No abstract

We describe how classical Floquet theory may be utilized, in a continuation framework, to construct an efficient Fourier spectral algorithm for approximating periodic orbits. At each continuation step, only a single square matrix, whose size equals the dimension of the phasespace, needs to be factorized; the rest of the required numerical linear algebra just consists of backsubstitutions with this matrix. The eigenvalues of this key matrix are the Floquet exponents, whose crossing of the imaginary axis indicates bifurcation and change-instability. Hence we also describe how the new periodic orbits created at a period-doubling bifurcation point may be efficiently computed using our approach.

The ubiquitous presence of inhibitory interneurons in the thalamus of primates contrasts with the sparsity of interneurons reported in mice. Here, we identify a larger than expected complexity and distribution of interneurons across the mouse thalamus, where all thalamic interneurons can be traced back to two developmental programs: one specified in the midbrain and the other in the forebrain. Interneurons migrate to functionally distinct thalamocrtical nuclei depending on their origin: the abundant, midbrain-generated class populates the first and higher order sensory thalamus while the rarer, forebrain-generated class is restricted to some higher order associative regions. We also observe that markers for the midbrain-born class are abundantly expressed throughout the thalamus of the New World monkey marmoset. These data therefore reveal that, despite the broad variability in interneuron density across mammalian species, the blueprint of the ontogenetic organisation of thalamic interneurons of larger-brained mammals exists and can be studied in mice.

In a recent paper, the present authors derived an automatic continuation algorithm for the solution of linear singular perturbation problems. The algorithm was incorporated into two general-purpose codes for solving boundary value problems, and it was shown to deal effectively with a large test set of linear problems. The present paper describes how the conintuation algorithm for linear problems can be extended to deal with the nonlinear case. The results of exstensive numerical testing on a set of nonlinear singular perturbation problems are given, and these clearly demonstrate the efficacy of continuation for solving such problems.

We develop new algorithms for computing invariant tori of autonomous systems and invariant curves of periodically forced systems. Our key idea is the choice of a two-stage parametrisation procedure. The manifolds are first computed using a parametrisation based on a known nearby manifold. This parametrisation is then relatively cheaply replaced by a conformal parametrisation in the case of invariant tori or by a form of arclength in the case of invariant curves.