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EXACT COHERENT STRUCTURES IN CARDIAC SYSTEMS

EXACT COHERENT STRUCTURES IN CARDIAC SYSTEMS. Shen, H. W., & Pang, A. (2007). Anisotropy based seeding for hyperstreamline. Biomedical Physics at MPI for Dynamics and Self-Organization http://www.bmp.ds.mpg.de/imaging-electro-mechanical-waves.html. THE HEART. Complicated geometries

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EXACT COHERENT STRUCTURES IN CARDIAC SYSTEMS

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  1. EXACT COHERENT STRUCTURES IN CARDIAC SYSTEMS

  2. Shen, H. W., & Pang, A. (2007). Anisotropy based seeding for hyperstreamline. Biomedical Physics at MPI for Dynamics and Self-Organization http://www.bmp.ds.mpg.de/imaging-electro-mechanical-waves.html THE HEART • Complicated geometries • orientation, dimensionality, anisotropy, defects • Electrical dynamics • Fluid dynamics • Mechanical dynamics

  3. THE HEART PROBLEM • pulse waves • spiral waves • turbulence Experiment and simulation: F. Fenton, E. Cherry thevirtualheart.org Can we understand these dynamics to control the system?

  4. MONODOMAIN • Effective field equation • Averages over the inside, membrane, and immediate outside of cardiac cells • Easy to analyze • Dynamics are weakly effected by geometry

  5. BIDOMAIN • Solve voltage over membrane, intracellular, and extracellular domains • Anisotropy effects are irreducible • Additional Poisson solve I don't solve bidomain field equations See Alessandro Veneziani at Emory Math

  6. IONIC CURRENT MODELING • Karma (2, 7) • Simitev-Biktashev (3, 14) • Bueno-Orovio–Cherry–Fenton (4, 28) • Beeler-Reuter (8, ??) • Iyer et al (67, ??) F. Fenton & E. Cherry: http://www.scholarpedia.org/article/Models_of_cardiac_cell

  7. IONIC CURRENT MODELING • Different regions of the heart have different properties and yield different qualitative dynamics • No Navier-Stokes equations for cellular action potential • Karma (2, 7) • Simitev-Biktashev (3, 14) • Bueno-Orovio–Cherry–Fenton (4, 28) • Beeler-Reuter (8, ??) • Iyer et al (67, ??)

  8. KARMA MODEL • Convective instability due to alternans • wavelength modulation • Minimal restitution length http://www.ibiblio.org/e-notes/html5/karma.html

  9. BUENO-OROVIO–CHERRY–FENTON • Reproduces qualitative dynamics from more complicated models • Reproduces dynamics from experimental data • Flexible • Simple – three ionic currents http://www.ibiblio.org/e-notes/html5/bcf.html We pay for realism with obfuscation through generality

  10. THE HEART SOLUTION (On the CPU) • Operator-Splitting • Semi-Implicit • Fourier basis, O(exp(-1/Δx)) • periodic, zero-field, or zero-derivative boundary conditions • Strang (ABA), O(Δt²) • Large time-steps • Easy (spatial) derivatives • Clever flipping restricts to odd/even modes, transforms scale well: Nlog(N)

  11. STRANG-SPLITTING • Most convergent operator-splitting method, without an a priori commutator [A, B] • Solve the pieces where they're best solved • Stitch it together

  12. THE HEART SOLUTION (On the GPU) • No operator-splitting • Fully explicit RK4 O(Δt³) • Stencil approximations • Evaluate entire RHS • Smaller stability window • Rotational symmetry O(Δx⁴) But it is fast

  13. THE GPU • Discretization of space into threads • Local terms (nondifferential) are easy • Nonlocal terms (differential) are hard • memory access patterns • register usage • local memory size • Potential efficiency improvements for operator splitting methods NVIDIA CUDA Programming Guide version 3.0 CC-BY-SA-3.0

  14. THE GPU • Segmenting the space breaks synchronization • Some effort to restore it • Compute the nonlinear update and the diffusion separately • Apply them together • Diffusion is computed by finite-difference stencil and stored apart from the state • time-update by Runge-Kutta

  15. COHERENT STRUCTURES • Generic chaotic trajectory visits the vicinity of unstable coherent structures • Build a map of phase space from the invariant structures • Know where the states are to know where to push them

  16. RECURRENCE • The “wait and see" method Integrate the system for a long time and look for large-scale recurrent structures.

  17. … and some time later… These nearly recurrent states serve as initial conditions for GMRES

  18. GMRES • Generalized Minimal Residual • Newton-Krylov (JFNK) • It's Newton, in Krylov • Solve small linear system instead of large nonlinear one • With an initial perturbation • iteratively build a basis • and an approximate Jacobian in that basis • to compute the correction to the initial guess

  19. GMRES • Find unstable structures with Newton-Raphson methods • The Jacobian is huge • N=128 ⇒ 20.25 GByte* • N=512 ⇒ 1 TByte* • Avoid forming the Jacobian explicitly * Assumes optimal structure using Arnoldi method for two-variable system

  20. ARNOLDI ALGORITHM • Builds an orthonormal basis which spans the least contracting subspace • Builds a small, approximate, and useful Jacobian • Relies only on forward-time integration, and some linear algebra

  21. PERIODIC ORBITS • State (u,v) maps back to (u,v) after some time T • Dynamically or time invariant • At least one marginal mode • Jacobian is uninvertible • Other marginal modes? E.T. Shea-Brown, http://www.scholarpedia.org/article/Periodic_orbit

  22. SYMMETRIES • Constraint equations in the GMRES system • translations in x, y • rotations are harder • Windowing suppresses boundary effects • Effective norm

  23. RESULTS • We got two*! • single pulse wave • relative equilibrium • single spiral core • relative equilibrium? *Families of un-/stable solutions

  24. JUST TWO? • Multi-core states present difficulties • Exponentially weak forcing • Local gauge invariance • local effects of global symmetries • this is hard to deal with

  25. WELL NOW WHAT • Symmetry reduction for a single core • Barkley, Biktashev • Co-moving frame • small set of ODE's which describes the dynamics of a single core

  26. PATHS TOWARD PROGRESS • Why periodic orbits? • Multi-core ⇒ multi-phase • quasi-periodic orbits? • n-core ⇒ n-tori? • Try to balance complexity and non-triviality • Cores by reduction • reduced ODE systems with core-core coupling • networked nonlinear oscillators • Is the PDE even reducible to cores? • Vorticity formulation?

  27. STATE OF THE PROGRAM • Phase space topology • ? Dynamical connections • Reduced order model of dynamics • ? Low-dimensional linear maps in Krylov subspaces • Feedback control • ? Local • ? Global • Efficient solvers • Numerical integration • Newton-Krylov iteration • Dominant unstable regular solutions • Traveling waves (relative equilibria) • Periodic solutions • Relative periodic solutions

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