Fig. 4.
Examples of dynamical and behavioral representations. a For C. elegans, a histogram of projections onto the first two postural modes, or “eigenworms” (the left two curves in Fig. 3b) shows a low-dimensional structure that can be parameterized by a single phase variable, ϕ. b Fitting the dynamics of this variable to a deterministic dynamical system yields this phase map, with forward and backward locomotion naturally emerging as traveling wave trajectories at the top and bottom, respectively, and two fixed points in the middle corresponding to two different pause states (a and b are adapted from [54]). c An alternative approach to represent C. elegans behavior is via motif-finding. Here, time-series of projections onto the eigenworms are scoured for repeated patterns (e.g. the blue and red curves here). These patterns are then catalogued and used as the basis for a behavioral representation (adapted from [55]). d Instead of using dynamical motifs directly, the worm’s behavior can be captured as a sequence of postures, as seen in this example from [77]. e The approach taken by [56] was to fit an autoregressive hidden markov model (AR-HMM) to postural data of mouse movements, generated in a similar, but not identical, manner to that seen in Fig. 3c. Here, each P t is a vector of the animal’s postural mode values at time t, and S t is an underlying state that affects the dynamics of postural outputs. Here, arrows imply direct dependence (i.e. P t is a stochastic function of S t , Pt−1, and Pt−2, and so on). It is assumed that the time scale for changes in P is much faster than that for changes in S. This latter time scale, a parameter in the model, sets the distribution for the length of time that an animal stays within a particular behavioral state. f Average behavioral usage frequencies using an AR-HMM for four different mouse genotypes: Wild type, C57/BL6, as well as homozygous (Mut) and heterozygous (Het) mutations in the retinoid-related orphan receptor 1 β (Ror1 β) gene. g Distinct walking gaits found in the Mut (top) and C57/BL6 (bottom) mice (e-g adapted from [56]. Neuron 88(6), Alexander B. Wiltschko, Matthew J. Johnson, Giuliano Iurilli, Ralph E. Peterson, Jesse M. Katon, Stan L. Pashkovski, Victoria E. Abraira, Ryan P. Adams, Sandeep Robert Datta, Mapping Sub-Second Structure in Mouse Behavior, 1121-1135., Copyright 2015, reprinted with permission from Elsevier.) h An example of a time-frequency analysis representation from freely-moving fruitflies, where each set of axes represents a mode, and the colormap values indicate the continuous wavelet transform amplitudes for at each point in time. This approach allows for multiple time scales to enter the dynamical representation. i Probability density resulting from embedding points into 2-d such that two instances when a fruit fly is moving similar parts of its body at similar speeds are mapped nearby. Note the peaks and valleys. Here, the peaks represent stereotyped behaviors. j Break-down of the behavioral representation in i, with names for the behaviors within each of these regions manually labelled. Black lines are proportional to the transition probability between moving from one coarse region to another, with right-handedness implying the direction of transmission. (h-j adapted from [53, 76])