EXACT COHERENT STRUCTURES IN CARDIAC SYSTEMS

THE HEART Complicated geometries orientation, dimensionality, anisotropy, defects Electrical dynamics Fluid dynamics Mechanical dynamics Shen, H. W., & Pang, A. (2007). Anisotropy based seeding for hyperstreamline. Biomedical Physics at MPI for Dynamics and Self-Organization waves.html

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

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

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

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:

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

KARMA MODEL Convective instability due to alternans wavelength modulation Minimal restitution length

BUENO-OROVIO–CHERRY– FENTON Reproduces qualitative dynamics from more complicated models Reproduces dynamics from experimental data Flexible Simple – three ionic currents We pay for realism with obfuscation through generality

THE HEART SOLUTION 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) (On the CPU)

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

THE HEART SOLUTION No operator-splitting Fully explicit RK4 O(Δt³) Stencil approximations Evaluate entire RHS Smaller stability window Rotational symmetry O(Δx ⁴ ) (On the GPU) But it is fast

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 CC-BY-SA-3.0

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

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

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

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

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

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

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

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,

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

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

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

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

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

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