108,957 research outputs found
Neural complexity: a graph theoretic interpretation
One of the central challenges facing modern neuroscience is to explain the ability of the nervous system to coherently integrate information across distinct functional modules in the absence of a central executive. To this end Tononi et al. [Proc. Nat. Acad. Sci. USA 91, 5033 (1994)] proposed a measure of neural complexity that purports to capture this property based on mutual information between complementary subsets of a system. Neural complexity, so defined, is one of a family of information theoretic metrics developed to measure the balance between the segregation and integration of a system's dynamics. One key question arising for such measures involves understanding how they are influenced by network topology. Sporns et al. [Cereb. Cortex 10, 127 (2000)] employed numerical models in order to determine the dependence of neural complexity on the topological features of a network. However, a complete picture has yet to be established. While De Lucia et al. [Phys. Rev. E 71, 016114 (2005)] made the first attempts at an analytical account of this relationship, their work utilized a formulation of neural complexity that, we argue, did not reflect the intuitions of the original work. In this paper we start by describing weighted connection matrices formed by applying a random continuous weight distribution to binary adjacency matrices. This allows us to derive an approximation for neural complexity in terms of the moments of the weight distribution and elementary graph motifs. In particular we explicitly establish a dependency of neural complexity on cyclic graph motifs
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions
We study multivariate integration of functions that are invariant under the
permutation (of a subset) of their arguments. Recently, in Nuyens,
Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the
authors derived an upper estimate for the th minimal worst case error for
such problems, and showed that under certain conditions this upper bound only
weakly depends on the dimension. We extend these results by proposing two
(semi-) explicit construction schemes. We develop a component-by-component
algorithm to find the generating vector for a shifted rank- lattice rule
that obtains a rate of convergence arbitrarily close to
, where denotes the smoothness of our
function space and is the number of cubature nodes. Further, we develop a
semi-constructive algorithm that builds on point sets which can be used to
approximate the integrands of interest with a small error; the cubature error
is then bounded by the error of approximation. Here the same rate of
convergence is achieved while the dependence of the error bounds on the
dimension is significantly improved
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