MS03.4: Computational Methods

The phase shift of an oscillator in response to an external stimulus has been used to characterise, understand and predict biological behaviours. Such phase shifts are studied mathematically by first modelling the oscillator as an attracting periodic orbit, and then considering its isochrons. Each isochron is an (n − 1)-dimensional submanifold defined as the set of points in the basin of attraction whose trajectories converge to the oscillator with the same phase. While methods exist to compute one- and even two-dimensional isochrons, the dimension of isochrons of biological models that motivate their computation are often much higher. One approach to this problem is to restrict the space of interest to points perturbed away from the periodic orbit for a perturbation with fixed form but where the amplitude and phase of application may vary. The slices of isochron that exist in this two-dimensional subspace are generally one-dimensional curves. We present our method to compute isochron slices via the continuation of a two-point boundary-value problem that relates the phase of a point resulting from a perturbation, with the amplitude of the perturbation and the phase in the cycle at which it is applied. We demonstrate our approach by computing isochrons of the four-dimensional Hodgkin–Huxley model, where we are able to resolve intricate geometrical detail.

Phase resetting investigates oscillatory systems by quantifying the phase shift between a point on a stable periodic orbit and its return position following a perturbation. While this approach is applicable to numerical simulations of mathematical models, computing phase resets with high precision poses challenges. We employ a recently developed high-precision computational method based on the numerical continuation of a multi-segment boundary value problem to compute phase resets. We focus on phase reset resulting from fixed perturbation amplitude applied at a fixed phase, while the perturbation angle varies. We call the graph of the map that relates the perturbation angle with the new phase the dircetional transition curve and show how characteristics of this map and curve changes as the perturbation amplitude increases. We illustrate our findings with the FitzHugh-Nagumo system.

We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analysed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and show three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

We present an implementation of Lin's method to compute codimension-one heteroclinic connections to infinity. Furthermore, we show how these heteroclinic bifurcations with infinity organise the limiting behaviour of a family of codimension-one homoclinic bifurcations that emanates from a codimension-two homoclinic flip bifurcation.