Proceedings of Neural, Parallel, and Scientific Computations 4 (2010) xxxx PHASE OSCILLATOR NETWORK WITH PIECEWISELINEAR DYNAMICS

 Alyson Bryan
 6 months ago
 Views:
Transcription
1 Proceedings of Neural, Parallel, and Scientific Computations 4 (2010) xxxx PHASE OSCILLATOR NETWORK WITH PIECEWISELINEAR DYNAMICS WALTER GALL, YING ZHOU, AND JOSEPH SALISBURY Department of Mathematics and Computer Science, Rhode Island College, Providence, Rhode Island 02908, USA. Department of Mathematics and Computer Science, Rhode Island College, Providence, Rhode Island 02908, USA. Neuroscience Program, Brandeis University, Waltham, Massachusetts 02454, USA. ABSTRACT. We consider a Filippov system formed from N coupled phase oscillators similar to the Kuramoto model but using a periodic sawtooth in place of the sine wave. A Hebbian rule based interaction provides for a phase oscillator associative memory network where the memorized patterns are stored as phaselocked limitcycle attractors. AMS (MOS) Subject Classification. 37N INTRODUCTION The Kuramoto model of coupled oscillators [5] was motivated by the phenomenon of collective synchronization in some networks with a large number of oscillators [11]. Here the frequencies of the oscillations can become locked to a common frequency despite the variations in the natural frequencies of the individual oscillators [8]. Generalizations of the model can be used to form phase oscillator associative memory networks, where the memorized patterns are stored as limitcycles, although there are problems with the stability of the limitcycle memories that must be overcome with the addition of higherorder terms [6]. The Kuramoto model describes the dynamics of a system of N phase oscillators θ i with natural frequencies ω i. The time evolution of the ith oscillator is given by (1.1) θi = ω i + k ij sin(θ j θ i ), i = 1,..., N, where k jk are parameters representing the coupling strengths from the jth to the ith oscillators. The sine function was chosen because it would be the firstorder term in the Fourier expansion of more general interactions. To visualize the phases, it is helpful to think of the oscillators as points on a unit circle moving with different angular velocities. In the case of synchronization, the points on the circle move around with constant phase differences. We approximate the sine wave interaction of the Kuramoto model by a periodic piecewiselinear (but discontinuous) sawtooth, which is interesting in its own right, and is more appropriate for some applications such as wireless sensor network synchronization [10]. We investigate the finiten case with a symmetric (selfadjoint) N N connection matrix, as occurs in some phaseoscillator network models of neurocomputers utilizing phaselock loops [3], and optical laser oscillator technologies [4], or any neural network that learns by a Received May 30, c Dynamic Publishers, Inc.
2 2 WALTER GALL, YING ZHOU, AND JOSEPH SALISBURY Hebbian rule. We will find that associative memories can be encoded by stable phaselocked limit cycles. 2. MODEL DESCRIPTION We propose a simplified model where the nonlinear sine function is replaced by a periodic piecewiselinear sawtoothlike function. Our finiten phase oscillators are modeled as (2.1) θi = ω i + k ij saw(θ j θ i ), i = 1,..., N, where x + π (2.2) saw(x) = x π π x π is defined using the floor and ceiling functions. Here ω i and k ij are the natural frequency and the coupling strength, respectively, for the oscillator θ i. Saw is a discontinuous function, and this is a Filippov dynamical system [2]. The function saw(x) = x is the firstorder term in the Taylor expansion of sin(x) on ( π, π), extended periodically so that saw(x) = x j when j π < x < j +π, j Z, and saw((2j ± 1)π) = 0, j Z. Our definition results in saw being an odd function and the average of its rightside and leftside limits at points of discontinuity. Hence, by Dirichlet s theorem, its Fourier sine series converges pointwise to saw for all x R [9], and we can also write (2.3) saw(x) = 2 (sin(x) 12 sin(2x) + 13 sin(3x) 14 ) sin(4x) As mentioned before, we will consider each phase component ω i S 1, and thus (2.1) as a dynamical system on the Ndimensional torus T N. The terms sin(θ j θ i ) and saw(θ j θ i ) are both positive whenever the oscillator θ j is slightly ahead of the oscillator θ i, so that the term saw(θ j θ i ) contributes to the acceleration of the oscillator θ i. Similarly, they are both negative when θ j is slightly behind θ i, so that θ i slows down. The terms are both zero when the phases differ by exactly half a period, as if diametrically opposed oscillators are equally undecided whether to speed up or slow down, as was the case with Buridan s donkey. See Figure 1(a). However, the saw function is increasing on ( π, π) and so becomes increasingly larger in absolute value as the phase difference approaches either ±π. This implies that as the absolute phase difference between the oscillators θ j θ i becomes larger, saw(θ j θ i ) (or π 1 saw(θ j θ i ) to compare similar amplitudes) contributes more to the catching up or slowing down than either sin(θ j θ i ) or a smaller absolute phase difference would, encouraging coherence in comparison with the original Kuramoto dynamics. One could perhaps imagine runners on a circular track connected by elastic bands. 3. SYNCHRONIZATION It turns out that in the symmetric case k ij = k ji, i, j, for N oscillators, the space of phasedeviations is actually a gradient system (in a generalized sense) which can endow the system (2.1) with neurocomputational properties.
3 PHASE OSCILLATOR NETWORK 3 Figure 1. (a) COMPARING THE PIECEWISELINEAR SAWTOOTH AND π SINE WAVES. (b) THE GENERALIZED ANTIDERIVATIVE I(X) OF SAW(X) The Generalized Antiderivative of Saw. The Fourier expansion of saw can be integrated termbyterm to obtain an antiderivative I(x) for saw: ( ) (3.1) I(x) = π2 6 2 cos 2x cos 3x cos x See Figure 1(b). The function I(x) is periodic, piecewisesmooth, and continuous, with (3.2) I(x) = 1 ( x + π ) 2 x for x (, ). 2 When x [ π, π], I(x) = x 2 /2. Now I (x) = saw(x), x R, with the derivative being taken in a generalized sense at the points {(2m + 1)π, m Z}. It is interesting to note that if we define a centeredderivative of a function f at a point x to be the limit of the centereddifference quotients: lim h 0 f(x + h) f(x h), 2h then saw(x) is the centeredderivative of I(x) for all x, including the points at which I is not differentiable in the classical sense. This can be made the basis of a simple multistep secondorder numerical scheme The Phase Deviation Equations. We let (3.3) ω = 1 N be the mean natural frequency, and let (3.4) ψ i = θ i ωt, i = 1,..., N ω i
4 4 WALTER GALL, YING ZHOU, AND JOSEPH SALISBURY be the phase deviations. Let (3.5) ν i = ω i ω be the natural frequency deviations. Note that i ν i = 0. The phase deviation equations become (3.6) ψ i = ν i + k ij saw(ψ j ψ i ), i = 1,..., N. Note that if (3.7) ψ = 1 N is the mean of the phase deviations, then (3.8) in the symmetric case. d ψ dt = A Potential For the Phase Deviation Equations. When the k ij s are symmetric, a (negative) potential for the phase deviation equations can be given by (3.9) U( ψ) = ν i ψ i 1 k ij I(ψ i ψ j ), 2 ψ i where ψ is the vector of phase deviations (ψ 1,..., ψ N ). equations (3.6) is just the gradient system (3.10) d ψ dt = U( ψ). The system of phase deviation Hence, the phase deviations ψ converge to an equilibrium ψ 0 on T N. Thus, the phases θ i = ωt + ψ i converge to a limitcycle attractor having frequency ω and phase relations ψ Hebbian Learning Rule and Neurocomputational Properties. This can form the basis of a neurocomputer with oscillatory associative memory. When N is large, there could be many such phaselocked limitcycle attractors corresponding to many memorized patterns which are encoded into (or decoded from) the symmetric connection matrix K = k ij using the spectral theorem. Each phaselocked limitcycle corresponds to an equilibrium of the phase deviation equations. And each equilibrium of the phase deviations corresponds to an extremum of U which corresponds to an eigenvector of K when all ν i = 0. For example, if ξ 1,..., ξ n are n orthogonal unit column Nvectors to be learned (n < N) with real weights µ 1,..., µ n, let n (3.11) K = µ j ξ j ξj t. Then, K is symmetric with eigenvectors ξ j and eigenvalues µ j on the orthogonal complement of the nullspace of K. Moreover, n (3.12) Kξ m = µ j ξ j ξj t ξ m = µ m ξ m. This results in the system asymptotically approaching the associated component at the bottom of a basin of attraction in which the initial input was located.
5 PHASE OSCILLATOR NETWORK 5 4. DISCUSSION Using the sawtooth function to model the symmetric interaction between N coupled phase oscillators, we obtain a piecewisesmooth, piecewiselinear system which can be analyzed explicitly. The phase deviations from the mean phase evolve according to a (generalized) gradient system. Stable phaselocking occurs at minimums of the potential energy function. One would expect that speed of convergence would be more rapid with sawtooth interactions compared to sine waves. Presented with a pattern close to one of the stored patterns, in the case that the ν i 0, the phase system with the symmetric connection matrix K and zero diagonal implementing a Hebbian learning rule will recall (or discriminant) by phaselocking with the closest stored pattern of phase deviations. Similarly, presented with a mixture of the stored patterns, the phase system will tend to phaselock (or abstract) with the pattern with the largest positive weight present in the mixture, just as in principal component analysis. The use of oscillatory memories is especially interesting to neurocomputing since phaselocking has been hypothesized [1] to be important in such things as in binding gestalts, multiplexing in the thalamus, and reaching consensus by winnertakeall in the cortex. Phase oscillator networks have also been used in modeling central pattern generators for locomotion [7]. Many questions regarding capacity, retrieval, effects of forcing, inputing faster oscillation rates, possible multiplexing, and practical applications remain to be investigated. REFERENCES [1] M.A. Arbib (Ed.), Handbook of Brain Theory and Neural Networks (2ed.), MIT Press, Cambridge, [2] M. dibernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewisesmooth Dynamical Systems: Theory and Applications, SpringerVerlag, London, [3] F. Hoppensteadt and E. Izhikevich, Pattern recognition via synchronization in phaselocked loop neural networks, IEEE Transaction on Neural Network, 1: , [4] F.C. Hoppensteadt and E.M. Izhikevich, Synchronization of laser oscillators, associative memory, and optical neurocomputing, Physical Review E, 62: , [5] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, SpringerVerlag, New York, [6] T. Nishikawa, F.C. Hoppensteadt and Y.C. Lai, Oscillatory associative memory networks with perfect retrieval, Physica D, 197: , [7] R.H. Rand, A.H. Cohen and P.J. Holmes, Systems of coupled oscillators as models of central pattern generators, in: A. Cohen (Ed.), Neural Control of Rhythmic Movements in Vertebrates, Wiley, New York, [8] S.H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143:1 20, [9] G.P. Tolstov, Fourier Series, Dover Publications, New York, [10] G. WernerAllen, G. Tewari, A. Patel, M. Welch and R. Nagpal, Fireflyinspired sensor network synchronicity with realistic radio effects, Proceedings of the 3rd International Conference on Embedded Networked Sensor Systems, , [11] A.T. Winfree, The Geometry of Biological Time, Springer, New York, 1980.
Plasticity and learning in a network of coupled phase oscillators
PHYSICAL REVIEW E, VOLUME 65, 041906 Plasticity and learning in a network of coupled phase oscillators Philip Seliger, Stephen C. Young, and Lev S. Tsimring Institute for onlinear Science, University of
More informationEffects of Interactive Function Forms in a SelfOrganized Critical Model Based on Neural Networks
Commun. Theor. Phys. (Beijing, China) 40 (2003) pp. 607 613 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Effects of Interactive Function Forms in a SelfOrganized Critical Model
More informationNonlinear systems, chaos and control in Engineering
Nonlinear systems, chaos and control in Engineering Module 1 block 3 Onedimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu http://www.fisica.edu.uy/~cris/ Schedule Flows on the
More informationIntroduction  Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction  Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationIn biological terms, memory refers to the ability of neural systems to store activity patterns and later recall them when required.
In biological terms, memory refers to the ability of neural systems to store activity patterns and later recall them when required. In humans, association is known to be a prominent feature of memory.
More informationPREMED COURSE, 14/08/2015 OSCILLATIONS
PREMED COURSE, 14/08/2015 OSCILLATIONS PERIODIC MOTIONS Mechanical Metronom Laser Optical Bunjee jumping Electrical Astronomical Pulsar Biological ECG AC 50 Hz Another biological exampe PERIODIC MOTIONS
More informationEntrainment and Chaos in the HodgkinHuxley Oscillator
Entrainment and Chaos in the HodgkinHuxley Oscillator Kevin K. Lin http://www.cims.nyu.edu/ klin Courant Institute, New York University Mostly Biomath  2005.4.5 p.1/42 Overview (1) Goal: Show that the
More informationDerivation of bordercollision maps from limit cycle bifurcations
Derivation of bordercollision maps from limit cycle bifurcations Alan Champneys Department of Engineering Mathematics, University of Bristol Mario di Bernardo, Chris Budd, Piotr Kowalczyk Gabor Licsko,...
More informationHopfield networks. Lluís A. Belanche Soft Computing Research Group
Lluís A. Belanche belanche@lsi.upc.edu Soft Computing Research Group Dept. de Llenguatges i Sistemes Informàtics (Software department) Universitat Politècnica de Catalunya 20102011 Introduction Contentaddressable
More informationAssociative Memories (I) Hopfield Networks
Associative Memories (I) Davide Bacciu Dipartimento di Informatica Università di Pisa bacciu@di.unipi.it Applied Brain Science  Computational Neuroscience (CNS) A Pun Associative Memories Introduction
More informationFourier Series. (Com S 477/577 Notes) YanBin Jia. Nov 29, 2016
Fourier Series (Com S 477/577 otes) YanBin Jia ov 9, 016 1 Introduction Many functions in nature are periodic, that is, f(x+τ) = f(x), for some fixed τ, which is called the period of f. Though function
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationNonlinear Optimization
Nonlinear Optimization (Com S 477/577 Notes) YanBin Jia Nov 7, 2017 1 Introduction Given a single function f that depends on one or more independent variable, we want to find the values of those variables
More informationFACTOR ANALYSIS AND MULTIDIMENSIONAL SCALING
FACTOR ANALYSIS AND MULTIDIMENSIONAL SCALING Vishwanath Mantha Department for Electrical and Computer Engineering Mississippi State University, Mississippi State, MS 39762 mantha@isip.msstate.edu ABSTRACT
More informationPhase Response Curves, Delays and Synchronization in Matlab
Phase Response Curves, Delays and Synchronization in Matlab W. Govaerts and B. Sautois Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281S9, B9000 Ghent, Belgium
More informationCSE 5526: Introduction to Neural Networks Hopfield Network for Associative Memory
CSE 5526: Introduction to Neural Networks Hopfield Network for Associative Memory Part VII 1 The basic task Store a set of fundamental memories {ξξ 1, ξξ 2,, ξξ MM } so that, when presented a new pattern
More informationSelforganized Criticality and Synchronization in a Pulsecoupled IntegrateandFire Neuron Model Based on Small World Networks
Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 466 470 c International Academic Publishers Vol. 43, No. 3, March 15, 2005 Selforganized Criticality and Synchronization in a Pulsecoupled IntegrateandFire
More informationExercise Sheet 1. 1 Probability revision 1: Studentt as an infinite mixture of Gaussians
Exercise Sheet 1 1 Probability revision 1: Studentt as an infinite mixture of Gaussians Show that an infinite mixture of Gaussian distributions, with Gamma distributions as mixing weights in the following
More informationAdjoint Orbits, Principal Components, and Neural Nets
Adjoint Orbits, Principal Components, and Neural Nets Some facts about Lie groups and examples Examples of adjoint orbits and a distance measure Descent equations on adjoint orbits Properties of the double
More informationStochastic WilsonCowan equations for networks of excitatory and inhibitory neurons II
Stochastic WilsonCowan equations for networks of excitatory and inhibitory neurons II Jack Cowan Mathematics Department and Committee on Computational Neuroscience University of Chicago 1 A simple Markov
More informationc Springer, Reprinted with permission.
Zhijian Yuan and Erkki Oja. A FastICA Algorithm for Nonnegative Independent Component Analysis. In Puntonet, Carlos G.; Prieto, Alberto (Eds.), Proceedings of the Fifth International Symposium on Independent
More informationBASIC WAVE CONCEPTS. Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, Giancoli?
1 BASIC WAVE CONCEPTS Reading: Main 9.0, 9.1, 9.3 GEM 9.1.1, 9.1.2 Giancoli? REVIEW SINGLE OSCILLATOR: The oscillation functions you re used to describe how one quantity (position, charge, electric field,
More informationPattern Association or Associative Networks. Jugal Kalita University of Colorado at Colorado Springs
Pattern Association or Associative Networks Jugal Kalita University of Colorado at Colorado Springs To an extent, learning is forming associations. Human memory associates similar items, contrary/opposite
More information7 Pendulum. Part II: More complicated situations
MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics
More informationExponential Controller for Robot Manipulators
Exponential Controller for Robot Manipulators Fernando Reyes Benemérita Universidad Autónoma de Puebla Grupo de Robótica de la Facultad de Ciencias de la Electrónica Apartado Postal 542, Puebla 7200, México
More informationFinite Difference and Finite Element Methods
Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element
More informationSynchronization and Bifurcation Analysis in Coupled Networks of DiscreteTime Systems
Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 871 876 c International Academic Publishers Vol. 48, No. 5, November 15, 2007 Synchronization and Bifurcation Analysis in Coupled Networks of DiscreteTime
More informationStorage capacity of hierarchically coupled associative memories
Storage capacity of hierarchically coupled associative memories Rogério M. Gomes CEFET/MG  LSI Av. Amazonas, 7675 Belo Horizonte, MG, Brasil rogerio@lsi.cefetmg.br Antônio P. Braga PPGEEUFMG  LITC Av.
More informationPolynomial Approximation: The Fourier System
Polynomial Approximation: The Fourier System Charles B. I. Chilaka CASA Seminar 17th October, 2007 Outline 1 Introduction and problem formulation 2 The continuous Fourier expansion 3 The discrete Fourier
More informationDiscriminative Direction for Kernel Classifiers
Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering
More informationMath Methods for Polymer Physics Lecture 1: Series Representations of Functions
Math Methods for Polymer Physics ecture 1: Series Representations of Functions Series analysis is an essential tool in polymer physics and physical sciences, in general. Though other broadly speaking,
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1D case: 3D case Time Total energy = Hamiltonian To find out about
More informationDamped harmonic motion
Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of underdamping,
More informationPart 1. The simple harmonic oscillator and the wave equation
Part 1 The simple harmonic oscillator and the wave equation In the first part of the course we revisit the simple harmonic oscillator, previously discussed in di erential equations class. We use the discussion
More informationPHYS 502 Lecture 3: Fourier Series
PHYS 52 Lecture 3: Fourier Series Fourier Series Introduction In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating
More informationWaves in a Shock Tube
Waves in a Shock Tube Ivan Christov c February 5, 005 Abstract. This paper discusses linearwave solutions and simplewave solutions to the Navier Stokes equations for an inviscid and compressible fluid
More informationPhysics 221A Fall 2005 Homework 8 Due Thursday, October 27, 2005
Physics 22A Fall 2005 Homework 8 Due Thursday, October 27, 2005 Reading Assignment: Sakurai pp. 56 74, 87 95, Notes 0, Notes.. The axis ˆn of a rotation R is a vector that is left invariant by the action
More informationEffects of Interactive Function Forms and Refractoryperiod in a SelfOrganized Critical Model Based on Neural Networks
Commun. Theor. Phys. (Beijing, China) 42 (2004) pp. 121 125 c International Academic Publishers Vol. 42, No. 1, July 15, 2004 Effects of Interactive Function Forms and Refractoryperiod in a SelfOrganized
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More informationA CAUCHY PROBLEM OF SINEGORDON EQUATIONS WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS 1. INTRODUCTION
Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 2, December 21, Pages 39 44 A CAUCHY PROBLEM OF SINEGORDON EQUATIONS WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS
More informationConsensus Protocols for Networks of Dynamic Agents
Consensus Protocols for Networks of Dynamic Agents Reza Olfati Saber Richard M. Murray Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 email: {olfati,murray}@cds.caltech.edu
More informationARTIFICIAL NEURAL NETWORK PART I HANIEH BORHANAZAD
ARTIFICIAL NEURAL NETWORK PART I HANIEH BORHANAZAD WHAT IS A NEURAL NETWORK? The simplest definition of a neural network, more properly referred to as an 'artificial' neural network (ANN), is provided
More informationFinal Exam Solution Dynamics :45 12:15. Problem 1 Bateau
Final Exam Solution Dynamics 2 191157140 31012013 8:45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat
More informationDifferentiating an Integral: Leibniz Rule
Division of the Humanities and Social Sciences Differentiating an Integral: Leibniz Rule KC Border Spring 22 Revised December 216 Both Theorems 1 and 2 below have been described to me as Leibniz Rule.
More informationViewpoint invariant face recognition using independent component analysis and attractor networks
Viewpoint invariant face recognition using independent component analysis and attractor networks Marian Stewart Bartlett University of California San Diego The Salk Institute La Jolla, CA 92037 marni@salk.edu
More informationChapter 2 Interpolation
Chapter 2 Interpolation Experiments usually produce a discrete set of data points (x i, f i ) which represent the value of a function f (x) for a finite set of arguments {x 0...x n }. If additional data
More informationResponse Selection Using Neural Phase Oscillators
Response Selection Using Neural Phase Oscillators J. Acacio de Barros 1 A symposium on the occasion of Patrick Suppes 90th birthday March 10, 2012 1 With Pat Suppes and Gary Oas (Suppes et al., 2012).
More informationDifferential Kinematics
Differential Kinematics Relations between motion (velocity) in joint space and motion (linear/angular velocity) in task space (e.g., Cartesian space) Instantaneous velocity mappings can be obtained through
More informationAOL Spring Wavefront Sensing. Figure 1: Principle of operation of the ShackHartmann wavefront sensor
AOL Spring Wavefront Sensing The Shack Hartmann Wavefront Sensor system provides accurate, highspeed measurements of the wavefront shape and intensity distribution of beams by analyzing the location and
More informationOscillations in Damped Driven Pendulum: A Chaotic System
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 10, October 2015, PP 1427 ISSN 2347307X (Print) & ISSN 23473142 (Online) www.arcjournals.org Oscillations
More informationLecture Notes. W. Tang. School of Mathematical & Statistical Sciences Arizona State University
Applied Math / Dynamical Systems Workshop Lecture Notes W. Tang School of Mathematical & Statistical Sciences Arizona State University May. 23 rd 25 th, 2011 Tang (ASU) AMDS Workshop Notes May. 23 rd
More informationHopfield Networks. (Excerpt from a Basic Course at IK 2008) Herbert Jaeger. Jacobs University Bremen
Hopfield Networks (Excerpt from a Basic Course at IK 2008) Herbert Jaeger Jacobs University Bremen Building a model of associative memory should be simple enough... Our brain is a neural network Individual
More informationLecture 15: Exploding and Vanishing Gradients
Lecture 15: Exploding and Vanishing Gradients Roger Grosse 1 Introduction Last lecture, we introduced RNNs and saw how to derive the gradients using backprop through time. In principle, this lets us train
More information1 What is numerical analysis and scientific computing?
Mathematical preliminaries 1 What is numerical analysis and scientific computing? Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations)
More informationThe method of lines (MOL) for the diffusion equation
Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just
More informationUsing a Hopfield Network: A Nuts and Bolts Approach
Using a Hopfield Network: A Nuts and Bolts Approach November 4, 2013 Gershon Wolfe, Ph.D. Hopfield Model as Applied to Classification Hopfield network Training the network Updating nodes Sequencing of
More informationEM Waves. From previous Lecture. This Lecture More on EM waves EM spectrum Polarization. Displacement currents Maxwell s equations EM Waves
EM Waves This Lecture More on EM waves EM spectrum Polarization From previous Lecture Displacement currents Maxwell s equations EM Waves 1 Reminders on waves Traveling waves on a string along x obey the
More informationA Discrete Robust Adaptive Iterative Learning Control for a Class of Nonlinear Systems with Unknown Control Direction
Proceedings of the International MultiConference of Engineers and Computer Scientists 16 Vol I, IMECS 16, March 1618, 16, Hong Kong A Discrete Robust Adaptive Iterative Learning Control for a Class of
More informationOil Field Production using Machine Learning. CS 229 Project Report
Oil Field Production using Machine Learning CS 229 Project Report Sumeet Trehan, Energy Resources Engineering, Stanford University 1 Introduction Effective management of reservoirs motivates oil and gas
More informationEquations. A body executing simple harmonic motion has maximum acceleration ) At the mean positions ) At the two extreme position 3) At any position 4) he question is irrelevant. A particle moves on the
More informationSynchronization of Limit Cycle Oscillators by Telegraph Noise. arxiv: v1 [condmat.statmech] 5 Aug 2014
Synchronization of Limit Cycle Oscillators by Telegraph Noise Denis S. Goldobin arxiv:148.135v1 [condmat.statmech] 5 Aug 214 Department of Physics, University of Potsdam, Postfach 61553, D14415 Potsdam,
More informationFractional Spectral and Spectral Element Methods
Fractional Calculus, Probability and Nonlocal Operators: Applications and Recent Developments Nov. 6th  8th 2013, BCAM, Bilbao, Spain Fractional Spectral and Spectral Element Methods (Based on PhD thesis
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationCITEC SummerSchool 2013 Learning From Physics to Knowledge Selected Learning Methods
CITEC SummerSchool 23 Learning From Physics to Knowledge Selected Learning Methods Robert Haschke Neuroinformatics Group CITEC, Bielefeld University September, th 23 Robert Haschke (CITEC) Learning From
More informationBloomsburg University Bloomsburg, Pennsylvania 17815
Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, Pennsylvania 17815 L Hôpital s Rule Summary Many its may be determined by direct substitution, using a geometric
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of ndof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More informationUSING COUPLED OSCILLATORS TO MODEL THE SINOATRIAL NODE IN THE HEART
USING COUPLED OSCILLATORS TO MODEL THE SINOATRIAL NODE IN THE HEART A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree
More informationNonlinear Control Lecture 1: Introduction
Nonlinear Control Lecture 1: Introduction Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 1 1/15 Motivation
More informationAPPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES
APPROXIMATING THE INVERSE FRAME OPERATOR FROM LOCALIZED FRAMES GUOHUI SONG AND ANNE GELB Abstract. This investigation seeks to establish the practicality of numerical frame approximations. Specifically,
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationWeakly PulseCoupled Oscillators: Heterogeneous Delays Lead to Homogeneous Phase
49th IEEE Conference on Decision and Control December 57, 2 Hilton Atlanta Hotel, Atlanta, GA, USA Weakly PulseCoupled Oscillators: Heterogeneous Delays Lead to Homogeneous Phase Enrique Mallada, Student
More informationNumerical Tests of Center Series
Numerical Tests of Center Series Jorge L. delyra Department of Mathematical Physics Physics Institute University of São Paulo April 3, 05 Abstract We present numerical tests of several simple examples
More information18.06 Problem Set 8  Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8  Solutions Due Wednesday, 4 November 2007 at 4 pm in 206 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationDisturbance Attenuation for a Class of Nonlinear Systems by Output Feedback
Disturbance Attenuation for a Class of Nonlinear Systems by Output Feedback Wei in Chunjiang Qian and Xianqing Huang Submitted to Systems & Control etters /5/ Abstract This paper studies the problem of
More informationSzalai, R., & Osinga, H. M. (2007). Unstable manifolds of a limit cycle near grazing.
Szalai, R., & Osinga, H. M. (2007). Unstable manifolds of a limit cycle near grazing. Early version, also known as preprint Link to publication record in Explore Bristol Research PDFdocument University
More informationComputability Theory for Neuroscience
Computability Theory for Neuroscience by Doug Rubino Abstract Neuronal circuits are ubiquitously held to be the substrate of computation in the brain, information processing in single neurons is though
More informationII. DIFFERENTIABLE MANIFOLDS. Washington Mio CENTER FOR APPLIED VISION AND IMAGING SCIENCES
II. DIFFERENTIABLE MANIFOLDS Washington Mio Anuj Srivastava and Xiuwen Liu (Illustrations by D. Badlyans) CENTER FOR APPLIED VISION AND IMAGING SCIENCES Florida State University WHY MANIFOLDS? Nonlinearity
More information2DVolterraLotka Modeling For 2 Species
Majalat AlUlum AlInsaniya wat  Tatbiqiya 2DVolterraLotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More information1, for s = σ + it where σ, t R and σ > 1
DIRICHLET LFUNCTIONS AND DEDEKIND ζfunctions FRIMPONG A. BAIDOO Abstract. We begin by introducing Dirichlet Lfunctions which we use to prove Dirichlet s theorem on arithmetic progressions. From there,
More information21 Symmetric and skewsymmetric matrices
21 Symmetric and skewsymmetric matrices 21.1 Decomposition of a square matrix into symmetric and skewsymmetric matrices Let C n n be a square matrix. We can write C = (1/2)(C + C t ) + (1/2)(C C t ) =
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationMultiscale Geometric Analysis: Thoughts and Applications (a summary)
Multiscale Geometric Analysis: Thoughts and Applications (a summary) Anestis Antoniadis, University Joseph Fourier Assimage 2005,Chamrousse, February 2005 Classical Multiscale Analysis Wavelets: Enormous
More informationDeep Feedforward Networks
Deep Feedforward Networks Yongjin Park 1 Goal of Feedforward Networks Deep Feedforward Networks are also called as Feedforward neural networks or Multilayer Perceptrons Their Goal: approximate some function
More informationVibrations and Waves Physics Year 1. Handout 1: Course Details
Vibrations and Waves JanFeb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office
More informationStability of flow past a confined cylinder
Stability of flow past a confined cylinder Andrew Cliffe and Simon Tavener University of Nottingham and Colorado State University Stability of flow past a confined cylinder p. 1/60 Flow past a cylinder
More informationDispersion relation for transverse waves in a linear chain of particles
Dispersion relation for transverse waves in a linear chain of particles V. I. Repchenkov* It is difficult to overestimate the importance that have for the development of science the simplest physical and
More informationTheory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati
Theory and Practice of Rotor Dynamics Prof. Dr. Rajiv Tiwari Department of Mechanical Engineering Indian Institute of Technology Guwahati Module  2 Simpul Rotors Lecture  2 Jeffcott Rotor Model In the
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda HolmesCerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.54.7 (eigenfunction methods and reversibility),
More informationNeural Networks. Prof. Dr. Rudolf Kruse. Computational Intelligence Group Faculty for Computer Science
Neural Networks Prof. Dr. Rudolf Kruse Computational Intelligence Group Faculty for Computer Science kruse@iws.cs.unimagdeburg.de Rudolf Kruse Neural Networks 1 Hopfield Networks Rudolf Kruse Neural Networks
More informationMath 215 HW #9 Solutions
Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith
More informationBoolean InnerProduct Spaces and Boolean Matrices
Boolean InnerProduct Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationStochastic Oscillator Death in Globally Coupled Neural Systems
Journal of the Korean Physical Society, Vol. 52, No. 6, June 2008, pp. 19131917 Stochastic Oscillator Death in Globally Coupled Neural Systems Woochang Lim and SangYoon Kim y Department of Physics, Kangwon
More informationOpening the Black Box: Lowdimensional dynamics in highdimensional recurrent neural networks
Opening the Black Box: Lowdimensional dynamics in highdimensional recurrent neural networks David Sussillo sussillo@stanford.edu Department of Electrical Engineering Neurosciences Program Stanford University
More informationAsynchronous and Synchronous Dispersals in Spatially Discrete Population Models
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 7, No. 2, pp. 284 310 c 2008 Society for Industrial and Applied Mathematics Asynchronous and Synchronous Dispersals in Spatially Discrete Population Models AbdulAziz
More informationarxiv: v1 [nlin.ao] 21 Feb 2018
Global synchronization of partially forced Kuramoto oscillators on Networks Carolina A. Moreira and Marcus A.M. de Aguiar Instituto de Física Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationOn the SIR s ( Signal to Interference Ratio) in. DiscreteTime Autonomous Linear Networks
On the SIR s ( Signal to Interference Ratio) in arxiv:93.9v [physics.dataan] 9 Mar 9 DiscreteTime Autonomous Linear Networks Zekeriya Uykan Abstract In this letter, we improve the results in [5] by
More informationA Novel Chaotic Neural Network Architecture
ESANN' proceedings  European Symposium on Artificial Neural Networks Bruges (Belgium),  April, DFacto public., ISBN , pp.  A Novel Neural Network Architecture Nigel Crook and Tjeerd olde Scheper
More informationON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT
ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT Received: 31 July, 2008 Accepted: 06 February, 2009 Communicated by: SIMON J SMITH Department of Mathematics and Statistics La Trobe University,
More informationPattern Recognition 2
Pattern Recognition 2 KNN,, Dr. Terence Sim School of Computing National University of Singapore Outline 1 2 3 4 5 Outline 1 2 3 4 5 The Bayes Classifier is theoretically optimum. That is, prob. of error
More information