In neurophysiology, extracellular signals-as measured by local field potentials (LFP) or electroencephalography-are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Planck equations of electrodiffusion. The propagation of an action potential along the axonal membrane is investigated by means of numerical simulations. Special attention is paid to the Debye layer, the region with strong concentration gradients close to the membrane, which is explicitly resolved by the computational mesh. We focus on the evolution of the extracellular electric potential. A characteristic up-down-up LFP waveform in the far-field is found. Close to the membrane, the potential shows a more intricate shape. A comparison with the widely used line source approximation reveals similarities and demonstrates the strong influence of membrane currents. However, the electrodiffusion model shows another signal component stemming directly from the intracellular electric field, called the action potential echo. Depending on the neuronal configuration, this might have a significant effect on the LFP. In these situations, electrodiffusion models should be used for quantitative comparisons with experimental data.

The geometric arrangement of interacting (magnetic) dipoles is a question of fundamental importance in physics, chemistry, and engineering. Motivated by recent progress concerning the self-assembly of magnetic structures, the equilibrium orientation of eight interacting dipoles in a cubic cluster is investigated in detail. Instead of discrete equilibria we find a type of ground state consisting of infinitely many orientations. This continuum of energetically degenerate states represents a yet unknown form of magnetic frustration. The corresponding dipole rotations in the flat potential valley of this Goldstone mode enable the construction of frictionless magnetic couplings. Using computer-assisted algebraic geometry methods, we moreover completely enumerate all equilibrium configurations. The seemingly simple cubic system allows for exactly 9536 unstable discrete equilibria falling into 183 distinct energy families. Magnetism has fascinated mankind for millenia [1]. Today, even the smallest magnets can hardly be overestimated in their relevance for magnetic storage technology. A fascinating example for the interplay of magnetic particles is their self-arrangement in cubic lattice clusters (see, e.g., [2,3]). Its macroscopic analog is the toy known as "magnetic cube puzzle" shown in Fig. 1(a), a stable arrangement of spherical magnets in a simple cubic cluster. How are these magnetic spheres oriented in such an ordered cluster? For the minimal arrangement within this class, namely, a cube consisting of eight magnets [see Fig. 1(b)], the answer is intriguing: There are infinitely many orientations. We find the ground state to be a continuum of energetically degenerate states-an extreme form of magnetic frustration. The phenomenon of frustration arises when the system cannot simultaneously minimize all dipole-dipole interaction energies (see [4] for a recent review). As this continuum is the ground state of the cube system, the question arises: Are there any other equilibrium orientations? Through our application of methods from numerical algebraic geometry (see Supplementary 4 [5]) we are able to construct and classify the complete set of equilibrium states. This set comprises thousands of unstable discrete dipole orientations in addition to the continuous states. We stress here that we find all equilibrium configurations (stable and unstable) unlike commonly used relaxation methods.The study of equilibrium states of dipolar hard and soft spheres has a long history in the context of magnetic colloids. Early works [6-8] on these so-called ferrofluids investigate the phase behavior (colloidal crystal structures, chain formation, string fluids, etc.). More recently, thermodynamic properties of two-dimensional (2D) monolayer systems have been studied [9] and a full phase diagram in terms of dipole strength and packing fraction was given [10]. Further, the self-assembly and transition from rings to chains controlled by an external field was investigated (see, e.g., [11][12][13]). Isolated planar dipole clusters in exte...

Billiards are idealizations for systems where particles or waves are confined to cavities, or to other homogeneous regions. In billiard systems a point particle moves freely except for specular reflections from rigid walls. However, billiard walls are not always completely reflective and measurements inside can also open the billiard. Since boundary openings have been studied extensively in the literature, we rather model leakages inside the billiard. In particular, we investigate the classical dynamics of a leakage for a continuous family of billiard systems, that is, the stadium-lemon-billiard family. With a single parameter the geometry of the billiard can be tuned from stadium (being fully hyperbolic) over circle (integrable) to the lemon-shaped billiard (mixed chaotic). For the stadium billiard we found an algebraically decaying mean escape time with the linear size of the leakage n(esc) approximately epsilon-1 together with an exponential decay of the survival probability distribution. The finding is nearly independent of the position and size of the leakage, as long as the leakage is much smaller than the system size, and it is in good agreement with a stochastic map approximation of the dynamics. Due to the mixed phase space for lemon billiards, the mean escape time depends both on the position and geometry of the leakage. For systems where quasiregular motion dominates, we found a linear dependence of the mean escape time, n(esc) approximately 1-epsilon, which we refer to as flooding law. Our findings are helpful in understanding dynamics of leaking Hamiltonian systems.

Aims. We investigate the influence of turbulent viscosity on the collapse of a rotating molecular cloud core with axial symmetry, in particular, on the first and second collapse phase, as well as the evolution of the second (protostellar) core during its first accretion period. By using extensive numerical calculations, we monitor the intricate interactions between the newly formed protostar and the surrounding accretion disk (the first core) in which the star is embedded. Methods. We use a grid-based radiation-hydrodynamics code with a spatial grid designed to meet the high resolution required to study the second core. The radiative transfer is treated in the flux-limited diffusion approximation. A slightly supercritical Bonnor-Ebert sphere of 1 M and uniform rotation according to a fixed centrifugal radius of 100 AU serves as the initial condition without exception. In a parameter study, we vary the β-viscosity driving the angular momentum transport. Results. Without viscosity (β = 0), a highly flattened accretion disk forms that fragments into several "cold" rings. For β = 10 −4 , a single "warm" ring forms that undergoes collapse due to hydrogen dissociation. For β = 10 −3 , ring formation is suppressed completely. The second collapse proceeds on the local thermal timescale, which is in contrast to the current view of a generally dynamical second collapse. During the first accretion period of the second core, the first core heats up globally and, as a consequence, a nearly spherical outflow occurs, destroying the structure of the former accretion disk completely. Finally, for β = 10 −2 , we see the classical dynamical second collapse and a shorter but more rapid accretion phase. The impact on the surrounding accretion disk is even more pronounced. We follow the resulting massive outflow up to several kyr after the second collapse, where the central parts (R < 0.7 AU) are now cut out and replaced with an appropriate inner boundary condition. Matter is found to turn back to the center at a radius of 500 AU after about 4 kyr and to reach the protostar again after approximately 7 kyr. The results suggest that the star formation process consists of short and rapid accretion phases (lasting on the order of 100 yr) between long and quiet outflow periods (lasting several kyr).

We present some important conclusions from models of the collapse of rotating molecular cloud cores with axial symmetry, corresponding to the evolution of young stellar objects from class 0 to the beginning of class I. There are three main findings of the calculations: (1) the typical timescale for building up a preplanetary disk, which was found to be of the order of one free-fall time decisively shorter than the widely assumed timescale related to the so-called "inside-out collapse"; (2) redistribution of angular momentum and the accompanying dissipation of kinetic (rotational) energy causing the growing disk to become more stable and strengthening the intrinsic meridional circulation pattern of the accretion flow; and (3) the origin of calcium-aluminium-rich inclusions (CAIs). Because of the persistent equatorial outflow, material that has undergone substantial chemical and mineralogical modifications in the hot ( > ∼ 900 K) interior of the protostellar core may have a good chance of being advectively transported outward into the cooler remote parts ( > ∼ 4 AU, say) of the growing disk and surviving there until it is incorporated into a meteoritic parent body.