579 research outputs found
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
Some fractal aspects of Self-Organized Criticality
The 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
We 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
Dynamics and spike trains statistics in conductance-based Integrate-and-Fire neural networks with chemical and electric synapses
We investigate the effect of electric synapses (gap junctions) on collective
neuronal dynamics and spike statistics in a conductance-based
Integrate-and-Fire neural network, driven by a Brownian noise, where
conductances depend upon spike history. We compute explicitly the time
evolution operator and show that, given the spike-history of the network and
the membrane potentials at a given time, the further dynamical evolution can be
written in a closed form. We show that spike train statistics is described by a
Gibbs distribution whose potential can be approximated with an explicit
formula, when the noise is weak. This potential form encompasses existing
models for spike trains statistics analysis such as maximum entropy models or
Generalized Linear Models (GLM). We also discuss the different types of
correlations: those induced by a shared stimulus and those induced by neurons
interactions.Comment: 42 pages, 1 figure, submitte
Spike train statistics and Gibbs distributions
This 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
This 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
We 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
Exact computation of the Maximum Entropy Potential of spiking neural networks models
Understanding how stimuli and synaptic connectivity in uence the statistics
of spike patterns in neural networks is a central question in computational
neuroscience. Maximum Entropy approach has been successfully used to
characterize the statistical response of simultaneously recorded spiking
neurons responding to stimuli. But, in spite of good performance in terms of
prediction, the fitting parameters do not explain the underlying mechanistic
causes of the observed correlations. On the other hand, mathematical models of
spiking neurons (neuro-mimetic models) provide a probabilistic mapping between
stimulus, network architecture and spike patterns in terms of conditional
proba- bilities. In this paper we build an exact analytical mapping between
neuro-mimetic and Maximum Entropy models.Comment: arXiv admin note: text overlap with arXiv:1309.587
Parameters estimation for spatio-temporal maximum entropy distributions: application to neural spike trains
We propose a numerical method to learn Maximum Entropy (MaxEnt) distributions
with spatio-temporal constraints from experimental spike trains. This is an
extension of two papers [10] and [4] who proposed the estimation of parameters
where only spatial constraints were taken into account. The extension we
propose allows to properly handle memory effects in spike statistics, for large
sized neural networks.Comment: 34 pages, 33 figure
Transmitting a signal by amplitude modulation in a chaotic network
We 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
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