121 research outputs found
Some fractal aspects of Self-Organized Criticality
Full text linkThe concept of Self-Organized Criticality (SOC) was proposed in an attempt to
explain the widespread appearance of power-law in nature. It describes a
mechanism in which a system reaches spontaneously a state where the
characteristic events (avalanches) are distributed according to a power law. We
present a dynamical systems approach to Self-Organized Criticality where the
dynamics is described either in terms of Iterated Function Systems, or as a
piecewise hyperbolic dynamical system of skew-product type. Some results
linking the structure of the attractor and some characteristic properties of
avalanches are discussed.Comment: 10 pages, proceeding of the conference "Fractales en progres", Paris
12-13 Novembe
Statistics of spike trains in conductance-based neural networks: Rigorous results
Get PDFWe consider a conductance based neural network inspired by the generalized
Integrate and Fire model introduced by Rudolph and Destexhe. We show the
existence and uniqueness of a unique Gibbs distribution characterizing spike
train statistics. The corresponding Gibbs potential is explicitly computed.
These results hold in presence of a time-dependent stimulus and apply therefore
to non-stationary dynamics.Comment: 42 pages, 1 figure, to appear in Journal of Mathematical Neuroscienc
Spike train statistics and Gibbs distributions
Get PDFThis paper is based on a lecture given in the LACONEU summer school,
Valparaiso, January 2012. We introduce Gibbs distribution in a general setting,
including non stationary dynamics, and present then three examples of such
Gibbs distributions, in the context of neural networks spike train statistics:
(i) Maximum entropy model with spatio-temporal constraints; (ii) Generalized
Linear Models; (iii) Conductance based Inte- grate and Fire model with chemical
synapses and gap junctions.Comment: 23 pages, submitte
Random Recurrent Neural Networks Dynamics
Full text linkThis paper is a review dealing with the study of large size random recurrent
neural networks. The connection weights are selected according to a probability
law and it is possible to predict the network dynamics at a macroscopic scale
using an averaging principle. After a first introductory section, the section 1
reviews the various models from the points of view of the single neuron
dynamics and of the global network dynamics. A summary of notations is
presented, which is quite helpful for the sequel. In section 2, mean-field
dynamics is developed.
The probability distribution characterizing global dynamics is computed. In
section 3, some applications of mean-field theory to the prediction of chaotic
regime for Analog Formal Random Recurrent Neural Networks (AFRRNN) are
displayed. The case of AFRRNN with an homogeneous population of neurons is
studied in section 4. Then, a two-population model is studied in section 5. The
occurrence of a cyclo-stationary chaos is displayed using the results of
\cite{Dauce01}. In section 6, an insight of the application of mean-field
theory to IF networks is given using the results of \cite{BrunelHakim99}.Comment: Review paper, 36 pages, 5 figure
On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses
Get PDFWe present a mathematical analysis of a networks with Integrate-and-Fire
neurons and adaptive conductances. Taking into account the realistic fact that
the spike time is only known within some \textit{finite} precision, we propose
a model where spikes are effective at times multiple of a characteristic time
scale , where can be \textit{arbitrary} small (in particular,
well beyond the numerical precision). We make a complete mathematical
characterization of the model-dynamics and obtain the following results. The
asymptotic dynamics is composed by finitely many stable periodic orbits, whose
number and period can be arbitrary large and can diverge in a region of the
synaptic weights space, traditionally called the "edge of chaos", a notion
mathematically well defined in the present paper. Furthermore, except at the
edge of chaos, there is a one-to-one correspondence between the membrane
potential trajectories and the raster plot. This shows that the neural code is
entirely "in the spikes" in this case. As a key tool, we introduce an order
parameter, easy to compute numerically, and closely related to a natural notion
of entropy, providing a relevant characterization of the computational
capabilities of the network. This allows us to compare the computational
capabilities of leaky and Integrate-and-Fire models and conductance based
models. The present study considers networks with constant input, and without
time-dependent plasticity, but the framework has been designed for both
extensions.Comment: 36 pages, 9 figure
Transmitting a signal by amplitude modulation in a chaotic network
Full text linkWe discuss the ability of a network with non linear relays and chaotic
dynamics to transmit signals, on the basis of a linear response theory
developed by Ruelle \cite{Ruelle} for dissipative systems. We show in
particular how the dynamics 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. This leads one to reconsider
notions such as ``hubs''. 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, \textit{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 \textit{specific} nodes. This opens the
possibility for encoding data in a way such that the recovery of the signal
requires the knowledge of the carrier frequency \textit{and} can be performed
only at some specific node.Comment: 19 pages, 13 figures, submitted (03-03-2005
What can one learn about Self-Organized Criticality from Dynamical Systems theory ?
Full text linkWe develop a dynamical system approach for the Zhang's 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
Hausdorff dimensions, and the system size are related to the probability
distribution of the avalanche size, via the Ledrappier-Young formula.Comment: 23 pages, 8 figures. to appear in Jour. of Stat. Phy
Self-Organized Criticality and Thermodynamic formalism
Full text linkWe introduce a dissipative version of the Zhang's model of Self-Organized
Criticality, where a parameter allows to tune the local energy dissipation. We
analyze the main dynamical features of the model and relate in particular the
Lyapunov spectrum with the transport properties in the stationary regime. We
develop a thermodynamic formalism where we define formal Gibbs measure,
partition function and pressure characterizing the avalanche distributions. We
discuss the infinite size limit in this setting. We show in particular that a
Lee-Yang phenomenon occurs in this model, for the only conservative case. This
suggests new connexions to classical critical phenomena.Comment: 35 pages, 15 Figures, submitte
Expectation-driven interaction: a model based on Luhmann's contingency approach
Full text linkWe introduce an agent-based model of interaction, drawing on the contingency
approach from Luhmann's theory of social systems. The agent interactions are
defined by the exchange of distinct messages. Message selection is based on the
history of the interaction and developed within the confines of the problem of
double contingency. We examine interaction strategies in the light of the
message-exchange description using analytical and computational methods.Comment: 37 pages, 16 Figures, to appear in Journal of Artificial Societies
and Social Simulation
- …
