Know thy neighbour: A normative theory of synaptic depression

Synapses exhibit an extraordinary degree of short-term malleability, with release probabilities and effective synaptic strengths changing markedly over multiple timescales. From the perspective of a fixed computational operation in a network, this seems like a most unacceptable degree of added variability. We suggest an alternative theory according to which short-term synaptic plasticity plays a normatively-justifiable role. This theory starts from the commonplace observation that the spiking of a neuron is an incomplete, digital, report of the analog quantity that contains all the critical information, namely its membrane potential. We suggest that a synapse solves the inverse problem of estimating the pre-synaptic membrane potential from the spikes it receives, acting as a recursive filter. We show that the dynamics of short-term synaptic depression closely resemble those required for optimal filtering, and that they indeed support high quality estimation. Under this account, the local postsynaptic potential and the level of synaptic resources track the (scaled) mean and variance of the estimated presynaptic membrane potential. We make experimentally testable predictions for how the statistics of subthreshold membrane potential fluctuations and the form of spiking non-linearity should be related to the properties of short-term plasticity in any particular cell type.

Information theory, inference, and learning algorithms

Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.

Classical plume theory: 1937-2010 and beyond

Developing a theoretical description of turbulent plumes, the likes of which may be seen rising above industrial chimneys, is a daunting thought. Plumes are ubiquitous on a wide range of scales in both the natural and the man-made environments. Examples that immediately come to mind are the vapour plumes above industrial smoke stacks or the ash plumes forming particle-laden clouds above an erupting volcano. However, plumes also occur where they are less visually apparent, such as the rising stream of warmair above a domestic radiator, of oil from a subsea blowout or, at a larger scale, of air above the so-called urban heat island. In many instances, not only the plume itself is of interest but also its influence on the environment as a whole through the process of entrainment. Zeldovich (1937, The asymptotic laws of freely-ascending convective flows. Zh. Eksp. Teor. Fiz., 7, 1463-1465 (in Russian)), Batchelor (1954, Heat convection and buoyancy effects in fluids. Q. J. R. Meteor. Soc., 80, 339-358) and Morton et al. (1956, Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A, 234, 1-23) laid the foundations for classical plume theory, a theoretical description that is elegant in its simplicity and yet encapsulates the complex turbulent engulfment of ambient fluid into the plume. Testament to the insight and approach developed in these early models of plumes is that the essential theory remains unchanged and is widely applied today. We describe the foundations of plume theory and link the theoretical developments with the measurements made in experiments necessary to close these models before discussing some recent developments in plume theory, including an approach which generalizes results obtained separately for the Boussinesq and the non-Boussinesq plume cases. The theory presented - despite its simplicity - has been very successful at describing and explaining the behaviour of plumes across the wide range of scales they are observed. We present solutions to the coupled set of ordinary differential equations (the plume conservation equations) that Morton et al. (1956) derived from the Navier-Stokes equations which govern fluid motion. In order to describe and contrast the bulk behaviour of rising plumes from general area sources, we present closed-form solutions to the plume conservation equations that were achieved by solving for the variation with height of Morton's non-dimensional flux parameter Γ - this single flux parameter gives a unique representation of the behaviour of steady plumes and enables a characterization of the different types of plume. We discuss advantages of solutions in this form before describing extensions to plume theory and suggesting directions for new research. © 2010 The Author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Gaussian Approximation Potential: an interatomic potential derived from first principles Quantum Mechanics

Simulation of materials at the atomistic level is an important tool in studying microscopic structure and processes. The atomic interactions necessary for the simulation are correctly described by Quantum Mechanics. However, the computational resources required to solve the quantum mechanical equations limits the use of Quantum Mechanics at most to a few hundreds of atoms and only to a small fraction of the available configurational space. This thesis presents the results of my research on the development of a new interatomic potential generation scheme, which we refer to as Gaussian Approximation Potentials. In our framework, the quantum mechanical potential energy surface is interpolated between a set of predetermined values at different points in atomic configurational space by a non-linear, non-parametric regression method, the Gaussian Process. To perform the fitting, we represent the atomic environments by the bispectrum, which is invariant to permutations of the atoms in the neighbourhood and to global rotations. The result is a general scheme, that allows one to generate interatomic potentials based on arbitrary quantum mechanical data. We built a series of Gaussian Approximation Potentials using data obtained from Density Functional Theory and tested the capabilities of the method. We showed that our models reproduce the quantum mechanical potential energy surface remarkably well for the group IV semiconductors, iron and gallium nitride. Our potentials, while maintaining quantum mechanical accuracy, are several orders of magnitude faster than Quantum Mechanical methods.

On the application of differential geometry to MDO

Multidisciplinary Design Optimization (MDO) is a methodology for optimizing large coupled systems. Over the years, a number of different MDO decomposition strategies, known as architectures, have been developed, and various pieces of analytical work have been done on MDO and its architectures. However, MDO lacks an overarching paradigm which would unify the field and promote cumulative research. In this paper, we propose a differential geometry framework as such a paradigm: Differential geometry comes with its own set of analysis tools and a long history of use in theoretical physics. We begin by outlining some of the mathematics behind differential geometry and then translate MDO into that framework. This initial work gives new tools and techniques for studying MDO and its architectures while producing a naturally arising measure of design coupling. The framework also suggests several new areas for exploration into and analysis of MDO systems. At this point, analogies with particle dynamics and systems of differential equations look particularly promising for both the wealth of extant background theory that they have and the potential predictive and evaluative power that they hold. © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

First principles interatomic potential for tungsten based on Gaussian process regression

An accurate description of atomic interactions, such as that provided by first principles quantum mechanics, is fundamental to realistic prediction of the properties that govern plasticity, fracture or crack propagation in metals. However, the computational complexity associated with modern schemes explicitly based on quantum mechanics limits their applications to systems of a few hundreds of atoms at most. This thesis investigates the application of the Gaussian Approximation Potential (GAP) scheme to atomistic modelling of tungsten - a bcc transition metal which exhibits a brittle-to-ductile transition and whose plasticity behaviour is controlled by the properties of $\frac{1}{2} \langle 111 \rangle$ screw dislocations. We apply Gaussian process regression to interpolate the quantum-mechanical (QM) potential energy surface from a set of points in atomic configuration space. Our training data is based on QM information that is computed directly using density functional theory (DFT). To perform the fitting, we represent atomic environments using a set of rotationally, permutationally and reflection invariant parameters which act as the independent variables in our equations of non-parametric, non-linear regression. We develop a protocol for generating GAP models capable of describing lattice defects in metals by building a series of interatomic potentials for tungsten. We then demonstrate that a GAP potential based on a Smooth Overlap of Atomic Positions (SOAP) covariance function provides a description of the $\frac{1}{2} \langle 111 \rangle$ screw dislocation that is in agreement with the DFT model. We use this potential to simulate the mobility of $\frac{1}{2} \langle 111 \rangle$ screw dislocations by computing the Peierls barrier and model dislocation-vacancy interactions to QM accuracy in a system containing more than 100,000 atoms.