43,569 research outputs found
Generalized cable formalism to calculate the magnetic field of single neurons and neuronal populations
Neurons generate magnetic fields which can be recorded with macroscopic
techniques such as magneto-encephalography. The theory that accounts for the
genesis of neuronal magnetic fields involves dendritic cable structures in
homogeneous resistive extracellular media. Here, we generalize this model by
considering dendritic cables in extracellular media with arbitrarily complex
electric properties. This method is based on a multi-scale mean-field theory
where the neuron is considered in interaction with a "mean" extracellular
medium (characterized by a specific impedance). We first show that, as
expected, the generalized cable equation and the standard cable generate
magnetic fields that mostly depend on the axial current in the cable, with a
moderate contribution of extracellular currents. Less expected, we also show
that the nature of the extracellular and intracellular media influence the
axial current, and thus also influence neuronal magnetic fields. We illustrate
these properties by numerical simulations and suggest experiments to test these
findings.Comment: Physical Review E (in press); 24 pages, 16 figure
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
Deformation of Singular Fibers of Genus Fibrations and Small Exotic Symplectic -Manifolds
We introduce the -nodal spherical deformation of certain singular fibers
of genus fibrations, and use such deformations to construct various
examples of simply connected minimal symplectic -manifolds with small
topology. More specifically, we construct new exotic minimal symplectic
-manifolds homeomorphic but not diffeomorphic to
,
, and
for
using combinations of such deformations, symplectic blowups, and (generalized)
rational blowdown surgery. We also discuss generalizing our constructions to
higher genus fibrations using -nodal spherical deformations of certain
singular fibers of genus fibrations.Comment: 38 pages, 14 figures. arXiv admin note: text overlap with
arXiv:math/0507006 by other author
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