1,101 research outputs found
Successive projections on hyperplanes
AbstractAny sequence of points in Rn obtained by successive projections of a point on elements of a finite set of hyperplanes is bounded
Convex Independence in Permutation Graphs
A set C of vertices of a graph is P_3-convex if every vertex outside C has at
most one neighbor in C. The convex hull \sigma(A) of a set A is the smallest
P_3-convex set that contains A. A set M is convexly independent if for every
vertex x \in M, x \notin \sigma(M-x). We show that the maximal number of
vertices that a convexly independent set in a permutation graph can have, can
be computed in polynomial time
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Predicting the effects of deep brain stimulation using a reduced coupled oscillator model
This is the final version. Available on open access from Public Library of Science via the DOI in this recordData Availability: The data analysed in this manuscript is available from MRC BNDU Data Sharing platform at: https://data.mrc.ox.ac.uk/data-set/tremor-data-measured-essential-tremor-patients-subjected-phase-locked-deep-brain DOI: 10.5287/bodleian:xq24eN2KmDeep brain stimulation (DBS) is known to be an effective treatment for a variety of neurological disorders, including Parkinson’s disease and essential tremor (ET). At
present, it involves administering a train of pulses with constant frequency via electrodes implanted into the brain. New ‘closed-loop’ approaches involve delivering
stimulation according to the ongoing symptoms or brain activity and have the potential to provide improvements in terms of efficiency, efficacy and reduction of side effects. The success of closed-loop DBS depends on being able to devise a stimulation strategy that minimizes oscillations in neural activity associated with symptoms of motor disorders. A useful stepping stone towards this is to construct a mathematical model, which can describe how the brain oscillations should change when stimulation is applied at a particular state of the system. Our work focuses on the use of coupled oscillators to represent neurons in areas generating pathological oscillations. Using a reduced form of the Kuramoto model, we analyse how a patient should respond to stimulation when neural oscillations have a given phase and amplitude, provided a number of conditions are satisfied. For such patients, we predict that the best stimulation strategy should be phase specific but also that stimulation should have a greater effect if applied when the amplitude of brain oscillations is lower. We compare this surprising prediction with data obtained from ET patients. In light of our predictions, we also propose a new hybrid strategy which effectively combines two of the closed-loop strategies found in the
literature, namely phase-locked and adaptive DBS
When do Bursts Matter in the Primary Motor Cortex? Investigating Changes in the Intermittencies of Beta Rhythms Associated With Movement States
Brain activity exhibits significant temporal structure that is not well captured in the power spectrum. Recently, attention has shifted to characterising the properties of intermittencies in rhythmic neural activity (i.e. bursts), yet the mechanisms regulating them are unknown. Here, we present evidence from electrocorticography recordings made from the motor cortex to show that the statistics of bursts, such as duration or amplitude, in beta frequency (14-30Hz) rhythms significantly aid the classification of motor states such as rest, movement preparation, execution, and imagery. These features reflect nonlinearities not detectable in the power spectrum, with states increasing in nonlinearity from movement execution to preparation to rest. Further, we show using a computational model of the cortical microcircuit, constrained to account for burst features, that modulations of laminar specific inhibitory interneurons are responsible for temporal organization of activity. Finally, we show that temporal characteristics of spontaneous activity can be used to infer the balance of cortical integration between incoming sensory information and endogenous activity. Critically, we contribute to the understanding of how transient brain rhythms may underwrite cortical processing, which in turn, could inform novel approaches for brain state classification, and modulation with novel brain-computer interfaces
- …