Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter

Chronic neural probe for simultaneous recording of single-unit, multi-unit, and local field potential activity from multiple brain sites

Drug resistant focal epilepsy can be treated by resecting the epileptic focus requiring a precise focus localization using stereoelectroencephalography (SEEG) probes. As commercial SEEG probes offer only a limited spatial resolution, probes of higher channel count and design freedom enabling the incorporation of macro and microelectrodes would help increasing spatial resolution and thus open new perspectives for investigating mechanisms underlying focal epilepsy and its treatment. This work describes a new fabrication process for SEEG probes with materials and dimensions similar to clinical probes enabling recording single neuron activity at high spatial resolution. Polyimide is used as a biocompatible flexible substrate into which platinum electrodes and leads are... The resulting probe features match those of clinically approved devices. Tests in saline solution confirmed the probe stability and functionality. Probes were implanted into the brain of one monkey (Macaca mulatta), trained to perform different motor tasks. Suitable configurations including up to 128 electrode sites allow the recording of task-related neuronal signals. Probes with 32 and 64 electrode sites were implanted in the posterior parietal cortex. Local field potentials and multi-unit activity were recorded as early as one hour after implantation. Stable single-unit activity was achieved for up to 26 days after implantation of a 64-channel probe. All recorded signals showed modulation during task execution. With the novel probes it is possible to record stable biologically relevant data over a time span exceeding the usual time needed for epileptic focus localization in human patients. This is the first time that single units are recorded along cylindrical polyimide probes chronically implanted 22 mm deep into the brain of a monkey, which suggests the potential usefulness of this probe for human applications

Methods and advanced Equipment for Simulation and Treatment in Radiation Oncology: programa MAESTRO

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Many-body symbolic dynamics of a classical oscillator chain

We study a certain type of the celebrated Fermi-Pasta-Ulam particle chain, namely the inverted FPU model, where the inter-particle potential has a form of a quartic double well. Numerical evidence is given in support of a simple symbolic description of dynamics (in the regime of sufficiently high potential barrier between the wells) in terms of an (approximate) Markov process. The corresponding transition matrix is formally identical to a ferromagnetic Heisenberg quantum spin-1/2 chain with long range coupling, whose diagonalization yields accurate estimates for a class of time correlation functions of the model.Comment: 22 pages in LaTeX with 14 figures; submitted to Nonlinearity ; corrected page offset proble

Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators

The largest Lyapunov exponent of a system composed by a heavy impurity embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators is numerically computed for various values of the impurity mass $M$. A crossover between weak and strong chaos is obtained at the same value $\epsilon_{_T}$ of the energy density $\epsilon$ (energy per degree of freedom) for all the considered values of the impurity mass $M$. The threshold \epsi lon_{_T} coincides with the value of the energy density $\epsilon$ at which a change of scaling of the relaxation time of the momentum autocorrelation function of the impurity ocurrs and that was obtained in a previous work ~[M. Romero-Bastida and E. Braun, Phys. Rev. E {\bf65}, 036228 (2002)]. The complete Lyapunov spectrum does not depend significantly on the impurity mass $M$. These results suggest that the impurity does not contribute significantly to the dynamical instability (chaos) of the chain and can be considered as a probe for the dynamics of the system to which the impurity is coupled. Finally, it is shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak to strong chaos at the same value of the energy density that the crossover value $\epsilon_{_T}$ of largest Lyapunov exponent. Implications of this result are discussed.Comment: 6 pages, 5 figures, revtex4 styl

Localized behavior in the Lyapunov vectors for quasi-one-dimensional many-hard-disk systems

We introduce a definition of a "localization width" whose logarithm is given by the entropy of the distribution of particle component amplitudes in the Lyapunov vector. Different types of localization widths are observed, for example, a minimum localization width where the components of only two particles are dominant. We can distinguish a delocalization associated with a random distribution of particle contributions, a delocalization associated with a uniform distribution and a delocalization associated with a wave-like structure in the Lyapunov vector. Using the localization width we show that in quasi-one-dimensional systems of many hard disks there are two kinds of dependence of the localization width on the Lyapunov exponent index for the larger exponents: one is exponential, and the other is linear. Differences, due to these kinds of localizations also appear in the shapes of the localized peaks of the Lyapunov vectors, the Lyapunov spectra and the angle between the spatial and momentum parts of the Lyapunov vectors. We show that the Krylov relation for the largest Lyapunov exponent $\lambda\sim-\rho\ln\rho$ as a function of the density $\rho$ is satisfied (apart from a factor) in the same density region as the linear dependence of the localization widths is observed. It is also shown that there are asymmetries in the spatial and momentum parts of the Lyapunov vectors, as well as in their $x$ and $y$-components.Comment: 41 pages, 21 figures, Manuscript including the figures of better quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm

Histological assessment of a chronically implanted cylindrically-shaped, polymer-based neural probe in the monkey

Objective. Previous studies demonstrated the possibility to fabricate stereo-electroencephalography probes with high channel count and great design freedom, which incorporate macro-electrodes as well as micro-electrodes offering potential benefits for the pre-surgical evaluation of drug resistant epileptic patients. These new polyimide probes allowed to record local field potentials, multi- and single-unit activity (SUA) in the macaque monkey as early as 1 h after implantation, and yielded stable SUA for up to 26 d after implantation. The findings opened new perspectives for investigating mechanisms underlying focal epilepsy and its treatment, but before moving to possible human application, safety data are needed. In the present study we evaluate the tissue response of this new neural interface by assessing post-mortem the reaction of brain tissue along and around the probe implantation site. Approach. Three probes were implanted, independently, in the brain of one monkey (Macaca mulatta) at different times. We used specific immunostaining methods for visualizing neuronal cells and astrocytes, for measuring the extent of damage caused by the probe and for relating it with the implantation time. Main results. The size of the region where neurons cannot be detected did not exceed the size of the probe, indicating that a complete loss of neuronal cells is only present where the probe was physically positioned in the brain. Furthermore, around the probe shank, we observed a slightly reduced number of neurons within a radius of 50 µm and a modest increase in the number of astrocytes within 100 µm. Significance. In the light of previous electrophysiological findings, the present data suggest the potential usefulness and safety of this probe for human applications