Scaling of Chaos in Strongly Nonlinear Lattices

Although it is now understood that chaos in complex classical systems is the foundation of thermodynamic behavior, the detailed relations between the microscopic properties of the chaotic dynamics and the macroscopic thermodynamic observations still remain mostly in the dark. In this work, we numerically analyze the probability of chaos in strongly nonlinear Hamiltonian systems and find different scaling properties depending on the nonlinear structure of the model. We argue that these different scaling laws of chaos have definite consequences for the macroscopic diffusive behavior, as chaos is the microscopic mechanism of diffusion. This is compared with previous results on chaotic diffusion [New J.\ Phys.\ 15, 053015 (2013)], and a relation between microscopic chaos and macroscopic diffusion is established.Comment: 6 pages, 4 figure

Odeint - Solving ordinary differential equations in C++

Many physical, biological or chemical systems are modeled by ordinary differential equations (ODEs) and finding their solution is an every-day-task for many scientists. Here, we introduce a new C++ library dedicated to find numerical solutions of initial value problems of ODEs: odeint (www.odeint.com). odeint is implemented in a highly generic way and provides extensive interoperability at top performance. For example, due to it's modular design it can be easily parallized with OpenMP and even runs on CUDA GPUs. Despite that, it provides a convenient interface that allows for a simple and easy usage.Comment: 4 pages, 1 figur

Spreading in Disordered Lattices with Different Nonlinearities

We study the spreading of initially localized states in a nonlinear disordered lattice described by the nonlinear Schr\"odinger equation with random on-site potentials - a nonlinear generalization of the Anderson model of localization. We use a nonlinear diffusion equation to describe the subdiffusive spreading. To confirm the self-similar nature of the evolution we characterize the peak structure of the spreading states with help of R\'enyi entropies and in particular with the structural entropy. The latter is shown to remain constant over a wide range of time. Furthermore, we report on the dependence of the spreading exponents on the nonlinearity index in the generalized nonlinear Schr\"odinger disordered lattice, and show that these quantities are in accordance with previous theoretical estimates, based on assumptions of weak and very weak chaoticity of the dynamics.Comment: 5 pages, 6 figure

Re-localization due to finite response times in a nonlinear Anderson chain

We study a disordered nonlinear Schr\"odinger equation with an additional relaxation process having a finite response time $\tau$. Without the relaxation term, $\tau=0$, this model has been widely studied in the past and numerical simulations showed subdiffusive spreading of initially localized excitations. However, recently Caetano et al.\ (EPJ. B \textbf{80}, 2011) found that by introducing a response time $\tau > 0$, spreading is suppressed and any initially localized excitation will remain localized. Here, we explain the lack of subdiffusive spreading for $\tau>0$ by numerically analyzing the energy evolution. We find that in the presence of a relaxation process the energy drifts towards the band edge, which enforces the population of fewer and fewer localized modes and hence leads to re-localization. The explanation presented here is based on previous findings by the authors et al.\ (PRE \textbf{80}, 2009) on the energy dependence of thermalized states.Comment: 3 pages, 4 figure

PySpike - A Python library for analyzing spike train synchrony

Understanding how the brain functions is one of the biggest challenges of our time. The analysis of experimentally recorded neural firing patterns (spike trains) plays a crucial role in addressing this problem. Here, the PySpike library is introduced, a Python package for spike train analysis providing parameter-free and time-scale independent measures of spike train synchrony. It allows to compute similarity and dissimilarity profiles, averaged values and distance matrices. Although mainly focusing on neuroscience, PySpike can also be applied in other contexts like climate research or social sciences. The package is available as Open Source on Github and PyPI.Comment: 7 pages, 6 figure

Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices

In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder.Comment: 7 pages, 7 figure

A guide to time-resolved and parameter-free measures of spike train synchrony

Measures of spike train synchrony have proven a valuable tool in both experimental and computational neuroscience. Particularly useful are time-resolved methods such as the ISI- and the SPIKE-distance, which have already been applied in various bivariate and multivariate contexts. Recently, SPIKE-Synchronization was proposed as another time-resolved synchronization measure. It is based on Event-Synchronization and has a very intuitive interpretation. Here, we present a detailed analysis of the mathematical properties of these three synchronization measures. For example, we were able to obtain analytic expressions for the expectation values of the ISI-distance and SPIKE-Synchronization for Poisson spike trains. For the SPIKE-distance we present an empirical formula deduced from numerical evaluations. These expectation values are crucial for interpreting the synchronization of spike trains measured in experiments or numerical simulations, as they represent the point of reference for fully randomized spike trains.Comment: 8 pages, 4 figure

Using spike train distances to identify the most discriminative neuronal subpopulation

Background: Spike trains of multiple neurons can be analyzed following the summed population (SP) or the labeled line (LL) hypothesis. Responses to external stimuli are generated by a neuronal population as a whole or the individual neurons have encoding capacities of their own. The SPIKE-distance estimated either for a single, pooled spike train over a population or for each neuron separately can serve to quantify these responses. New Method: For the SP case we compare three algorithms that search for the most discriminative subpopulation over all stimulus pairs. For the LL case we introduce a new algorithm that combines neurons that individually separate different pairs of stimuli best. Results: The best approach for SP is a brute force search over all possible subpopulations. However, it is only feasible for small populations. For more realistic settings, simulated annealing clearly outperforms gradient algorithms with only a limited increase in computational load. Our novel LL approach can handle very involved coding scenarios despite its computational ease. Comparison with Existing Methods: Spike train distances have been extended to the analysis of neural populations interpolating between SP and LL coding. This includes parametrizing the importance of distinguishing spikes being fired in different neurons. Yet, these approaches only consider the population as a whole. The explicit focus on subpopulations render our algorithms complimentary. Conclusions: The spectrum of encoding possibilities in neural populations is broad. The SP and LL cases are two extremes for which our algorithms provide correct identification results.Comment: 14 pages, 9 Figure

Scaling of energy spreading in strongly nonlinear disordered lattices

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.Comment: 4 pages, 4 figure