85,999 research outputs found
Storing cycles in Hopfield-type networks with pseudoinverse learning rule: admissibility and network topology
Cyclic patterns of neuronal activity are ubiquitous in animal nervous
systems, and partially responsible for generating and controlling rhythmic
movements such as locomotion, respiration, swallowing and so on. Clarifying the
role of the network connectivities for generating cyclic patterns is
fundamental for understanding the generation of rhythmic movements. In this
paper, the storage of binary cycles in neural networks is investigated. We call
a cycle admissible if a connectivity matrix satisfying the cycle's
transition conditions exists, and construct it using the pseudoinverse learning
rule. Our main focus is on the structural features of admissible cycles and
corresponding network topology. We show that is admissible if and only
if its discrete Fourier transform contains exactly nonzero
columns. Based on the decomposition of the rows of into loops, where a
loop is the set of all cyclic permutations of a row, cycles are classified as
simple cycles, separable or inseparable composite cycles. Simple cycles contain
rows from one loop only, and the network topology is a feedforward chain with
feedback to one neuron if the loop-vectors in are cyclic permutations
of each other. Composite cycles contain rows from at least two disjoint loops,
and the neurons corresponding to the rows in from the same loop are
identified with a cluster. Networks constructed from separable composite cycles
decompose into completely isolated clusters. For inseparable composite cycles
at least two clusters are connected, and the cluster-connectivity is related to
the intersections of the spaces spanned by the loop-vectors of the clusters.
Simulations showing successfully retrieved cycles in continuous-time
Hopfield-type networks and in networks of spiking neurons are presented.Comment: 48 pages, 3 figure
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations
We find an explicit combinatorial interpretation of the coefficients of Kerov
character polynomials which express the value of normalized irreducible
characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,...
of the corresponding Young diagram. Our interpretation is based on counting
certain factorizations of a given permutation
Linear versus spin: representation theory of the symmetric groups
We relate the linear asymptotic representation theory of the symmetric groups
to its spin counterpart. In particular, we give explicit formulas which express
the normalized irreducible spin characters evaluated on a strict partition
with analogous normalized linear characters evaluated on the double
partition . We also relate some natural filtration on the usual
(linear) Kerov-Olshanski algebra of polynomial functions on the set of Young
diagrams with its spin counterpart. Finally, we give a spin counterpart to
Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable)
map
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