Existence and stability of standing waves for nonlinear fractional Schrödinger equations J. Math. Phys. 53, 083702 (2012) N-fold Darboux transformations and soliton solutions of three nonlinear equations J. Math. Phys. 53, 083502 (2012) Some algebro-geometric solutions for the coupled modified Kadomtsev-Petviashvili equations arising from the Neumann type systems

Techniques for recording large-scale neuronal spiking activity are developing very fast. This leads to an increasing demand for algorithms capable of analyzing large amounts of experimental spike train data. One of the most crucial and demanding tasks is the identification of similarity patterns with a very high temporal resolution and across different spatial scales. To address this task, in recent years three time-resolved measures of spike train synchrony have been proposed, the ISI-distance, the SPIKE-distance, and event synchronization. The Matlab source codes for calculating and visualizing these measures have been made publicly available. However, due to the many different possible representations of the results the use of these codes is rather complicated and their application requires some basic knowledge of Matlab. Thus it became desirable to provide a more user-friendly and interactive interface. Here we address this need and present SPIKY, a graphical user interface that facilitates the application of time-resolved measures of spike train synchrony to both simulated and real data. SPIKY includes implementations of the ISI-distance, the SPIKE-distance, and the SPIKE-synchronization (an improved and simplified extension of event synchronization) that have been optimized with respect to computation speed and memory demand. It also comprises a spike train generator and an event detector that makes it capable of analyzing continuous data. Finally, the SPIKY package includes additional complementary programs aimed at the analysis of large numbers of datasets and the estimation of significance levels.

We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization, characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength.PACS numbers: 05.45.-a, 63.50.-x, 63.70.+h The studies of ergodicity and dynamical thermalization in regular nonlinear lattices have a long history initiated by the Fermi-Pasta-Ulam problem [1] but they are still far from being complete (see, e.g., [2] for thermal transport in nonlinear chains and [3] for thermalization in a Bose-Hubbard model). In this letter, we study how the dynamical thermalization appears in nonlinear disordered chains where all linear modes are exponentially localized. Such modes appear due to the Anderson localization, introduced in the context of electron transport in disordered solids [4,5,6] and describing various physical situations like wave propagation in a random medium [7], localization of a Bose-Einstein condensate [8] and quantum chaos [9].Effects of nonlinearity on localization properties have attracted large interest recently. Indeed, nonlinearity naturally appears for localization of a Bose-Einstein condensate, as its evolution is described by the nonlinear Gross-Pitaevskii equation [10]. An interplay of disorder, localization, and nonlinearity is also important in other physical systems like wave propagation in nonlinear disordered media [11,12] and chains of nonlinear oscillators with randomly distributed frequencies [13].The main question here is whether the localization is destroyed by nonlinearity. It has been addressed recently using two physical setups. In refs. [14,15]it was demonstrated that an initially concentrated wavepacket spreads apparently indefinitely, although subdiffusively, in a disordered nonlinear lattice. For a transmission through a nonlinear disordered layer [16,17], chaotic destruction of localization leads to a drastically enhanced transparency.Here we study the thermalization properties of the dynamics of a nonlinear disordered lattice -discrete Anderson nonlinear Schrödinger equation (DANSE). We describe in details the features of the time-evolution of an initially localized excitation towards a statistical equilibrium in a finite lattice (we stress that this evolution is purely deterministic -and the relaxation to equilibrium is due to determinsitc chaos.). Below we argue that a statistically stationary state is characterized by the Gibbs energy equipartition across the linear eigenmodes (Eq. (5)) and call a relaxation to such an equilibrium state thermalization. Because thermalization is due to deterministic chaos, its rate heavily dependes on the statistical properties of the chaos. As is typical for nonlinear Hamiltonian systems, depending on initial conditions one can obtain solutions belonging to a "chaotic sea" or to "regul...

To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law.