A dynamic, spatially periodic, micro-pattern of HES5 underlies neurogenesis in the mouse spinal cord.

Ultradian oscillations of HES Transcription Factors (TFs) at the single-cell level enable cell state transitions. However, the tissue-level organisation of HES5 dynamics in neurogenesis is unknown. Here, we analyse the expression of HES5 ex vivo in the developing mouse ventral spinal cord and identify microclusters of 4-6 cells with positively correlated HES5 level and ultradian dynamics. These microclusters are spatially periodic along the dorsoventral axis and temporally dynamic, alternating between high and low expression with a supra-ultradian persistence time. We show that Notch signalling is required for temporal dynamics but not the spatial periodicity of HES5. Few Neurogenin 2 cells are observed per cluster, irrespective of high or low state, suggesting that the microcluster organisation of HES5 enables the stable selection of differentiating cells. Computational modelling predicts that different cell coupling strengths underlie the HES5 spatial patterns and rate of differentiation, which is consistent with comparison between the motoneuron and interneuron progenitor domains. Our work shows a previously unrecognised spatiotemporal organisation of neurogenesis, emergent at the tissue level from the synthesis of single-cell dynamics.

Key questions for modelling COVID-19 exit strategies.

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

Classification of boundary equilibrium bifurcations in planar Filippov systems.

If a family of piecewise smooth systems depending on a real parameter is defined on two different regions of the plane separated by a switching surface, then a boundary equilibrium bifurcation occurs if a stationary point of one of the systems intersects the switching surface at a critical value of the parameter. We derive the leading order terms of a normal form for boundary equilibrium bifurcations of planar systems. This makes it straightforward to derive a complete classification of the bifurcations that can occur. We are thus able to confirm classic results of Filippov [Differential Equations with Discontinuous Right Hand Sides (Kluwer, Dordrecht, 1988)] using different and more transparent methods, and explain why the 'missing' cases of Hogan et al. [Piecewise Smooth Dynamical Systems: The Case of the Missing Boundary Equilibrium Bifurcations (University of Bristol, 2015)] are the only cases omitted in more recent work.

Auto-regressive moving average analysis of linear and discontinuous models of human balance during quiet standing.

Linear Time Invariant (LTI) processes can be modelled by means of Auto-Regressive Moving Average (ARMA) model systems. In this paper, we examine whether an ARMA model can be fitted to a process characterised by switched nonlinearities. In particular, we conduct the following test: we generate data from known LTI and nonlinear (threshold/dead-zone) models of human balance and analyse the output using ARMA. We show that both these known systems can be fitted, according to standard criteria, with low order ARMA models. To check if there are some obvious effects of the dead-zone, we compare the power spectra of both systems with the power spectra of their ARMA models. We then examine spectral properties of three posturographic data sets and their ARMA models and compare them with the power spectra of our model systems. Finally, we examine the dynamics of our model systems in the absence of noise to determine what is the effect of the switching threshold (dead-zone) on the asymptotic dynamics.

Dynamic switching of lateral inhibition spatial patterns.

Hes genes are transcriptional repressors activated by Notch. In the developing mouse neural tissue, HES5 expression oscillates in neural progenitors (Manning et al. 2019 Nat. Commun. 10, 1-19 (doi:10.1038/s41467-019-10734-8)) and is spatially organized in small clusters of cells with synchronized expression (microclusters). Furthermore, these microclusters are arranged with a spatial periodicity of three-four cells in the dorso-ventral axis and show regular switching between HES5 high/low expression on a longer time scale and larger amplitude than individual temporal oscillators (Biga et al. 2021 Mol. Syst. Biol. 17, e9902 (doi:10.15252/msb.20209902)). However, our initial computational modelling of coupled HES5 could not explain these features of the experimental data. In this study, we provide theoretical results that address these issues with biologically pertinent additions. Here, we report that extending Notch signalling to non-neighbouring progenitor cells is sufficient to generate spatial periodicity of the correct size. In addition, introducing a regular perturbation of Notch signalling by the emerging differentiating cells induces a temporal switching in the spatial pattern, which is longer than an individual cell's periodicity. Thus, with these two new mechanisms, a computational model delivers outputs that closely resemble the complex tissue-level HES5 dynamics. Finally, we predict that such dynamic patterning spreads out differentiation events in space, complementing our previous findings whereby the local synchronization controls the rate of differentiation.

Border collision bifurcations, snap-back repellers, and chaos.

The normal form for codimension 1 border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems is considered in the unstable case. We show that in appropriate parameter regions there is a snap-back repeller immediately after the bifurcation, and hence that the bifurcation creates chaos. Although the chaotic solutions are repellers they may explain observations, and this is illustrated through an example.

The mathematics of motion camouflage.

Motion camouflage is a strategy whereby an aggressor moves towards a target while appearing stationary to the target except for the inevitable change in perceived size of the aggressor as it approaches. The strategy has been observed in insects, and mathematical models using discrete time or neural-network control have been used to simulate the behaviour. Here, the differential equations for motion camouflage are derived and some simple cases are analysed. These equations are easy to simulate numerically, and simulations indicate that motion camouflage is more efficient than the classical pursuit strategy ('move directly towards the target').

Non-smooth Hopf-type bifurcations arising from impact-friction contact events in rotating machinery.

We analyse the novel dynamics arising in a nonlinear rotor dynamic system by investigating the discontinuity-induced bifurcations corresponding to collisions with the rotor housing (touchdown bearing surface interactions). The simplified Föppl/Jeffcott rotor with clearance and mass unbalance is modelled by a two degree of freedom impact-friction oscillator, as appropriate for a rigid rotor levitated by magnetic bearings. Two types of motion observed in experiments are of interest in this paper: no contact and repeated instantaneous contact. We study how these are affected by damping and stiffness present in the system using analytical and numerical piecewise-smooth dynamical systems methods. By studying the impact map, we show that these types of motion arise at a novel non-smooth Hopf-type bifurcation from a boundary equilibrium bifurcation point for certain parameter values. A local analysis of this bifurcation point allows us a complete understanding of this behaviour in a general setting. The analysis identifies criteria for the existence of such smooth and non-smooth bifurcations, which is an essential step towards achieving reliable and robust controllers that can take compensating action.

Mechanisms of intermittent state transitions in a coupled heterogeneous oscillator model of epilepsy.

We investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural mass model of epilepsy. A low dimensional model is constructed, which preserves two key features of the neural mass model, namely (i) coupling between oscillators and (ii) heterogeneous proximity of these oscillators to a bifurcation between distinct limit cycles. We demonstrate that state transitions due to intermittency occur in the abstract model. This suggests that there is a general bifurcation mechanism responsible for this behaviour and that this is independent of the precise form of the evolution equations. Such abstractions of neural mass models allow a deeper insight into underlying dynamic and physiological mechanisms, and also allow the more efficient exploration of large scale brain dynamics in disease.

Modelling human balance using switched systems with linear feedback control.

We are interested in understanding the mechanisms behind and the character of the sway motion of healthy human subjects during quiet standing. We assume that a human body can be modelled as a single-link inverted pendulum, and the balance is achieved using linear feedback control. Using these assumptions, we derive a switched model which we then investigate. Stable periodic motions (limit cycles) about an upright position are found. The existence of these limit cycles is studied as a function of system parameters. The exploration of the parameter space leads to the detection of multi-stability and homoclinic bifurcations.

Two-ball Newton's cradle.

Newton's cradle for two balls with Hertzian interactions is considered as a hybrid system, and this makes it possible to derive return maps for the motion between collisions in an exact form despite the fact that the three-halves interaction law cannot be solved in closed form. The return maps depend on a constant whose value can only be determined numerically, but solutions can be written down explicitly in terms of this parameter, and we compare this with the results of simulations. The results are in fact independent of the details of the interaction potential.

Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows.

Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow, namely a Rössler's oscillator. We show that spirals associated with periodicity hubs emerge and accumulate at the folding of certain fractal-like sheaves of Shilnikov homoclinic bifurcations of a common saddle-focus equilibrium. The specific organization of hub networks is found to depend strongly on the interaction between the homoclinic orbits and the global structure of the underlying attractor.

Visual and vestibular determinants of the translational vestibulo-ocular reflex.

Prior studies indicate that the human translational vestibulo-ocular reflex (tVOR) generates eye rotations approximately half the magnitude required to keep the line of sight pointed at a stationary object--a compensation ratio (CR) of ∼0.5. We asked whether changes of visual or vestibular stimuli could increase the CR of tVOR. First, subjects viewed their environment through an optical device that required eye movements to increase by ∼50% to maintain fixation of a stationary visual target. During vertical translation, eye movements did increase, but tVOR CR remained at ∼0.5. Second, subjects viewed through LCD goggles providing 4 Hz strobe vision that minimized retinal image motion; this reduced tVOR CR. Finally, subjects were rotated in roll while they translated vertically; no increase in tVOR occurred. Taken with prior studies, we conclude that tVOR is optimally set to generate eye rotations that are about 50% of those required to stabilize the line of sight.