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

Recently, the SPIKE-distance has been proposed as a parameter-free and timescale-independent measure of spike train synchrony. This measure is time resolved since it relies on instantaneous estimates of spike train dissimilarity. However, its original definition led to spuriously high instantaneous values for eventlike firing patterns. Here we present a substantial improvement of this measure that eliminates this shortcoming. The reliability gained allows us to track changes in instantaneous clustering, i.e., time-localized patterns of (dis)similarity among multiple spike trains. Additional new features include selective and triggered temporal averaging as well as the instantaneous comparison of spike train groups. In a second step, a causal SPIKE-distance is defined such that the instantaneous values of dissimilarity rely on past information only so that time-resolved spike train synchrony can be estimated in real time. We demonstrate that these methods are capable of extracting valuable information from field data by monitoring the synchrony between neuronal spike trains during an epileptic seizure. Finally, the applicability of both the regular and the real-time SPIKE-distance to continuous data is illustrated on model electroencephalographic (EEG) recordings.

Using a numerical implementation of the ADHMN construction, we compute the fields and energy densities of a charge three monopole with tetrahedral symmetry and a charge four monopole with octahedral symmetry. We then construct a one parameter family of spectral curves and Nahm data which represent charge four monopoles with tetrahedral symmetry, which includes the monopole with octahedral symmetry as a special case. In the moduli space approximation, this family describes a novel kind of four monopole scattering and we use our numerical scheme to construct the energy density at various times during the motion.

The Victor-Purpura spike train metric has recently been extended to a family of multineuron metrics and used to analyze spike trains recorded simultaneously from pairs of proximate neurons. The metric is one of the two metrics commonly used for quantifying the distance between two spike trains; the other is the van Rossum metric. Here, we suggest an extension of the van Rossum metric to a multineuron metric. We believe this gives a metric that is both natural and easy to calculate. Both types of multineuron metric are applied to simulated data and are compared.

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.

By imposing certain combined inversion and rotation symmetries on the rational maps for SU(2) BPS monopoles we construct geodesics in the monopole moduli space. In the moduli space approximation these geodesics describe a novel kind of monopole scattering. During these scattering processes axial symmetry is instantaneously attained and, in some, monopoles with the symmetries of the regular solids are formed. The simplest example corresponds to a charge three monopole invariant under a combined inversion and 90° rotation symmetry. In this example three well-separated collinear unit charge monopoles coalesce to form first a tetrahedron, then a torus, then the dual tetrahedron and finally separate again along the same axis of motion. We explicitly construct the spectral curves in this case and use a numerical ADHMN construction to compute the energy density at various times during the motion. We find that the dynamics of the zeros of the Higgs field is extremely rich and we discover a new phenomenon; there exist charge kSU(2) BPS monopoles with more than k zeros of the Higgs field.