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Quantifying diversity is of central importance for the study of structure, function and evolution of microbial communities. The estimation of microbial diversity has received renewed attention with the advent of large-scale metagenomic studies. Here, we consider what the diversity observed in a sample tells us about the diversity of the community being sampled. First, we argue that one cannot reliably estimate the absolute and relative number of microbial species present in a community without making unsupported assumptions about species abundance distributions. The reason for this is that sample data do not contain information about the number of rare species in the tail of species abundance distributions. We illustrate the difficulty in comparing species richness estimates by applying Chao’s estimator of species richness to a set of in silico communities: they are ranked incorrectly in the presence of large numbers of rare species. Next, we extend our analysis to a general family of diversity metrics (‘Hill diversities’), and construct lower and upper estimates of diversity values consistent with the sample data. The theory generalizes Chao’s estimator, which we retrieve as the lower estimate of species richness. We show that Shannon and Simpson diversity can be robustly estimated for the in silico communities. We analyze nine metagenomic data sets from a wide range of environments, and show that our findings are relevant for empirically-sampled communities. Hence, we recommend the use of Shannon and Simpson diversity rather than species richness in efforts to quantify and compare microbial diversity.

Active brain stimulation to abate epileptic seizures has shown mixed success. In spike-wave (SW) seizures, where the seizure and background state were proposed to coexist, single-pulse stimulations have been suggested to be able to terminate the seizure prematurely. However, several factors can impact success in such a bistable setting. The factors contributing to this have not been fully investigated on a theoretical and mechanistic basis. Our aim is to elucidate mechanisms that influence the success of single-pulse stimulation in noise-induced SW seizures. In this work, we study a neural population model of SW seizures that allows the reconstruction of the basin of attraction of the background activity as a four dimensional geometric object. For the deterministic (noise-free) case, we show how the success of response to stimuli depends on the amplitude and phase of the SW cycle, in addition to the direction of the stimulus in state space. In the case of spontaneous noise-induced seizures, the basin becomes probabilistic introducing some degree of uncertainty to the stimulation outcome while maintaining qualitative features of the noise-free case. Additionally, due to the different time scales involved in SW generation, there is substantial variation between SW cycles, implying that there may not be a fixed set of optimal stimulation parameters for SW seizures. In contrast, the model suggests an adaptive approach to find optimal stimulation parameters patient-specifically, based on real-time estimation of the position in state space. We discuss how the modelling work can be exploited to rationally design a successful stimulation protocol for the abatement of SW seizures using real-time SW detection.

The threat of nuclear winter from a regional nuclear war is an existential hazard that must be actively addressed by policy makers to ensure the shared future of humanity. Here a cross-cultural analysis of 20 societies that experienced the Late Antique Little Ice Age (ca. 536-556CE) is performed in the hope of providing security policy makers with an empirical example of social resilience mechanisms. The climatic conditions of the Late Antique Little Ice Age are strikingly similar to those modelled as resulting from a regional nuclear war employing low-yield nuclear weapons, and thus provides a context in which mechanisms of resilience to nuclear winter might be empirically identified. It is argued that broad political participation fostering bridging ties between communities, agencies, and organisations was a key element of social resilience to the Late Antique Little Ice Age, and may indicate a means to foster resilience to nuclear winter today.

Recent experimental and clinical studies have provided diverse insight into the mechanisms of human focal seizure initiation and propagation. Often these findings exist at different scales of observation, and are not reconciled into a common understanding. Here we develop a new, multiscale mathematical model of cortical electric activity with realistic mesoscopic connectivity. Relating the model dynamics to experimental and clinical findings leads us to propose three classes of dynamical mechanisms for the onset of focal seizures in a unified framework. These three classes are: (i) globally induced focal seizures; (ii) globally supported focal seizures; (iii) locally induced focal seizures. Using model simulations we illustrate these onset mechanisms and show how the three classes can be distinguished. Specifically, we find that although all focal seizures typically appear to arise from localised tissue, the mechanisms of onset could be due to either localised processes or processes on a larger spatial scale. We conclude that although focal seizures might have different patient-specific aetiologies and electrographic signatures, our model suggests that dynamically they can still be classified in a clinically useful way. Additionally, this novel classification according to the dynamical mechanisms is able to resolve some of the previously conflicting experimental and clinical findings.

Epilepsy is a neurological disorder in which patients have recurrent seizures. Seizures occur in conjunction with abnormal electrical brain activity which can be recorded by the electroencephalogram (EEG). Often, this abnormal brain activity consists of high amplitude regular spike-wave oscillations as opposed to low amplitude irregular oscillations in the non-seizure state. Active brain stimulation has been proposed as a method to terminate seizures prematurely, however, a general and widely-applicable approach to optimal stimulation protocols is still lacking. In this study we use a computational model of epileptic spike-wave dynamics to evaluate the effectiveness of a pseudospectral method to simulated seizure abatement. We incorporate brain connectivity derived from magnetic resonance imaging of a subject with idiopathic generalized epilepsy. We find that the pseudospectral method can successfully generate time-varying stimuli that abate simulated seizures, even when including heterogeneous patient specific brain connectivity. The strength of the stimulus required varies in different brain areas. Our results suggest that seizure abatement, modeled as an optimal control problem and solved with the pseudospectral method, offers an attractive approach to treatment for in vivo stimulation techniques. Further, if optimal brain stimulation protocols are to be experimentally successful, then the heterogeneity of cortical connectivity should be accounted for in the development of those protocols and thus more spatially localized solutions may be preferable.

Multi-functional urban green infrastructure (GI) can deliver nature-based solutions that help address climate change, while providing wider benefits for human health and biodiversity. However, this will only be achieved effectively, sustainably and equitably if GI is carefully planned, implemented and maintained to a high standard, in partnership with stakeholders. This paper draws on original research into the design of a menu of GI standards for England, commissioned by Natural England—a United Kingdom Government agency. It describes the evolution of the standards within the context of United Kingdom government policy initiatives for nature and climate. We show how existing standards and guidelines were curated into a comprehensive framework consisting of a Core Menu and five Headline Standards. This moved beyond simplistic metrics such as total green space, to deliver GI that meets five key ‘descriptive principles’: accessible , connected , locally distinctive , multi-functional and varied, and thus delivers 5 ‘benefits principles’: places that are nature rich and beautiful, active and healthy, thriving and prosperous, resilient and climate positive, and with improved water management . It also builds in process guidance, bringing together stakeholders to co-ordinate GI development strategically across different sectors. Drawing on stakeholder feedback, we evaluate the strengths and weaknesses of the standards and discuss how they provide clarity and consistency while balancing tensions between top-down targets and the need for flexibility to meet local needs. A crucial factor is the delivery of the standards within a framework of supporting tools, advice and guidance, to help planners with limited resources deliver more effective and robust green infrastructure with multiple benefits.

The cerebral cortex displays a bewildering diversity of shapes and sizes across and within species. Despite this diversity, we present a universal multi-scale description of primate cortices. We show that all cortical shapes can be described as a set of nested folds of different sizes. As neighbouring folds are gradually merged, the cortices of 11 primate species follow a common scale-free morphometric trajectory, that also overlaps with over 70 other mammalian species. Our results indicate that all cerebral cortices are approximations of the same archetypal fractal shape with a fractal dimension of d f = 2.5. Importantly, this new understanding enables a more precise quantification of brain morphology as a function of scale. To demonstrate the importance of this new understanding, we show a scale-dependent effect of ageing on brain morphology. We observe a more than fourfold increase in effect size (from two standard deviations to eight standard deviations) at a spatial scale of approximately 2 mm compared to standard morphological analyses. Our new understanding may, therefore, generate superior biomarkers for a range of conditions in the future. Many of the brain’s essential functions – from decision-making to movement – take place in its outer layer known as the cerebral cortex. The shape of the cerebral cortex varies significantly between species. For instance, in humans, it is folded in to grooves and ridges, whereas in other animals, including mice, it is completely smooth. The structure of the cortex can also differ within a species, and be altered by aging and certain diseases. This vast variation can make it difficult it to characterize and compare the structure of the cortex between different species, ages and diseases. To address this, Wang et al. developed a new mathematical model for describing the shape of the cortex. The model uses a method known as coarse graining to erase, or ‘melt away’, any cortical folds or structures smaller than a given threshold size. As this threshold increases, the cortex becomes progressively smoother. The relationship between surface areas and threshold sizes indicates the fractal dimension – that is, how fragmented the cortex is across different scales. Wang et al. applied their model to the brain scans of eleven primates, including humans, and found the fractal dimension of the cortex was almost exactly 2.5 for all eleven species . This suggests that the cortices of the different primates follow a single fractal shape, which means the folds of each cortex have a similar branching pattern. Although there were distinctions between the species, they were mainly due to the different ranges of fold sizes in each cortex. The model revealed that the broader the range of fold sizes, the more folded the brain – but the fractal pattern remains the same. The brain melting method created by Wang et al. provides a new way to characterise cortical shape. Besides revealing a hitherto hidden regularity of nature, they hope that in the future their new method will be useful in assessing brain changes during human development and ageing, and in diseases like Alzheimer’s and epilepsy.

Almost all epidemic models make the assumption that infection is driven by the interaction between pairs of individuals, one of whom is infectious and the other of whom is susceptible. However, in society individuals mix in groups of varying sizes, at varying times, allowing one or more infectives to be in close contact with one or more susceptible individuals at a given point in time. In this paper we study the effect of mixing groups beyond pairs on the transmission of an infectious disease in an SIR (susceptible → infective → recovered) model, both through a branching process approximation for the initial stages of an epidemic with few initial infectives and a functional central limit theorem for the trajectories of the numbers of infectives and susceptibles over time for epidemics with many initial infectives. We also derive central limit theorems for the final size of (i) an epidemic with many initial infectives and (ii) a major outbreak triggered by few initial infectives. We show that, for a given basic reproduction number R0, the distribution of the size of mixing groups has a significant impact on the probability and final size of a major epidemic outbreak. Moreover, the standard pair-based homogeneously mixing epidemic model is shown to represent the worst case scenario, with both the highest probability and the largest final size of a major epidemic.