BackgroundDelirium is a common and serious problem among acutely unwell persons. Alhough linked to higher rates of mortality, institutionalisation and dementia, it remains underdiagnosed. Careful consideration of its phenomenology is warranted to improve detection and therefore mitigate some of its clinical impact. The publication of the fifth edition of the Diagnostic and Statistical Manual of the American Psychiatric Association (DSM-5) provides an opportunity to examine the constructs underlying delirium as a clinical entity.DiscussionAltered consciousness has been regarded as a core feature of delirium; the fact that consciousness itself should be physiologically disrupted due to acute illness attests to its clinical urgency. DSM-5 now operationalises ‘consciousness’ as ‘changes in attention’. It should be recognised that attention relates to content of consciousness, but arousal corresponds to level of consciousness. Reduced arousal is also associated with adverse outcomes. Attention and arousal are hierarchically related; level of arousal must be sufficient before attention can be reasonably tested.SummaryOur conceptualisation of delirium must extend beyond what can be assessed through cognitive testing (attention) and accept that altered arousal is fundamental. Understanding the DSM-5 criteria explicitly in this way offers the most inclusive and clinically safe interpretation.

The well-established method of phase reduction neglects information about a limit-cycle oscillator's approach towards its periodic orbit. Consequently, phase reduction suffers in practicality unless the magnitude of the Floquet multipliers of the underlying limit cycle are small in magnitude. By defining isostable coordinates of a periodic orbit, we present an augmentation to classical phase reduction which obviates this restriction on the Floquet multipliers. This framework allows for the study and understanding of periodic dynamics for which standard phase reduction alone is inadequate. Most notably, isostable reduction allows for a convenient and self-contained characterization of the dynamics near unstable periodic orbits.

The applicability of phase models is generally limited by the constraint that the dynamics of a perturbed oscillator must stay near its underlying periodic orbit. Consequently, external perturbations must be sufficiently weak so that these assumptions remain valid. Using the notion of isostables of periodic orbits to provide a simplified coordinate system from which to understand the dynamics transverse to a periodic orbit, we devise a strategy to correct for changing phase dynamics for locations away from the limit cycle. Consequently, these corrected phase dynamics allow for perturbations of larger magnitude without invalidating the underlying assumptions of the reduction. The proposed reduction strategy yields a closed set of equations and can be applied to periodic orbits embedded in arbitrarily high dimensional spaces. We illustrate the utility of this strategy in two models with biological relevance. In the first application, we find that an optimal control strategy for modifying the period of oscillation can be improved with the corrected phase reduction. In the second, the corrected phase reduced dynamics are used to understand adaptation and memory effects resulting from past perturbations.