Search

The widely cited Haken-Kelso-Bunz (HKB) model of motor coordination is used in an enormous range of applications. In this paper, we show analytically that the weakly damped, weakly coupled HKB model of two oscillators depends on only two dimensionless parameters; the ratio of the linear damping coefficient and the linear coupling coefficient and the ratio of the combined nonlinear damping coefficients and the combined nonlinear coupling coefficients. We illustrate our results with a mechanical analogue. We use our analytic results to predict behaviours in arbitrary parameter regimes and show how this led us to explain and extend recent numerical continuation results of the full HKB model. The key finding is that the HKB model contains a significant amount of behaviour in biologically relevant parameter regimes not yet observed in experiments or numerical simulations. This observation has implications for the development of virtual partner interaction and the human dynamic clamp, and potentially for the HKB model itself., (© 2021. The Author(s).)

We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies., Competing Interests: We have no competing interests.

An account is provided of a UK national seminar series on Arts, Health and Wellbeing funded by the Economic and Social Research Council during 2012-13. Four seminars were organised addressing current issues and challenges facing the field. Details of the programme and its outputs are available online. A central concern of the seminar programme was to provide a foundation for creating a UK national network for researchers in the field to help promote evidence-based policy and practice. With funding from Lankelly Chase Foundation, and the support of the Royal Society for Public Health, a Special interest Group for Arts, Health and Wellbeing was launched in 2015.

A simple mechanical model of the skateboard-skater system is analysed, in which the effect of human control is considered by means of a linear proportional-derivative (PD) controller with delay. The equations of motion of this non-holonomic system are neutral delay-differential equations. A linear stability analysis of the rectilinear motion is carried out analytically. It is shown how to vary the control gains with respect to the speed of the skateboard to stabilize the uniform motion. The critical reflex delay of the skater is determined as the function of the speed. Based on this analysis, we present an explanation for the linear instability of the skateboard-skater system at high speed. Moreover, the advantages of standing ahead of the centre of the board are demonstrated from the viewpoint of reflex delay and control gain sensitivity., (© 2016 The Author(s).)

In Parkinson's disease, an increase in beta oscillations within the basal ganglia nuclei has been shown to be associated with difficulty in movement initiation. An important role in the generation of these oscillations is thought to be played by the motor cortex and by a network composed of the subthalamic nucleus (STN) and the external segment of globus pallidus (GPe). Several alternative models have been proposed to describe the mechanisms for generation of the Parkinsonian beta oscillations. However, a recent experimental study of Tachibana and colleagues yielded results which are challenging for all published computational models of beta generation. That study investigated how the presence of beta oscillations in a primate model of Parkinson's disease is affected by blocking different connections of the STN-GPe circuit. Due to a large number of experimental conditions, the study provides strong constraints that any mechanistic model of beta generation should satisfy. In this paper we present two models consistent with the data of Tachibana et al. The first model assumes that Parkinsonian beta oscillation are generated in the cortex and the STN-GPe circuits resonates at this frequency. The second model additionally assumes that the feedback from STN-GPe circuit to cortex is important for maintaining the oscillations in the network. Predictions are made about experimental evidence that is required to differentiate between the two models, both of which are able to reproduce firing rates, oscillation frequency and effects of lesions carried out by Tachibana and colleagues. Furthermore, an analysis of the models reveals how the amplitude and frequency of the generated oscillations depend on parameters.

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

A key pathology in the development of Parkinson's disease is the occurrence of persistent beta oscillations, which are correlated with difficulty in movement initiation. We investigated the network model composed of the subthalamic nucleus (STN) and globus pallidus (GP) developed by A. Nevado Holgado et al. [(2010) Journal of Neuroscience, 30, 12340-12352], who identified the conditions under which this circuit could generate beta oscillations. Our work extended their analysis by deriving improved analytic stability conditions for realistic values of the synaptic transmission delay between STN and GP neurons. The improved conditions were significantly closer to the results of simulations for the range of synaptic transmission delays measured experimentally. Furthermore, our analysis explained how changes in cortical and striatal input to the STN-GP network influenced oscillations generated by the circuit. As we have identified when a system of mutually connected populations of excitatory and inhibitory neurons can generate oscillations, our results may also find applications in the study of neural oscillations produced by assemblies of excitatory and inhibitory neurons in other brain regions., (© 2012 The Authors. European Journal of Neuroscience © 2012 Federation of European Neuroscience Societies and Blackwell Publishing Ltd.)

We constructed and modeled a novel synthetic network which may be able to exhibit bistable expression of a reporter gene in mammalian cells. This network is based on an aptamer-fused short-hairpin RNA (shRNA) directed against a single mRNA encoding both a EGFP reporter gene and the repressor tTR-KRAB, which, in turn, represses transcription of the shRNA. The activity of the shRNA can be controlled by an inducer molecule (theophylline) which prevents the aptamer-fused shRNA to be properly processed. Repression of the tTR-KRAB can be relieved by treatment with doxycyline. This reciprocal negative feed-back loop can exhibit a bistable response, as shown through the mathematical analysis performed here. Specifically, the network can be controlled to induce sustained expression of a shRNA, or the reporter gene, with a transient input of two different inducer molecules.

This paper presents an overview of the current state of the art in the analysis of discontinuity-induced bifurcations (DIBs) of piecewise smooth dynamical systems, a particularly relevant class of hybrid dynamical systems. Firstly, we present a classification of the most common types of DIBs involving non-trivial interactions of fixed points and equilibria of maps and flows with the manifolds in phase space where the system is non-smooth. We then analyse the case of limit cycles interacting with such manifolds, presenting grazing and sliding bifurcations. A description of possible classification strategies to predict and analyse the scenarios following such bifurcations is also discussed, with particular attention to those methodologies that can be applied to generic n-dimensional systems.

A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. A radically different approach is to replace the Hill functions by piecewise-linear approximations [Casey, R., de Jong, H., Gouze , J.-L., 2006. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52 (1), 27-56]. A further modelling approach is the use of discrete-time maps [Coutinho, R., Fernandez, B., Lima, R., Meyroneinc, A., 2006. Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52, 524-570] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics.

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a stable traveling pulse, the control is able to advance stationary Turing patterns or yield the above-mentioned bistability regime. In each case, the stability boundary is found in the parameter space of the control strength and the time delay, and numerical simulations suggest that diagonal control fails to control the spatiotemporal chaos.

In planar nematic electrohydrodynamic convection (EHC), a microscopic liquid crystal cell is driven by a homogeneous ac electric field, which, if strong enough, causes the fluid to destabilize into a regular pattern-forming state. We consider asymmetric electric fields E(t)=E(t+T) not equal-E(t+T2) , which leads to the possibility of three different types of instabilities at onset: conductive, dielectric, and subharmonic. The first two are already well known as they are easily produced when the system is driven by symmetric electric fields; the third can only occur when the electric field symmetry is broken. We present theoretical results on EHC using linear stability analysis and Floquet theory. We consider rigid and free boundary conditions, extending the model to two Fourier modes in the vertical plane, the inclusion of flexoelectricity, and using standard (nematic electric conductivity sigma&#95;{a}>0 and dielectric anisotorpy &#95;{a}<0 ) and nonstandard (sigma&#95;{a}<0) material parameters. We make full use of a three-dimensional linear model where two mutually perpendicular planar wave numbers q and p can be varied. Our results show that there is a qualitative difference between the boundary conditions used, which is also dependent on how many vertical Fourier modes were used in the model. We have obtained threshold values favoring oblique rolls in subharmonic and dielectric regimes in parameter space. For the nonstandard EHC parameter values, both conduction and subharmonic regimes disappear and only the dielectric threshold exists.

In the case of low immersion high-speed milling, the ratio of time spent cutting to not cutting can be considered as a small parameter. In this case the classical regenerative vibration model of machine tool vibrations reduces to a simplified discrete mathematical model. The corresponding stability charts contain stability boundaries related to period doubling and Neimark-Sacker bifurcations. The subcriticality of both types of bifurcations is proved in this paper. Further, global period-2 orbits are found and analyzed. In connection with these orbits, the existence of chaotic motion is demonstrated for realistic high-speed milling parameters.

Tumour growth rates have been measured in metastatic non-seminomatous germ cell testicular tumours (NSGCTT) by estimating the rate of rise of tumour marker production (TMP). TMP was calculated for the time between orchidectomy and the start of chemotherapy in a group of 58 patients with metastatic NSGCTT treated with BEP combination chemotherapy (bleomycin, etoposide and cisplatin). Calculation of TMP (iu/l/day) took account of the continuing clearance of marker from the serum. TMP increased with time in 51 patients and this rise generally appeared to be exponential. The rate of this increase was expressed as the marker production doubling time (MPDT) and is a measure of the tumour growth rate. MPDT varied from 0.5 to greater than 80 days (45 cases) for AFP + ve patients and from 1.8 to greater than 80 days (34 cases) for HCG + ve patients. Patients who failed BEP first line therapy had shorter MPDTs than those who responded (AFP P = 0.08, HCG P = 0.003). It was found that patients with a MPDT less than or equal to 4 days were more likely to fail treatment than those who had a MPDT greater than 4 days (AFP P = 0.009, HCG P = 0.005). MPDTs were independent of initial serum marker concentration. Patients with small volume disease had longer MPDTs than patients with large volume disease (AFP P = 0.02, HCG P = 0.04). Rapid tumour growth rate reflected by short MPDT carries a poor prognosis in patients with NSGCTT treated by BEP chemotherapy.

Changes in serum tumour marker levels in non-seminomatous germ cell testicular tumours (NSGCTT) are used to monitor tumour growth and response to treatment. A novel method of calculating the actual tumour marker production (TMP) per day is reported; estimation of the rate of change of TMP is a measure of the tumour growth or regression rate. TMP is calculated mathematically from the rate of increase in serum marker level and its natural half life. TMP is assumed to be proportional to the number of marker producing cells in the tumour. TMP was calculated over the time between orchidectomy and the start of chemotherapy. The rate of increase in TMP with time is expressed as the marker production doubling time (MPDT) and is a measure of the growth rate. In a group of 51 patients with metastatic NSGCTT, TMP varied from 0.012 to 5985 iu/l/day (AFP) and 0.08-5404 iu/l/day (HCG). MPDT varied from 0.5 to greater than 80 days (45 cases) for AFP + ve patients and from 1.8 to greater than 80 days (34 cases) for HCG + ve patients; greater than 80% of cases had a MPDT less than or equal to 32 days. In 45/51 (88%) patients, there was no discrepancy in MPDT between markers. The use of changes in serum marker level to follow tumour progression and regression is simple, but the calculation of actual TMP provides clearer information about the change in number of marker producing cells and can be used as non-invasive method for measuring the tumour growth rate of metastatic disease and response to treatment.