4
$\begingroup$

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:

Maucher, Fabian, and Paul Sutcliffe. "Untangling knots via reaction-diffusion dynamics of vortex strings." (arXiv abs (2016).)


          VortexKnotFig2


If anyone could connect the dots (at a high level) between heart-muscle dynamics, vortices, and knots, I would greatly appreciate it—Thanks!

$\endgroup$
5
  • $\begingroup$ The FitzHugh-Nagumo model is usually presented as a two dimensional dynamical system. This differs slightly from the version used in the pre-print you cited (which has a diffusion term thrown in making it a PDE instead of ODE system). Not being an expert in mathematical biology, I am not sure how much the diffusion term changes things (but being somewhat of an expert in PDEs, I would expect the diffusion term, in the presence of decaying boundary conditions at infinity, to make things quite different). $\endgroup$ Apr 30, 2016 at 2:32
  • 2
    $\begingroup$ If you ignore the "connection" to heart muscles, and take the particular reaction diffusion equation as given, the mathematical heuristics is actually quite well explained in the article. Roughly speaking if you take the PDE and solve it in 2D, it supports vortex-like solutions (things that look like it is spinning around about a center). Plugging in data with two different vortices it looks like that the two vortices will repel each other. By a "vortex string" the authors mean a solution of the 3D equation that in a tubular nbhd of a embedded circle looks like the 2D vortex times the circle. $\endgroup$ Apr 30, 2016 at 2:38
  • 2
    $\begingroup$ Movies can be found at the publication page $\endgroup$ Apr 30, 2016 at 7:31
  • $\begingroup$ @WillieWong: Thanks for your explanation! $\endgroup$ Apr 30, 2016 at 11:07
  • 2
    $\begingroup$ Possibly Related (in the opposite sense...): Brenier's recent work: Brenier, Yann. "Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation." Communications in Mathematical Physics 330.2 (2014): 757-770. $\endgroup$ Apr 30, 2016 at 12:02

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.