A constructive mean field analysis of multi population neural networks with random synaptic weights and stochastic inputs

We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit \cite{jansen-rit:95}: their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.Comment: 55 pages, 4 figures, to appear in "Frontiers in Neuroscience

Sounds of silence: an interview with Rolf de Heer

Rolf de Heer's twelfth feature film, "Dr. Plonk" (starring Nigel Lunghi, Paul Blackwell and Magda Szubanski), premiered on closing night of the 2007 Adelaide Film Festival recently. Already feted as South Australian of the Year, De Heer received the Don Dunstan award on opening night to rapturous applause from his home town crowd. D. Bruno Starrs interviewed Australia's most successful non-mainstream film-maker about the black and white, silent slap-stick comedy three days before its inaugural screening

Spectral Features of Magnetic Fluctuations at Proton Scales from Fast to Slow Solar Wind

This Letter investigates the spectral characteristics of the interplanetary magnetic field fluctuations at proton scales during several time intervals chosen along the speed profile of a fast stream. The character of the fluctuations within the first frequency decade, beyond the high frequency break located between the fluid and the kinetic regime, strongly depends on the type of wind. While the fast wind shows a clear signature of both right handed and left handed polarized fluctuations, possibly associated with KAW and Ion-Cyclotron waves, respectively, the rarefaction region, where the wind speed and the Alfv\'{e}nicity of low frequency fluctuations decrease, shows a rapid disappearance of the ion-cyclotron signature followed by a more gradual disappearance of the KAWs. Moreover, also the power associated to perpendicular and parallel fluctuations experiences a rapid depletion, keeping, however, the power anisotropy in favour of the perpendicular spectrum.Comment: 10 pages, 5 figures, to be published in ApJ

Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures

Quasi-logarithmic combinatorial structures are a class of decomposable combinatorial structures which extend the logarithmic class considered by Arratia, Barbour and Tavar\'{e} (2003). In order to obtain asymptotic approximations to their component spectrum, it is necessary first to establish an approximation to the sum of an associated sequence of independent random variables in terms of the Dickman distribution. This in turn requires an argument that refines the Mineka coupling by incorporating a blocking construction, leading to exponentially sharper coupling rates for the sums in question. Applications include distributional limit theorems for the size of the largest component and for the vector of counts of the small components in a quasi-logarithmic combinatorial structure.Comment: 22 pages; replaces earlier paper [arXiv:math/0609129] with same title by Bruno Nietlispac

It\u27s Fun, But Is It Science? Goals and Strategies in a Problem-Based Learning Course

All students at Hampshire College must complete a science requirement in which they demonstrate their understanding of how science is done, examine the work of science in larger contexts, and communicate their ideas effectively. Human Biology: Selected Topics in Medicine is one of 18-20 freshman seminars designed to move students toward completing this requirement. Students work in cooperative groups of 4-6 people to solve actual medical cases about which they receive information progressively. Students assign themselves homework tasks to bring information back for group deliberation. The goal is for case teams to work cooperatively to develop a differential diagnosis and recommend treatment. Students write detailed individual final case reports. Changes observed in student work over six years of developing this course include: increased motivation to pursue work in depth, more effective participation on case teams, increase in critical examination of evidence, and more fully developed arguments in final written reports. As part of a larger study of eighteen introductory science courses in two institutions, several types of pre- and post-course assessments were used to evaluate how teaching approaches might have influenced students’ attitudes about science, their ability to learn science, and their understanding of how scientific knowledge is developed [1]. Preliminary results from interviews and Likert-scale measures suggest improvements in the development of some students’ views of epistemology and in the importance of cooperative group work in facilitating that development

On the Hamiltonian formulation of Yang--Mills gauge theories

The Hamiltonian formulation of the theory of J-bundles is given both in the Hamilton--De Donder and in the Multimomentum Hamiltonian geometrical approaches. (3+3) Yang-Mills gauge theories are dealt with explicitly in order to restate them in terms of Einstein-Cartan like field theories.Comment: 18 Pages, Submitted to International Journal of Geometric Methods in Modern Physic