Spatiotemporal dynamics pervade the natural sciences, from the morphogen dynamics underlying patterning in animal pigmentation to the protein waves controlling cell division. A central challenge lies in understanding how controllable parameters induce qualitative changes in system behavior called bifurcations. This endeavor is particularly difficult in realistic settings where governing partial differential equations (PDEs) are unknown and data is limited and noisy. To address this challenge, we propose TRENDy (Temporal Regression of Effective Nonlinear Dynamics), an equation-free approach to learning low-dimensional, predictive models of spatiotemporal dynamics. TRENDy first maps input data to a low-dimensional space of effective dynamics through a cascade of multiscale filtering operations. Our key insight is the recognition that these effective dynamics can be fit by a neural ordinary differential equation (NODE) having the same parameter space as the input PDE. The preceding filtering operations strongly regularize the phase space of the NODE, making TRENDy significantly more robust to noise compared to existing methods. We train TRENDy to predict the effective dynamics of synthetic and real data representing dynamics from across the physical and life sciences. We then demonstrate how we can automatically locate both Turing and Hopf bifurcations in unseen regions of parameter space. We finally apply our method to the analysis of spatial patterning of the ocellated lizard through development. We found that TRENDy's predicted effective state not only accurately predicts spatial changes over time but also identifies distinct pattern features unique to different anatomical regions, such as the tail, neck, and body--an insight that highlights the potential influence of surface geometry on reaction-diffusion mechanisms and their role in driving spatially varying pattern dynamics.

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

BACKGROUND: Charting cellular trajectories over gene expression is key to understanding dynamic cellular processes and their underlying mechanisms. While advances in single-cell RNA-sequencing technologies and computational methods have pushed forward the recovery of such trajectories, trajectory inference remains a challenge due to the noisy, sparse, and high-dimensional nature of single-cell data. This challenge can be alleviated by increasing either the number of cells sampled along the trajectory (breadth) or the sequencing depth, i.e. the number of reads captured per cell (depth). Generally, these two factors are coupled due to an inherent breadth-depth tradeoff that arises when the sequencing budget is constrained due to financial or technical limitations. RESULTS: Here we study the optimal allocation of a fixed sequencing budget to optimize the recovery of trajectory attributes. Empirical results reveal that reconstruction accuracy of internal cell structure in expression space scales with the logarithm of either the breadth or depth of sequencing. We additionally observe a power law relationship between the optimal number of sampled cells and the corresponding sequencing budget. For linear trajectories, non-monotonicity in trajectory reconstruction across the breadth-depth tradeoff can impact downstream inference, such as expression pattern analysis along the trajectory. We demonstrate these results for five single-cell RNA-sequencing datasets encompassing differentiation of embryonic stem cells, pancreatic beta cells, hepatoblast and multipotent hematopoietic cells, as well as induced reprogramming of embryonic fibroblasts into neurons. By addressing the challenges of single-cell data, our study offers insights into maximizing the efficiency of cellular trajectory analysis through strategic allocation of sequencing resources.