368 research outputs found
Probing the basins of attraction of a recurrent neural network
A recurrent neural network is considered that can retrieve a collection of
patterns, as well as slightly perturbed versions of this `pure' set of patterns
via fixed points of its dynamics. By replacing the set of dynamical
constraints, i.e., the fixed point equations, by an extended collection of
fixed-point-like equations, analytical expressions are found for the weights
w_ij(b) of the net, which depend on a certain parameter b. This so-called basin
parameter b is such that for b=0 there are, a priori, no perturbed patterns to
be recognized by the net. It is shown by a numerical study, via probing sets,
that a net constructed to recognize perturbed patterns, i.e., with values of
the connections w_ij(b) with b unequal zero, possesses larger basins of
attraction than a net made with the help of a pure set of patterns, i.e., with
connections w_ij(b=0). The mathematical results obtained can, in principle, be
realized by an actual, biological neural net.Comment: 17 pages, LaTeX, 2 figure
A recurrent neural network with ever changing synapses
A recurrent neural network with noisy input is studied analytically, on the
basis of a Discrete Time Master Equation. The latter is derived from a
biologically realizable learning rule for the weights of the connections. In a
numerical study it is found that the fixed points of the dynamics of the net
are time dependent, implying that the representation in the brain of a fixed
piece of information (e.g., a word to be recognized) is not fixed in time.Comment: 17 pages, LaTeX, 4 figure
Controlling chaos in diluted networks with continuous neurons
Diluted neural networks with continuous neurons and nonmonotonic transfer
function are studied, with both fixed and dynamic synapses. A noisy stimulus
with periodic variance results in a mechanism for controlling chaos in neural
systems with fixed synapses: a proper amount of external perturbation forces
the system to behave periodically with the same period as the stimulus.Comment: 11 pages, 8 figure
Through-membrane electron-beam lithography for ultrathin membrane applications
We present a technique to fabricate ultrathin (down to 20 nm) uniform
electron transparent windows at dedicated locations in a SiN membrane for in
situ transmission electron microscopy experiments. An electron-beam (e-beam)
resist is spray-coated on the backside of the membrane in a KOH- etched cavity
in silicon which is patterned using through-membrane electron-beam lithography.
This is a controlled way to make transparent windows in membranes, whilst the
topside of the membrane remains undamaged and retains its flatness. Our
approach was optimized for MEMS-based heating chips but can be applied to any
chip design. We show two different applications of this technique for (1)
fabrication of a nanogap electrode by means of electromigration in thin
free-standing metal films and (2) making low-noise graphene nanopore devices
Derivation of Hebb's rule
On the basis of the general form for the energy needed to adapt the
connection strengths of a network in which learning takes place, a local
learning rule is found for the changes of the weights. This biologically
realizable learning rule turns out to comply with Hebb's neuro-physiological
postulate, but is not of the form of any of the learning rules proposed in the
literature.
It is shown that, if a finite set of the same patterns is presented over and
over again to the network, the weights of the synapses converge to finite
values.
Furthermore, it is proved that the final values found in this biologically
realizable limit are the same as those found via a mathematical approach to the
problem of finding the weights of a partially connected neural network that can
store a collection of patterns. The mathematical solution is obtained via a
modified version of the so-called method of the pseudo-inverse, and has the
inverse of a reduced correlation matrix, rather than the usual correlation
matrix, as its basic ingredient. Thus, a biological network might realize the
final results of the mathematician by the energetically economic rule for the
adaption of the synapses found in this article.Comment: 29 pages, LaTeX, 3 figure
Damage spreading in the mode-coupling equations for glasses
We examine the problem of damage spreading in the off-equilibrium mode
coupling equations. The study is done for the spherical -spin model
introduced by Crisanti, Horner and Sommers. For we show the existence of
a temperature transition well above any relevant thermodynamic transition
temperature. Above the asymptotic damage decays to zero while below
it decays to a finite value independent of the initial damage. This transition
is stable in the presence of asymmetry in the interactions. We discuss the
physical origin of this peculiar phase transition which occurs as a consequence
of the non-linear coupling between the damage and the two-time correlation
functions.Comment: 5 pages, 2 figures, Revtex fil
Dynamical TAP approach to mean field glassy systems
The Thouless, Anderson, Palmer (TAP) approach to thermodynamics of mean field
spin-glasses is generalised to dynamics. A method to compute the dynamical TAP
equations is developed and applied to the p-spin spherical model. In this
context we show to what extent the dynamics can be represented as an evolution
in the free energy landscape. In particular the relationship between the
long-time dynamics and the local properties of the free energy landscape shows
up explicitly within this approach. Conversely, by an instantaneous normal
modes analysis we show that the local properties of the energy landscape seen
by the system during its dynamical evolution do not change qualitatively at the
dynamical transition.Comment: final version, 21 pages, 1 eps figur
Hierarchical Self-Programming in Recurrent Neural Networks
We study self-programming in recurrent neural networks where both neurons
(the `processors') and synaptic interactions (`the programme') evolve in time
simultaneously, according to specific coupled stochastic equations. The
interactions are divided into a hierarchy of groups with adiabatically
separated and monotonically increasing time-scales, representing sub-routines
of the system programme of decreasing volatility. We solve this model in
equilibrium, assuming ergodicity at every level, and find as our
replica-symmetric solution a formalism with a structure similar but not
identical to Parisi's -step replica symmetry breaking scheme. Apart from
differences in details of the equations (due to the fact that here
interactions, rather than spins, are grouped into clusters with different
time-scales), in the present model the block sizes of the emerging
ultrametric solution are not restricted to the interval , but are
independent control parameters, defined in terms of the noise strengths of the
various levels in the hierarchy, which can take any value in [0,\infty\ket.
This is shown to lead to extremely rich phase diagrams, with an abundance of
first-order transitions especially when the level of stochasticity in the
interaction dynamics is chosen to be low.Comment: 53 pages, 19 figures. Submitted to J. Phys.
Diluted neural networks with adapting and correlated synapses
We consider the dynamics of diluted neural networks with clipped and adapting
synapses. Unlike previous studies, the learning rate is kept constant as the
connectivity tends to infinity: the synapses evolve on a time scale
intermediate between the quenched and annealing limits and all orders of
synaptic correlations must be taken into account. The dynamics is solved by
mean-field theory, the order parameter for synapses being a function. We
describe the effects, in the double dynamics, due to synaptic correlations.Comment: 6 pages, 3 figures. Accepted for publication in PR
Damage spreading transition in glasses: a probe for the ruggedness of the configurational landscape
We consider damage spreading transitions in the framework of mode-coupling
theory. This theory describes relaxation processes in glasses in the mean-field
approximation which are known to be characterized by the presence of an
exponentially large number of meta-stable states. For systems evolving under
identical but arbitrarily correlated noises we demonstrate that there exists a
critical temperature which separates two different dynamical regimes
depending on whether damage spreads or not in the asymptotic long-time limit.
This transition exists for generic noise correlations such that the zero damage
solution is stable at high-temperatures being minimal for maximal noise
correlations. Although this dynamical transition depends on the type of noise
correlations we show that the asymptotic damage has the good properties of an
dynamical order parameter such as: 1) Independence on the initial damage; 2)
Independence on the class of initial condition and 3) Stability of the
transition in the presence of asymmetric interactions which violate detailed
balance. For maximally correlated noises we suggest that damage spreading
occurs due to the presence of a divergent number of saddle points (as well as
meta-stable states) in the thermodynamic limit consequence of the ruggedness of
the free energy landscape which characterizes the glassy state. These results
are then compared to extensive numerical simulations of a mean-field glass
model (the Bernasconi model) with Monte Carlo heat-bath dynamics. The freedom
of choosing arbitrary noise correlations for Langevin dynamics makes damage
spreading a interesting tool to probe the ruggedness of the configurational
landscape.Comment: 25 pages, 13 postscript figures. Paper extended to include
cross-correlation
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