The Input Signal Step Function (ISSF), a standard method to encode input signals in SBML models with software support, applied to circadian clock models.

Time-dependent light input is an important feature of computational models of the circadian clock. However, publicly available models encoded in standard representations such as the Systems Biology Markup Language (SBML) either do not encode this input or use different mechanisms to do so, which hinders reproducibility of published results as well as model reuse. The authors describe here a numerically continuous function suitable for use in SBML for models of circadian rhythms forced by periodic light-dark cycles. The Input Signal Step Function (ISSF) is broadly applicable to encoding experimental manipulations, such as drug treatments, temperature changes, or inducible transgene expression, which may be transient, periodic, or mixed. It is highly configurable and is able to reproduce a wide range of waveforms. The authors have implemented this function in SBML and demonstrated its ability to modify the behavior of publicly available models to accurately reproduce published results. The implementation of ISSF allows standard simulation software to reproduce specialized circadian protocols, such as the phase-response curve. To facilitate the reuse of this function in public models, the authors have developed software to configure its behavior without any specialist knowledge of SBML. A community-standard approach to represent the inputs that entrain circadian clock models could particularly facilitate research in chronobiology.

Analysis of signal propagation in an elastic-tube flow model.

We combine linear and nonlinear signal analysis techniques to investigate the transmission of pressure signals along a one-dimensional model of fluid flow in an elastic tube. We derive a simple, generally applicable measure for the robustness of a simulated vessel against in vivo pressure fluctuations, based on quantifying the degree of synchronization between proximal and distal pressure pulses. The practical use of this measure will be in its application to simulated pulses generated in response to a stochastic forcing term mimicking biological variations of root pressure in arterial blood flow.

Nonlinear time series analysis of jerk congenital nystagmus.

Nonlinear dynamics provides a complementary framework to control theory for the quantitative analysis of the oculomotor control system. This paper presents a number of findings relating to the aetiology and mechanics of the pathological ocular oscillation jerk congenital nystagmus (jerk CN). A range of time series analysis techniques were applied to recorded jerk CN waveforms, and also to simulated jerk waveforms produced by an established model in which the oscillations are a consequence of an unstable neural integrator. The results of the analysis were then interpreted within the framework of a generalised model of the unforced oculomotor system.This work suggests that for jerk oscillations, the origin of the instability lies in one of the five oculomotor subsystems, rather than in the final common pathway (the neural integrator and muscle plant). Additionally, experimental estimates of the linearised foveation dynamics imply that a refixating fast phase induced by a near-homoclinic trajectory will result in periodic oscillations. Local dimension calculations show that the dimension of the experimental jerk CN data increases during the fast phase, indicating that the oscillations are not periodic, and hence that the refixation mechanism is of greater complexity than a homoclinic reinjection. The dimension increase is hypothesised to result either from a signal-dependent noise process in the saccadic system, or the activation of additional oculomotor components at the beginning of the fast phase. The modification of a recent saccadic system model to incorporate biologically realistic signal-dependent noise is suggested, in order to test the first of these hypotheses.

Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system.

The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus. The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards. The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution.