In numerosity comparison, performance is faster, more accurate, and less noisy with the ratio of compared numbers.Whereas the ratio-dependency has been intensively studied in relation to internal noise, processes of numerosity com-parison that may increase internal noise have not been fully understood. In this paper, we propose a process theory thataccounts for non-numerical, visuo-spatial processes in numerosity comparison. Consistent with the theory, we found thatas required processes decreased, performance improved significantly, to the extent that there were no differences betweennon-symbolic and symbolic number comparison in reaction time, accuracy, and internal noise. The findings suggest thatcomparing numerosities requires multiple processes homogenizing ancillary stimulus dimensions and that the homoge-nization processes are the major source of fuzziness in approximate number comparison.

Perceptual judgments result from a dynamic process, but little is known about the dynamics of numerosity estimation. Arecent study proposed a computational model (D-MLLM) that combined a model of trial-to-trial changes with a modelfor the internal scaling of discrete number. Here, we tested a surprising prediction of the model – a situation in whichchildren’s estimates of numerosity would be better than those of adults. Consistent with the model simulations, taskcontexts led to a clear developmental reversal: children made more adult-like, linear estimates when to-be-estimatednumbers were descending over trials (backward), whereas adults became more like children with log estimates whennumbers were ascending (forward). In addition, adults’ estimates were subject to inter-trial differences regardless ofstimulus order. In contrast, children were not able to use the trial-to-trial dynamics unless task contexts were salient(forward or backward order), indicating the limited memory capacity for dynamic updates. Together, the model adequatelypredicts both developmental and trial-to-trial changes in number-line tasks.

Despite a heated debate regarding a cognitive mechanism of magnitude representation, little has been done to di-rectly compare numerical and non-numerical estimation and provide a unified account of the two processes. In the current study,we examined estimation of numerical and non-numerical quantities on a continuum using various psychophysical functions.Inconsistent with the proportion reasoning and measurement skills accounts, estimates of both numerical and non-numericalquantities were better predicted by the logarithmic-linear model than by cyclic power models. Furthermore, individual dif-ferences in the degree of logarithmic compression was highly correlated over tasks, whereas bias measures from competingmodels did not show such associations. These findings suggest that estimation of both numerical and non-numerical magnitudeis processed via shared representation systems that are logarithmically or linearly constructed.

Representations of numerical value have been assessed usingbounded (e.g., 0-1000) and unbounded (e.g., 0-?) number-linetasks, with considerable debate regarding whether one or bothtasks elicit unique cognitive strategies (e.g., addition orsubtraction) and require unique cognitive models. To test this,we examined 86 5- to 9-year-olds' addition, subtraction, andestimation skill (bounded and unbounded). Against themeasurement-skills hypothesis, estimates were even morelogarithmic on unbounded than bounded number lines andwere better described by conventional log-linear models thanby alternative cognitive models. Moreover, logarithmic indexvalues reliably predicted arithmetic scores, whereas modelparameters of alternative models failed to do so. Resultssuggest that the logarithmic-to-linear shift theory provides aunified framework for numerical estimation with highdescriptive adequacy and yields uniquely accurate predictionsfor children’s early math proficiency.

Young children’s estimates of numerosity increase approximately logarithmically with actual set size. The conventional interpretation of this finding is that children’s estimates reflect an innate logarithmic encoding of number. A recent set of findings, however, suggest logarithmic number-line estimates could emerge via a dynamic encoding mechanism that is sensitive to the prior distribution of stimuli. Here we test this idea by examining trial-to-trial changes in logarithmicity of numerosity estimates. Against the dynamic encoding hypothesis, first trial estimates in both adults (Study 1) and adults and children (Study 2) were strongly logarithmic, despite there being zero previous stimuli. Additionally, although numerosity of a previous trial affected adult estimates of numerosity, the nature of this effect varied across experiments, yet always resulted in a logarithmic-to-linear shift from trial-to-trial. These results suggest that a dynamic encoding mechanism is neither necessary nor sufficient to elicit logarithmic estimates of numerosity

Animals harbor diverse microbial communities that play important roles in their health and fitness. The gut microbiome is complex, with its composition shaped by a complex interplay of diet, host genetics, microbial interactions, environmental factors, and stochasticity. Previous work in the lab has highlighted the central role of genetic factors, identifying the DBL-1/BMP immune signaling pathway in shaping gut microbiome composition. My dissertation work started with the goal of identifying the downstream mediators of the DBL-1/BMP pathway that regulated gut commensal abundance, but went on subsequently to study additional forces that shaped commensal abundance, in particular environmental stress, exploring the participation of host genetic factors in reshaping the microbiome in response to toxin exposure.Chapter 1 of this dissertation reviews the use of Caenorhabditis elegans as a model organism to study the role of host genetics in shaping gut microbiome composition. With a characteristic gut microbiome that is distinct in its composition from its environment and as a genetically tractable system, C. elegans provides a valuable platform for dissecting the interactions between host genes and microbial communities. This chapter focuses on the influence of TGFβ/BMP signaling, particularly the DBL-1/BMP immune pathway, in regulating microbiome composition. This pathway plays a key role in controlling Enterobacter abundance, preventing overproliferation which can change this normally beneficial commensal into a harmful pathogen. Notably, this regulation occurs in the anterior intestine but is mediated by DBL-1 signaling in extra-intestinal tissues. Thus, chapter 1 highlights C. elegans as a robust model for identifying genetic factors that shape the gut microbiome and further introduces cross-tissue regulation of host-microbiome interactions.Chapter 2 examines how DBL-1/BMP signaling pathway shapes the gut microbiome composition in C. elegans. Through RNA sequencing, I identified gene targets regulated by DBL-1/BMP signaling, which were expressed in the intestine, and predicted to be secreted. Among the verified targets were genes encoding C-type lectins and lysozymes, which are associated with immune responses, but also nuc-1 and scl-2 which are not thought to affect the gut microbiome and whose function remains to be elucidated. Disruption of these genes led to increased colonization by bacteria of the Enterobacteriaceae family. Furthermore, intestine-specific disruption of these genes and epistasis analysis demonstrated that these genes were regulated by DBL-1 signaling in the intestine. The results describe a network of intestinal effectors mediating the effects of DBL-1/BMP signaling on gut microbiome composition, expanding our understanding of host-microbiome interactions.Chapter 3 examines the role of gut bacteria in host adaptation to changing environments, particularly to environmental toxins. I found that gut microbiome provides protection from the antibiotic neomycin, which is toxic to worms and variably toxic to its gut bacteria. Neomycin exposure led to delayed worm development and reduced survival. However, colonization with neomycin-resistant bacterial strains conferred protection. Among four protective strains, Stenotrophomonas indicatrix consistently dominated the gut microbiome under toxin exposure, providing protection via its aminoglycoside-3′-phosphotransferase (APH(3’)) enzyme. Using 16S rRNA gene sequencing, bacterial quantification, and behavioral assays, I identified two distinct protective mechanisms: (1) microbiome remodeling through altered microbial competition that enriched S. indicatrix in the gut, and (2) host-driven avoidance behavior favoring colonization by APH(3’)-expressing bacteria. This behavioral response was driven by the stress-activated KGB-1/JNK MAPK signaling pathway. These findings suggest that microbiome remodeling is a simple yet effective strategy for hosts to cope with novel environmental stressors. However, this adaptation comes with trade-offs, including increased vulnerability to infections and metabolic shifts. My results suggest the dynamic interplay between host behavior and microbial communities in adapting to environmental challenges, while also emphasizing the potential trade-offs associated with such adaptations.

There is an ongoing debate over the psychophysical functionsthat best fit human data from numerical estimation tasks. Totest whether one psychophysical function could account fordata across diverse tasks, we examined 40 kindergartners, 38first graders, 40 second graders and 40 adults’ estimates usingtwo fully crossed 2 × 2 designs, crossing symbol (symbolic,non-symbolic) and boundedness (bounded, unbounded) onfree number-line tasks (Experiment 1) and crossing the samefactors on anchored tasks (Experiment 2). Across all 8 tasks,88.84% of participants provided estimates best fit by a mixedlog-linear model, and the weight of the logarithmiccomponent (λ) decreased with age. After controlling for age,the λ significantly predicted arithmetic skills, whereasparameters of other models failed to do so. Results suggestthat the logarithmic-to-linear shift theory provides a unifiedaccount of numerical estimation and provides uniquelyaccurate predictions for mathematical proficiency.

East Asian students routinely outperform American peers in mathematics. One source of this learning gap may be linguis-tic, such as explicitly naming part-whole relations in fractions (e.g., of four parts, one in Korean vs. one-fourth in English).Our study examined whether adopting such language would improve American children’s number-line estimates. To testthis, 83 10-year-olds were read fractions using either Korean-style or English names over pretest, training, and posttest. Inboth conditions, number-line problems either had no landmarks, landmarks that matched the denominator, or landmarksthat did not match the denominator. As expected, we observed a session by problem type interaction (F=2.71, p¡.05),indicating that feedback improved accuracy most for problems involving matching landmarks. Surprisingly, the effect ofKorean naming was to reduce accuracy across all problems and test phases (ps ¡ .01). Results offer an important warningagainst linguistic reform that may be harmful for American students.

Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs 3 of 4 apples) but have considerabledifficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., 1/5 vs 3/4). Theoretically, reducingthe gap between representations of implicit vs explicit proportions would improve understanding of fractions, but littleis known about how the representations develop and interact with one another. To address this, we asked 163 third tofifth graders to estimate the position of proportionally-equivalent integers and fractions on number lines (e.g., 3 on 0-8number line vs 3/8 on 0-1 number line). We found that, with increasing age, children were more accurate and linear inrepresenting both integers and fractions. More importantly, childrens estimates of implicit and explicit proportions becamemore coherent, such that a childs estimates of fractions on a 0-1 number-line was a linear function of the same childsestimates of equivalent integers. This representational coherence independently predicted childrens fraction proficiency inother tasks, suggest- ing that building a coherent understanding of proportions is an educationally-important goal.