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The discrepancy method is the glue that binds randomness and complexity. It is the bridge between randomized computation and discrepancy theory, the area of mathematics concerned with irregularities in distributions. The discrepancy method has played a major role in complexity theory; in particular, it has caused a mini-revolution of sorts in computational geometry. This book tells the story of the discrepancy method in a few short independent vignettes. It is a varied tale which includes such topics as communication complexity, pseudo-randomness, rapidly mixing Markov chains, points on the sphere and modular forms, derandomization, convex hulls, Voronoi diagrams, linear programming and extensions, geometric sampling, VC-dimension theory, minimum spanning trees, linear circuit complexity, and multidimensional searching. The mathematical treatment is thorough and self-contained. In particular, background material in discrepancy theory is supplied as needed. Thus the book should appeal to students and researchers in computer science, operations research, pure and applied mathematics, and engineering.
For decades Robotica has had a characteristic look and page layout that has served our readership well. However, recent advances in publishing technology make now an excellent time to experiment with some new ideas. From January 2014 we will be using an exciting new cover design concept that will include unique artwork featuring one or more of the topics covered in each issue. Moreover, we will be changing from our traditional double-column format to a new single-column one that will facilitate larger figures, tables and equations. At the end of this month's issue, we will include a sneak peek at the new page format via an article typeset in this new single-column style. We hope that our readership embraces this change. And we look forward to continuing to publish papers of high quality and broad international appeal.
Erdős asked in 1962 about the value of f(n,k,l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k,l)=(3,3) was solved by Lorden. Here we solve the problem (for all large n) for (3,l) with 4 ≤ l ≤ 7 and (k,3) with 4 ≤ k ≤ 7. Independently, Das, Huang, Ma, Naves and Sudakov resolved the cases (k,l)=(3,4) and (4,3).
The purpose of this paper is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We then apply this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.
and study some basic properties of these operators where ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} ,(0\leq k\leq n), u\in [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$. Also, we establish the order of approximation by using weighted modulus of continuity.
In this chapter, we explore the range of applications for BCI technology. We have already touched upon some medical applications such as restoration of lost motor and sensory function when we examined invasive and noninvasive BCIs in previous chapters. Here we briefly review these applications before exploring applications in other areas such as entertainment, robotic control, gaming, security, and art.
Medical Applications
The field of brain-computer interfacing originated with the goal of helping the paralyzed and the disabled. It is therefore not surprising that some of the major applications of BCIs to date have been in medical technology, particularly restoring sensory and motor function.
Sensory Restoration
One of the most widely used commercial BCIs is the cochlear implant for the deaf,discussed in Section 10.1.1. The cochlear implant is an example of a BCI for sensoryrestoration, as are retinal implants being developed for the blind (Section 10.1.2).
There has not been much research on two other possible types of purely sensoryBCIs, namely, BCIs for somatosensation and BCIs for olfaction and taste. In the caseof the former, the need for a BCI is minimized because it is ot en possible to restoretactile sensation through skin grafting. However, as we saw in Chapter 11, there isconsiderable interest in somatosensory stimulation as a component of bidirectionalBCIs for allowing paralyzed individuals and amputees to, for example, sense objectsbeing grasped or touched by prosthetic devices.
In this chapter, we review the signal-processing methods applied to recorded brain signals in BCIs for tasks ranging from extracting spikes from the raw signals recorded from invasive electrodes to extracting features for classification. For many of the techniques, we use EEG as the noninvasive recording modality to illustrate the concepts involved, although the techniques could be applied to signals from other sources as well such as ECoG and MEG.
Spike Sorting
Invasive approaches to brain-computer interfacing typically rely on recording spikes from an array of microelectrodes. The goal of signal-processing methods for such an input signal is to reliably isolate and extract the spikes being emitted by a single neuron per recording electrode. This procedure is usually called spike sorting.
The signal recorded by an extracellular electrode implanted in the brain is typicallya mixture of signals from several neighboring neurons, with spikes from closerneurons producing larger amplitude del ections in the recorded signal. h is signalis ot en referred to as multiunit hash or neural hash (Figure 4.1A). Although hashcould also potentially be used as input to brain- computer interfaces, the more traditionalform of input is spikes from individual neurons. Spike sorting methods allowspikes from a single neuron to be extracted from hash.
Our brains evolved to control a complex biological device: our body. As we are finding out today, many millennia of evolutionary tinkering has made the brain a surprisingly versatile and adaptive system, to the extent that it can learn to control devices that are radically different from our body. Brain-computer interfacing, the subject of this book, is a new interdisciplinary field that seeks to explore this idea by leveraging recent advances in neuroscience, signal processing, machine learning, and information technology.
The idea of brains controlling devices other than biological bodies has long been a staple of science-fiction novels and Hollywood movies. However, this idea is fast becoming a reality: in the past decade, rats have been trained to control the delivery of a reward to their mouths, monkeys have moved robotic arms, and humans have controlled cursors and robots, all directly through brain activity.
What aspects of neuroscience research have made these advances possible? Whatare the techniques in computing and machine learning that are allowing brains tocontrol machines? What is the current state-of-the-art in brain-computer interfaces(BCIs)? What limitations still need to be overcome to make BCIs more commonplaceand useful for day-to-day use? What are the ethical, moral, and societal implicationsof BCIs? These are some of the questions that this book addresses.
We have thus far focused on BCIs that record signals from the brain and transform those signals to a control signal for an external device. In this chapter, we reverse the direction of control and discuss BCIs that can be used to stimulate and control specific brain circuits. Some of these BCIs have made the transition from the lab to the clinic and are currently being used by human subjects, such as cochlear implants and deep brain stimulators (DBS), while others are still in experimental stages. We divide these BCIs broadly into two classes: BCIs for sensory restoration and BCIs for motor restoration. We also consider the possibility of sensory augmentation.
Restoring Hearing: Cochlear Implants
One of the most successful BCI devices to date is the cochlear implant for restoring or enabling hearing in the deaf. The implant is a good example of how one can convert knowledge of information processing in a neural system, in this case the cochlea, into building a working BCI that can benefit people.
A holy grail of BCI research is to be able to control complex devices using noninvasive recordings of brain signals at high spatial and temporal resolution. Current noninvasive recording techniques capture changes in blood flow or fluctuations in electric/magnetic fields caused by the activity of large populations of neurons, but we are still far from a recording technique that can capture neural activity at the level of spikes noninvasively. In the absence of such a recording technique, researchers have focused on noninvasive techniques such as EEG, MEG, fMRI, and fNIR, and studied how the large-scale population-level brain signals recorded by these techniques can be used for BCI.
Electroencephalographic (EEG) BCIs
The technique of EEG involves recording electrical signals from the scalp (Section 3.1.2). The idea of using EEG to build a BCI was first suggested by Vidal (1973), but progress was limited until the 1990s when the advent of fast and cheap processors sparked a surge of interest in this area, leading to the development of a variety of EEG-based BCI techniques.
“Bionic vision: Amazing new eye chip helps two blind Brits to see again”
(Mirror, May 3, 2012)
“Paralyzed, moving a robot with their minds”
(New York Times, May 16, 2012)
“Stephen Hawking trials device that reads his mind”
(New Scientist, July 12, 2012)
These headlines, from just a few weeks of news stories in 2012, illustrate the growing fascination of the media and the public with the idea of interfacing minds with machines. What is not clear amid all this hype is: (a) What exactly can and cannot be achieved with current brain-computer interfaces (BCIs) (sometimes also called brain-machine interfaces or BMIs)? (b) What techniques and advances in neuroscience and computing are making these BCIs possible? (c) What are the available types of BCIs? and (d) What are their applications and ethical implications? The goal of this book is to answer these questions and provide the reader with a working knowledge of BCIs and BCI techniques.
Among the most important aspects of brain-computer interfacing are ethical issues – issues pertaining to the medical use of BCIs, the use of BCIs for human augmentation and other applications, and the potential for their misuse. Some of these issues fall under the rubric of neuroethics, but other issues are specific to technological aspects of BCIs.
BCI conferences and workshops sometimes include sessions on ethics, and there have been several articles discussing ethical aspects of BCIs and neural interfaces (e.g., Clausen, 2009; Haselager et al., 2009; Tamburrini, 2009; Salvini et al., 2008; Warwick, 2003). However, there are currently no official regulations or guidelines on BCI use, aside from conventional laws regarding medical and legal ethics. As with other technologies in the past, one can expect that as BCIs become more prevalent in society, laws and ethics pertaining to BCI use will likely be codified by medical and governmental regulatory agencies. In the meantime, this chapter surveys the variety of ethical issues and dilemmas surrounding BCI research and BCI use.