Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials, in parameter regimes characterised by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime, and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.Comment: 36 page

Convergence to equilibrium for many particle systems

The goal of this paper is to give a short review of recent results of the authors concerning classical Hamiltonian many particle systems. We hope that these results support the new possible formulation of Boltzmann's ergodicity hypothesis which sounds as follows. For almost all potentials, the minimal contact with external world, through only one particle of $N$, is sufficient for ergodicity. But only if this contact has no memory. Also new results for quantum case are presented

Corrosion Grade on Anchor Rods of Guyed Transmission Towers Applying Machine Committee / Grau de Corrosão em Hastes de Âncora de Torres de Transmissão Guiadas Comitê de Aplicação de Máquinas

The use of guyed structures in electric power transmission lines is a growing practice because of their cost efficiency. However, the anchor systems are subject to corrosion, which can lead to their rupture and loss of tower support. Monitoring the evolution of the corrosion of the anchor rods by visual inspection is a destructive and costly method; therefore, there is considerable interest in developing methods and tools that are capable of generating a maintenance diagnosis of the system. This work aimed to propose a classification tool for guyed towers in terms of the corrosion degree by a machine committee with neural networks and applied it to the Paraiso-Açu line located in Rio Grande do Norte in Brazil. Thirty-eight samples were collected and 33 variables related to the soil corrosion along the line were analyzed. The targets for training the networks were obtained from the inspection of anchor rods taken from the field. A simplification of the problem's dimension was proposed by principal component analysis, describing the phenomenon with 6 variables instead of 33, simplifying the practical application by massively reducing the requirements for data sampling in the field. Several network typologies were trained and the best ones in terms of their generalist and specialist capacities were combined in a machine committee for the final proposal of this work. The classification obtained by the application of the committee for 10 towers was compared with the classification from non-destructive impulse reflectometry tests and showed an 80% correlation

Richardson's pair diffusion and the stagnation point structure of turbulence

DNS and laboratory experiments show that the spatial distribution of straining stagnation points in homogeneous isotropic 3D turbulence has a fractal structure with dimension D_s = 2. In Kinematic Simulations the time exponent gamma in Richardson's law and the fractal dimension D_s are related by gamma = 6/D_s. The Richardson constant is found to be an increasing function of the number of straining stagnation points in agreement with pair duffusion occuring in bursts when pairs meet such points in the flow.Comment: 4 pages; Submitted to Phys. Rev. Let

Large deviation principle for Benedicks-Carleson quadratic maps

Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of arbitrarily small scale and associated towers, to which the absolutely continuous measures can be lifted. These various lifts together enable us to obtain a control of recurrence that is sufficient to establish a level 2 large deviation principle, for the absolutely continuous measures. This result encompasses dynamics far from equilibrium, and thus significantly extends presently known local large deviations results for quadratic maps.Comment: 23 pages, no figure, former title: Full large deviation principle for Benedicks-Carleson quadratic map

Boundaries from Inhomogeneous Bernoulli Trials

The boundary problem is considered for inhomogeneous increasing random walks on the square lattice \mathbbZ2+Z2+ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles