A binary state on a graph means an assignment of binary values to its vertices. A time dependent sequence of binary states is referred to as binary dynamics. We describe a method for the classification of binary dynamics of digraphs, using particular choices of closed neighbourhoods. Our motivation and application comes from neuroscience, where a directed graph is an abstraction of neurons and their connections, and where the simplification of large amounts of data is key to any computation. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. We consider existing and introduce new real-valued functions on closed neighbourhoods, comparing them by their ability to accurately classify different binary dynamics. We describe a classification algorithm that uses two parameters and sets up a machine learning pipeline. We demonstrate the effectiveness of the method on simulated activity on a digital reconstruction of cortical tissue of a rat, and on a non-biological random graph with similar density.

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Animals regulate their diet in order to maximise the expression of fitness traits that often have different nutritional needs. These nutritional trade-offs have been experimentally uncovered using the Geometric framework for nutrition (GF). However, current analytical methods to measure such responses rely on either visual inspection or complex models applied to multidimensional performance landscapes, making these approaches subjective, or conceptually difficult, computationally expensive, and in some cases inaccurate. This limits our ability to understand how animal nutrition evolved to support life-histories within and between species. Here, we introduce a simple trigonometric model to measure nutritional trade-offs in multidimensional landscapes (‘Nutrigonometry’). Nutrigonometry is both conceptually and computationally easier than current approaches, as it harnesses the trigonometric relationships of right-angle triangles instead of vector calculations. Using landmark GF datasets, we first show how polynomial (Bayesian) regressions can be used for precise and accurate predictions of peaks and valleys in performance landscapes, irrespective of the underlying structure of the data (i.e., individual food intakes vs fixed diet ratios). Using trigonometric relationships, we then identified the known nutritional trade-off between lifespan and reproductive rate both in terms of nutrient balance and concentration. Nutrigonometry enables a fast, reliable and reproducible quantification of nutritional trade-offs in multidimensional performance landscapes, thereby broadening the potential for future developments in comparative research on the evolution of animal nutrition.

Nutrition is one of the underlying factors necessary for the expression of life-histories and fitness across the tree of life. In recent decades, the geometric framework (GF) has become a powerful framework to obtain biological insights through the construction of multidimensional performance landscapes. However, to date, many properties of these multidimensional landscapes have remained inaccessible due to our lack of mathematical and statistical frameworks for GF analysis. This has limited our ability to understand, describe and estimate parameters which may contain useful biological information from GF multidimensional performance landscapes. Here, we propose a new model to investigate the curvature of GF multidimensional landscapes by calculating the parameters from differential geometry known as Gaussian and mean curvatures. We also estimate the surface area of multidimensional performance landscapes as a way to measure landscape deviations from flat. We applied the models to a landmark dataset in the field, where we also validate the assumptions required for the calculations of curvature. In particular, we showed that linear models perform as well as other models used in GF data, enabling landscapes to be approximated by quadratic polynomials. We then introduced the Hausdorff distance as a metric to compare the similarity of multidimensional landscapes.

Living organisms are limited in the range of values of ecological factors they can explore. This defines where animals exist (or could exist) and forms an ecological fingerprint that explains species’ distribution at global scales. Species’ ecological fingerprints can be represented as a n-dimensional hypervolume – known as Hutchinson’s niche hypervolume. This concept has enabled significant progress in our understanding of species’ ecological needs and distributions across environmental gradients. Nevertheless, the properties of Hutchinson’s n-dimensional hypervolumes can be challenging to calculate and several methods have been proposed to extract meaningful measurements of hypervolumes’ properties. One key property of hypervolumes are holes, which provide important information about the ecological occupancy of species. However, to date, current methods rely on volume estimates and set operations to identify holes in hypervolumes. Yet, this approach can be problematic because in high-dimensions, the volume of region enclosing a hole tends to zero. We propose the use of persistence homology (PH) to identify holes in hypervolumes and in ecological datasets more generally. PH allows for the estimates of topological properties in n-dimensional niche hypervolumes independent of the volume estimates of the hypervolume. We demonstrate the application of PH to canonical datasets and to the identification of holes in the hypervolumes of five vertebrate species with diverse niches, highlighting the potential benefits of this approach to gain further insights into animal ecology. Overall, our approach enables the study of a yet unexplored property of Hutchinson’s hypervolumes, and thus, have important implications to our understanding of animal ecology.

Hutchinsons niche hypervolume concept has enabled significant progress in our understanding of species ecological needs and distributions across environmental gradients. Nevertheless, the properties of Hutchinsons n-dimensional hypervolumes can be challenging to calculate and several methods have been proposed to extract meaningful measurements of hypervolumes properties (e.g., volume). One key property of hypervolumes are holes, which provide important information about the ecological occupancy of species. However, to date, current methods rely on volume estimates and set operations to identify holes in hypervolumes. Yet, this approach can be problematic because in high-dimensions, the volume of region enclosing a hole tends to zero. Here, we propose the use of the topological concept of persistence homology (PH) to identify holes in hypervolumes and in ecological datasets more generally. PH allows for the estimates of topological properties in n-dimensional niche hypervolumes and is independent of the volume estimates of the hypervolume. We demonstrate the application of PH to canonical datasets and to the identification of holes in the hypervolumes of five vertebrate species with diverse niches, highlighting the potential benefits of this approach to gain further insights into animal ecology. Overall, our approach enables the study of an yet unexplored property of Hutchinsons hypervolumes (i.e., holes), and thus, have important implications to our understanding of animal ecology.