552 research outputs found
Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder
We study the statistical properties of the stationary firing-rate states of a
neural network model with quenched disorder. The model has arbitrary size,
discrete-time evolution equations and binary firing rates, while the topology
and the strength of the synaptic connections are randomly generated from known,
generally arbitrary, probability distributions. We derived semi-analytical
expressions of the occurrence probability of the stationary states and the mean
multistability diagram of the model, in terms of the distribution of the
synaptic connections and of the external stimuli to the network. Our
calculations rely on the probability distribution of the bifurcation points of
the stationary states with respect to the external stimuli, which can be
calculated in terms of the permanent of special matrices, according to extreme
value theory. While our semi-analytical expressions are exact for any size of
the network and for any distribution of the synaptic connections, we also
specialized our calculations to the case of statistically-homogeneous
multi-population networks. In the specific case of this network topology, we
calculated analytically the permanent, obtaining a compact formula that
outperforms of several orders of magnitude the
Balasubramanian-Bax-Franklin-Glynn algorithm. To conclude, by applying the
Fisher-Tippett-Gnedenko theorem, we derived asymptotic expressions of the
stationary-state statistics of multi-population networks in the
large-network-size limit, in terms of the Gumbel (double exponential)
distribution. We also provide a Python implementation of our formulas and some
examples of the results generated by the code.Comment: 30 pages, 6 figures, 2 supplemental Python script
Inferring hidden states in Langevin dynamics on large networks: Average case performance
We present average performance results for dynamical inference problems in
large networks, where a set of nodes is hidden while the time trajectories of
the others are observed. Examples of this scenario can occur in signal
transduction and gene regulation networks. We focus on the linear stochastic
dynamics of continuous variables interacting via random Gaussian couplings of
generic symmetry. We analyze the inference error, given by the variance of the
posterior distribution over hidden paths, in the thermodynamic limit and as a
function of the system parameters and the ratio {\alpha} between the number of
hidden and observed nodes. By applying Kalman filter recursions we find that
the posterior dynamics is governed by an "effective" drift that incorporates
the effect of the observations. We present two approaches for characterizing
the posterior variance that allow us to tackle, respectively, equilibrium and
nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals
average spectral properties of the inference error and typical posterior
relaxation times, the second is based on dynamical functionals and yields the
inference error as the solution of an algebraic equation.Comment: 20 pages, 5 figure
The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models
In this contribution we give a pedagogic introduction to the newly introduced
adaptive interpolation method to prove in a simple and unified way replica
formulas for Bayesian optimal inference problems. Many aspects of this method
can already be explained at the level of the simple Curie-Weiss spin system.
This provides a new method of solution for this model which does not appear to
be known. We then generalize this analysis to a paradigmatic inference problem,
namely rank-one matrix estimation, also refered to as the Wigner spike model in
statistics. We give many pointers to the recent literature where the method has
been succesfully applied
Instability of frozen-in states in synchronous Hebbian neural networks
The full dynamics of a synchronous recurrent neural network model with Ising
binary units and a Hebbian learning rule with a finite self-interaction is
studied in order to determine the stability to synaptic and stochastic noise of
frozen-in states that appear in the absence of both kinds of noise. Both, the
numerical simulation procedure of Eissfeller and Opper and a new alternative
procedure that allows to follow the dynamics over larger time scales have been
used in this work. It is shown that synaptic noise destabilizes the frozen-in
states and yields either retrieval or paramagnetic states for not too large
stochastic noise. The indications are that the same results may follow in the
absence of synaptic noise, for low stochastic noise.Comment: 14 pages and 4 figures; accepted for publication in J. Phys. A: Math.
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