The distinction between the chaotic and regular behaviour of orbits in galactic models is an important issue and can help our understanding of galact

We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis on understanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and their immediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is still much affected (``dragged'') by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic approach rests upon the following applications: the GALI method to detect the regular and chaotic regions and thus to describe in detail the structure of the phase space, the description of the momentum distribution in terms of the L\'evy stable distributions, and the numerical calculation of the diffusion exponent and of the corresponding diffusion constant. We use this approach to analyze in detail and systematically the standard map at all values of the kick parameter $K$, up to $K=70$. All complex features of the anomalous diffusion are well understood in terms of the role of the accelerator modes, mainly of period 1 at large $K$\ge${}2$\pi${}$, but also of higher periods ($2,3,4,...$) at smaller values of $K$\le${}2$\pi${}$.

Several brain diseases are characterized by abnormally strong neuronal synchrony. Coordinated Reset (CR) stimulation was computationally designed to specifically counteract abnormal neuronal synchronization processes by desynchronization. In the presence of spike-timing-dependent plasticity (STDP) this may lead to a decrease of synaptic excitatory weights and ultimately to an anti-kindling, i.e. unlearning of abnormal synaptic connectivity and abnormal neuronal synchrony. The long-lasting desynchronizing impact of CR stimulation has been verified in pre-clinical and clinical proof of concept studies. However, as yet it is unclear how to optimally choose the CR stimulation frequency, i.e. the repetition rate at which the CR stimuli are delivered. This work presents the first computational study on the dependence of the acute and long-term outcome on the CR stimulation frequency in neuronal networks with STDP. For this purpose, CR stimulation was applied with Rapidly Varying Sequences (RVS) as well as with Slowly Varying Sequences (SVS) in a wide range of stimulation frequencies and intensities. Our findings demonstrate that acute desynchronization, achieved during stimulation, does not necessarily lead to long-term desynchronization after cessation of stimulation. By comparing the long-term effects of the two different CR protocols, the RVS CR stimulation turned out to be more robust against variations of the stimulation frequency. However, SVS CR stimulation can obtain stronger anti-kindling effects. We revealed specific parameter ranges that are favorable for long-term desynchronization. For instance, RVS CR stimulation at weak intensities and with stimulation frequencies in the range of the neuronal firing rates turned out to be effective and robust, in particular, if no closed loop adaptation of stimulation parameters is (technically) available. From a clinical standpoint, this may be relevant in the context of both invasive as well as non-invasive CR stimulation. Abnormally strong neuronal synchronization is found in a number of brain disorders. To specifically counteract abnormal neuronal synchrony and, hence, related symptoms, Coordinated Reset (CR) stimulation was developed. CR stimulation employs basic plasticity and dynamic self-organization principles of the nervous system. Its fundamental goal is to induce long-lasting desynchronizing effects that persist cessation of stimulation. The latter are key to reducing side effects of invasive therapies such as deep brain stimulation. Furthermore, sustained stimulation effects pave the way for non-invasive neuromodulation treatments, where a few hours of stimulation delivered regularly or occasionally may provide substantial relief. Long-lasting CR-induced desynchronizing therapeutic effects have been verified in several pre-clinical and clinical studies. However, we here present the first computational study that systematically investigates the impact of key stimulation parameters on the stimulation outcome. Our results provide experimentally testable predictions that are relevant for pre-clinical and clinical studies. Furthermore, our results may contribute to stimulation techniques that enable to probe the functional role of brain rhythms in general.

We study the kicked rotator in the classically fully chaotic regime using Izrailev's $N$-dimensional model for various $N\le 4000$, which in the limit $N\to \infty$ tends to the quantized kicked rotator. We do treat not only the case $K=5$, as studied previously, but also many different values of the classical kick parameter $5\le K\le 35$ and many different values of the quantum parameter $k\in [5,60]$. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable $\Lambda =l_{\infty }/N$ to the case of anomalous diffusion in the classical phase space by deriving the localization length $l_{\infty }$ for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents ${\beta }_{\mathrm{BR}}$. (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by ${\beta }_{\mathrm{loc}}$ in the interval $[0,1]$. The level repulsion parameters ${\beta }_{\mathrm{BR}}$ and ${\beta }_{\mathrm{loc}}$ are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between ${\beta }_{\mathrm{loc}}$ and the relative localization length $\Lambda$, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistić and Robnik, J. Phys. A: Math. Gen. 43, 215101 (2010); arXiv:1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).

Several brain disorders are characterized by abnormal neuronal synchronization. To specifically counteract abnormal neuronal synchrony and, hence, related symptoms, coordinated reset (CR) stimulation was computationally developed. In principle, successive epochs of synchronizing and desynchronizing stimulation may reversibly move neural networks with plastic synapses back and forth between stable regimes with synchronized and desynchronized firing. Computationally derived predictions have been verified in pre-clinical and clinical studies, paving the way for novel therapies. However, as yet, computational models were not able to reproduce the clinically observed increase of desynchronizing effects of regularly administered CR stimulation intermingled by long stimulation-free epochs. We show that this clinically important phenomenon can be computationally reproduced by taking into account structural plasticity (SP), a mechanism that deletes or generates synapses in order to homeostatically adapt the firing rates of neurons to a set point-like target firing rate in the course of days to months. If we assume that CR stimulation favorably reduces the target firing rate of SP, the desynchronizing effects of CR stimulation increase after long stimulation-free epochs, in accordance with clinically observed phenomena. Our study highlights the pivotal role of stimulation- and dosing-induced modulation of homeostatic set points in therapeutic processes.