4,085 research outputs found
PySpike - A Python library for analyzing spike train synchrony
Understanding how the brain functions is one of the biggest challenges of our
time. The analysis of experimentally recorded neural firing patterns (spike
trains) plays a crucial role in addressing this problem. Here, the PySpike
library is introduced, a Python package for spike train analysis providing
parameter-free and time-scale independent measures of spike train synchrony. It
allows to compute similarity and dissimilarity profiles, averaged values and
distance matrices. Although mainly focusing on neuroscience, PySpike can also
be applied in other contexts like climate research or social sciences. The
package is available as Open Source on Github and PyPI.Comment: 7 pages, 6 figure
Efficient spatial modelling using the SPDE approach with bivariate splines
Gaussian fields (GFs) are frequently used in spatial statistics for their
versatility. The associated computational cost can be a bottleneck, especially
in realistic applications. It has been shown that computational efficiency can
be gained by doing the computations using Gaussian Markov random fields (GMRFs)
as the GFs can be seen as weak solutions to corresponding stochastic partial
differential equations (SPDEs) using piecewise linear finite elements. We
introduce a new class of representations of GFs with bivariate splines instead
of finite elements. This allows an easier implementation of piecewise
polynomial representations of various degrees. It leads to GMRFs that can be
inferred efficiently and can be easily extended to non-stationary fields. The
solutions approximated with higher order bivariate splines converge faster,
hence the computational cost can be alleviated. Numerical simulations using
both real and simulated data also demonstrate that our framework increases the
flexibility and efficiency.Comment: 26 pages, 7 figures and 3 table
A guide to time-resolved and parameter-free measures of spike train synchrony
Measures of spike train synchrony have proven a valuable tool in both
experimental and computational neuroscience. Particularly useful are
time-resolved methods such as the ISI- and the SPIKE-distance, which have
already been applied in various bivariate and multivariate contexts. Recently,
SPIKE-Synchronization was proposed as another time-resolved synchronization
measure. It is based on Event-Synchronization and has a very intuitive
interpretation. Here, we present a detailed analysis of the mathematical
properties of these three synchronization measures. For example, we were able
to obtain analytic expressions for the expectation values of the ISI-distance
and SPIKE-Synchronization for Poisson spike trains. For the SPIKE-distance we
present an empirical formula deduced from numerical evaluations. These
expectation values are crucial for interpreting the synchronization of spike
trains measured in experiments or numerical simulations, as they represent the
point of reference for fully randomized spike trains.Comment: 8 pages, 4 figure
Total Generalized Variation for Manifold-valued Data
In this paper we introduce the notion of second-order total generalized
variation (TGV) regularization for manifold-valued data in a discrete setting.
We provide an axiomatic approach to formalize reasonable generalizations of TGV
to the manifold setting and present two possible concrete instances that
fulfill the proposed axioms. We provide well-posedness results and present
algorithms for a numerical realization of these generalizations to the manifold
setup. Further, we provide experimental results for synthetic and real data to
further underpin the proposed generalization numerically and show its potential
for applications with manifold-valued data
Computational Aspects of Optional P\'{o}lya Tree
Optional P\'{o}lya Tree (OPT) is a flexible non-parametric Bayesian model for
density estimation. Despite its merits, the computation for OPT inference is
challenging. In this paper we present time complexity analysis for OPT
inference and propose two algorithmic improvements. The first improvement,
named Limited-Lookahead Optional P\'{o}lya Tree (LL-OPT), aims at greatly
accelerate the computation for OPT inference. The second improvement modifies
the output of OPT or LL-OPT and produces a continuous piecewise linear density
estimate. We demonstrate the performance of these two improvements using
simulations
Approximation and geometric modeling with simplex B-splines associated with irregular triangles
Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud
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With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud
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If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-Bézier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-Bézier polynomials with respect to the entire domain. From the degenerate Bernstein-Bézier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud
Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud
Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud
An example for least squares approximation with simplex splines is presented
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