A comment on "A fast L_p spike alignment metric" by A. J. Dubbs, B. A. Seiler and M. O. Magnasco [arXiv:0907.3137]

Measuring the transmitted information in metric-based clustering has become something of a standard test for the performance of a spike train metric. In this comment, the recently proposed L_p Victor-Purpura metric is used to cluster spiking responses to zebra finch songs, recorded from field L of anesthetized zebra finch. It is found that for these data the L_p metrics with p>1 modestly outperform the standard, p=1, Victor-Purpura metric. It is argued that this is because for larger values of p, the metric comes closer to performing windowed coincidence detection.Comment: 9 pages, 3 figures included as late

Pulsation and Precession of the Resonant Swinging Spring

When the frequencies of the elastic and pendular oscillations of an elastic pendulum or swinging spring are in the ratio two-to-one, there is a regular exchange of energy between the two modes of oscillation. We refer to this phenomenon as pulsation. Between the horizontal excursions, or pulses, the spring undergoes a change of azimuth which we call the precession angle. The pulsation and stepwise precession are the characteristic features of the dynamics of the swinging spring. The modulation equations for the small-amplitude resonant motion of the system are the well-known three-wave equations. We use Hamiltonian reduction to determine a complete analytical solution. The amplitudes and phases are expressed in terms of both Weierstrass and Jacobi elliptic functions. The strength of the pulsation may be computed from the invariants of the equations. Several analytical formulas are found for the precession angle. We deduce simplified approximate expressions, in terms of elementary functions, for the pulsation amplitude and precession angle and demonstrate their high accuracy by numerical experiments. Thus, for given initial conditions, we can describe the envelope dynamics without solving the equations. Conversely, given the parameters which determine the envelope, we can specify initial conditions which, to a high level of accuracy, yield this envelope.Comment: 33 pages, 9 eps figure

Nahm Data and the Mass of 1/4-BPS States

The mass of 1/4-BPS dyonic configurations in N=4 D=4 supersymmetric Yang-Mills theories is calculated within the Nahm formulation. The SU(3) example, with two massive monopoles and one massless monopole, is considered in detail. In this case, the massless monopole is attracted to the massive monopoles by a linear potential.Comment: 23 pages, 2 Postscript figures, v2 references added and typos correcte

SU(N) Monopoles and Platonic Symmetry

We discuss the ADHMN construction for SU(N) monopoles and show that a particular simplification arises in studying charge N-1 monopoles with minimal symmetry breaking. Using this we construct families of tetrahedrally symmetric SU(4) and SU(5) monopoles. In the moduli space approximation, the SU(4) one-parameter family describes a novel dynamics where the monopoles never separate, but rather, a tetrahedron deforms to its dual. We find a two-parameter family of SU(5) tetrahedral monopoles and compute some geodesics in this submanifold numerically. The dynamics is rich, with the monopoles scattering either once or twice through octahedrally symmetric configurations.Comment: 14pp, RevTex, two figures made of six Postscript files. To appear in the Journal of Mathematical Physic

Octahedral and Dodecahedral Monopoles

It is shown that there exists a charge five monopole with octahedral symmetry and a charge seven monopole with icosahedral symmetry. A numerical implementation of the ADHMN construction is used to calculate the energy density of these monopoles and surfaces of constant energy density are displayed. The charge five and charge seven monopoles look like an octahedron and a dodecahedron respectively. A scattering geodesic for each of these monopoles is presented and discussed using rational maps. This is done with the aid of a new formula for the cluster decomposition of monopoles when the poles of the rational map are close together.Comment: uuencoded latex, 20 pages, 2 figures To appear in Nonlinearit

Rational Maps, Monopoles and Skyrmions

We discuss the similarities between BPS monopoles and Skyrmions, and point to an underlying connection in terms of rational maps between Riemann spheres. This involves the introduction of a new ansatz for Skyrme fields. We use this to construct good approximations to several known Skyrmions, including all the minimal energy configurations up to baryon number nine, and some new solutions such as a baryon number seventeen Skyrme field with the truncated icosahedron structure of a buckyball. The new approach is also used to understand the low-lying vibrational modes of Skyrmions, which are required for quantization. Along the way we discover an interesting Morse function on the space of rational maps which may be of use in understanding the Sen forms on the monopole moduli spaces.Comment: 35pp including four figures, typos corrected, appearing in Nuclear Physics

A Kernel-Based Calculation of Information on a Metric Space

Kernel density estimation is a technique for approximating probability distributions. Here, it is applied to the calculation of mutual information on a metric space. This is motivated by the problem in neuroscience of calculating the mutual information between stimuli and spiking responses; the space of these responses is a metric space. It is shown that kernel density estimation on a metric space resembles the k-nearest-neighbor approach. This approach is applied to a toy dataset designed to mimic electrophysiological data

Two monopoles of one type and one of another

The metric on the moduli space of charge (2,1) SU(3) Bogomolny-Prasad-Sommerfield monopoles is calculated and investigated. The hyperKahler quotient construction is used to provide an alternative derivation of the metric. Various properties of the metric are derived using the hyperKahler quotient construction and the correspondence between BPS monopoles and rational maps. Several interesting limits of the metric are also considered.Comment: 48 pages, LaTeX, 2 figures. Typos corrected. Version in JHE

Neural processing of sentences

Slides for a talk at the Exeter Dynamics seminar 2019-02-18 Abstact: How the human cognitive system is able to comprehend language has been a matter of recent debate. On the one hand the brain may make use of learned grammatical rules to decompose sentences into a hierarchy of syntactic structures to generate meaning. On the other hand the brain may rely on simpler, statistical methods where the generation of meaning relies on sequential processing. To examine this we use human EEG to record from participants as they listen to linguistic stimuli. In this talk I will outline our experiments and explain what we think they tell us about the neural processing of sentences

Estimating mutual information for spike trains: a bird song example

Zebra finch are a model animal used in the study of audition. They are adept at recognizing zebra finch songs and the neural pathway involved in song recognition is well studied. Here, this example is used to illustrate the estimation of mutual information between stimulus and response using a Kozachenko-Leonenko estimator. The challenge in calculating mutual information for spike trains is that there are no obvious coordinates for the data. The Kozachenko-Leonenko estimator does not require coordinates, it relies only on the distance between data points. In the case of bird song, estimating the mutual information demonstrates that the information content of spiking does not diminish as the song progresses.Comment: 11 pages, 4 figures, submitted to Entrop