Efficient detection of periodic orbits in high dimensional systems

This paper is concerned with developing a method for detecting unstable periodic orbits (UPOs) by stabilising transformations. Here the strategy is to transform the system of interest in such away that the orbits become stable. However, the number of such transformations becomes overwhelming as we move to higher dimensions [5, 16, 17]. We have recently proposed a set of stabilising transformations which is constructed from a small set of already found UPOs [1]. The real value of the set is that its cardinality depends on the dimension of the unstable manifold at the UPO rather than the dimension of the system. Here we extend this approach to high dimensional systems of ODEs and apply it to the model example of a chaotic spatially extended system - the Kuramoto-Sivashinsky equation

Network motif frequency vectors reveal evolving metabolic network organisation

At the systems level many organisms of interest may be described by their patterns of interaction, and as such, are perhaps best characterised via network or graph models. Metabolic networks, in particular, are fundamental to the proper functioning of many important biological processes, and thus, have been widely studied over the past decade or so. Such investigations have revealed a number of shared topological features, such as a short characteristic path-length, large clustering coefficient and hierarchical modular structure. However, the extent to which evolutionary and functional properties of metabolism manifest via this under- lying network architecture remains unclear. In this paper, we employ a novel graph embedding technique, based upon low-order network motifs, to compare metabolic network structure for 383 bacterial species categorised according to a number of biological features. In particular, we introduce a new global significance score which enables us to quantify important evolutionary relationships that exist between organisms and their physical environments. Using this new approach, we demonstrate a number of significant correlations between environmental factors, such as growth conditions and habitat variability, and network motif structure, providing evidence that organism adaptability leads to increased complexities in the resultant metabolic network

Hypergraph models of metabolism

In this paper, we employ a directed hypergraph model to investigate the extent to which environmental variability influences the set of available biochemical reactions within a living cell. Such an approach avoids the limitations of the usual complex network formalism by allowing for the multilateral relationships (i.e. connections involving more than two nodes) that naturally occur within many biological processes. More specifically, we extend the concept of network reciprocity to complex hyper-networks, thus enabling us to characterise a network in terms of the existence of mutual hyper-connections, which may be considered a proxy for metabolic network complexity. To demonstrate these ideas, we study 115 metabolic hyper-networks of bacteria, each of which can be classified into one of 6 increasingly varied habitats. In particular, we found that reciprocity increases significantly with increased environmental variability, supporting the view that organism adaptability leads to increased complexities in the resultant biochemical networks

The topology of connections between rat prefrontal and temporal cortices

Understanding the structural organization of the prefrontal cortex (PFC) is an important step toward determining its functional organization. Here we investigated the organization of PFC using different neuronal tracers. We injected retrograde (Fluoro-Gold, 100 nl) and anterograde [Biotinylated dextran amine (BDA) or Fluoro-Ruby, 100 nl] tracers into sites within PFC subdivisions (prelimbic, ventral orbital, ventrolateral orbital, dorsolateral orbital) along a coronal axis within PFC. At each injection site one injection was made of the anterograde tracer and one injection was made of the retrograde tracer. The projection locations of retrogradely labeled neurons and anterogradely labeled axon terminals were then analyzed in the temporal cortex: area Te, entorhinal and perirhinal cortex. We found evidence for an ordering of both the anterograde (anterior-posterior, dorsal-ventral, and medial-lateral axes: p < 0.001) and retrograde (anterior-posterior, dorsal-ventral, and medial-lateral axes: p < 0.001) connections of PFC. We observed that anterograde and retrograde labeling in ipsilateral temporal cortex (i.e., PFC inputs and outputs) often occurred reciprocally (i.e., the same brain region, such as area 35d in perirhinal cortex, contained anterograde and retrograde labeling). However, often the same specific columnar temporal cortex regions contained only either labeling of retrograde or anterograde tracer, indicating that PFC inputs and outputs are frequently non-matched

Tracking vibrational energy on curved shell structures of variable thickness in the mid-to-high frequency - a ray tracing approach

Modelling the vibro-acoustic properties of mechanical built-up structures is a challenging task. Commonly employed techniques, such as finite element methods, are robust only in the low frequency regime. Recently, Discrete Flow Mapping has been forwarded as a cost efficient alternative method for mid- to high-frequency vibro-acoustic modelling. Discrete Flow Mapping employs local ray tracing approximations, providing a good model of the ray dynamics in homogeneous, isotropic flat plates or on curved shells in the geodesic high-frequency limit. However, in the mid-frequency case when the wavelength approaches the shell’s local radius of curvature, the resulting ray dynamics depend on the curvature in a non-trivial way. In this work, we consider ray-tracing approaches for modelling vibrational energy transport in curved shells of variable thickness at mid-to-high frequencies. In particular, we analyse mid-frequency effects on the dispersion curves for curved shells of variable thickness, and identify novel reflection/transmission behaviour

Complexity and robustness in hypernetwork models of metabolism

Metabolic reaction data is commonly modelled using a complex network approach, whereby nodes represent the chemical species present within the organism of interest, and connections are formed between those nodes participating in the same chemical reaction. Unfortunately, such an approach provides an inadequate description of the metabolic process in general, as a typical chemical reaction will involve more than two nodes, thus risking over-simplification of the the system of interest in a potentially significant way. In this paper, we employ a complex hypernetwork formalism to investigate the robustness of bacterial metabolic hypernetworks by extending the concept of a percolation process to hypernetworks. Importantly, this provides a novel method for determining the robustness of these systems and thus for quantifying their resilience to random attacks/errors. Moreover, we performed a site percolation analysis on a large cohort of bacterial metabolic networks and found that hypernetworks that evolved in more variable enviro nments displayed increased levels of robustness and topological complexity

Can linear collocation ever beat quadratic?

Computational approaches are becoming increasingly important in neuroscience, where complex, nonlinear systems modelling neural activity across multiple spatial and temporal scales are the norm. This paper considers collocation techniques for solving neural field models, which typically take the form of a partial integro-dfferential equation. In particular, we investigate and compare the convergence properties of linear and quadratic collocation on both regular grids and more general meshes not fixed to the regular Cartesian grid points. For regular grids we perform a comparative analysis against more standard techniques, in which the convolution integral is computed either by using Fourier based methods or via the trapezoidal rule. Perhaps surprisingly, we find that on regular, periodic meshes, linear collocation displays better convergence properties than quadratic collocation, and is in fact comparable with the spectral convergence displayed by both the Fourier based and trapezoidal techniques. However, for more general meshes we obtain superior convergence of the convolution integral using higher order methods, as expected

A numerical simulation of neural fields on curved geometries

Despite the highly convoluted nature of the human brain, neural field models typically treat the cortex as a planar two-dimensional sheet of neurons. Here, we present an approach for solving neural field equations on surfaces more akin to the cortical geometries typically obtained from neuroimaging data. Our approach involves solving the integral form of the partial integro-differential equation directly using collocation techniques alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods, we study localised activity patterns in a two-dimensional neural field equation posed on a periodic square domain, the curved surface of a torus, and the cortical surface of a rat brain, the latter of which is constructed using neuroimaging data. Our results are twofold: Firstly, we find that collocation techniques are able to replicate solutions obtained using more standard Fourier based methods on a flat, periodic domain, independent of the underlying mesh. This result is particularly significant given the highly irregular nature of the type of meshes derived from modern neuroimaging data. And secondly, by deploying efficient numerical schemes to compute geodesics, our approach is not only capable of modelling macroscopic pattern formation on realistic cortical geometries, but can also be extended to include cortical architectures of more physiological relevance. Importantly, such an approach provides a means by which to investigate the influence of cortical geometry upon the nucleation and propagation of spatially localised neural activity and beyond. It thus promises to provide model-based insights into disorders like epilepsy, or spreading depression, as well as healthy cognitive processes like working memory or attention

Mechanisms and points of control in the spread of inflammation: a mathematical investigation

Understanding the mechanisms that control the body’s response to inflammation is of key importance, due to its involvement in myriad medical conditions, including cancer, arthritis, Alzheimer’s disease and asthma. While resolving inflammation has historically been considered a passive process, since the turn of the century the hunt for novel therapeutic interventions has begun to focus upon active manipulation of constituent mechanisms, particularly involving the roles of apoptosing neutrophils, phagocytosing macrophages and anti-inflammatory mediators. Moreover, there is growing interest in how inflammatory damage can spread spatially due to the motility of inflammatory mediators and immune cells. For example, impaired neutrophil chemotaxis is implicated in causing chronic inflammation under trauma and in ageing, while neutrophil migration is an attractive therapeutic target in ailments such as chronic obstructive pulmonary disease. We extend an existing homogeneous model that captures interactions between inflammatory mediators, neutrophils and macrophages to incorporate spatial behaviour. Through bifurcation analysis and numerical simulation, we show that spatially inhomogeneous outcomes can present close to the switch from bistability to guaranteed resolution in the corresponding homogeneous model. Finally, we show how aberrant spatial mechanisms can play a role in the failure of inflammation to resolve and discuss our results within the broader context of seeking novel inflammatory treatments

Towards In Silico identification of genes contributing to similarity of patients’ multi-omics profiles: a case study of acute myeloid leukemia

We propose a computational framework for selecting biologically plausible genes identified by clustering of multi-omics data that reveal patients' similarity, thus giving researchers a more comprehensive view on any given disease. We employ spectral clustering of a similarity network created by fusion of three similarity networks, based on mRNA expression of immune genes, miRNA expression and DNA methylation data, using SNF_v2.1 software. For each cluster, we rank multi-omics features, ensuring the best separation between clusters, and select the top-ranked features that preserve clustering. To find genes targeted by DNA methylation and miRNAs found in the top-ranked features, we use chromosome-conformation capture data and miRNet 2.0 software, respectively. To identify informative genes, these combined sets of target genes are analyzed in terms of their enrichment in somatic/germline mutations, GO biological processes/pathways terms and known sets of genes considered to be important in relation to a given disease, as recorded in the Molecular Signature Database from GSEA. The protein-protein interaction (PPI) networks were analyzed to identify genes that are hubs of PPI networks. We used data recorded in The Cancer Genome Atlas for patients with acute myeloid leukemia to demonstrate our approach, and discuss our findings in the context of results in the literature