Mathematics & Statistics Faculty WorksCopyright (c) 2024 Swarthmore College All rights reserved.
https://works.swarthmore.edu/fac-math-stat
Recent documents in Mathematics & Statistics Faculty Worksen-usWed, 14 Feb 2024 01:38:08 PST3600An Exploration Of Parameter Duality In Statistical Inference
https://works.swarthmore.edu/fac-math-stat/311
https://works.swarthmore.edu/fac-math-stat/311Mon, 12 Feb 2024 12:37:54 PST
Well-known debates among statistical inferential paradigms emerge from conflicting views on the notion of probability. One dominant view understands probability as a representation of sampling variability; another prominent view understands probability as a measure of belief. The former generally describes model parameters as fixed values, in contrast to the latter. We propose that there are actually two versions of a parameter within both paradigms: a fixed unknown value that generated the data and a random version to describe the uncertainty in estimating the unknown value. An inferential approach based on CDs deciphers seemingly conflicting perspectives on parameters and probabilities.
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Suzanne Thornton et al.On The Third Moment Of <em>L</em>(½, <em>χ<sub>d</sub></em>) {II}: The Number Field Case
https://works.swarthmore.edu/fac-math-stat/312
https://works.swarthmore.edu/fac-math-stat/312Mon, 12 Feb 2024 12:37:54 PST
We establish a smoothed asymptotic formula for the third moment of quadratic Dirichlet L-functions at the central value. In addition to the main term, which is known, we prove the existence of a secondary term of size x^{¾}. The error term in the asymptotic formula is on the order of O(x^{⅔+δ}) for every δ > 0.
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A. Diaconu et al.Bridging Bayesian, Frequentist And Fiducial Inferences Using Confidence Distributions
https://works.swarthmore.edu/fac-math-stat/310
https://works.swarthmore.edu/fac-math-stat/310Mon, 12 Feb 2024 12:37:53 PST
Bayesian, frequentist, and fiducial (BFF) inferences are much more congruous than have been perceived historically in the scientific community (e.g., Reid and Cox (2015); Kass (2011); Efron (1998)). Most practitioners are probably more familiar with the two dominant statistical inferential paradigms, Bayesian inference and frequentist inference. The third, lesser known fiducial inference paradigm was pioneered by R.A. Fisher in an attempt to define an inversion procedure for inference as an alternative to Bayes’ theorem. Although each paradigm has its own strengths and limitations subject to their different philosophical underpinnings, this chapter intends to bridge these different inferential methodologies by calling upon confidence distribution theory and Monte-Carlo simulation procedures, thereby increasing the range of possible techniques available to both statistical theorists and practitioners across all fields.
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Suzanne Thornton et al.Statistics Education And Reconsidering The Status Quo
https://works.swarthmore.edu/fac-math-stat/308
https://works.swarthmore.edu/fac-math-stat/308Mon, 12 Feb 2024 12:37:52 PSTSuzanne ThorntonStatistics For Equity: Capturing, Not Masking, Intersectional Dynamics In Data
https://works.swarthmore.edu/fac-math-stat/309
https://works.swarthmore.edu/fac-math-stat/309Mon, 12 Feb 2024 12:37:52 PSTS. H. Cook et al.Towards Statistical Best Practices For Gender And Sex Data
https://works.swarthmore.edu/fac-math-stat/307
https://works.swarthmore.edu/fac-math-stat/307Mon, 12 Feb 2024 12:37:51 PST
Suzanne Thornton, Dooti Roy, Stephen Parry, Donna LaLonde, Wendy Martinez, Renee Ellis and David Corliss call for a more inclusive – and informative – approach to collecting data on human gender and sex.
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Suzanne Thornton et al.Approximate Confidence Distribution Computing
https://works.swarthmore.edu/fac-math-stat/306
https://works.swarthmore.edu/fac-math-stat/306Mon, 12 Feb 2024 12:37:50 PST
Approximate confidence distribution computing (ACDC) offers a new take on the rapidly developing field of likelihood-free inference from within a frequentist framework. The appeal of this computational method for statistical inference hinges upon the concept of a confidence distribution, a special type of estimator which is defined with respect to the repeated sampling principle. An ACDC method provides frequentist validation for computational inference in problems with unknown or intractable likelihoods. The main theoretical contribution of this work is the identification of a matching condition necessary for frequentist validity of inference from this method. In addition to providing an example of how a modern understanding of confidence distribution theory can be used to connect Bayesian and frequentist inferential paradigms, we present a case to expand the current scope of so-called approximate Bayesian inference to include non-Bayesian inference by targeting a confidence distribution rather than a posterior. The main practical contribution of this work is the development of a data-driven approach to drive ACDC in both Bayesian or frequentist contexts. The ACDC algorithm is data-driven by the selection of a data-dependent proposal function, the structure of which is quite general and adaptable to many settings. We explore three numerical examples that both verify the theoretical arguments in the development of ACDC and suggest instances in which ACDC outperform approximate Bayesian computing methods computationally.
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Suzanne Thornton et al.A Global Ecological Signal Of Extinction Risk In Terrestrial Vertebrates
https://works.swarthmore.edu/fac-math-stat/305
https://works.swarthmore.edu/fac-math-stat/305Mon, 12 Feb 2024 12:37:50 PST
To determine the distribution and causes of extinction threat across functional groups of terrestrial vertebrates, we assembled an ecological trait data set for 18,016 species of terrestrial vertebrates and utilized phylogenetic comparative methods to test which categories of habitat association, mode of locomotion, and feeding mode best predicted extinction risk. We also examined the individual categories of the International Union for Conservation of Nature Red List extinction drivers (e.g., agriculture and logging) threatening each species and determined the greatest threats for each of the four terrestrial vertebrate groups. We then quantified the sum of extinction drivers threatening each species to provide a multistressor perspective on threat. Cave dwelling amphibians (p < 0.01), arboreal quadrupedal mammals (all of which are primates) (p < 0.01), aerial and scavenging birds (p < 0.01), and pedal (i.e., walking) squamates (p < 0.01) were all disproportionately threatened with extinction in comparison with the other assessed ecological traits. Across all threatened vertebrate species in the study, the most common risk factors were agriculture, threatening 4491 species, followed by logging, threatening 3187 species, and then invasive species and disease, threatening 2053 species. Species at higher risk of extinction were simultaneously at risk from a greater number of threat types. If left unabated, the disproportionate loss of species with certain functional traits and increasing anthropogenic pressures are likely to disrupt ecosystem functions globally. A shift in focus from species- to trait-centric conservation practices will allow for protection of at-risk functional diversity from regional to global scales.
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M. J. Munstermann et al.On Fossil Recovery Potential In The <em>Australopithecus anamensis–Australopithecus afarensis</em> Lineage: A Reply To Žliobaitė (2020)
https://works.swarthmore.edu/fac-math-stat/304
https://works.swarthmore.edu/fac-math-stat/304Mon, 12 Feb 2024 12:37:49 PST
We thank Žliobaitė (2020) for the interest in our article (Du et al., 2020) and the resulting dialogue. In this response, we briefly summarize Du and colleagues' analysis and Žliobaitė's commentary, respond to the points raised by Žliobaitė, and conclude with some thoughts on why the distribution of Australopithecus anamensis–Australopithecus afarensis fossil horizons does not conform to a wax–wane pattern.
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A. Du et al.Generalized Sasakian Structures From A Poisson Geometry Viewpoint
https://works.swarthmore.edu/fac-math-stat/303
https://works.swarthmore.edu/fac-math-stat/303Mon, 12 Feb 2024 12:37:48 PST
In this paper we define a canonical Poisson structure on a normal generalized contact metric space and use this structure to define a generalized Sasakian structure. We show also that this canonical Poisson structure enables us to distinguish generalized Sasakian structures from generalized coKähler structures.
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Janet TalvacchiaOn Best Practices For The Recruitment, Retention, And Flourishing Of LGBTQ+ Mathematicians
https://works.swarthmore.edu/fac-math-stat/301
https://works.swarthmore.edu/fac-math-stat/301Mon, 12 Feb 2024 12:37:47 PSTR. Buckmire et al.Spectra Of Variants Of Distance Matrices Of Graphs And Digraphs: A Survey
https://works.swarthmore.edu/fac-math-stat/302
https://works.swarthmore.edu/fac-math-stat/302Mon, 12 Feb 2024 12:37:47 PST
Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on comparing and contrasting techniques and results for the four types of distance matrices. Digraphs are treated separately after the discussion of graphs, including discussion of similarities and differences between graphs and digraphs. New results are presented that complement existing results, including results for some the matrices on unimodality of characteristic polynomials for graphs, preservation of parameters by cospectrality for graphs, and bounds on spectral radii for digraphs.
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L. Hogben et al.Recent Activities And Progress By Spectra And The LGBTQ+ Mathematics Community
https://works.swarthmore.edu/fac-math-stat/300
https://works.swarthmore.edu/fac-math-stat/300Mon, 12 Feb 2024 12:37:46 PSTJoseph NakaoStrongly Invertible Knots, Equivariant Slice Genera, And An Equivariant Algebraic Concordance Group
https://works.swarthmore.edu/fac-math-stat/299
https://works.swarthmore.edu/fac-math-stat/299Mon, 12 Feb 2024 07:33:39 PST
We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let K be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum #^{nK is at least n/4. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.}
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Allison N. Miller et al.Branched Covers Bounding Rational Homology Balls
https://works.swarthmore.edu/fac-math-stat/298
https://works.swarthmore.edu/fac-math-stat/298Mon, 12 Feb 2024 07:33:38 PST
Prime power–fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. We give a new construction of nonslice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.
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P. Aceto et al.Embedding Spheres In Knot Traces
https://works.swarthmore.edu/fac-math-stat/297
https://works.swarthmore.edu/fac-math-stat/297Mon, 12 Feb 2024 07:33:37 PST
The trace of the n-framed surgery on a knot in S³ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded 2-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable 3-dimensional knot invariants. For each n, this provides conditions that imply a knot is topologically n-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.
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P. Feller et al.Amphichiral Knots With Large 4-Genus
https://works.swarthmore.edu/fac-math-stat/295
https://works.swarthmore.edu/fac-math-stat/295Mon, 12 Feb 2024 07:33:36 PST
For each we give g > 0 infinitely many knots that are strongly negative amphichiral, hence rationally slice and representing 2-torsion in the smooth concordance group, yet which do not bound any locally flatly embedded surface in the 4-ball with genus less than or equal to g. Our examples also allow us to answer a question about the four-dimensional clasp number of knots.
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Allison N. MillerThe Topological Slice Genus Of Satellite Knots
https://works.swarthmore.edu/fac-math-stat/296
https://works.swarthmore.edu/fac-math-stat/296Mon, 12 Feb 2024 07:33:36 PST
We present evidence supporting the conjecture that, in the topological category, the slice genus of a satellite knot P(K) is bounded above by the sum of the slice genera of K and P(U). Our main result establishes this conjecture for a variant of the topological slice genus, the ℤ–slice genus. Notably, the conjectured upper bound does not involve the algebraic winding number of the pattern P. This stands in stark contrast with the smooth category, where, for example, there are many genus 1 knots whose (n,1)–cables have arbitrarily large smooth 4–genera. As an application, we show that the (n,1)–cable of any knot of 3–genus 1 (eg the figure-eight or trefoil knot) has topological slice genus at most 1, regardless of the value of n∈N. Further, we show that the lower bounds on the slice genus coming from the Tristram–Levine and Casson–Gordon signatures cannot be used to disprove the conjecture.
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P. Feller et al.A Note on the Concordance ℤ-Genus
https://works.swarthmore.edu/fac-math-stat/294
https://works.swarthmore.edu/fac-math-stat/294Mon, 12 Feb 2024 07:33:35 PST
We show that the difference between the topological 4-genus of a knot and the minimal genus of a surface bounded by that knot that can be decomposed into a smooth concordance followed by an algebraically simple locally flat surface can be arbitrarily large. This extends work of Hedden, Livingston, and Ruberman showing that there are topologically slice knots which are not smoothly concordant to any knot with trivial Alexander polynomial.
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Allison N. Miller et al.Homomorphism Obstructions For Satellite Maps
https://works.swarthmore.edu/fac-math-stat/293
https://works.swarthmore.edu/fac-math-stat/293Mon, 12 Feb 2024 07:33:34 PST
A knot in a solid torus defines a map on the set of (smooth or topological) concordance classes of knots in S³. This set admits a group structure, but a conjecture of Hedden suggests that satellite maps never induce interesting homomorphisms: we give new evidence for this conjecture in both categories. First, we use Casson-Gordon signatures to give the first obstruction to a slice pattern inducing a homomorphism on the topological concordance group, constructing examples with every winding number besides ± 1. We then provide subtle examples of satellite maps which map arbitrarily deep into the n-solvable filtration of Cochran, Orr, and Teichner [Ann. of Math. (2) 157 (2003), pp. 433–519], act like homomorphisms on arbitrary finite sets of knots, and yet which still do not induce homomorphisms. Finally, we verify Hedden’s conjecture in the smooth category for all small crossing number satellite operators but one.
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Allison N. MillerSteklov Spectrum And Elliptic Problems With Nonlinear Boundary Conditions
https://works.swarthmore.edu/fac-math-stat/292
https://works.swarthmore.edu/fac-math-stat/292Mon, 12 Feb 2024 07:33:34 PST
Problems with nonlinear boundary conditions arise naturally in many applications. For instance, in population dynamics where an impact of habitat-edges (boundary) on the dispersal pattern of species as they reach the boundary takes place in spatial ecology CC06. They occur when the biochemical reactions take place at or near the boundary, for example, in the limb bud development of a chick in which a chemical reaction produces outgrowth due to cell growth and division, and interactions between morphogens produced in several zones of the limb bud DO99. They also appear in noninvasive testing methods to locate defects in a medium by using boundary data measurements (see, e.g., CCMM16). In cryosurgery (a minimally invasive treatment used to treat some types of cancers and some conditions that may become cancer), a highly exothermic reaction takes place in a thin layer around the boundary in order to destroy abnormal tissue LOS98. These examples are not exhaustive.
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Nsoki MavingaFucik Spectrum With Weights And Existence Of Solutions For Nonlinear Elliptic Equations With Nonlinear Boundary Conditions
https://works.swarthmore.edu/fac-math-stat/291
https://works.swarthmore.edu/fac-math-stat/291Mon, 12 Feb 2024 07:33:33 PST
We consider the boundary value problem −Δu + c(x)u = αm(x)u^{+} − βm(x)u^{−} + f(x,u), x∈Ω, (∂u)/(∂η) + σ(x)u = αρ(x)u^{+} − βρ(x)u^{−} + g(x,u), x∈∂Ω, where (α,β) ∈R2, c, m ∈ L^{∞}(Ω), σ, ρ ∈ L^{∞}(∂Ω), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum.
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Nsoki Mavinga et al.Bifurcation From Infinity With Oscillatory Nonlinearity For Neumann Problems
https://works.swarthmore.edu/fac-math-stat/289
https://works.swarthmore.edu/fac-math-stat/289Mon, 12 Feb 2024 07:33:32 PST
We consider a sublinear perturbation of an elliptic eigenvalue problem with Neumann boundary condition. We give sufficient conditions on the nonlinear perturbation which guarantee that the unbounded continuum, bifurcating from infinity at the first eigenvalue, contains an unbounded sequence of turning points as well as an unbounded sequence of resonant solutions. We prove our result by using bifurcation theory combined with a careful analysis of the oscillatory behavior of the continuum near the bifurcation point.
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M. Chhetri et al.Maximal And Minimal Weak Solutions For Elliptic Problems With Nonlinearity On The Boundary
https://works.swarthmore.edu/fac-math-stat/290
https://works.swarthmore.edu/fac-math-stat/290Mon, 12 Feb 2024 07:33:32 PST
This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.
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S. Bandyopadhyay et al.Decision-Making In Forensic Identification Tasks
https://works.swarthmore.edu/fac-math-stat/288
https://works.swarthmore.edu/fac-math-stat/288Mon, 12 Feb 2024 07:33:31 PSTAmanda LubyModeling Covarying Responses In Complex Tasks
https://works.swarthmore.edu/fac-math-stat/286
https://works.swarthmore.edu/fac-math-stat/286Mon, 12 Feb 2024 07:33:30 PST
In testing situations, participants are often asked for supplementary responses in addition to the primary response of interest, which may include quantities like confidence or reported difficulty. These additional responses can be incorporated into a psychometric model either as a predictor of the main response or as a secondary response. In this paper we explore both of these approaches for incorporating participant’s reported difficulty into a psychometric model using an error rate study of fingerprint examiners. Participants were asked to analyze print pairs and make determinations about the source, which can be scored as correct or incorrect decisions. Additionally, participants were asked to report the difficulty of the print pair on a five point scale. In this paper, we model (a) the responses of individual examiners without incorporating reported difficulty using a Rasch model, (b) the responses using their reported difficulty as a predictor, and (c) the responses and their reported difficulty as a multivariate response variable. We find that approach (c) results in more balanced classification errors, but incorporating reported difficulty using either approach does not lead to substantive changes in proficiency or difficulty estimates. These results suggest that, while there are individual differences in reported difficulty, these differences appear to be unrelated to examiners’ proficiency in correctly distinguishing matched from non-matched fingerprints.
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Amanda Luby et al.Psychometrics For Forensic Fingerprint Comparisons
https://works.swarthmore.edu/fac-math-stat/287
https://works.swarthmore.edu/fac-math-stat/287Mon, 12 Feb 2024 07:33:30 PST
Forensic science often involves the evaluation of crime-scene evidence to determine whether it matches a known-source sample, such as whether a fingerprint or DNA was left by a suspect or if a bullet was fired from a specific firearm. Even as forensic measurement and analysis tools become increasingly automated and objective, final source decisions are often left to individual examiners’ interpretation of the evidence. Furthermore, forensic analyses often consist of a series of steps. While some of these steps may be straightforward and relatively objective, substantial variation may exist in more subjective decisions. The current approach to characterizing uncertainty in forensic decision-making has largely centered around conducting error rate studies (in which examiners evaluate a set of items consisting of known-source comparisons) and calculating error rates aggregated across examiners and identification tasks. We propose a new approach using Item Response Theory (IRT) and IRT-like models to account for differences in examiner behavior and for varying difficulty among identification tasks. There are, however, substantial differences between forensic decision-making and traditional IRT applications such as educational testing. For example, the structure of the response process must be considered, “answer keys” for comparison tasks do not exist, and information about participants and items is not available due to privacy constraints. In this paper, we provide an overview of forensic decision-making, outline challenges in applying IRT in practice, and survey some recent advances in the application of Bayesian psychometric models to fingerprint examiner behavior.
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Amanda Luby et al.Analyzing Spatial Responses: A Comparison Of IRT-Based Approaches
https://works.swarthmore.edu/fac-math-stat/285
https://works.swarthmore.edu/fac-math-stat/285Mon, 12 Feb 2024 07:33:29 PST
We investigate two approaches for analyzing spatial coordinate responses using models inspired by Item Response Theory (IRT). In the first, we use a two-stage approach to first construct a pseudo-response matrix using the spatial information and then apply standard IRT techniques to estimate proficiency and item parameters. In the second approach, we introduce the Spatial Error Model and use the spatial coordinates directly to infer information about the true locations and participant precision. As a motivating example, we use a study from forensic science designed to measure how fingerprint examiners use minutiae (small details in the fingerprint that form the basis for uniqueness) to come to an identification decision. The study found substantial participant variability, as different participants tend to focus on different areas of the image and some participants mark more minutiae than others. Using simulated data, we illustrate the relative strengths and weaknesses of each modeling approach, and demonstrate the advantages of modeling the spatial coordinates directly in the Spatial Error Model.
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Amanda Luby et al.Psychometric Analysis Of Forensic Examiner Behavior
https://works.swarthmore.edu/fac-math-stat/284
https://works.swarthmore.edu/fac-math-stat/284Mon, 12 Feb 2024 07:33:29 PST
Forensic science often involves the comparison of crime-scene evidence to a known-source sample to determine if the evidence and the reference sample came from the same source. Even as forensic analysis tools become increasingly objective and automated, final source identifications are often left to individual examiners’ interpretation of the evidence. Each source identification relies on judgements about the features and quality of the crime-scene evidence that may vary from one examiner to the next. The current approach to characterizing uncertainty in examiners’ decision-making has largely centered around the calculation of error rates aggregated across examiners and identification tasks, without taking into account these variations in behavior. We propose a new approach using IRT and IRT-like models to account for differences among examiners and additionally account for the varying difficulty among source identification tasks. In particular, we survey some recent advances (Luby 2019a) in the application of Bayesian psychometric models, including simple Rasch models as well as more elaborate decision tree models, to fingerprint examiner behavior.
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Amanda Luby et al.Opening Up The Court: Analyzing Player Performance Across Tennis Grand Slams
https://works.swarthmore.edu/fac-math-stat/283
https://works.swarthmore.edu/fac-math-stat/283Mon, 12 Feb 2024 07:33:28 PST
In tennis, the Australian Open, French Open, Wimbledon, and US Open are the four most prestigious events (Grand Slams). These four Grand Slams differ in the composition of the court surfaces, when they are played in the year, and which city hosts the players. Individual Grand Slams come with different expectations, and it is often thought that some players achieve better results at some Grand Slams than others. It is also thought that differences in results may be attributed, at least partially, to surface type of the courts. For example, Rafael Nadal, Roger Federer, and Serena Williams have achieved their best results on clay, grass, and hard courts, respectively. This paper explores differences among Grand Slams, while adjusting for confounders such as tour, competitor strength, and player attributes. More specifically, we examine the effect of the Grand Slam on player performance for matches from 2013 to 2019. We take two approaches to modeling these data: (1) a mixed-effects model accounting for both player and tournament features and (2) models that emphasize individual performance. We identify differences across the Grand Slams at both the tournament and individual player level.
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S. K. Gallagher et al.A Probabilistic Formalisation Of Contextual Bias: From Forensic Analysis To Systemic Bias In The Criminal Justice System
https://works.swarthmore.edu/fac-math-stat/281
https://works.swarthmore.edu/fac-math-stat/281Mon, 12 Feb 2024 07:33:27 PST
Researchers have found evidence of contextual bias in forensic science, but the discussion of contextual bias is currently qualitative. We formalise existing empirical research and show quantitatively how biases can be propagated throughout the legal system, all the way up to the final determination of guilt in a criminal trial. We provide a probabilistic framework for describing how information is updated in a forensic analysis setting by using the ratio form of Bayes’ rule. We analyse results from empirical studies using this framework and employ simulations to demonstrate how bias can be compounded where experiments do not exist. We find that even minor biases in the earlier stages of forensic analysis can lead to large, compounded biases in the final determination of guilt in a criminal trial.
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M. Cuellar et al.Think-Aloud Interviews: A Tool For Exploring Student Statistical Reasoning
https://works.swarthmore.edu/fac-math-stat/282
https://works.swarthmore.edu/fac-math-stat/282Mon, 12 Feb 2024 07:33:27 PST
Think-aloud interviews have been a valuable but underused tool in statistics education research. Think-alouds, in which students narrate their reasoning in real time while solving problems, differ in important ways from other types of cognitive interviews and related education research methods. Beyond the uses already found in the statistics literature—mostly validating the wording of statistical concept inventory questions and studying student misconceptions—we suggest other possible use cases for think-alouds and summarize best-practice guidelines for designing think-aloud interview studies. Using examples from our own experiences studying the local student body for our introductory statistics courses, we illustrate how research goals should inform study-design decisions and what kinds of insights think-alouds can provide. We hope that our overview of think-alouds encourages more statistics educators and researchers to begin using this method. Supplementary materials for this article are available online.
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A. Reinhart et al.A Method For Quantifying Individual Decision Thresholds Of Latent Print Examiners
https://works.swarthmore.edu/fac-math-stat/280
https://works.swarthmore.edu/fac-math-stat/280Mon, 12 Feb 2024 07:33:26 PST
In recent years, ‘black box’ studies in forensic science have emerged as the preferred way to provide information about the overall validity of forensic disciplines in practice. These studies provide aggregated error rates over many examiners and comparisons, but errors are not equally likely on all comparisons. Furthermore, inconclusive responses are common and vary across examiners and comparisons, but do not fit neatly into the error rate framework. This work introduces Item Response Theory (IRT) and variants for the forensic setting to account for these two issues. In the IRT framework, participant proficiency and item difficulty are estimated directly from the responses, which accounts for the different subsets of items that participants often answer. By incorporating a decision-tree framework into the model, inconclusive responses are treated as a distinct cognitive process, which allows inter-examiner differences to be estimated directly. The IRT-based model achieves superior predictive performance over standard logistic regression techniques, produces item effects that are consistent with common sense and prior work, and demonstrates that most of the variability among fingerprint examiner decisions occurs at the latent print evaluation stage and as a result of differing tendencies to make inconclusive decisions.
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Amanda LubyThe Complexity Threshold For The Emergence Of Kakutani Inequivalence
https://works.swarthmore.edu/fac-math-stat/279
https://works.swarthmore.edu/fac-math-stat/279Mon, 12 Feb 2024 07:33:25 PST
We show that linear complexity is the threshold for the emergence of Kakutani inequivalence for measurable systems supported on a minimal subshift. In particular, we show that there are minimal subshifts of arbitrarily low superlinear complexity that admit both loosely Bernoulli and non-loosely Bernoulli ergodic measures and that no minimal subshift with linear complexity can admit inequivalent measures.
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V. Cyr et al.Topological Speedups Of ℤ<sup>d</sup>-Actions
https://works.swarthmore.edu/fac-math-stat/278
https://works.swarthmore.edu/fac-math-stat/278Mon, 12 Feb 2024 07:33:24 PST
We study minimal ℤ^{d}-Cantor systems and the relationship between their speedups, their collections of invariant Borel measures, their associated unital dimension groups, and their orbit equivalence classes. In the particular case of minimal ℤ^{d}-odometers, we show that their bounded speedups must again be odometers but, contrary to the 1-dimensional case, they need not be conjugate, or even isomorphic, to the original. Furthermore, we give examples of speedups of ℤ^{d}-odometers which show the significant role played by a choice of ‘cone’ associated to the speedup.
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Aimee S.A. Johnson et al.SET With A Twist
https://works.swarthmore.edu/fac-math-stat/277
https://works.swarthmore.edu/fac-math-stat/277Mon, 12 Feb 2024 07:33:24 PSTCathy Hsu et al.Berglund–Hübsch Transpose Rule And Sasakian Geometry
https://works.swarthmore.edu/fac-math-stat/276
https://works.swarthmore.edu/fac-math-stat/276Mon, 12 Feb 2024 07:33:23 PST
We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an n−1-dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension 2n+1 which are n−1-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four seven-dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein.
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Ralph R. GomezDynamics Of The Auditory Continuity Illusion
https://works.swarthmore.edu/fac-math-stat/275
https://works.swarthmore.edu/fac-math-stat/275Mon, 12 Feb 2024 07:33:22 PST
Illusions give intriguing insights into perceptual and neural dynamics. In the auditory continuity illusion, two brief tones separated by a silent gap may be heard as one continuous tone if a noise burst with appropriate characteristics fills the gap. This illusion probes the conditions under which listeners link related sounds across time and maintain perceptual continuity in the face of sudden changes in sound mixtures. Conceptual explanations of this illusion have been proposed, but its neural basis is still being investigated. In this work we provide a dynamical systems framework, grounded in principles of neural dynamics, to explain the continuity illusion. We construct an idealized firing rate model of a neural population and analyze the conditions under which firing rate responses persist during the interruption between the two tones. First, we show that sustained inputs and hysteresis dynamics (a mismatch between tone levels needed to activate and inactivate the population) can produce continuous responses. Second, we show that transient inputs and bistable dynamics (coexistence of two stable firing rate levels) can also produce continuous responses. Finally, we combine these input types together to obtain neural dynamics consistent with two requirements for the continuity illusion as articulated in a well-known theory of auditory scene analysis: responses persist through the noise-filled gap if noise provides sufficient evidence that the tone continues and if there is no evidence of discontinuities between the tones and noise. By grounding these notions in a quantitative model that incorporates elements of neural circuits (recurrent excitation, and mutual inhibition, specifically), we identify plausible mechanisms for the continuity illusion. Our findings can help guide future studies of neural correlates of this illusion and inform development of more biophysically-based models of the auditory continuity illusion.
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Qianyi Cao , '22 et al.Glycinergic Axonal Inhibition Subserves Acute Spatial Sensitivity To Sudden Increases In Sound Intensity
https://works.swarthmore.edu/fac-math-stat/274
https://works.swarthmore.edu/fac-math-stat/274Mon, 12 Feb 2024 07:33:21 PST
Locomotion generates adventitious sounds which enable detection and localization of predators and prey. Such sounds contain brisk changes or transients in amplitude. We investigated the hypothesis that ill-understood temporal specializations in binaural circuits subserve lateralization of such sound transients, based on different time of arrival at the ears (interaural time differences, ITDs). We find that Lateral Superior Olive (LSO) neurons show exquisite ITD-sensitivity, reflecting extreme precision and reliability of excitatory and inhibitory postsynaptic potentials, in contrast to Medial Superior Olive neurons, traditionally viewed as the ultimate ITD-detectors. In vivo, inhibition blocks LSO excitation over an extremely short window, which, in vitro, required synaptically evoked inhibition. Light and electron microscopy revealed inhibitory synapses on the axon initial segment as the structural basis of this observation. These results reveal a neural vetoing mechanism with extreme temporal and spatial precision and establish the LSO as the primary nucleus for binaural processing of sound transients.
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T. P. Franken et al.Structure And Dynamics That Specialize Neurons For High-Frequency Coincidence Detection In The Barn Owl Nucleus Laminaris
https://works.swarthmore.edu/fac-math-stat/273
https://works.swarthmore.edu/fac-math-stat/273Mon, 12 Feb 2024 07:33:20 PST
A principal cue for sound source localization is the difference in arrival times of sounds at an animal’s two ears (interaural time difference, ITD). Neurons that process ITDs are specialized to compare the timing of inputs with submillisecond precision. In the barn owl, ITD processing begins in the nucleus laminaris (NL) region of the auditory brain stem. Remarkably, NL neurons are sensitive to ITDs in high-frequency sounds (kilohertz-range). This contrasts with ITD-based sound localization in analogous regions in mammals where ITD sensitivity is typically restricted to lower-frequency sounds. Guided by previous experiments and modeling studies of tone-evoked responses of NL neurons, we propose NL neurons achieve high-frequency ITD sensitivity if they respond selectively to the small-amplitude, high-frequency oscillations in their inputs, and remain relatively non-responsive to mean input level. We use a biophysically based model to study the effects of soma–axon coupling on dynamics and function in NL neurons. First, we show that electrical separation of the soma from the axon region in the neuron enhances high-frequency ITD sensitivity. This soma–axon coupling configuration promotes linear subthreshold dynamics and rapid spike initiation, making the model more responsive to input oscillations, rather than mean input level. Second, we provide new evidence for the essential role of phasic dynamics for high-frequency neural coincidence detection. Transforming our model to the phasic firing mode further tunes the model to respond selectively to the oscillating inputs that carry ITD information. Similar structural and dynamical mechanisms specialize mammalian auditory brain stem neurons for ITD sensitivity, and thus, our work identifies common principles of ITD processing and neural coincidence detection across species and for sounds at widely different frequencies.
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Ben Drucker , '22 et al.The Explorer–Director Game On Graphs
https://works.swarthmore.edu/fac-math-stat/272
https://works.swarthmore.edu/fac-math-stat/272Mon, 12 Feb 2024 07:33:19 PST
The Explorer-Director game, first introduced by Nedev and Muthukrishnan, can be described as a game where two players—Explorer and Director—determine the movement of a token that is positioned on the vertices of some given graph. At each time step, the Explorer specifies a distance that the token must move with an aim to maximize the total number of vertices ultimately visited. However, the Director adversarially chooses where to move token in an effort to minimize this number. The game ends when no new vertices can be visited. Given a graph G and a starting vertex v, the number of vertices that are visited under optimal play is denoted by f_{d}(G,v). In this paper, we first reduce the study of f_{d}(G,v) to the determination of the minimum sets of vertices that are closed in a certain combinatorial sense, thus providing a structural understanding of each player’s optimal strategies. As an application, we provide some exact results as well as more general bounds when G is a square lattice or tree. In the case of trees, we also provide a complete solution even in the more restrictive setting where the strategy used by the Explorer is not allowed to depend on their opponent’s responses. In addition to this paper, a supplementary companion note will be posted to arXiv providing additional results about the game in a variety of specific graph families.
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Pat Devlin et al.On The Finiteness Of Quantum K-Theory Of A Homogeneous Space
https://works.swarthmore.edu/fac-math-stat/271
https://works.swarthmore.edu/fac-math-stat/271Mon, 12 Feb 2024 07:33:18 PST
We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.
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D. Anderson et al.Motivic Classes Of Degeneracy Loci And Pointed Brill-Noether Varieties
https://works.swarthmore.edu/fac-math-stat/269
https://works.swarthmore.edu/fac-math-stat/269Mon, 12 Feb 2024 07:33:17 PST
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern–Schwartz–MacPherson classes, K-theory classes, and Cappell–Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χ_{y}-genus of classical and one-pointed Brill–Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
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D. Anderson et al.An Affine Approach To Peterson Comparison
https://works.swarthmore.edu/fac-math-stat/270
https://works.swarthmore.edu/fac-math-stat/270Mon, 12 Feb 2024 07:33:17 PST
The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov’s strange duality isomorphism.
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Linda Chen et al.Quantum Hooks And Mirror Symmetry For Flag Varieties
https://works.swarthmore.edu/fac-math-stat/268
https://works.swarthmore.edu/fac-math-stat/268Mon, 12 Feb 2024 07:33:16 PST
Given a flag variety F1 (n; r_{1},...,r_{p}), there is natural ring morphism from the symmetric polynomial ring in r_{1} variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions λ, the image of s_{λ} under the ring homomorphism is a Schubert class which is described by partitioning λ into a quantum hook (or q-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpotential, and the relation between superpotentials of flag varieties and those of Grassmannians (where the superpotential was introduced by Marsh–Rietsch).
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Linda Chen et al.New Insights Into Binocular Rivalry From The Reconstruction Of Evolving Percepts Using Model Network Dynamics
https://works.swarthmore.edu/fac-math-stat/267
https://works.swarthmore.edu/fac-math-stat/267Mon, 12 Feb 2024 07:33:15 PST
When the two eyes are presented with highly distinct stimuli, the resulting visual percept generally switches every few seconds between the two monocular images in an irregular fashion, giving rise to a phenomenon known as binocular rivalry. While a host of theoretical studies have explored potential mechanisms for binocular rivalry in the context of evoked model dynamics in response to simple stimuli, here we investigate binocular rivalry directly through complex stimulus reconstructions based on the activity of a two-layer neuronal network model with competing downstream pools driven by disparate monocular stimuli composed of image pixels. To estimate the dynamic percept, we derive a linear input-output mapping rooted in the non-linear network dynamics and iteratively apply compressive sensing techniques for signal recovery. Utilizing a dominance metric, we are able to identify when percept alternations occur and use data collected during each dominance period to generate a sequence of percept reconstructions. We show that despite the approximate nature of the input-output mapping and the significant reduction in neurons downstream relative to stimulus pixels, the dominant monocular image is well-encoded in the network dynamics and improvements are garnered when realistic spatial receptive field structure is incorporated into the feedforward connectivity. Our model demonstrates gamma-distributed dominance durations and well obeys Levelt's four laws for how dominance durations change with stimulus strength, agreeing with key recurring experimental observations often used to benchmark rivalry models. In light of evidence that individuals with autism exhibit relatively slow percept switching in binocular rivalry, we corroborate the ubiquitous hypothesis that autism manifests from reduced inhibition in the brain by systematically probing our model alternation rate across choices of inhibition strength. We exhibit sufficient conditions for producing binocular rivalry in the context of natural scene stimuli, opening a clearer window into the dynamic brain computations that vary with the generated percept and a potential path toward further understanding neurological disorders.
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Kenneth Barkdoll , '24 et al.Neural Network Learning Of Improved Compressive Sensing Sampling And Receptive Field Structure
https://works.swarthmore.edu/fac-math-stat/266
https://works.swarthmore.edu/fac-math-stat/266Mon, 12 Feb 2024 07:33:14 PST
While the theory of compressive sensing (CS) in modern signal processing typically indicates that uniformly random sampling facilitates the efficient recovery of sparse signals, such measurements are infeasible in many engineering applications and are not well reflected by the constraints of natural systems, including neuronal networks in the brain. Uniformly random sampling also does not leverage the underlying structure of many classes of signals, and may therefore be suboptimal in these cases. We address these issues by formulating a novel neural network framework for learning improved CS sampling based on the intrinsic structure present in classes of training signals. Beyond sparsity in an appropriate domain, this approach does not assume knowledge of any specific signal statistics and is purely data-driven. The learning methodology is biologically realistic in that it utilizes (1) asymmetric feedback and feedforward connections in the neural network and (2) only information from adjacent layers in training the CS measurement matrix. Observing a broad spectrum of learned sampling paradigms that improve CS signal reconstructions relative to uniformly random sampling, our learned sampling is widely applicable across logistical constraints. Motivated by the receptive field structure of sensory systems, we specifically analyze natural scene inputs and demonstrate improved CS reconstruction as a result of training across several choices of penalization schemes on the sampling weights. Considering this learning is effective even under sparse and spatially localized constraints, as commonly observed in the brain, we hypothesize that neuronal connectivity may have manifested with the aim of providing a compressive encoding of data by leveraging its sparse structure, thereby achieving efficient signal transmission.
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Victor J. BarrancaFunctional Implications Of Dale's Law In Balanced Neuronal Network Dynamics And Decision Making
https://works.swarthmore.edu/fac-math-stat/264
https://works.swarthmore.edu/fac-math-stat/264Mon, 12 Feb 2024 07:33:13 PST
The notion that a neuron transmits the same set of neurotransmitters at all of its post-synaptic connections, typically known as Dale's law, is well supported throughout the majority of the brain and is assumed in almost all theoretical studies investigating the mechanisms for computation in neuronal networks. Dale's law has numerous functional implications in fundamental sensory processing and decision-making tasks, and it plays a key role in the current understanding of the structure-function relationship in the brain. However, since exceptions to Dale's law have been discovered for certain neurons and because other biological systems with complex network structure incorporate individual units that send both positive and negative feedback signals, we investigate the functional implications of network model dynamics that violate Dale's law by allowing each neuron to send out both excitatory and inhibitory signals to its neighbors. We show how balanced network dynamics, in which large excitatory and inhibitory inputs are dynamically adjusted such that input fluctuations produce irregular firing events, are theoretically preserved for a single population of neurons violating Dale's law. We further leverage this single-population network model in the context of two competing pools of neurons to demonstrate that effective decision-making dynamics are also produced, agreeing with experimental observations from honeybee dynamics in selecting a food source and artificial neural networks trained in optimal selection. Through direct comparison with the classical two-population balanced neuronal network, we argue that the one-population network demonstrates more robust balanced activity for systems with less computational units, such as honeybee colonies, whereas the two-population network exhibits a more rapid response to temporal variations in network inputs, as required by the brain. We expect this study will shed light on the role of neurons violating Dale's law found in experiment as well as shared design principles across biological systems that perform complex computations.
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Victor J. Barranca et al.Data-Driven Reconstruction And Encoding Of Sparse Stimuli Across Convergent Sensory Layers From Downstream Neuronal Network Dynamics
https://works.swarthmore.edu/fac-math-stat/265
https://works.swarthmore.edu/fac-math-stat/265Mon, 12 Feb 2024 07:33:13 PST
Neuronal networks vary dramatically in size, connectivity structure, and functionality across downstream layers of the brain. This raises the question of whether information is lost as it is re-encoded along compressive and expansive pathways. In this work, we develop a potential data-driven mechanism for the preservation of information in the activity of neuronal networks across downstream layers, which uses the widespread linearity of individual neuronal responses to sufficiently strong ramped artificial inputs to fit a linear input-output mapping across the network. We analyze the dynamics of several families of two-layer neuronal network models, where the input components far outnumber the downstream neurons, as in compressive pathways, and apply the fitted mapping in conjunction with compressive sensing theory to reconstruct stimuli with sparse structure. The input-output mapping facilitates stimulus reconstructions that only use measurements of downstream neuronal firing rates in response to inputs over a short time duration, furnishing stimulus recovery even when theoretical analysis is intractable or the governing equations of the dynamical system are unknown as in experiment. Similarly accurate stimulus reconstructions are obtained across different single-neuron models, network coupling functions, and image classes. Improved reconstructions are yielded when uniformly random feedforward connectivity is replaced by spatially localized feedforward connectivity akin to receptive fields. We expect that similar principles could be leveraged experimentally in prosthetics as well as in the reconstruction of large-scale network connectivity.
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Victor J. Barranca et al.Reconstruction Of Sparse Recurrent Connectivity And Inputs From The Nonlinear Dynamics Of Neuronal Networks
https://works.swarthmore.edu/fac-math-stat/263
https://works.swarthmore.edu/fac-math-stat/263Mon, 12 Feb 2024 07:33:12 PST
Reconstructing the recurrent structural connectivity of neuronal networks is a challenge crucial to address in characterizing neuronal computations. While directly measuring the detailed connectivity structure is generally prohibitive for large networks, we develop a novel framework for reverse-engineering large-scale recurrent network connectivity matrices from neuronal dynamics by utilizing the widespread sparsity of neuronal connections. We derive a linear input-output mapping that underlies the irregular dynamics of a model network composed of both excitatory and inhibitory integrate-and-fire neurons with pulse coupling, thereby relating network inputs to evoked neuronal activity. Using this embedded mapping and experimentally feasible measurements of the firing rate as well as voltage dynamics in response to a relatively small ensemble of random input stimuli, we efficiently reconstruct the recurrent network connectivity via compressive sensing techniques. Through analogous analysis, we then recover high dimensional natural stimuli from evoked neuronal network dynamics over a short time horizon. This work provides a generalizable methodology for rapidly recovering sparse neuronal network data and underlines the natural role of sparsity in facilitating the efficient encoding of network data in neuronal dynamics.
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Victor J. Barranca