The shape of worm behavior
Why study worm behavior?
These particular worms (C. elegans) are widely used model organisms, meaning they are convenient for experiments:
They are translucent, so you can see many of their internal systems with regular microscopes and cameras.
It is easy to manipulate their genes, which you may do to simulate a genetic disorder that occurs in humans.
You can produce many clones with the exact same genetics, so you can create large, well-controlled experiments.
They have only 302 neurons (compare to 70,000,000 in mice and 86,000,000,000 in humans) but still exhibit complex neurological processes (e.g., learning), so it is easier to decipher how those neurological processes work.
A large percentage of their genes correspond directly to human genes. For example, C. elegans have a gene related to their coordination which is very similar to a gene in humans that affects the inner ear and is linked with some balance disorders.
These worms are convenient for experimentation and can give insight into the functioning of the human body and various human disorders and diseases, so C. elegans research is an efficient stepping stone for creating and iterating on theories and treatments for human problems.
Topological data analysis is based on understanding the "shape" of a data set. Here, the data set is videos of single worms (C. elegans) moving around various environments. A worm's "skeleton" is drawn for each frame of a video and can be recorded as a 100-dimensional vector (list of 100 numbers). This skeleton is a mathematical representation of the posture of the worm for that point in time. The skeletons from a series of frames (say, 20 frames/skeletons in a row) gives a movement of the worm. All of these movements can be plotted together in a "movement space," like the one below.
A movement space: movements of length 20 from a ~13 second video of a worm (ie, the first two principal components of a time delay embedding of delay 1 and length 20). The movements are color-coded in temporal order from purple to yellow.
One type of shape that can be extracted from the movement space are loops. The biggest loops in the figure are the purple/blue loop and the green/yellow loop. These loops represent specific behaviors of the worm: crawling forward and crawling backward (via undulation, the way a snake moves). Other types of movements (or aberrations from "normal" movements) create other loops or amend existing ones.
The size and shape of these loops change when the movements change: for example, a "smaller" movement -- like crawling with smaller amplitude of body bends -- makes a smaller loop. We do statistics and machine learning on these loops to create a wholistic measure of movements and variation between them.
Below: videos of worms (C. elegans) from an experiment on the effect of environment on behavior. The worms are immersed in liquids/gels of varying viscosity.