A set of nonlinear continuum field equations is presented which describes the dynamics of neural activity in cortex. These take into account the most pertinent anatomical and physiological features found in cortex with all parameter values obtainable from independent experiment. Derivation of a white noise fluctuation spectrum from a linearized set of equations shows the presence of strong resonances that correspond to electroencephalographically observed 0.3–4 Hz (mammalian delta), 4–8 Hz (mammalian theta), 8–13 Hz (mammalian alpha) and >13 Hz (mammalian beta) activity. Numerical solutions of a full set of one-dimensional nonlinear equations include properties analogous to cortical evoked potentials, travelling waves at experimentally observed velocities, threshold type spike activity and limit cycle, chaotic and noise driven oscillations at the frequency of the mammalian alpha rhythm. All these types of behaviour are generated with parameters that are within ranges reported experimentally. The strong dependence of the phenomena observed on inhibitory–inhibitory interactions is demonstrated. These results suggest that the classically described alpha may be instantiated in a number of qualitatively distinct dynamical regimes, all of which depend on the integrity of inhibitory– inhibitory population interactions.

A spatially continuous mean field theory of electrocortical activity

The Mariner and Venera probes sent to Venus in the 1960s and 1970s revealed many differences between the venusian atmosphere and that on Earth. One of the hardest to account for is the preponderance of noble gases on Venus, in particular an argon-36 level 50 times higher than on Earth. A new theory tracks the cause of this difference to around 4.5 billion years ago, when Earth and Venus are thought to have grown as a result of collisions between several Mars-sized planets. Numerical simulations show that when a giant impact occurs, the presence of an ocean drastically increases the rate at which atmosphere is lost. On Earth, almost all the proto-atmosphere accrued during planet formation would have been stripped away during collisions. Venus, nearer the Sun, is unlikely to have had a major ocean, and its proto-atmosphere would have survived. The atmospheric compositions of Venus and Earth differ significantly, with the venusian atmosphere containing about 50 times as much 36Ar as the atmosphere on Earth1. The different effects of the solar wind on planet-forming materials for Earth and Venus have been proposed to account for some of this difference in atmospheric composition2,3, but the cause of the compositional difference has not yet been fully resolved. Here we propose that the absence or presence of an ocean at the surface of a protoplanet during the giant impact phase could have determined its subsequent atmospheric amount and composition. Using numerical simulations, we demonstrate that the presence of an ocean significantly enhances the loss of atmosphere during a giant impact owing to two effects: evaporation of the ocean, and lower shock impedance of the ocean compared to the ground. Protoplanets near Earth's orbit are expected to have had oceans, whereas those near Venus’ orbit are not, and we therefore suggest that remnants of the noble-gas rich proto-atmosphere survived on Venus, but not on Earth. Our proposed mechanism explains differences in the atmospheric contents of argon, krypton and xenon on Venus and Earth, but most of the neon must have escaped from both planets’ atmospheres later to yield the observed ratio of neon to argon.

Enhanced atmospheric loss on protoplanets at the giant impact phase in the presence of oceans

IS B N 0-523108-5) C opright (C ) 19-1992 by C am bidge U nirsity P rss. P rogram s C opright (C ) 19-1992 by N um eical R eipes S ftw are. P rm ission is grnted or inrnet uers to m ke ne pper cpy or teir ow n peonal use. F uther repruction, or ny coying of m acineredable fles (inluding his one) to ny srver om pter, is sictly proibited. T o oder N um eical R eipes boks, disettes, or C D R O M s visit w esite hp://w w w .n.com or call 1-8072-7423 (N orth A m erica oly), or snd em il to trde@ cu.cam .ac.uk (otside N orth A m eca). Numerical Recipes in C

/pdf/the-art-of-scientific-computing-second-edition-3zekttp7cu.pdf

The Art of Scientific Computing Second Edition

SUMMARY This paper analyses the numerical stability of coupling procedures in modelling the thermal diffusion in a solid and a fluid with continuity of temperature and heat flux at the interface. A simple one-dimensional model is employed with uniform material properties and grid density in each domain. A number of different explicit and implicit algorithms are considered for both the interior equations and the boundary conditions. The analysis shows that in general these are stable provided that Dirichlet boundary conditions are imposed on the fluid and Neumann boundary conditions are imposed on the solid; in each case the imposed values are obtained from the other domains. # 1997 by John Wiley & Sons, Ltd.

Stability analysis of numerical interface conditions in fluid-structure thermal analysis

http://www.math.le.ac.uk/people/ag153/homepage/PhysRepCorr.pdf

Constructive methods of invariant manifolds for kinetic problems

R. D. Richtmyer and K. W. Morton Chichester: John Wiley. 1967 Pp. xiv + 405. Price £6. In this second edition the opportunity has been taken of considerably revising the first edition, published 10 years ago, with the help of K. W. Morton of the United Kingdom Atomic Energy Authority as the second author. The book is still in two parts, the first dealing with the underlying general theoretical considerations and the second with applications.

https://iopscience.iop.org/article/10.1088/0031-9112/19/12/017/pdf

J Crank

Papers

Difference Methods for Initial-Value Problems 2nd edn