Canard explosion in delayed equations with multiple timescales

We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of `classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold.Comment: arXiv admin note: substantial text overlap with arXiv:1404.584

On the dynamics of random neuronal networks

We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the limit in distribution of the sample path of the state of a neuron of the network when its size gets large. The invariant distributions of this limiting stochastic process are analyzed as well as their stability properties. We show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.Comment: 37 pages, 3 figure

A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.Comment: 55 pages, 4 figures, to appear in "Frontiers in Neuroscience

Noise-induced synchronization and anti-resonance in excitable systems; Implications for information processing in Parkinson's Disease and Deep Brain Stimulation

We study the statistical physics of a surprising phenomenon arising in large networks of excitable elements in response to noise: while at low noise, solutions remain in the vicinity of the resting state and large-noise solutions show asynchronous activity, the network displays orderly, perfectly synchronized periodic responses at intermediate level of noise. We show that this phenomenon is fundamentally stochastic and collective in nature. Indeed, for noise and coupling within specific ranges, an asymmetry in the transition rates between a resting and an excited regime progressively builds up, leading to an increase in the fraction of excited neurons eventually triggering a chain reaction associated with a macroscopic synchronized excursion and a collective return to rest where this process starts afresh, thus yielding the observed periodic synchronized oscillations. We further uncover a novel anti-resonance phenomenon: noise-induced synchronized oscillations disappear when the system is driven by periodic stimulation with frequency within a specific range. In that anti-resonance regime, the system is optimal for measures of information capacity. This observation provides a new hypothesis accounting for the efficiency of Deep Brain Stimulation therapies in Parkinson's disease, a neurodegenerative disease characterized by an increased synchronization of brain motor circuits. We further discuss the universality of these phenomena in the class of stochastic networks of excitable elements with confining coupling, and illustrate this universality by analyzing various classical models of neuronal networks. Altogether, these results uncover some universal mechanisms supporting a regularizing impact of noise in excitable systems, reveal a novel anti-resonance phenomenon in these systems, and propose a new hypothesis for the efficiency of high-frequency stimulation in Parkinson's disease

Finite-size and correlation-induced effects in Mean-field Dynamics

The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finite-size networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such large-scale networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of the new mean-field equations, the stability properties of limit cycles are modified by the presence of correlations, and additional non-trivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finite-size networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinite-size system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system

Widespread tungsten isotope anomalies and W mobility in crustal and mantle rocks of the Eoarchean Saglek Block, northern Labrador, Canada: Implications for early Earth processes and W recycling

Well-resolved 182W isotope anomalies, relative to the present mantle, in Hadean–Archean terrestrial rocks have been interpreted to reflect the effects of variable late accretion and early mantle differentiation processes. To further explore these early Earth processes, we have carried out W concentration and isotopic measurements of Eoarchean ultramafic rocks, including lithospheric mantle rocks, meta-komatiites, a layered ultramafic body and associated crustal gneisses and amphibolites from the Uivak gneiss terrane of the Saglek Block, northern Labrador, Canada. These analyses are augmented by in situ W concentration measurements of individual phases in order to examine the major hosts of W in these rocks. Although the W budget in some rocks can be largely explained by a combination of their major phases, W in other rocks is hosted mainly in secondary grain-boundary assemblages, as well as in cryptic, unidentified W-bearing ‘nugget’ minerals. Whole rock W concentrations in the ultramafic rocks show unexpected enrichments relative, to elements with similar incompatibilities. By contrast, W concentrations are low in the Uivak gneisses. These data, along with the in situ W concentration data, suggest metamorphic transport/re-distribution of W from the regional felsic rocks, the Uivak gneiss precursors, to the spatially associated ultramafic rocks. All but one sample from the lithologically varied Eoarchean Saglek suite is characterized by generally uniform enrichments in 182W relative to Earth's modern mantle. Modeling shows that the W isotopic enrichments in the ultramafic rocks were primarily inherited from the surrounding 182W-rich felsic precursor rocks, and that the W isotopic composition of the original ultramafic rocks cannot be determined. The observed W isotopic composition of mafic to ultramafic rocks in intimate contact with ancient crust should be viewed with caution in order to plate constraints on the early Hf–W isotopic evolution of the Earth's mantle with regard to late accretionary processes. Although 182W anomalies can be erased via mixing in the convective mantle, recycling of 182W-rich crustal rocks into the mantle can produce new mantle sources with anomalous W isotopic compositions that can be tapped at much later times and, hence, this process should be considered as a mechanism for the generation of 182W-rich rocks at any subsequent time in Earth history.The NSERC Discovery Grants program to DGP U.S. NSF-CSEDI grant EAR1265169 (to RJW)

Correlation between Progetto Cuore risk score and early cardiovascular damage in never treated subjects

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