144 research outputs found
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.
Ge
Symmetric sequence processing in a recurrent neural network model with a synchronous dynamics
The synchronous dynamics and the stationary states of a recurrent attractor
neural network model with competing synapses between symmetric sequence
processing and Hebbian pattern reconstruction is studied in this work allowing
for the presence of a self-interaction for each unit. Phase diagrams of
stationary states are obtained exhibiting phases of retrieval, symmetric and
period-two cyclic states as well as correlated and frozen-in states, in the
absence of noise. The frozen-in states are destabilised by synaptic noise and
well separated regions of correlated and cyclic states are obtained. Excitatory
or inhibitory self-interactions yield enlarged phases of fixed-point or cyclic
behaviour.Comment: Accepted for publication in Journal of Physics A: Mathematical and
Theoretica
Mean-field analysis of the majority-vote model broken-ergodicity steady state
We study analytically a variant of the one-dimensional majority-vote model in
which the individual retains its opinion in case there is a tie among the
neighbors' opinions. The individuals are fixed in the sites of a ring of size
and can interact with their nearest neighbors only. The interesting feature
of this model is that it exhibits an infinity of spatially heterogeneous
absorbing configurations for whose statistical properties we
probe analytically using a mean-field framework based on the decomposition of
the -site joint probability distribution into the -contiguous-site joint
distributions, the so-called -site approximation. To describe the
broken-ergodicity steady state of the model we solve analytically the
mean-field dynamic equations for arbitrary time in the cases n=3 and 4. The
asymptotic limit reveals the mapping between the statistical
properties of the random initial configurations and those of the final
absorbing configurations. For the pair approximation () we derive that
mapping using a trick that avoids solving the full dynamics. Most remarkably,
we find that the predictions of the 4-site approximation reduce to those of the
3-site in the case of expectations involving three contiguous sites. In
addition, those expectations fit the Monte Carlo data perfectly and so we
conjecture that they are in fact the exact expectations for the one-dimensional
majority-vote model
A Population Genetic Approach to the Quasispecies Model
A population genetics formulation of Eigen's molecular quasispecies model is
proposed and several simple replication landscapes are investigated
analytically. Our results show a remarcable similarity to those obtained with
the original kinetics formulation of the quasispecies model. However, due to
the simplicity of our approach, the space of the parameters that define the
model can be explored. In particular, for the simgle-sharp-peak landscape our
analysis yelds some interesting predictions such as the existence of a maximum
peak height and a mini- mum molecule length for the onset of the error
threshold transition.Comment: 16 pages, 4 Postscript figures. Submited to Phy. Rev.
Error threshold in the evolution of diploid organisms
The effects of error propagation in the reproduction of diploid organisms are
studied within the populational genetics framework of the quasispecies model.
The dependence of the error threshold on the dominance parameter is fully
investigated. In particular, it is shown that dominance can protect the
wild-type alleles from the error catastrophe. The analysis is restricted to a
diploid analogue of the single-peaked landscape.Comment: 9 pages, 4 Postscript figures. Submitted to J. Phy. A: Mat. and Ge
The Little-Hopfield model on a Random Graph
We study the Hopfield model on a random graph in scaling regimes where the
average number of connections per neuron is a finite number and where the spin
dynamics is governed by a synchronous execution of the microscopic update rule
(Little-Hopfield model).We solve this model within replica symmetry and by
using bifurcation analysis we prove that the spin-glass/paramagnetic and the
retrieval/paramagnetictransition lines of our phase diagram are identical to
those of sequential dynamics.The first-order retrieval/spin-glass transition
line follows by direct evaluation of our observables using population dynamics.
Within the accuracy of numerical precision and for sufficiently small values of
the connectivity parameter we find that this line coincides with the
corresponding sequential one. Comparison with simulation experiments shows
excellent agreement.Comment: 14 pages, 4 figure
Statistical mechanics of temporal association in neural networks with transmission delays
We study the representation of static patterns and temporal sequences in neural networks with signal delays and a stochastic parallel dynamics. For a wide class of delay distributions, the asymptotic network behavior can be described by a generalized Gibbs distribution, generated by a novel Lyapunov functional for the determination dynamics. We extend techniques of equilibrium statistical mechanics so as to deal with time-dependent phenomena, derive analytic results for both retrieval quality and storage capacity, and compare them with numerical simulations
Random replicators with high-order interactions
We use tools of the equilibrium statistical mechanics of disordered systems
to study analytically the statistical properties of an ecosystem composed of N
species interacting via random, Gaussian interactions of order p >= 2, and
deterministic self-interactions u <= 0. We show that for nonzero u the effect
of increasing the order of the interactions is to make the system more
cooperative, in the sense that the fraction of extinct species is greatly
reduced. Furthermore, we find that for p > 2 there is a threshold value which
gives a lower bound to the concentration of the surviving species, preventing
then the existence of rare species and, consequently, increasing the robustness
of the ecosystem to external perturbations.Comment: 7 pages, 4 Postscript figure
Instance Space of the Number Partitioning Problem
Within the replica framework we study analytically the instance space of the
number partitioning problem. This classic integer programming problem consists
of partitioning a sequence of N positive real numbers \{a_1, a_2,..., a_N}
(the instance) into two sets such that the absolute value of the difference of
the sums of over the two sets is minimized. We show that there is an
upper bound to the number of perfect partitions (i.e. partitions
for which that difference is zero) and characterize the statistical properties
of the instances for which those partitions exist. In particular, in the case
that the two sets have the same cardinality (balanced partitions) we find
. Moreover, we show that the disordered model resulting from hte
instance space approach can be viewed as a model of replicators where the
random interactions are given by the Hebb rule.Comment: 7 page
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