David A W Barton

Tuning a resonant energy harvester using a generalized electrical load

A fundamental drawback of vibration-based energy harvesters is that they typically feature a resonant mass/spring mechanical system to amplify the small source vibrations; the limited bandwidth of the mechanical amplifier restricts the effectiveness of the energy harvester considerably. By extending the range of input frequencies over which a vibration energy harvester can generate useful power, e.g. through adaptive tuning, it is not only possible to open up a wider range of applications, such as those where the source frequency changes over time, but also possible to relax the requirements for precision manufacture or the need for mechanical adjustment in situ. In this paper, a vibration-based energy harvester connected to a generalized electrical load (containing both real and reactive impedance) is presented. It is demonstrated that the reactive component of the electrical load can be used to tune the harvester system to significantly increase the output power away from the resonant peak of the device. An analytical model of the system is developed, which includes non-ideal components arising from the physical implementation, and the results are confirmed by experiment. The − 3 dB (half-power) bandwidth of the prototype energy harvester is shown to be over three times greater when presented with an optimized load impedance compared to that for the same harvester presented with an optimized resistive only load.

Uncertainty in performance for linear and nonlinear energy harvesting strategies

Vibrational energy harvesters are often linear mass–spring–damper-type devices, which have their resonant frequency tuned to the dominant vibration frequency of their host environment. As such, they can be highly sensitive to uncertainties, which may arise from the imprecise characterization of the host environment or, alternatively, from manufacturing defects and tolerances. It has previously been claimed that the use of nonlinear energy harvesters may be one way to alleviate the problems of these uncertainties. This article presents a systematic uncertainty propagation study of a prototypical electromagnetic energy harvester. More specifically, the response of a linear harvester in the presence of parametric uncertainty is compared to the response of harvesters containing some common forms of nonlinearity, that is, hardening, softening, or bistability. Analytical solutions are used in combination with presumed levels of parametric uncertainty to quantify the resulting uncertainty in the power output. Consequently, these studies can determine the regions in the parameter space where a nonlinear strategy may outperform a more traditional linear approach.

Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model

In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human electroencephalogram (EEG) activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from inter-ictal activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of delay differential equations (DDEs). We demonstrate that the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures.

Vibration energy harvesters with non-linear compliance

Vibration powered electrical generators typically feature a mass/spring resonant system to amplify small background vibrations. The compliance element in these resonant systems can become non-linear as a result of manufacturing limitations, physical operating constraints, or by deliberate design. The characteristics of mass/spring resonant systems with non-linear compliance elements are well known but they have not been widely applied within the field of energy harvesting. In this paper analysis of non-linear system behaviour using the harmonic balance method is presented, giving an insight into the potential benefits of non-linearities in energy harvesting applications. The design of a vibration powered energy harvester is reviewed and it is shown how the deliberate incorporation of non-linear behaviour within a design can be beneficial in improving magnetic loading and also in extending the range of frequencies over which the device can generate useful power.

How to Turn a Genetic Circuit into a Synthetic Tunable Oscillator, or a Bistable Switch

Systems and Synthetic Biology use computational models of biological pathways in order to study in silico the behaviour of biological pathways. Mathematical models allow to verify biological hypotheses and to predict new possible dynamical behaviours. Here we use the tools of non-linear analysis to understand how to change the dynamics of the genes composing a novel synthetic network recently constructed in the yeast Saccharomyces cerevisiae for In-vivo Reverse-engineering and Modelling Assessment (IRMA). Guided by previous theoretical results that make the dynamics of a biological network depend on its topological properties, through the use of simulation and continuation techniques, we found that the network can be easily turned into a robust and tunable synthetic oscillator or a bistable switch. Our results provide guidelines to properly re-engineering in vivo the network in order to tune its dynamics.

Collocation schemes for periodic solutions of neutral delay differential equations

We introduce two collocation schemes for the computation of periodic solutions of neutral delay differential equations (NDDEs): one based on a direct discretisation of the underlying NDDE, and one based on a discretisation of a related delay differential difference equation (i.e. a delay differential equation (DDE) coupled with a difference equation). Numerical examples are used to demonstrate these schemes and their respective orders of convergence. Both collocation schemes are implemented in DDE-BIFTOOL, a numerical continuation tool for delay equations. Their use in a continuation setting is shown with one- and two-parameter bifurcation studies of a transmission line model.

Data-driven stochastic modelling of zebrafish locomotion

In this work, we develop a data-driven modelling framework to reproduce the locomotion of fish in a confined environment. Specifically, we highlight the primary characteristics of the motion of individual zebrafish (Danio rerio), and study how these can be suitably encapsulated within a mathematical framework utilising a limited number of calibrated model parameters. Using data captured from individual zebrafish via automated visual tracking, we develop a model using stochastic differential equations and describe fish as a self propelled particle moving in a plane. Based on recent experimental evidence of the importance of speed regulation in social behaviour, we extend stochastic models of fish locomotion by introducing experimentally-derived processes describing dynamic speed regulation. Salient metrics are defined which are then used to calibrate key parameters of coupled stochastic differential equations, describing both speed and angular speed of swimming fish. The effects of external constraints are also included, based on experimentally observed responses. Understanding the spontaneous dynamics of zebrafish using a bottom-up, purely data-driven approach is expected to yield a modelling framework for quantitative investigation of individual behaviour in the presence of various external constraints or biological assays.

Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation

We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control. We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems.

Explicit periodic solutions in a model of a relay controller with delay and forcing

In this paper we use a combination of numerical and analytical methods to find and construct solutions of a cameo model of relay control, formulated as a piecewise-constant delay differential equation (DDE). Numerical solutions of a related equation, where the discontinuities of the original DDE are smoothed out, are used to guide the construction of explicit solutions of the original DDE. On the other hand, the construction of explicit solutions provides starting data for numerical continuation of the smoothed equation. The stability of the explicit solutions can also be inferred from the numerical approach.

A method for detecting false bifurcations in dynamical systems: application to neural-field models

In this article, we present a method for tracking changes in curvature of limit cycle solutions that arise due to inflection points. In keeping with previous literature, we term these changes false bifurcations, as they appear to be bifurcations when considering a Poincaré section that is tangent to the solution, but in actual fact the deformation of the solution occurs smoothly as a parameter is varied. These types of solutions arise commonly in electroencephalogram models of absence seizures and correspond to the formation of spikes in these models. Tracking these transitions in parameter space allows regions to be defined corresponding to different types of spike and wave dynamics, that may be of use in clinical neuroscience as a means to classify different subtypes of the more general syndrome.