Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

Geometric Singular Perturbation Theory and Averaging: Analysing Torus Canards in Neural Models

Neuronal bursting, an oscillatory pattern of repeated spikes interspersed with periods of rest, is a pervasive phenomenon in brain function which is used to relay information in the body. Mathematical models of bursting typically consist of singularly perturbed systems of ordinary differential equations, which are well suited to analysis by geometric singular perturbation theory (GSPT). There are numerous types of bursting models, which are classified by a slow/fast decomposition and identification of fast subsystem bifurcation structures. Of interest are so-called fold/fold-cycle bursters, where burst initiation (termination) occurs at a fold of equilibria (periodic orbits), respectively. Such bursting models permit torus canards, special solutions which track a repelling fast subsystem manifold of periodic orbits. In this thesis we analyse the Wilson-Cowan-Izhikevich (WCI) and Butera models, two fold/fold-cycle bursters. Using numerical averaging and GSPT, we construct an averaged slow subsystem and identify the bifurcations corresponding to the transitions between bursting and spiking activity patterns. In both models we find that the transition involves toral folded singularities (TFS), averaged counterparts of folded singularities. In the WCI model, we show that the transition occurs at a degenerate TFS, resulting in a torus canard explosion, reminiscent of a classic canard explosion in the van der Pol oscillator. The TFS identified in the Butera model are generic, and using numerical continuation methods, we continue them and construct averaged bifurcation diagrams. We find three types of folded-saddle node (FSN) bifurcations which mediate transitions between activity patterns: FSN type I, II, and III. Type III is novel and studied here for the first time. We utilise the blow-up technique and dynamic bifurcation theory to extend current canard theory to the FSN III

Dynamical systems and their applications in neuroscience

This thesis deals with dynamical systems, numerical software for the continuation study of dynamical systems, and some important neurobiological applications. First there are two introductory chapters, in which a background is given in dynamical systems and neuroscience. We elucidate what the problems are with some existing classifications of neural models, and suggest an improved version. We introduce the Phase Response Curve (PRC), which is a curve that describes the effect of an input on a periodic orbit. We derive an efficient method to compute this PRC. The extended functionalities of MatCont, a software package for the study of dynamical systems and their bifurcations, are explained: the user can compute the PRC of a limit cycle and its derivative, he can detect and continue homoclinic bifurcations, initiate these curves from different bifurcations and detect many codim 2 bifurcations on these curves. The speed of the software was improved by introducing C-code among the matlab-routines. We have for the first time made a complete bifurcation diagram of the Morris-Lecar neural model. We show that PRCs can be used to determine the synchronizing and/or phase-locking abilities of neural networks, and how the connection delay plays a role in this, and demonstrate some phenomena to do with PRCs and bifurcations. In collaboration with biologists at the University of Bristol, we have built detailed models of the neurons in the spinal cord of the hatchling Xenopus laevis. The biological background and the equations and parameters for the models of individual neurons and synapses are listed elaborately. These models are used to construct biologically realistic networks of neurons. The first network was used to simulate the swimming behaviour of the tadpole and to show that to disregard some important differences in the models for different neurons, will result in breakdown of the good network output. Then we have used the individual models to study a hypothesis regarding synaptogenesis, which states that the specificity in connection between neurons could be purely based on the anatomical organization of the neurons, instead of the ability of growing synapses to make a distinction between the different neurons