A dynamical system approach to SOC models of Zhang's type

We discuss Zhang's model of SOC in the framework of hyperbolic dynamical systems with singularities. The fractal structure of the invariant energy distribution, correlation decay-like phenomena, and symbolic coding are discussed.     

Self-Organized Criticality and Thermodynamic Formalism

We develop a thermodynamic formalism for a dissipative version of the Zhang model of Self-Organized Criticality, where a parameter allows us to tune the local energy dissipation. By constructing a suitable Markov partition we define Gibbs measures (in the sense of Sinai, Ruelle, and Bowen), partition functions, and topological pressure allowing the analysis of probability distributions of avalanches. We discuss the infinite-size limit in this setting. In particular, we show that a Lee–Yang phenomenon occurs in the conservative case. This suggests new connections to classical critical phenomena.     

Linear Response of General Observables in Spiking Neuronal Network Models

We establish a general linear response relation for spiking neuronal networks, based on chains with unbounded memory. This relation allow us to predict the influence of a weak amplitude time dependent external stimuli on spatio-temporal spike correlations, from the spontaneous statistics (without stimulus) in a general context where the memory in spike dynamics can extend arbitrarily far in the past. Using this approach, we show how the linear response is explicitly related to the collective effect of the stimuli, intrinsic neuronal dynamics, and network connectivity on spike train statistics. We illustrate our results with numerical simulations performed over a discrete time integrate and fire model.     

Transmitting a signal by amplitude modulation in a chaotic network

We discuss the ability of a model of network with nonlinear units and chaotic dynamics to transmit signals, on the basis of a linear response theory developed by Ruelle [D. Ruelle, J. Stat. Phys. 95, 393 (1999)] for dissipative systems. We discuss in particular how the dynamics may interfere with the graph topology to produce an effective transmission network, whose topology depends on the signal, and cannot be directly read on the "wired" network. Then, we show examples where, with a suitable choice of the carrier frequency (resonance), one can transmit a signal from a node to another one by amplitude modulation, in spite of chaos. Also, we give an example where a signal, transmitted to any node via different paths, can only be recovered by a couple of specific nodes. This opens up the possibility for encoding data in a way such that the recovery of the signal requires the knowledge of the carrier frequency and can be performed only at some specific node.     

How Gibbs Distributions May Naturally Arise from Synaptic Adaptation Mechanisms. A Model-Based Argumentation

This paper addresses two questions in the context of neuronal networks dynamics, using methods from dynamical systems theory and statistical physics: (i) How to characterize the statistical properties of sequences of action potentials (“spike trains”) produced by neuronal networks? and; (ii) what are the effects of synaptic plasticity on these statistics? We introduce a framework in which spike trains are associated to a coding of membrane potential trajectories, and actually, constitute a symbolic coding in important explicit examples (the so-called gIF models). On this basis, we use the thermodynamic formalism from ergodic theory to show how Gibbs distributions are natural probability measures to describe the statistics of spike trains, given the empirical averages of prescribed quantities. As a second result, we show that Gibbs distributions naturally arise when considering “slow” synaptic plasticity rules where the characteristic time for synapse adaptation is quite longer than the characteristic time for neurons dynamics.     

What Can One Learn About Self-Organized Criticality from Dynamical Systems Theory?

We develop a dynamical system approach for the Zhang model of self-organized criticality, for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Haussdorf dimensions, and the system size are related to the probability distribution of the avalanche size via the Ledrappier–Young formula.     

Raman scattering in uniaxially oriented samples of planar zigzag poly(vinylidiene fluoride)

Band assignments of phase-I PVF₂ have been Revised by using additional data obtained by laser Raman spectra of oriented samples. A set of least-squares refined force constants was obtained which reproduce the experimental data to an average error in frequencies of 1.3 cm^?1. In order to determine the effect of electrical polarization on the spectra, a Gaussian distribution of the dipole axis was assumed. The calculation shows that polarizations of less than 60% will not significantly affect the Raman spectra.