Detecting chaos in lineage-trees: A deep learning approach

Abstract

Many complex phenomena, from weather systems to heartbeat rhythm patterns, are effectively modeled as low-dimensional dynamical systems. Such systems may behave chaotically under certain conditions, and so the ability to detect chaos based on empirical measurement is an important step in understanding these processes. Classifying a system as chaotic entails estimating its largest Lyapunov exponent (LLE), which quantifies the average rate of convergence or divergence of initially close trajectories in state space. Estimating the largest Lyapunov exponent from observations of a process is especially challenging in low-dimensional systems affected by a small amount of dynamical noise, which are used to model many real-world processes, and in particular biological systems. We describe a method for accurately estimating the largest Lyapunov exponent from noisy data, based on training deep learning models on synthetically generated trajectories. Once the deep learning models are trained, they can be used to detect the LLE from empirical data without any assumption on the underlying dynamics. Our method naturally extends to different input topologies, allowing us to analyze tree-shaped data, characteristic of inheritance dynamics, where near-identical replication offers a unique and hitherto unstudied avenue to detect chaos. We also characterize the input information extracted by our models for their predictions, allowing for a deeper understanding into the different ways by which chaos can be analyzed in different topologies.