A Primer on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal Processin

Stochastic firing rate models

We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network equations in a general setting, and we express here the results in a more customary and simpler framework. We propose a heuristic argument to derive these equations providing a more intuitive understanding of their origin. These equations are characterized by a strong coupling between the different moments of the solutions. We analyse the equations, present an algorithm to simulate the solutions of these mean-field equations, and investigate numerically the equations. In particular, we build a bridge between these equations and Sompolinsky and collaborators approach (1988, 1990), and show how the coupling between the mean and the covariance function deviates from customary approaches

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker- Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure

Equilibrium composition between liquid and clathrate reservoirs on Titan

Hundreds of lakes and a few seas of liquid hydrocarbons have been observed by the Cassini spacecraft to cover the polar regions of Titan. A significant fraction of these lakes or seas could possibly be interconnected with subsurface liquid reservoirs of alkanes. In this paper, we investigate the interplay that would happen between a reservoir of liquid hydrocarbons located in Titan's subsurface and a hypothetical clathrate reservoir that progressively forms if the liquid mixture diffuses throughout a preexisting porous icy layer. To do so, we use a statistical-thermodynamic model in order to compute the composition of the clathrate reservoir that forms as a result of the progressive entrapping of the liquid mixture. This study shows that clathrate formation strongly fractionates the molecules between the liquid and the solid phases. Depending on whether the structure I or structure II clathrate forms, the present model predicts that the liquid reservoirs would be mainly composed of either propane or ethane, respectively. The other molecules present in the liquid are trapped in clathrates. Any river or lake emanating from subsurface liquid reservoirs that significantly interacted with clathrate reservoirs should present such composition. On the other hand, lakes and rivers sourced by precipitation should contain higher fractions of methane and nitrogen, as well as minor traces of argon and carbon monoxide.Comment: Accepted for publication in Icaru

Repentir: Digital exploration beneath the surface of an oil painting

Repentir is a mobile application that employs marker-less tracking and augmented reality to enable gallery visitors to explore the under drawing and successive stages of pigment beneath an oil painting's surface. Repentir recognises the position and orientation of a specific painting within a photograph and precisely overlays images that were captured during that painting's creation. The viewer may then browse through the work's multiple states and closely examine its painted surface in one of two ways: sliding or rubbing. Our current prototype recognises realist painter Nathan Walsh's most recent work, "Transamerica". Repentir enables the viewer to explore intermediary stages in the painting's development and see what is usually lost within the materially additive painting process. The prototype offers an innovative approach to digital reproduction and provides users with unique insights into the painter's working method

Event-driven mathematical framework for noisy integrate-and-fire neuron networks: spike trains statistics via stochastic calculus, network analysis inspired by queuing theory and an event-driven simulator

International audienc

Carbon-rich planet formation in a solar composition disk

The C--to--O ratio is a crucial determinant of the chemical properties of planets. The recent observation of WASP 12b, a giant planet with a C/O value larger than that estimated for its host star, poses a conundrum for understanding the origin of this elemental ratio in any given planetary system. In this paper, we propose a mechanism for enhancing the value of C/O in the disk through the transport and distribution of volatiles. We construct a model that computes the abundances of major C and O bearing volatiles under the influence of gas drag, sublimation, vapor diffusion, condensation and coagulation in a multi--iceline 1+1D protoplanetary disk. We find a gradual depletion in water and carbon monoxide vapors inside the water's iceline with carbon monoxide depleting slower than water. This effect increases the gaseous C/O and decreases the C/H ratio in this region to values similar to those found in WASP 12b's day side atmosphere. Giant planets whose envelopes were accreted inside the water's iceline should then display C/O values larger than those of their parent stars, making them members of the class of so-called ``carbon-rich planets''.Comment: 8 pages, 4 figures, accepted for publication Ap