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Abstract Timelapse microscopy has recently been employed to study the metabolism and physiology of cyanobacteria at the single-cell level. However, the identification of individual cells in brightfield images remains a significant challenge. Traditional intensity-based segmentation algorithms perform poorly when identifying individual cells in dense colonies due to a lack of contrast between neighboring cells. Here, we describe a newly developed software package called Cypose which uses machine learning (ML) models to solve two specific tasks: segmentation of individual cyanobacterial cells, and classification of cellular phenotypes. The segmentation models are based on the Cellpose framework, while classification is performed using a convolutional neural network named Cyclass. To our knowledge, these are the first developed ML-based models for cyanobacteria segmentation and classification. When compared to other methods, our segmentation models showed improved performance and were able to segment cells with varied morphological phenotypes, as well as differentiate between live and lysed cells. We also found that our models were robust to imaging artifacts, such as dust and cell debris. Additionally, the classification model was able to identify different cellular phenotypes using only images as input. Together, these models improve cell segmentation accuracy and enable high-throughput analysis of dense cyanobacterial colonies and filamentous cyanobacteria. 

Abstract In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $$\sigma _{c}$$. We provide explicit bounds on $$\sigma _{c}$$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE. 

Abstract We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of$$C^{\infty }$$ $C^{\infty }$bump functions of varying support sizes.We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available athttps://github.com/MathBioCU/WENDy. 

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the ℓ0-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions. 

Interacting particle system (IPS) models have proven to be highly successful for describing the spatial movement of organisms. However, it is challenging to infer the interaction rules directly from data. In the field of equation discovery, the weak-form sparse identification of nonlinear dynamics (WSINDy) methodology has been shown to be computationally efficient for identifying the governing equations of complex systems from noisy data. Motivated by the success of IPS models to describe the spatial movement of organisms, we develop WSINDy for the second-order IPS to learn equations for communities of cells. Our approach learns the directional interaction rules for each individual cell that in aggregate govern the dynamics of a heterogeneous population of migrating cells. To sort a cell according to the active classes present in its model, we also develop a novel ad hoc classification scheme (which accounts for the fact that some cells do not have enough evidence to accurately infer a model). Aggregated models are then constructed hierarchically to simultaneously identify different species of cells present in the population and determine best-fit models for each species. We demonstrate the efficiency and proficiency of the method on several test scenarios, motivated by common cell migration experiments.