencies. In spite of the analogies with the Kuramoto setup, a much richer scenario is observed. The "synchronised phase", which emerges upon increasing the coupling strength, is characterized by highly-irregular fluctuations: a time-series analysis reveals that the dynamics of the order parameter is indeed high-dimensional. The complex dynamics appears to be the result of the non-perturbative action of a suitably shaped phase-response curve. Such mechanism differs from the often invoked balance between excitation and inhibition and might provide an alternative basis to account for the self-sustained brain activity in the resting state. The potential interest of this dynamical regime is further strengthened by its (microscopic) linear stability, which makes it quite suited for computational tasks. The overall study has been performed by combining analytical and numerical studies, starting from the linear stability analysis of the asynchronous regime, to include the Fourier analysis of the Kuramoto order parameter, the computation of various types of Lyapunov exponents, and a microscopic study of the inter-spike intervals., Comment: 11 pages, 10 figures

A quantitative comparison of various classes of oscillators (integrate-and-fire, Winfree, and Kuramoto-Daido type) is performed in the weak-coupling limit for a fully connected network of identical units. An almost perfect agreement is found, with only tiny differences among the models. We also show that the regime of self-consistent partial synchronization [SCPS] is rather general and can be observed for arbitrarily small coupling strength in any model class. As a by-product of our study, we are able to show that an integrate-and-fire model with a generic pulse shape can be always transformed into a similar model with $\delta$-pulses and a suitable phase response curve., Comment: 28 pages; 8 figures - accepted in PRE

A novel class of nonequilibrium phase-transitions at zero temperature is found in chains of nonlinear oscillators.For two paradigmatic systems, the Hamiltonian XY model and the discrete nonlinear Schr\"odinger equation, we find that the application of boundary forces induces two synchronized phases, separated by a non-trivial interfacial region where the kinetic temperature is finite. Dynamics in such supercritical state displays anomalous chaotic properties whereby some observables are non-extensive and transport is superdiffusive. At finite temperatures, the transition is smoothed, but the temperature profile is still non-monotonous., Comment: Published version, minor changes

Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase-space. A recent numerical study of spatially-extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a non-trivial scaling behavior, D(L) ~ L^{-\gamma}, with the system size L. Here, we show that the wandering exponent \gamma can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov-surface". Our theoretical predictions are supported by the numerical analysis of several spatially-extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, \gamma can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified., Comment: 8 pages

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 Neuroscience

We investigate the coarsening evolution occurring in a simplified stochastic model of the Discrete NonLinear Schr\"odinger (DNLS) equation in the so-called negative-temperature region. We provide an explanation of the coarsening exponent $n=1/3$, by invoking an analogy with a suitable exclusion process. In spite of the equivalence with the exponent observed in other known universality classes, this model is certainly different, in that it refers to a dynamics with two conservation laws.

The recent years have witnessed a growing interest for covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a "dynamical" algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of H\'enon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).

The microscopic and macroscopic dynamics of random networks is investigated in the strong-dilution limit (i.e. for sparse networks). By simulating chaotic maps, Stuart-Landau oscillators, and leaky integrate-and-fire neurons, we show that a finite connectivity (of the order of a few tens) is able to sustain a nontrivial collective dynamics even in the thermodynamic limit. Although the network structure implies a non-additive dynamics, the microscopic evolution is extensive (i.e. the number of active degrees of freedom is proportional to the number of network elements)., Comment: 5 pages, 4 figures, to appear in Physical Review Letters

We analytically investigate the stability of {\it splay states} in networks of $N$ pulse-coupled phase-like models of neurons. By developing a perturbative technique, we find that, in the limit of large $N$, the Floquet spectrum scales as $1/N^2$ for generic discontinuous velocity fields. Moreover, the stability of the so-called short-wavelength component is determined by the sign of the jump at the discontinuity. Altogether, the form of the spectrum depends on the pulse shape but is independent of the velocity field., Comment: 22 pages, no figures and 120 equations

We generalize the concept of convective (or velocity-dependent) Lyapunov exponent $\Lambda(v)$ to an entire spectrum $\Lambda(v,n)$. Our results are supported by the consistency between the outcome of the chronotopic approach [{\it S. Lepri et al. J. Stat. Phys., 82 5/6 (1996) 1429}] and a more direct method. There exists a critical integrated density $n=n_c$, beyond which the convective exponent exhibits a discontinuous dependence on the velocity, which originates from the appearance of multiple branches. This phenomenon can be traced back to a change of concavity of the so-called {\it temporal} Lyapunov spectrum for $n>n_c$, which is therefore a dynamical invariant., Comment: 5 pages, 5 figures, Submitted to Europhysics Letters