Dimensionality and Dynamics in the Behavior of C. elegans
(A) The covariance matrix of fluctuations in angle C(s, s′). The inhomogeneity along the diagonal shows that the normal modes of the motion are not sinusoidal but the smooth structure of C(s, s′) means that a small number of modes are significant. (B) We find the eigenvalues of C(s, s′) and compute σ2K, the fraction of the total variance (integrated along the body of the worm) captured by keeping K modes (see Materials and Methods). (C) Associated with each dominant mode is an eigenvector and we refer to these as eigenworms uμ(s). The population-mean eigenworms (red) are highly reproducible across individual worms (black). (D) The fraction of variance, σ̃2K, at each point along the body curve captured by keeping K modes (K = 1 to 4, from bottom to top curve). The overall error in reconstruction of the worm body curve decreases as the number of modes increases, but does so inhomogeneously. (E) In response to strong thermal stimuli, reconstructions using the eigenworms of spontaneous crawling continue to account for most of the shape variance. Worm images are recorded at times synchronized to a heat pulse and we display σ2K aligned with this pulse (red line). (K = 1 to 4, from bottom to top curve).