Learning spatio-temporal patterns with Neural Cellular Automata.

Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and Partial Differential Equation (PDE) trajectories. Our method is designed to identify underlying local rules that govern large scale dynamic emergent behaviours. Previous work on NCA focuses on learning rules that give stationary emergent structures. We extend NCA to capture both transient and stable structures within the same system, as well as learning rules that capture the dynamics of Turing pattern formation in nonlinear PDEs. We demonstrate that NCA can generalise very well beyond their PDE training data, we show how to constrain NCA to respect given symmetries, and we explore the effects of associated hyperparameters on model performance and stability. Being able to learn arbitrary dynamics gives NCA great potential as a data driven modelling framework, especially for modelling biological pattern formation.

Coupled differentiation and division of embryonic stem cells inferred from clonal snapshots.

The deluge of single-cell data obtained by sequencing, imaging and epigenetic markers has led to an increasingly detailed description of cell state. However, it remains challenging to identify how cells transition between different states, in part because data are typically limited to snapshots in time. A prerequisite for inferring cell state transitions from such snapshots is to distinguish whether transitions are coupled to cell divisions. To address this, we present two minimal branching process models of cell division and differentiation in a well-mixed population. These models describe dynamics where differentiation and division are coupled or uncoupled. For each model, we derive analytic expressions for each subpopulation's mean and variance and for the likelihood, allowing exact Bayesian parameter inference and model selection in the idealised case of fully observed trajectories of differentiation and division events. In the case of snapshots, we present a sample path algorithm and use this to predict optimal temporal spacing of measurements for experimental design. We then apply this methodology to an in vitro dataset assaying the clonal growth of epiblast stem cells in culture conditions promoting self-renewal or differentiation. Here, the larger number of cell states necessitates approximate Bayesian computation. For both culture conditions, our inference supports the model where cell state transitions are coupled to division. For culture conditions promoting differentiation, our analysis indicates a possible shift in dynamics, with these processes becoming more coupled over time.

Assembly of microbial communities in replicate nutrient-cycling model ecosystems follows divergent trajectories, leading to alternate stable states.

We studied in detail the reproducibility of community development in replicate nutrient-cycling microbial microcosms that were set up identically and allowed to develop under the same environmental conditions. Multiple replicate closed microcosms were constructed using pond sediment and water, enriched with cellulose and sulphate, and allowed to develop over several months under constant environmental conditions, after which their microbial communities were characterized using 16S rRNA gene sequencing. Our results show that initially similar microbial communities can follow alternative - yet stable - trajectories, diverging in time in a system size-dependent manner. The divergence between replicate communities increased in time and decreased with larger system size. In particular, notable differences emerged in the heterotrophic degrader communities in our microcosms; one group of steady state communities was enriched with Firmicutes, while the other was enriched with Bacteroidetes. The communities dominated by these two phyla also contained distinct populations of sulphate-reducing bacteria. This biomodality in community composition appeared to arise during recovery from a low-diversity state that followed initial cellulose degradation and sulphate reduction.

Spatiotemporally complete condensation in a non-Poissonian exclusion process.

We investigate a non-Poissonian version of the asymmetric simple exclusion process, motivated by the observation that coarse graining the interactions between particles in complex systems generically leads to a stochastic process with a non-Markovian (history-dependent) character. We characterize a large family of one-dimensional hopping processes using a waiting-time distribution for individual particle hops. We find that when its variance is infinite, a real-space condensate forms that is complete in space (involves all particles) and time (exists at almost any given instant) in the thermodynamic limit. The mechanism for the onset and stability of the condensate is rather subtle and depends on the microscopic dynamics subsequent to a failed particle hop attempt.

Stochastic dynamics of lexicon learning in an uncertain and nonuniform world.

We study the time taken by a language learner to correctly identify the meaning of all words in a lexicon under conditions where many plausible meanings can be inferred whenever a word is uttered. We show that the most basic form of cross-situational learning--whereby information from multiple episodes is combined to eliminate incorrect meanings--can perform badly when words are learned independently and meanings are drawn from a nonuniform distribution. If learners further assume that no two words share a common meaning, we find a phase transition between a maximally efficient learning regime, where the learning time is reduced to the shortest it can possibly be, and a partially efficient regime where incorrect candidate meanings for words persist at late times. We obtain exact results for the word-learning process through an equivalence to a statistical mechanical problem of enumerating loops in the space of word-meaning mappings.

Self-contained Beta-with-Spikes approximation for inference under a Wright-Fisher model.

We construct a reliable estimation method for evolutionary parameters within the Wright-Fisher model, which describes changes in allele frequencies due to selection and genetic drift, from time-series data. Such data exist for biological populations, for example via artificial evolution experiments, and for the cultural evolution of behavior, such as linguistic corpora that document historical usage of different words with similar meanings. Our method of analysis builds on a Beta-with-Spikes approximation to the distribution of allele frequencies predicted by the Wright-Fisher model. We introduce a self-contained scheme for estimating parameters in the approximation, and demonstrate its robustness with synthetic data, especially in the strong-selection and near-extinction regimes where previous approaches fail. We further apply the method to allele frequency data for baker's yeast (Saccharomyces cerevisiae), finding a significant signal of selection in cases where independent evidence supports such a conclusion. We further demonstrate the possibility of detecting time points at which evolutionary parameters change in the context of a historical spelling reform in the Spanish language.

How do communication systems emerge?

Communication involves a pair of behaviours--a signal and a response--that are functionally interdependent. Consequently, the emergence of communication involves a chicken-and-egg problem: if signals and responses are dependent on one another, then how does such a relationship emerge in the first place? The empirical literature suggests two solutions to this problem: ritualization and sensory manipulation; and instances of ritualization appear to be more common. However, it is not clear from a theoretical perspective why this should be the case, nor if there are any other routes to communication. Here, we develop an analytical model to examine how communication can emerge. We show that: (i) a state of non-interaction is evolutionarily stable, and so communication will not necessarily emerge even when it is in both parties' interest; (ii) the conditions for sensory manipulation are more stringent than for ritualization, and hence ritualization is likely to be more common; and (iii) communication can arise by a third route, when the intention to communicate can itself be communicated, but this may be limited to humans. More generally, our results demonstrate the utility of a functional approach to communication.

How individuals change language.

Languages emerge and change over time at the population level though interactions between individual speakers. It is, however, hard to directly observe how a single speaker's linguistic innovation precipitates a population-wide change in the language, and many theoretical proposals exist. We introduce a very general mathematical model that encompasses a wide variety of individual-level linguistic behaviours and provides statistical predictions for the population-level changes that result from them. This model allows us to compare the likelihood of empirically-attested changes in definite and indefinite articles in multiple languages under different assumptions on the way in which individuals learn and use language. We find that accounts of language change that appeal primarily to errors in childhood language acquisition are very weakly supported by the historical data, whereas those that allow speakers to change incrementally across the lifespan are more plausible, particularly when combined with social network effects.

Conceptual Similarity and Communicative Need Shape Colexification: An Experimental Study.

Colexification refers to the phenomenon of multiple meanings sharing one word in a language. Cross-linguistic lexification patterns have been shown to be largely predictable, as similar concepts are often colexified. We test a recent claim that, beyond this general tendency, communicative needs play an important role in shaping colexification patterns. We approach this question by means of a series of human experiments, using an artificial language communication game paradigm. Our results across four experiments match the previous cross-linguistic findings: all other things being equal, speakers do prefer to colexify similar concepts. However, we also find evidence supporting the communicative need hypothesis: when faced with a frequent need to distinguish similar pairs of meanings, speakadjust their colexification preferences to maintain communicative efficiency and avoid colexifying those similar meanings which need to be distinguished in communication. This research provides further evidence to support the argument that languages are shaped by the needs and preferences of their speakers.

Solvable model of a many-filament Brownian ratchet.

We construct and exactly solve a model of an extended Brownian ratchet. The model comprises an arbitrary number of heterogeneous, growing and shrinking filaments which together move a rigid membrane by a ratchet mechanism. The model draws parallels with the dynamics of actin filament networks at the leading edge of the cell. In the model, the filaments grow and contract stochastically. The model also includes forces which derive from a potential dependent on the separation between the filaments and the membrane. These forces serve to attract the filaments to the membrane or generate a surface tension that prevents the filaments from dispersing. We derive an N-dimensional diffusion equation for the N filament-membrane separations, which allows the steady-state probability distribution function to be calculated exactly under certain conditions. These conditions are fulfilled by the physically relevant cases of linear and quadratic interaction potentials. The exact solution of the diffusion equation furnishes expressions for the average velocity of the membrane and critical system parameters for which the system stalls and has zero net velocity. In the case of a restoring force, the membrane velocity grows as the square root of the force constant, whereas it decreases once a surface tension is introduced.

Word learning under infinite uncertainty.

Language learners must learn the meanings of many thousands of words, despite those words occurring in complex environments in which infinitely many meanings might be inferred by the learner as a word's true meaning. This problem of infinite referential uncertainty is often attributed to Willard Van Orman Quine. We provide a mathematical formalisation of an ideal cross-situational learner attempting to learn under infinite referential uncertainty, and identify conditions under which word learning is possible. As Quine's intuitions suggest, learning under infinite uncertainty is in fact possible, provided that learners have some means of ranking candidate word meanings in terms of their plausibility; furthermore, our analysis shows that this ranking could in fact be exceedingly weak, implying that constraints which allow learners to infer the plausibility of candidate word meanings could themselves be weak. This approach lifts the burden of explanation from 'smart' word learning constraints in learners, and suggests a programme of research into weak, unreliable, probabilistic constraints on the inference of word meaning in real word learners.

Executive functions predict conceptual learning of science.

We examined the relationship between executive functions and both factual and conceptual learning of science, specifically chemistry, in early adolescence. Sixty-three pupils in their second year of secondary school (aged 12-13 years) participated. Pupils completed tasks of working memory (Spatial Working Memory), inhibition (Stop-Signal), attention set-shifting (ID/ED), and planning (Stockings of Cambridge), from the CANTAB. They also participated in a chemistry teaching session, practical, and assessment on the topic of acids and alkalis designed specifically for this study. Executive function data were related to (1) the chemistry assessment which included aspects of factual and conceptual learning and (2) a recent school science exam. Correlational analyses between executive functions and both the chemistry assessment and science grades revealed that science achievements were significantly correlated with working memory. Linear regression analysis revealed that visuospatial working memory ability was predictive of chemistry performance. Interestingly, this relationship was observed solely in relation to the conceptual learning condition of the assessment highlighting the role of executive functions in understanding and applying knowledge about what is learned within science teaching.

Formal solution of a class of reaction-diffusion models: reduction to a single-particle problem.

We consider the trapping reaction A+B-->B in space dimension d0 or B, in d=2.

Simulating the real origins of communication.

How communication systems emerge is a topic of relevance to several academic disciplines. Numerous existing models, both mathematical and computational, study this emergence. However, with few exceptions, these models all build some form of communication into their initial specification. Consequently, what these models study is how communication systems transition from one form to another, and not how communication itself emerges in the first place. Here we present a new computational model of the emergence of communication which, unlike previous models, does not pre-specify the existence of communication. We conduct two experiments using this model, in order to derive general statements about how communication systems emerge. The two main routes to communication that we identify correspond with findings from the empirical literature on the evolution of animal signals. We use this finding to explain when and why we should expect communication to emerge in nature. We also compare our model to experimental research on the origins of human communication systems, and hence show that humans are an important exception to the general trends we observe. We argue that this is because humans, and probably only humans, are able to 'signal signalhood', i.e. to express communicative intentions.

Why is combinatorial communication rare in the natural world, and why is language an exception to this trend?

In a combinatorial communication system, some signals consist of the combinations of other signals. Such systems are more efficient than equivalent, non-combinatorial systems, yet despite this they are rare in nature. Why? Previous explanations have focused on the adaptive limits of combinatorial communication, or on its purported cognitive difficulties, but neither of these explains the full distribution of combinatorial communication in the natural world. Here, we present a nonlinear dynamical model of the emergence of combinatorial communication that, unlike previous models, considers how initially non-communicative behaviour evolves to take on a communicative function. We derive three basic principles about the emergence of combinatorial communication. We hence show that the interdependence of signals and responses places significant constraints on the historical pathways by which combinatorial signals might emerge, to the extent that anything other than the most simple form of combinatorial communication is extremely unlikely. We also argue that these constraints can be bypassed if individuals have the socio-cognitive capacity to engage in ostensive communication. Humans, but probably no other species, have this ability. This may explain why language, which is massively combinatorial, is such an extreme exception to nature's general trend for non-combinatorial communication.

Fast fixation with a generic network structure.

We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model), and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may covary, and that the results are robust to correlations in the network or the initial condition.

Learning times for large lexicons through cross-situational learning.

Cross-situational learning is a mechanism for learning the meaning of words across multiple exposures, despite exposure-by-exposure uncertainty as to a word's true meaning. Doubts have been expressed regarding the plausibility of cross-situational learning as a mechanism for learning human-scale lexicons in reasonable timescales under the levels of referential uncertainty likely to confront real word learners. We demonstrate mathematically that cross-situational learning facilitates the acquisition of large vocabularies despite significant levels of referential uncertainty at each exposure, and we provide estimates of lexicon learning times for several cross-situational learning strategies. This model suggests that cross-situational word learning cannot be ruled out on the basis that it predicts unreasonably long lexicon learning times. More generally, these results indicate that there is no necessary link between the ability to learn individual words rapidly and the capacity to acquire a large lexicon.

Building social cognitive models of language change.

Studies of language change have begun to contribute to answering several pressing questions in cognitive sciences, including the origins of human language capacity, the social construction of cognition and the mechanisms underlying culture change in general. Here, we describe recent advances within a new emerging framework for the study of language change, one that models such change as an evolutionary process among competing linguistic variants. We argue that a crucial and unifying element of this framework is the use of probabilistic, data-driven models both to infer change and to compare competing claims about social and cognitive influences on language change.

Exact asymptotics for one-dimensional diffusion with mobile traps.

We consider a diffusing particle, with diffusion constant D', moving in one dimension in an infinite sea of noninteracting mobile traps with diffusion constant D and density rho. We show that the asymptotic behavior of the survival probability, P(t), satisfies lim([-ln(P(t)]/sqrt[rho2Dt]=4/sqrt[pi], independent of D'. The result comes from obtaining upper and lower bounds on P(t), and showing that they coincide asymptotically. We also obtain exact results for P(t) to first order in D(')/D for an arbitrary finite number of traps.