We propose a high-dimensional generalisation of the standard Klein bottle extending beyond those considered previously. We address the problem of generating continuous scalar fields (distributions) and dynamical systems (flows) on such state spaces, which can provide a rich source of examples for future investigations. We consider a class of high-dimensional dynamical systems that model distributed information processing within the human cortex, which may be capable of exhibiting some Klein bottle symmetries. We deploy topological data analytic methods in order to analyse their resulting dynamical behaviour and suggesting future challenges.

PurposeConsider an information processing algorithm that is designed to process an input data object onto an output data object via a number of successive internal {\it layers} and mappings between them. The possible activation state within each layer can be represented as a cube within Euclidean space of a high dimension (e.g. equal to the number of artificial neurons at that level). Multiple instances of such input objects produce a point cloud within each layer’s cube: this is the “representation of the reality” at that layer, as sampled by the set of input objects.Design/methodology/approachMost neural networks reduce the dimension of each layer’s cube from layer to successive layer. This gives the false impression of refining the inner representations of reality, distilling it down to fewer dimensions from which to discriminate or to infer outcomes (whatever is the aim). However, the representation of reality realised within each layer’s cube is a manifold, a curved subset embedded within it and of much lower dimension. Investigations show that such manifolds may not always be reducing in their local dimension. Instead, the manifold may become folded over and over, filling up further dimensions and creating non-realistic (unforeseeable) proximities.FindingsWe discuss some of the likely consequences of these relatively unforeseen characteristics and, in particular, the possible vulnerability of such algorithms to non-realistic perturbations. We consider a possible response to this issue.Practical implicationsNew forms of calibration are necessary, using geometric/topological loss functions, as opposed to simple (variation-limiting) regularisation terms.Originality/valueWe apply persistent homology methods to understand how the images of the point cloud (representing the sampled reality) change as they pass from layer to layer.

Generative artificial intelligence (AI) refers to algorithms that create synthetic but realistic output. Diffusion models currently offer state of the art performance in generative AI for images. They also form a key component in more general tools, including text-to-image generators and large language models. Diffusion models work by adding noise to the available training data and then learning how to reverse the process. The reverse operation may then be applied to new random data in order to produce new outputs. We provide a brief introduction to diffusion models for applied mathematicians and statisticians. Our key aims are (a) to present illustrative computational examples, (b) to give a careful derivation of the underlying mathematical formulas involved, and (c) to draw a connection with partial differential equation (PDE) diffusion models. We provide code for the computational experiments. We hope that this topic will be of interest to advanced undergraduate students and postgraduate students. Portions of the material may also provide useful motivational examples for those who teach courses in stochastic processes, inference, machine learning, PDEs or scientific computing.

We provide a constructive response to the recent National Risk Register 2023, highlighting a number of generic issues. These primarily include reflections on the deployed methodology, and its possible biases. We suggest that the National Risk Register should be extended to adopt a whole systems perspective, reasoning that it is unlikely that the acute risks will each occur in an isolated way, and additionally there may be (non-risk) precursor contexts and events which might amplify many acute risks and/or nullify planned responses. We discuss some potential causes of a lack of national resilience; and some possible generic failures in acute risk response. We make a number of recommendations for consideration by HM Government.

This report investigates the modular organisation of the Central region in the Drosophila connectome. We identify groups of neurones amongst which information circulates rapidly before spreading to the rest of the network using Infomap. We find that information flows along pathways linking distant neurones, forming modules that span across the brain. Remarkably, these modules, derived solely from neuronal connectivity patterns, exhibit a striking left-right symmetry in their spatial distribution as well as in their connections. We also identify a hierarchical structure at the coarse-grained scale of these modules, demonstrating the directional nature of information flow in the system.

What is the dimension of a network? Here, we view it as the smallest dimension of Euclidean space into which nodes can be embedded so that pairwise distances accurately reflect the connectivity structure. We show that a recently proposed and extremely efficient algorithm for data clouds, based on computing first- and second-nearest neighbour distances, can be used as the basis of an approach for estimating the dimension of a network with weighted edges. We also show how the algorithm can be extended to unweighted networks when combined with spectral embedding. We illustrate the advantages of this technique over the widely used approach of characterizing dimension by visually searching for a suitable gap in the spectrum of the Laplacian.

We contrast three distinct mathematical approaches to the hard problem of consciousness: quantum consciousness, integrated information theory, and the very large scale dynamical systems simulation of a network of networks. We highlight their features and their associated hypotheses, and we discuss how they are aligned or in conflict. We suggest some challenges for the future theories, in considering how they might apply to the human brain as it develops both cognitive and conscious sophistication, from infancy to adulthood; and how an evolutionary perspective challenges the distinct approaches to aver performance advantages and physiological surrogates for consciousness.

We consider a large cloud of vectors formed at each layer of a standard neural network, corresponding to a large number of separate inputs which were presented independently to the classifier. Although the embedding dimension (the total possible degrees of freedom) reduces as we pass through successive layers, from input to output, the actual dimensionality of the point clouds that the layers contain does not necessarily reduce. We argue that this phenomenon may result in a vulnerability to (universal) adversarial attacks (which are small specific perturbations). This analysis requires us to estimate the intrinsic dimension of point clouds (with values between 20 and 200) within embedding spaces of dimension 1000 up to 800,000. This needs some care. If the cloud dimension actually increases from one layer to the next it implies there is some ‘volume filling’ over-folding, and thus there exist possible small directional perturbations in the latter space that are equivalent to shifting large distances within the former space, thus inviting possibility of universal and imperceptible attacks.

We argue that cognition (information processing) and internal phenomenological sensations, including emotions, are intimately related and are not separable. We aver that phenomenological sensations are dynamical "modes" of firing behaviour that (i) exist over time and over large parts of the cortex's neuron-to-neuron network and (ii) are consequences of the network-of-networks architecture, coupling the individual neuronal dynamics and the necessary time delay incurred by neuron-to-neuron transmission: if you possess those system properties, then you will have the dynamical modes and, thus, the phenomenological sensations. These modes are consequences of incoming external stimuli and are competitive within the system, suppressing and locking-out one another. On the other hand, the presence of any such mode acts as a preconditioner for the immediate (dynamic) cognitive processing of information. Thus, internal phenomenological sensations, including emotions, reduce the immediate decision set (of feasible interpretations) and hence the cognitive load. For organisms with such a mental inner life, there would clearly be a large cognitive evolutionary advantage, resulting in the well-known "thinking fast, thinking slow" phenomena. We call this the entwinement hypothesis: how latent conscious phenomena arise from the dynamics of the cognitive processing load, and how these precondition the cognitive tasks immediately following. We discuss how internal dynamical modes, which are candidates for emotions down to single qualia, can be observed by reverse engineering large sets of simulations of system's stimulated responses, either using vast supercomputers (with full 10B neuronal network analyses) or else using laptops to do the same for appropriately generalised Kuramoto models (networks of k-dimensional clocks, each representing the 10,000 neurons within a single neural column). We explain why such simplifications are appropriate. We also discuss the consequent cognitive advantages for information-processing systems exhibiting internal sensations and the exciting implications for next-generation (non-binary) computation and for AI.

There is a growing interest in novelty search : that is, in sampling a parameter space to search for radical or unexpected behaviour(s), occurring as a consequence of parameter choice, being input to some downstream complex system, process, or service that will not yield to analysis, without imposing any specific pre-ordained objective function, or fitness function to be optimised. We mean “parameter” in the widest sense, including system learnables, non-autonomous forcing, sequencing and all inputs.Depending upon the nature of the underlying parameter space of interest one may adopt a rather wide range of search algorithms. We do consider that this search activity has meta-objectives , though: one is of achieving diversity (efficiently reaching out across the space in some way); and one is of achieving some minimum density (not leaving out large unexplored holes). These are in tension. In general, the computational costs of both of these qualities become restrictive as the di- mension of the parameter spaces increase; and consequently their balance is harder to maintain. We may also wish for a substantial random element of search to provide some luck in discovery and to avoid any naive preset sampling patterns.We consider archive-based methods within a range of spaces: finite discrete spaces, where the problem is straightforward (provided we are patient with the random element); Euclidean spaces, of increasing dimension, that become very lonely places; and infinite dimensional spaces. Our aim is to discuss a raft of distinctive search concepts, that respond to identified challenges, and rely on a rather diverse range of mathematical ideas. This arms practitioners with a range of highly practical methods.However applications requiring novelty search arise, one should avoid rushing to code-up a standard evolving search algorithm and instead give some thought to the nature and requirements of the search: there is a range of effective options available. We give some considered advice.

We consider undirected graphs that grow through the successive combination of component sub-graphs. For any well-behaved functions defined for such graphs, taking values in a Banach space, we show that there must exist a scaling law applicable when successive copies of the same component graph are combined. Crucially, we extend the approach introduced in previous work to the successive combination of component random sub-graphs. We illustrate this by generalizing the preferential attachment operation for the combination of stochastic block models. We discuss a further wide range of random graph combination operators to which this theory now applies, indicating the ubiquity of growth scaling laws (and asymptotic decay scaling laws) within applications, where the modules are quite distinct, yet may be considered as instances drawn from the same random graph. This is a type of statistically self-similar growth process, as opposed to a deterministic growth process incorporating exact copies of the same motif, and it represents a natural, partially random, growth processes for graphs observed in the analysis of social and technology contexts.

The instant messaging platform Telegram has become popular among the far-right movements in the US and UK in recent years. These groups use public Telegram channels and group chats to disseminate hate speech, disinformation, and conspiracy theories. Recent works revealed that the far-right Telegram network structure is decentralized and formed of several communities divided mostly along ideological and national lines. Here, we investigated the UK far-right network on Telegram and are interested in understanding the different roles of different channels and their influence relations. We apply a community detection method, based on the clustering of a flow of random walkers, that allows us to uncover the organization of the Telegram network in communities with different roles. We find three types of communities: (1) upstream communities contain mostly group chats that comment on content from channels in the rest of the network; (2) core communities contain broadcast channels tightly connected to each other and can be seen as forming echo chambers; (3) downstream communities contain popular channels that are highly referenced by other channels. We find that the network is composed of two main sub-networks: one containing mainly channels related to the English-speaking far-right movements and one with channels in Russian. We analyze the dynamics of the different communities and the most shared external links in the different types of communities over a period going from 2015 to 2020. We find that different types of communities have different dynamics and share links to different types of websites. We finish by discussing several directions for further work.

We describe our approach to the Western Power Distribution (WPD) Presumed Open Data (POD) 6 MWh battery storage capacity forecasting competition, in which we finished second. The competition entails two distinct forecasting aims to maximise the daily evening peak reduction and using as much solar photovoltaic energy as possible. For the latter, we combine a Bayesian (MCMC) linear regression model with an average generation distribution. For the former, we introduce a new error metric that allows even a simple weighted average combined with a simple linear regression model to score very well using the competition performance metric.

We consider cortex-like complex systems in the form of strongly connected, directed networks-of-networks . In such a network, there are spiking dynamics at each of the nodes (modelling neurones), together with non-trivial time-lags associated with each of the directed edges (modelling synapses). The connections of the outer network are sparse, while the many inner networks, called modules, are dense. These systems may process various incoming stimulations by producing whole-system dynamical responses. We specifically discuss a generic class of systems with up to 10 billion nodes simulating the human cerebral cortex. It has recently been argued that such a system’s responses to a wide range of stimulations may be classified into a number of latent, internal dynamical modes . The modes might be interpreted as focussing and biasing the system’s short-term dynamical system responses to any further stimuli. In this work, we illustrate how latent modes may be shown to be both present and significant within very large-scale simulations for a wide and appropriate class of complex systems. We argue that they may explain the inner experience of the human brain.

The European Space Agency (ESA) and Roscosmos ExoMars mission will launch the “Rosalind Franklin” rover in 2022 for a landing on Mars in 2023.The goals of the mission are to search for signs of past and present life on Mars, investigate the water/geochemical environment as a function of depth in the shallow subsurface, and characterize the surface environment. To meet these scientific objectives while minimizing the risk for landing, a 5-year-long landing site selection process was conducted by ESA, during which eight candidate sites were down selected to one: Oxia Planum. Oxia Planum is a 200 km-wide low-relief terrain characterized by hydrous clay-bearing bedrock units located at the southwest margin of Arabia Terra. This region exhibits Noachian-aged terrains. We show in this study that the selected landing site has recorded at least two distinct aqueous environments, both of which occurred during the Noachian: (1) a first phase that led to the deposition and alteration of ∼100 m of layered clay-rich deposits and (2) a second phase of a fluviodeltaic system that postdates the widespread clay-rich layered unit. Rounded isolated buttes that overlie the clay-bearing unit may also be related to aqueous processes. Our study also details the formation of an unaltered mafic-rich dark resistant unit likely of Amazonian age that caps the other units and possibly originated from volcanism. Oxia Planum shows evidence for intense erosion from morphology (inverted features) and crater statistics. Due to these erosional processes, two types of Noachian sedimentary rocks are currently exposed. We also expect rocks at the surface to have been exposed to cosmic bombardment only recently, minimizing organic matter damage.

Small-scale lobes (SSL) observed at the surface of Mars are thought to be the result of solifluction. Hence, their existence has important implications for our understanding of recent climate history, the distribution of thaw liquids and its geomorphic effects, as water is believed not to have been recently stable at the martian surface. These small-scale lobes are limited to sloping terrains and often occur alongside polygonal patterned ground attesting to the presence of ice in the environment. In this study, we perform a comparative 3D morphometric analysis of the lobes on Mars with terrestrial analogues. We use data from High Resolution Imaging Science Experiment (HiRISE) camera in order to create Digital Terrain Models (DTMs) on Mars at 1 m/pix. For the terrestrial analogues, we use DTMs and orthophotos from five periglacial environments with solifluction lobes and from one site with lobate features found in the Atacama Desert. The lobate features in the Atacama Desert are found within features called "zebra stripes", which are apparently unique to the Atacama and are now generally accepted to be the result of seismic shaking. We show that SSL on Mars overlap morphometrically with terrestrial solifluction lobes, and are less similar to zebra stripe lobes. The majority of martian SSL occur below the angle of repose which demonstrates that a dry fall mechanism cannot be responsible for their formation and that lubrication of the subsurface is required. We found that martian SSL only occur on slopes greater than similar to 10 degrees, a constraint not observed for terrestrial solifluction lobes nor zebra stripe lobes in Atacama. We hypothesise that this apparent constraint does not result from the lowered gravity on Mars, but instead from differing soil properties and/or thermal regime between Mars and Earth which affects the slope at which creep can initiate. We cannot rule out a role of CO2 sublimation in lobe-formation based on our terrestrial measurements.

Abstract Aeolian systems are active across much of the surface of Mars and quantifying the activity of bedforms is important for understanding the modern and recent Martian environment. Recently, the migration rates and sand fluxes of dunes and ripples have been precisely measured using repeat High Resolution Imaging Science Experiment (HiRISE) images. However, the limited areal extent of HiRISE coverage means that only a small area can be targeted for repeat coverage. Context Camera (CTX) images, although lower in spatial resolution, have wider spatial coverage, meaning that dune migration can potentially be monitored over larger areas. We used time series, coregistered CTX images and digital elevation models to measure dune migration rates and sand fluxes at six sites: Nili Patera, Meroe Patera, two sites at Herschel crater, McLaughlin crater, and Hellespontus Montes. We observed dune displacement in the CTX images over long‐term baselines (7.5–11 Earth years; 4–6 Mars years). Bedform activity has previously been measured at all these sites using HiRISE, which we used to validate our results. Our dune migration rates (0.2–1.1 m/EY) and sand fluxes (2.4–11.6 m3 m−1 EY−1) compare well to measurements made with HiRISE. The use of CTX in monitoring dune migration has advantages (wider spatial coverage, faster processing time) and disadvantages (ripples not resolved, digital elevation model dune heights may be underestimates); the future combined use of HiRISE and CTX is likely to be beneficial.

To understand Martian palaeoclimatic conditions and the role of volatiles therein, the spatiotemporal evolution of gullies must be deciphered. While the spatial distribution of gullies has been extensively studied, their temporal evolution is poorly understood. We show that gully size is similar in very young and old craters. Gullies on the walls of very young impact craters (less than a few myr) typically cut into bedrock and are free of latitude-dependent mantle (LDM) and glacial deposits, while such deposits become increasingly evident in older craters. These observations suggest that gullies go through obliquity-driven degradation-accumulation cycles over time, controlled by: (1) LDM emplacement and degradation; and (2) glacial emplacement and removal. In glacially-influenced craters, the distribution of gullies on crater walls coincides with the extent of glacial deposits, which suggests that the melting of snow and ice played a role in the formation of these gullies. Yet, present-day activity is observed in some gullies on formerly glaciated crater walls. Moreover, in very young craters, extensive gullies have formed in the absence of LDM and glacial deposits, showing that gully formation can also be unrelated to these deposits. The Martian climate varied substantially over time, and the gully-forming mechanisms are likely to have varied accordingly.

One of the most contentious hypotheses in the geological history of Mars is whether the northern lowlands ever contained an oceanic water body. Arguably, the best evidence for an ocean comes from the presence of sedimentary fans around Mars' dichotomy boundary, which separates the northern lowlands from the southern highlands. Here we describe the palaeogeomorphology of the Hypanis Valles sediment fan, the largest sediment fan complex reported on Mars (area >970 km(2)). This has an extensive catchment (4.6 x 10(5) km(2)) incorporating Hypanis and Nanedi Valles, that we show was active during the late-Noachian/early-Hesperian period (similar to 3.7 Ga). The fan comprises a series of lobe shaped sediment bodies, connected by multiple bifurcating flat-topped ridges. We interpret the latter as former fluvial channel belts now preserved in inverted relief. Meter-scale-thick, sub-horizontal layers that are continuous over tens of kilometres are visible in scarps and the inverted channel margins. The inverted channel branches and lobes are observed to occur up to at least 140 km from the outlet of Hypanis Valles and descend similar to 500 m in elevation. The progressive basinward advance of the channellobe transition records deposition and avulsion at the margin of a retreating standing body of water, assuming the elevation of the northern plains basin floor is stable. We interpret the Hypanis sediment fan to represent an ancient delta as opposed to a fluvial fan system. At its location at the dichotomy boundary, the Hypanis Valles fan system is topographically open to Chryse Planitia - an extensive plain that opens in turn into the larger northern lowlands basin. We conclude that the observed progradation of fan bodies was due to basinward shoreline retreat of an ancient body of water which extended across at least Chryse Planitia. Given the open topography, it is plausible that the Hypanis fan system records the existence, last highstand, and retreat of a large sea in Chryse Planitia and perhaps even an ocean that filled the northern plains of Mars. (C) 2018 The Authors. Published by Elsevier B.V.

Stepped fan deposits and phyllosilicate mineralogies are relatively common features on Mars but have not previously been found in association with each other. Both of these features are widely accepted to be the result of aqueous processes, but the assumed role and nature of any water varies. In this study we have investigated two stepped fan deposits in Coprates Catena, Mars, which have a genetic link to light toned material that is rich in Fe-Mg phyllosilicate phases. Although of different sizes and in separate, but adjacent, trough-like depressions, we identify similar features at these stepped fans and phyllosilicates that are indicative of similar formation conditions and processes. Our observations of the overall geomorphology, mineralogy and chronology of these features are consistent with a two stage formation process, whereby deposition in the troughs first occurs into shallow standing water or playas, forming fluvial or alluvial fans that terminate in delta deposits and interfinger with interpreted lacustrine facies, with a later period of deposition under sub-aerial conditions, forming alluvial fan deposits. We suggest that the distinctive stepped appearance of these fans is the result of aeolian erosion, and is not a primary depositional feature. This combined formation framework for stepped fans and phyllosilicates can also explain other similar features on Mars, and adds to the growing evidence of fluvial activity in the equatorial region of Mars during the Hesperian and Amazonian. (C) 2017 The Authors. Published by Elsevier Inc.