The uncoupling limit of identical Hopf bifurcations with an application to perceptual bistability

We study the dynamics arising when two identical oscillators are coupled near a Hopf bifurcation where we assume a parameter $\epsilon$ uncouples the system at $\epsilon=0$. Using a normal form for $N=2$ identical systems undergoing Hopf bifurcation, we explore the dynamical properties. Matching the normal form coefficients to a coupled Wilson-Cowan oscillator network gives an understanding of different types of behaviour that arise in a model of perceptual bistability. Notably, we find bistability between in-phase and anti-phase solutions that demonstrates the feasibility for synchronisation to act as the mechanism by which periodic inputs can be segregated (rather than via strong inhibitory coupling, as in existing models). Using numerical continuation we confirm our theoretical analysis for small coupling strength and explore the bifurcation diagrams for large coupling strength, where the normal form approximation breaks down

The effects of delay on the HKB model of human motor coordination

Understanding human motor coordination holds the promise of developing diagnostic methods for mental illnesses such as schizophrenia. In this paper, we analyse the celebrated Haken-Kelso-Bunz (HKB) model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behaviour and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and anti-phase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and centre manifold reduction. Moreover, we uncover further details on the global dynamic behaviour by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.Comment: Submitted to the SIAM Journal on Applied Dynamical Systems. 27 pages, 8 figure

A Dynamical Systems Analysis of Movement Coordination Models

In this thesis, we present a dynamical systems analysis of models of movement coordination, namely the Haken-Kelso-Bunz (HKB) model and the Jirsa-Kelso excitator (JKE). The dynamical properties of the models that can describe various phenomena in discrete and rhythmic movements have been explored in the models' parameter space. The dynamics of amplitude-phase approximation of the single HKB oscillator has been investigated. Furthermore, an approximated version of the scaled JKE system has been proposed and analysed. The canard phenomena in the JKE system has been analysed. A combination of slow-fast analysis, projection onto the Poincare sphere and blow-up method has been suggested to explain the dynamical mechanisms organising the canard cycles in JKE system, which have been shown to have different properties comparing to the classical canards known for the equivalent FitzHugh-Nagumo (FHN) model. Different approaches to de fining the maximal canard periodic solution have been presented and compared. The model of two HKB oscillators coupled by a neurologically motivated function, involving the effect of time-delay and weighted self- and mutual-feedback, has been analysed. The periodic regimes of the model have been shown to capture well the frequency-induced drop of oscillation amplitude and loss of anti-phase stability that have been experimentally observed in many rhythmic movements and by which the development of the HKB model has been inspired. The model has also been demonstrated to support a dynamic regime of stationary bistability with the absence of periodic regimes that can be used to describe discrete movement behaviours.This work was supported by The Higher Committee For Education Development in Iraq (HCED) and the University of Mosul

Dynamics of neural systems with discrete and distributed time delays

In real-world systems, interactions between elements do not happen instantaneously, due to the time required for a signal to propagate, reaction times of individual elements, and so forth. Moreover, time delays are normally nonconstant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In order to analyze the fundamental properties of neural networks with time-delayed connections, we consider a system of two coupled two-dimensional nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. An exciting feature of the model under consideration is the combination of both discrete and distributed delays, where distributed time delays represent the neural feedback between the two subsystems, and the discrete delays describe the neural interaction within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding results on stability for networks with no distributed delays. It is shown how approximations of the boundary of the stability region of a trivial equilibrium can be obtained analytically for the cases of delta, uniform, and weak gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system, and they fully confirm all analytical findings

Clustering behaviour in networks with time delayed all-to-all coupling

Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches. We first use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specfi c examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies

Stochastic analysis of nonlinear dynamics and feedback control for gene regulatory networks with applications to synthetic biology

The focus of the thesis is the investigation of the generalized repressilator model (repressing genes ordered in a ring structure). Using nonlinear bifurcation analysis stable and quasi-stable periodic orbits in this genetic network are characterized and a design for a switchable and controllable genetic oscillator is proposed. The oscillator operates around a quasi-stable periodic orbit using the classical engineering idea of read-out based control. Previous genetic oscillators have been designed around stable periodic orbits, however we explore the possibility of quasi-stable periodic orbit expecting better controllability. The ring topology of the generalized repressilator model has spatio-temporal symmetries that can be understood as propagating perturbations in discrete lattices. Network topology is a universal cross-discipline transferable concept and based on it analytical conditions for the emergence of stable and quasi-stable periodic orbits are derived. Also the length and distribution of quasi-stable oscillations are obtained. The findings suggest that long-lived transient dynamics due to feedback loops can dominate gene network dynamics. Taking the stochastic nature of gene expression into account a master equation for the generalized repressilator is derived. The stochasticity is shown to influence the onset of bifurcations and quality of oscillations. Internal noise is shown to have an overall stabilizing effect on the oscillating transients emerging from the quasi-stable periodic orbits. The insights from the read-out based control scheme for the genetic oscillator lead us to the idea to implement an algorithmic controller, which would direct any genetic circuit to a desired state. The algorithm operates model-free, i.e. in principle it is applicable to any genetic network and the input information is a data matrix of measured time series from the network dynamics. The application areas for readout-based control in genetic networks range from classical tissue engineering to stem cells specification, whenever a quantitatively and temporarily targeted intervention is required