Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks Drift & Meander of Spiral Waves Andy Foulkes & Vadim Biktashev 12 March 2009 Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Theory of Drift and Meander of Spiral Waves Spiral Wave Solutions in the Comoving Frame of Reference Concluding Remarks Table of contents 1 Spiral Waves Introduction Spiral Wave Motion Models Studied General Comments 2 Theory of Drift and Meander of Spiral Waves General Approach Example: Electrophoresis Induced Drift Drift & Meander 3 Spiral Wave Solutions in the Comoving Frame of Reference General Idea & Motivation Rigid Rotation- Stationary Solutions Meandering Waves - Periodic Solutions Applications - 1:1 Resonance & Large Core Spirals 4 Concluding Remarks Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves - Introduction Spiral waves occur naturally in physical (galaxies, liquid crystals), chemical (rusting, BZ reaction) and biological systems (cardiac arrhythmias, Dictyostelium Discoideum) One of the most famous examples of spiral waves is in the Belousov-Zhabotinski (BZ) reactions (circa 1960) Popularised in the West by Arthur Winfree in the 1970's First finite automata model was derived in 1946 by Wiener & Rosenblueth Since then, they have been the subject of intense research involving many interdisiplinary teams. Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves - BZ Reaction Zhabotinsky A. M. and Zaikin, 1971 Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves - (Barkley model) Andy FoulkesFigure: & Vadim Snapshot Biktashev ofDrift a & spiral Meander wave of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Project Overview Dynamical Systems project - not much Biology My project commenced as a project to study the unification of the two main types of motion of spiral waves - Drift and Meander Frequency locking Numerical technique to study spiral waves in a comoving frame of reference (came from studies into frequency locking) Used response functions Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves - Motion Three main types of motion of the spiral wave that we consider: Rigid Rotation - shape of the arm of the spiral is fixed Meander - quasiperiodic motion, arm changes shape Drift - motion about a moving center of rotation Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves: kinetics models used - f(u) For the theory, we first consider just a general Reaction-Diffusion system of equations: @u 2 n = D u + f(u) + h(u; u; r; t) u = (u1; u2;:::; un) R @t r r 2 Numerical work involves Barkleys and FitzHugh-Nagumo (FHN) models: Barkley's FHN 3 1 u2+b 1 u1 fu1 (u1; u2) = " u1(1 − u1)(u1 − a ) fu1 (u1; u2) = " (u1 − 3 − u2) fu2 (u1; u2) = (u1 − u2) fu2 (u1; u2) = "(u1 + β − γu2) Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves Introduction Theory of Drift and Meander of Spiral Waves Spiral Wave Motion Spiral Wave Solutions in the Comoving Frame of Reference Models Studied Concluding Remarks General Comments Spiral Waves: General Comments Reaction-Diffusion equations generating spiral wave solutions are equivariant under Euclidean Symmetry (group SE(2) of rotations and translations) Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Approach There are two separate theories for Meander of Spiral Waves and Drift of Spiral Waves: 1995 - Biktashev & Holden - \Resonant Drift of Autowave Vortices and the Effects of Boundaries and Inhomogeneities" 1996 - Biktashev, Holden & Nikolaev - \Spiral Wave Meander and Symmetry of the Plane" The aim of this work is to unify the two theories and determine the dynamics of spiral waves which both drift and meander simultaneously by studying the equations of motion of the tip of the spiral wave Problem - Theories written using different techniques Solution - Rewrite one theory using techniques similar to the the other Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Approach Once theory of drift had been rewritten, study meander Note that in the space of group orbits we have limit cycle solutions Use Floquet Theory to study these limit cycles Find explicit analytical solutions in the space of group orbits Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Group Theory We will exploit the fact that the system generating Spiral Wave solution is equivariant under Euclidean Symmetry We therefore use Group Theory techniques and consider group elements g SE(2), where g = (R; Θ) such that R is a translation in2 the (x; y)-plane with R = (X ; Y ) and Θ is a rotation. We will also choose R and Θ such that they are the tip coordinates and orientation. Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Step 1: Rewritting the Theory of Drift We first consider a spiral wave solution to a Reaction-Diffusion system with a symmetry breaking perturbation: @u 2 n = D u + f(u) + h(u; u; r; t); u = u(r; t) R (1) @t r r 2 Represent this in a Banach space - space of all group orbits Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves U ′ B G U g g′ V M V ′ Figure: Representation in a Banach Space Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves 1 Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.) Split out motion along group orbits and Representative Manifold We find that the associated Reaction-Diffusion-Advection system is: @v 2 @v n = D v + f(v) + (c; )v + ! + h~(v; u; r; t; g); v R (2) @t r r @θ r 2 v1(0; 0) = u∗ (3) v2(0; 0) = v∗ (4) @v 1 (0; 0) = 0 (5) @x h~(v; u; r; t; g) = T(g −1)h(T(g)v; T(g) u; r; t) r r Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.) The solutions to this system of equations are ( = (v; c;!)), where v is the spiral wave solution in a frameM of reference comoving with the tip of the spiral wave. c is the translation velocity of the comoving frame of reference, and ! is its rotational velocity. Frame of reference fixed to the tip of the spiral wave via the tip pinning conditions (Eqns.(3)-(5)). The tip equations of motion are: dR = ceiΘ (6) dt dΘ = ! (7) dt Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral Waves Spiral Waves General Approach Theory of Drift and Meander of Spiral Waves Example: Electrophoresis Induced Drift Spiral Wave Solutions in the Comoving Frame of Reference Drift & Meander Concluding Remarks Drift of a rigidly rotating spiral waves (cont.) We now apply perturbation techniques as follows: 2 v = v0 + v1 + O( ) (8) 2 cx = c0x + c1x + O( ) (9) 2 cy = c0y + c1y + O( ) (10) 2 ! = !0 + ω1 + O( ) (11) which yields the following: @v0 2 = 0 = D v0 + f(v0) + (c0; )v0 + !0@θv0 ) @t r r @v1 2 df(v0) = 0 = D v1 + v1 + (c0; )v1 + !0@θv1 6 ) @t r dv · r +(c1; )v0 + !1v0 + h~(v0; r; t) + O() r Andy Foulkes & Vadim Biktashev Drift & Meander of Spiral