Zvi Rosen

What Makes a Neural Code Convex

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

Parameter-free methods distinguish Wnt pathway models and guide design of experiments

Significance The canonical Wnt/β-catenin signaling pathway is important for essential cellular functions and is implicated in many diseases. We introduce a new mathematical model that focuses on β-catenin degradation and protein shuttling between cellular compartments. We compare our model to others and show that all fit to time-dependent experimental data. To evade this parameter problem, we use algebraic methods and characterize model features that are independent of the choice of parameter values. We find that multiple responses to Wnt are feasible under certain conditions for the new model, but not for the others; moreover, we provide dependencies between species (variables) that inform future experiments and model discrimination. We also highlight the wide applicability of these tools across problems in systems biology. The canonical Wnt signaling pathway, mediated by β-catenin, is crucially involved in development, adult stem cell tissue maintenance, and a host of diseases including cancer. We analyze existing mathematical models of Wnt and compare them to a new Wnt signaling model that targets spatial localization; our aim is to distinguish between the models and distill biological insight from them. Using Bayesian methods we infer parameters for each model from mammalian Wnt signaling data and find that all models can fit this time course. We appeal to algebraic methods (concepts from chemical reaction network theory and matroid theory) to analyze the models without recourse to specific parameter values. These approaches provide insight into aspects of Wnt regulation: the new model, via control of shuttling and degradation parameters, permits multiple stable steady states corresponding to stem-like vs. committed cell states in the differentiation hierarchy. Our analysis also identifies groups of variables that should be measured to fully characterize and discriminate between competing models, and thus serves as a guide for performing minimal experiments for model comparison.

Algebraic Matroids with Graph Symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroids with symmetries we introduce combinatorial invariants capturing structural properties of the rank function and its limit behavior, and obtain proofs which are purely combinatorial and do not assume algebraicity of the matroid; these imply and generalize known results in some specific cases where the matroid is also algebraic. These results are motivated by, and readily applicable to framework rigidity, low-rank matrix completion and determinantal varieties, which lie in the intersection of (a) and (b) where additional results can be derived. We study the corresponding matroids and their associated invariants, and for selected cases, we characterize the matroidal structure and the circuit polynomials completely.

The Geometry of Rank-One Tensor Completion

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format

Geometry of the Sample Frequency Spectrum and the Perils of Demographic Inference

d\times\dots\times d

Computing Algebraic Matroids

, the complete semialgebraic description of the completable region is found.

Matrix Completion for the Independence Model

Numerous studies in population genetics have been based on analyzing the sample frequency spectrum (SFS) summary statistic. Most SFS-based inference methods can display pathological behavior in optimization: some demographic model parameters can degenerate to 0... The sample frequency spectrum (SFS), which describes the distribution of mutant alleles in a sample of DNA sequences, is a widely used summary statistic in population genetics. The expected SFS has a strong dependence on the historical population demography and this property is exploited by popular statistical methods to infer complex demographic histories from DNA sequence data. Most, if not all, of these inference methods exhibit pathological behavior, however. Specifically, they often display runaway behavior in optimization, where the inferred population sizes and epoch durations can degenerate to zero or diverge to infinity, and show undesirable sensitivity to perturbations in the data. The goal of this article is to provide theoretical insights into why such problems arise. To this end, we characterize the geometry of the expected SFS for piecewise-constant demographies and use our results to show that the aforementioned pathological behavior of popular inference methods is intrinsic to the geometry of the expected SFS. We provide explicit descriptions and visualizations for a toy model, and generalize our intuition to arbitrary sample sizes using tools from convex and algebraic geometry. We also develop a universal characterization result which shows that the expected SFS of a sample of size n under an arbitrary population history can be recapitulated by a piecewise-constant demography with only κn epochs, where κn is between n/2 and 2n−1. The set of expected SFS for piecewise-constant demographies with fewer than κn epochs is open and nonconvex, which causes the above phenomena for inference from data.