The importance of understanding the nonlinear dynamics of neural systems, and the relation
to cognitive systems more generally, has been recognised for a long time. Approaches that …

Today, based on brain imaging analyses, we can consider the brilliant metaphor about
event discreteness of the conscious process by William James (1890) to be an experimental …

Our goal in this paper is to review the existing literature on homoclinic and heteroclinic
bifurcation theory for flows. More specifically, we shall focus on bifurcations from homoclinic …

Key features of biological activity can often be captured by transitions between a finite
number of semi-stable states that correspond to behaviors or decisions. We present here a …

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically
coupled or otherwise constrained dynamical systems. These networks may have a …

Thomas and Lindner [PJ Thomas and B. Lindner, Phys. Rev. Lett. 113, 254101 (2014).
PRLTAO 0031-9007 10.1103/PhysRevLett. 113.254101], defined an asymptotic phase for …

Heteroclinic networks are a mathematical concept in dynamic systems theory that is suited to
describe metastable states and switching events in brain dynamics. The framework is …

We study the dynamical behaviour of a smooth vector field on a three-manifold near a
heteroclinic network. Under some generic assumptions on the network, we prove that every …

We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic
network. We show that under the logarithmic time rescaling the diffusion converges in …

Stochastic oscillations are ubiquitous in many systems. For deterministic systems, the
oscillator's phase has been widely used as an effective one-dimensional description of a …