Israeli study detects chaos in biological systems with new ‘deep learning’ model

Mathematical models have been created to successfully describe the behavior of complex phenomena ranging from weather systems to heartbeat patterns, and are being extended to describe the biological growth process.

A problem with such models is the appearance of a phenomenon known as chaos, that is, irregular behavior that is extremely dependent on very small changes in the initial conditions of the system. 

Knowing when chaotic conditions are present is a challenge for all such models. 

A paper authored by researchers at the Hebrew University of Jerusalem, addresses the difficulties of identifying chaos in biological models known as lineage trees, using the new mathematical techniques of deep learning. Their paper, “Detecting chaos in lineage-trees: A deep learning approach,” was published March 23 in Physical Review Research. 

Kicked cell cycle

To study chaos in biological systems, the authors looked at a growth model known as the kicked cell cycle (KCC). This model takes into account systems that grow and reproduce in a discrete form such as cell division which can be represented as tree-like. The KCC model has been applied to analyzing cell-cycle duration in successive generations of cancer cell lineages. 

According to the authors, the KCC model can correctly predict the observed distributions of cell-division times in experimentally observed populations of dividing cells, “as well as specific nontrivial correlation patterns within lineages.”

If cells do follow KCC dynamics, they note, “one immediate question is whether they likewise display different qualitative dynamical behavior.” In particular they seek to know if organisms can transition into chaotic dynamics as a response to environmental or local stresses––for example bacteria trying to adapt to antibiotic exposure. 

Detecting chaos with Lyapunov exponent

One of the ways to detect chaos is to try to estimate the value of a mathematical construct known as the Lyapunov exponent, a concept developed by 19th Century Russian mathematician Aleksandr Lyapunov in his study of dynamical systems. The Lyapunov exponent is a measure of the rate at which a dynamical system varies from a projected trajectory. 

Of interest is the largest value of the Lyapunov exponent (LLE). A negative value of the exponent indicates a steady state. When its value rises above 0, the system enters into a chaotic state. Thus, for detecting the presence of chaos, one need only know if the LLE is above or below zero. 

Deep learning

Estimating the largest Lyapunov exponent can be very challenging, and the authors have looked at ways to utilize machine learning programs to aid this effort. In particular, they examined the class of learning algorithms known as deep learning, which have contributed to recent major breakthroughs in such fields as computer vision and natural language processing.

The flexibility inherent in deep learning techniques allowed them to adapt existing models to their task of studying processes of biological growth such as cell division, in particular what they call tree trajectories. 

After training their model on kicked cell cycle-generated time series with a fixed noise level, they composed 11 increasingly noisy test sets of 2000 KCC parameter combinations. They then compared the prediction of largest Lyapunov exponent obtained by their model to predictions generated by standard methods of LLE estimation. 

A distinct advantage

Analysis of their results revealed a distinct advantage to their deep learning approach when the noise levels are low. 

The authors summarized, “[W]hile for purely deterministic inputs our method is slightly inferior as a classifier, its performance is stable under increasing levels of dynamical noise, and for relatively mild levels of noise (<0.1%) is superior in classification ability.” 

In concluding their report, however, the authors warn that certain limitations, some theoretical and some practical, remain before wide application of their model could be implemented. 

“On the theoretical front, it is not always clear when and to what extent mathematical properties of a phenomenological model actually exist in the process being modeled," the authors wrote. "But even if we believe that a given real world process can be chaotic for certain values of its control parameters, it is not obvious that we can extrapolate the success of our chaos detection algorithm on synthetic data and expect it to perform equally well on real data.  

“If an algorithm such as ours outputs a positive LLE estimation for an experimental time series, it would definitely count as evidence for chaotic behavior, but not necessarily strong evidence,” they warn.  “One way to circumvent this problem is to estimate the LLE for many different values of the control parameters. If the emerging pattern of LLE values is consistent with the one expected from the model, this may count as strong evidence of chaotic behavior. 

"These sorts of experiments could also boost our confidence in the ability of the algorithm to detect chaos in any single given experimental data set,” they conclude.

Hagai Rappeport et al. Detecting chaos in lineage-trees: A deep learning approach. Physical Review Research 4 (23 March 2022). 

DOI: 10.1103/PhysRevResearch.4.013223