EXACT COHERENT
STRUCTURES IN CARDIAC
SYSTEMS
THE HEART
•
Complicated geometries
•
orientation,
dimensionality,
anisotropy, defects
•
Electrical dynamics
•
Fluid dynamics
•
Shen, H. W., & Pang, A.
(2007).
Anisotropy based seeding for
hyperstreamline.
Mechanical dynamics
Biomedical Physics at MPI for Dynamics and Self-Organization
http://www.bmp.ds.mpg.de/imaging-electro-mechanicalwaves.html
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?
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
•
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
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
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
http://www.ibiblio.org/e-notes/html5/karma.html
BUENO-OROVIO–CHERRY–
FENTON
•
Reproduces qualitative
dynamics from more
complicated models
•
Reproduces dynamics
from experimental data
http://www.ibiblio.org/e-notes/html5/bcf.html
•
Flexible
•
Simple – three ionic
currents
We pay for realism with
obfuscation through generality
THE HEART SOLUTION
(On the CPU)
•
Operator-Splitting
•
Strang (ABA), O(Δt²)
•
Semi-Implicit
•
Large time-steps
•
Fourier basis, O(exp(1/Δx))
•
Easy (spatial) derivatives
•
Clever flipping restricts to
odd/even modes,
transforms scale well:
Nlog(N)
•
periodic, zero-field, or
zero-derivative boundary
conditions
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
(On the GPU)
•
No operator-splitting
•
Evaluate entire RHS
•
Fully explicit RK4 O(Δt³)
•
Smaller stability window
•
Stencil approximations
•
Rotational symmetry O(Δx⁴)
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
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 finitedifference 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.
… and some time later…
These nearly recurrent states serve as initial conditions for GMRES
GMRES
•
•
Generalized Minimal
Residual
With an initial perturbation
•
iteratively build a basis
•
and an approximate
Jacobian in that basis
•
to compute the
correction to the initial
guess
Newton-Krylov (JFNK)
•
•
•
It's Newton, in Krylov
Solve small linear system
instead of large nonlinear
one
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
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, http://www.scholarpedia.org/article/Periodic_orbit
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
Why periodic orbits?
•
Multi-core ⇒ multi-phase
•quasi-periodic
networked nonlinear
oscillators
•
Is the PDE even
reducible to cores?
•
Vorticity formulation?
•n-core
•
orbits?
⇒ 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