Zebrafish backbone shapes while free swimming are described by just a few orthonormal basis functions

How can one quantify the free swimming behavior of zebrafish without a priori criteria to classify the observed behavior? To answer this question, we applied singular value decomposition (SVD) [24] to the sequence of zebrafish backbone shapes as they evolve during swimming. SVD is a parameter-free linear algebra technique that, in our application, reduces all shapes to a small number of linearly independent “eigenshapes.” We recorded video of individual larvae swimming freely in quasi-2D in a petri dish at a rate of 500 frames per second (fps) using a high-speed camera (S1 Movie and Materials and Methods). Each video typically had 4–5 swimming episodes of duration ~250 ms each, separated by pauses. The video was divided into movies of individual swimming bouts which were analyzed separately. The movies consisted of a sequence of m frames at times t_i. Fig 2A shows still images from a representative swimming bout (see Fig 2A).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Decomposition of free swimming zebrafish larval backbone shapes into orthonormal basis.

(A) Still images of a swim bout from a representative movie of free swimming zebrafish larva, recorded at 500 fps. (B) Spline fit of fish backbone (cyan). Tangent vectors (black arrow) at 10 evenly spaced segments along the backbone from head (s₀ = 0) to tail (s₉ = 9) point along a direction θ(s_j). (C) Swim bout from A parameterized by Δθ(s_j,t_i) = θ(s_j,t_i)–θ(0,t_i). (D) Singular value decomposition of Δθ into eigenshapes V_k(s_j) (j = 1, 2… 9), the first three of which are plotted (red, blue, and green, respectively). The light colors correspond to eigenshapes determined from analyzing individual fish from a population, and dark colors are the collective eigenshapes from analyzing the entire population at once. (E) Bar plot of % weights (, where S_kk are the singular values) of each eigenfunction V_k(s_j). The right axis in cyan shows the cumulative contribution of each eigenshape. The first three eigenshapes contribute 96% of the total variance in Δθ.

https://doi.org/10.1371/journal.pone.0128668.g002

Next, we applied background subtraction algorithms and threshold procedures to the movies to extract the backbone shape of each fish as a sequence in time (see S1A–S1F Fig). The backbone shape in each frame was fitted to a cubic spline (see Materials and Methods and S1G Fig). We sampled the spline at 2 to 100 points (S2 Fig), and found that n = 10 points were sufficient to converge the spline to camera pixel resolution of the backbone. This procedure allows an accurate but very compact representation of the backbone trajectories, enabling low-dimensional visualization. As shown in Fig 2B, the 10-points spline curve was converted into a one-dimensional array of 10 spine angles θ(s_j,t_i), measured along the normalized arc lengths of the fish from head (s₀ = 0) to tail (s₉ = 1). All spine angles are given relative to the head angle, Δθ(s_j,t_i) = θ(s_j,t_i)–θ(s₀,t_i).

Δθ(s_j,t_i)is an m × n matrix in which each row is the spine angles from head to tail, and successive rows down the matrix represent snapshots at different times. In Fig 2C, Δθ = 0 at the head and swings out to maximum values towards the tip of the tail. The matrix Δθ(s_j,t_i) was then decomposed into a set of n = 10 linearly independent, orthonormal basis functions V_k by SVD [24], as given by the relation (1) The basis functions represent eigenshapes of the zebrafish backbone (Fig 2D). Their linear combination reconstructs the spline-fitted backbone shapes exactly, according to Eq (1). Each U_k(t_i) represents the amplitude of the k-th basis function V_k(s_j) at each time point t_i. S_kk is an n × n diagonal matrix of singular values. S_kk is conventionally normalized to 1, such that each singular value represents the fractional contribution a basis function makes to the overall swimming behavior. The basis functions are sorted from most important (largest singular value; k = 1) to least important (smallest singular value; k = n). The key element of this analysis is that many of the singular values in the matrix S_kk may be small, and thus many basis functions can be left out of the sum in Eq (1) with negligible effect. As shown in Fig 2E, performing SVD on all movies (N = 115) from a population of 20 fish reveals that 96% of the variation in Δθ is accounted for by the first three eigenshapes only, i.e. taking the summation in Eq (1) only up to n = 3. The residual error was ~7% between the spine angles Δθ(s_j,t_i) and reconstructed spine angles Δθ^r(s_j,t_i) using the first three eigenshapes (see S3 Fig).

Plotting the three eigenshapes (light blue, red, and green in Fig 2D) from individually analyzed movies shows that the basis functions were similar across the population of fish. The collective eigenshapes shown in dark colors (dark blue, red, and green) were obtained by analyzing all movies together. Translated into real space (S4 Fig), the three basis functions correspond to zebrafish shapes with the following features: a single bend at the midpoint (blue), bends at the head and at the tail (red), and bends at the head, midpoint, and tail (green).

Stereotyped behaviors are visualized in a quantitative low-dimensional postural space

Based on this analysis method, any zebrafish free swimming bout can be represented as a trajectory in the low-dimensional postural space spanned by the three collective eigenshapes described above. Fig 3 shows an example of a zebrafish turning bout and how it can be visualized as a trajectory in the postural space (S5 Fig shows a corresponding scooting bout). At each time point t_i of the movie, the zebrafish backbone shape is represented by a set of three amplitudes {U_k(t_i)} with k = 1, 2 and 3. Fig 3B plots these three amplitudes U_k vs. time, and in Fig 3C the three amplitudes define the coordinates of this fish’s trajectory in the three-dimensional postural space. Represented in this manner, a swimming bout appears as a sequence of cycles of shapes. For this particular example the swim bout has four cycles as shown by dashed lines in Fig 3B that explore different regions of the postural space in Fig 3C.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Representation of free swimming zebrafish in low-dimensional postural space.

(A) Still images of a representative turning bout during free swimming. As discussed in the text, it is convenient to divide the bout into a “turn” region (t = 50–100 ms) followed by a “scoot” region (100–250 ms). (B) Plot of the amplitudes U₁(t), U₂(t), and U₃(t) of the three collective eigenshapes corresponding to the movie in A. Each amplitude undergoes multiple oscillation cycles before returning to zero. The regions marked by dashed lines and labeled as cycles (1–4) in U₁(t), U₂(t), and U₃(t) are obtained from the oscillation cycles in U₁(t). The colored dots mark time points corresponding to the still images in A. (C) Representation of a turn bout in postural space. The three-dimensional coordinates of the trajectory are the amplitudes U₁(t), U₂(t), and U₃(t) in B. In this space, the bout involves a turn region (t = 50–100 ms), represented as a bent ellipse (cycle 1), followed by a scoot region (t = 100–250 ms) represented as multiple cycles (2–4) along the flat ellipses, and a final return to the rest behavior. Throughout, time (0–250 ms) is represented by the black—magenta colormap. An analysis of a representative scooting bout is shown in S5 Fig.

https://doi.org/10.1371/journal.pone.0128668.g003

Previous studies have categorized free swimming of zebrafish into two behavioral states, the “R-turn” and “scoot”, based on a limiting angle of the backbone [9, 10]. Turns correspond to a major change in swimming direction and are specifically defined as a swimming bout at the start of which the tail makes an angle >30–40° relative to the head. In contrast, scoots are swimming bouts with no significant change in direction and are defined by the tail making an angle relative to the head <30–40° [9, 10]. As shown in S5 Fig, a scoot bout is represented in postural space by oscillation cycles predominantly in U₂ and U₃, returning to a fixed point corresponding to a “resting” fish. In contrast, the turn bout in Fig 3C takes an excursion along the U₁–U₃ plane lasting between one half and one whole oscillation of cycle 1, before shifting to cycles 2–4 in U₂ and U₃ similar to those in the scoot trajectory (see cycles 1–5 in S5C Fig) and eventually returning to the resting behavior. (See S2 and S3 Movies) for demonstrations of a turn and a scoot bout in 3-D postural space, respectively.) The sign of U₁ corresponds to the direction of the turn, with U₁ < 0 corresponding to a right turn and U₁ > 0 to a left turn, and we consider these cases to be mirror images of the same behavior. To distinguish the different cycles of a trajectory in postural space, we found it useful to name the excursion in the U₁–U₃ plane “turn region”, the oscillation cycles in the U₂–U₃ plane “scoot region” and finally decaying to zero amplitude as “resting region.” Thus a swimming bout can progress through these regions in postural space in succession. Time is color coded from black to magenta on the trajectory in Fig 3C, corresponding to the dots in Fig 3A.

Multi-dimensional scaling of trajectories from postural space reveals the behavioral space of zebrafish free swimming

How does this approach extend to an ensemble of swimming bouts? To determine if multiple trajectories can be similarly categorized, we compared trajectories from different fish, normalizing their time axes by matching the duration of the first oscillation period in U₁(t) termed “cycle 1;”(see S1 File and S6A Fig). The histogram of normalized time was in the range of 15–33 ms of data set as shown in S6B Fig We also normalized the sign of the trajectory amplitudes, so that handedness is ignored in our analysis. Our data set included both younger (7–8 dpf) and older larvae (9–10 dpf) to determine if factors such as age alter free swimming behavior significantly.

We next used multidimensional scaling (MDS) [25] to quantify the differences between trajectories (and between the cycles of trajectories), and to determine whether they cluster into discrete behavioral patterns. Past behavioral studies have used both non-linear and linear clustering methods [19, 20], and here we applied a linear method to ensure robustness. MDS takes as input the Euclidean “distance” d_αβ between every pair of postural space trajectories α and β over one oscillation cycle (see Materials and Methods and S1 File), and determines the optimal low-dimensional space required to embed the distance data. In this space, shown in Fig 4A, each oscillation cycle of a trajectory is represented by a point, and the distance between two points represents how similar the trajectories in the pair are. Thus, MDS generates a “behavioral space” in which distinct behavioral patterns would appear as well-separated clusters of points.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Behavioral space of free swimming zebrafish.

(A–B) Ensemble of swim bout trajectories (N = 115) embedded by multi-dimensional scaling (MDS) into low-dimensional “behavioral” space. Each cycle of every trajectory is represented by a single point (filled circles for younger larvae (N = 8; 7–8 dpf), open circles for older larvae (N = 12; 9–10 dpf)). The distances between points reflect their behavioral similarities/differences. MDS dimensions 1 and 2 (A) reveal behavioral regions corresponding to turns (red points), scoots (green) and rests (blue). The shaded ellipses demarcate the points from cycles 1–3 of the trajectory (orange, light green, green, respectively. S8A Fig displays the cycles individually). MDS dimensions 2 and 3 (B) reveal differences with zebrafish age (solid ellipse for younger larvae; dotted ellipse for older larvae). The ellipses were determined using PCA analysis of the behavioral space in MDS dimension 1 and dimension 2 for each cycle in (A) and MDS dimension 2 and dimension 3 for cycle 1 in (B). The principal axes of the ellipses are the singular values from PCA. Simulated trajectories from the neuro-kinematic model in Figs 5 and 6 and its training data are shown in diamonds and circles with black outline, respectively. (C) Amplitudes U₁, U₂, and U₃ vs. normalized time of the trajectories in A and B from younger larvae. (D) Same plot for older larvae. Shaded areas demarcate each cycle of the trajectory. (E–F) Postural space representation of the trajectories in C and D, respectively. Throughout, color represents the location in the behavioral space and cycle number: in the RGB colormap, red and green channels correspond to position along MDS dimension 1 and 2, respectively, and the blue channel to the cycle number.

https://doi.org/10.1371/journal.pone.0128668.g004

In our analysis of an ensemble of swimming bouts, three dimensions (not to be confused with the three axes of postural space) captured ~85% of the behavioral variability contained in the set of all trajectories (S7A Fig). In Fig 4A and 4B we plot coordinate pairs 1–2 and 2–3, respectively. Points are coded as follows: filled circles are younger (7–8 dpf) and open circles older larvae (9–10 dpf). Color corresponds to position in the behavioral space and time (i.e. oscillation cycle labeled in Fig 4C and 4D): the red hue represents position along MDS dimension 1 (< 0 for less red, > 0 for more red), green hue to dimension 2 (< 0 for less green, > 0 for more green), and blue hue for cycle number (less blue for early oscillation cycles and more blue for later oscillation cycles).

The behavioral space in Fig 4A represents graphically both the temporal sequence of patterns during a swim bout and the variability in behavior among different bouts. The first oscillation cycle (points under red shaded area) captures most of this variability, spanning almost the full range of MDS dimensions 1 and 2. As seen in S7B and S7C Fig, MDS dimensions 1 and 2 correlate well with turning and scooting. Turn-like trajectories have large positive values of MDS dimension 1 and negative values of MDS dimension 2 whereas scoot-like trajectories have small values of MDS dimension 1 and positive values of MDS dimension 2. In contrast, the second oscillation cycle (points under green shaded area; see also S8A Fig) displays much less variability, with all points localized in one region. This region of behavioral space also contains the scoot-like points in cycle 1, indicating that scooting fish undergoing a scoot simply repeat the same pattern of postures over multiple cycles. Turning fish, on the other hand, generate a different oscillation cycle 1 before going to the scoot-like cycles. Further oscillation cycles (e.g. cycle 3 points under light green shaded area in Fig 4A; green, purple, blue, and gray shaded areas; see also S8A Fig) either overlap with cycle 2 or shift to a third locus of points close to the origin in behavioral space. Inspection of the data traces indicates that the latter correspond to “terminated” cycles, i.e. oscillations decaying to the resting behavior.

These observations confirm the qualitative picture formed from individual trajectories. Fig 4C and 4D color-code the amplitudes of the three eigenshapes for every recorded swim bout as in the MDS analysis in Fig 4A (trajectories from younger and older fish are plotted separately for clarity). Most of the fish-to-fish variability lies along U₃ in the first cycle, as seen by inspection. Although the contribution of U₃ to the zebrafish shape is less than the other two modes (<5%, see Fig 2E), it is essential to quantify fish-to-fish variation. In Fig 4E and 4F, the same set of trajectories is plotted in postural space (see also S4 and S5 Movies). As for the single trajectories in Fig 3C and S5 Fig, the postural spaces in Fig 4E and 4F show some trajectories (red) with excursions along U₁–U₃ (a so-called “turn region”) and others (green) lying mostly along the U₂–U₃ plane (a “scoot region”).

Fig 4A suggests that each cycle of a swim bout can be classified according to a region in the behavioral space. Applying a k-means clustering algorithm to the data in Fig 4A and using the gap method to evaluate the optimal number of clusters (see S9 Fig) reveals that behaviors can be grouped into three clusters corresponding to turn-like behavior, scoot-like behavior, and resting behavior. The first two groupings partially overlaps with classifications of scoots and R-turns based on bend angle <40° and >40° used in the literature (see S7B and S7C Fig). Nevertheless, it is important to note that the clustering is weak, with no strong boundaries in the data, and points occupying regions of behavioral space between these groupings. To assess the strength of the clustering more quantitatively, we constructed a behavioral probability distribution from the data in Fig 4A. While scoot-like behavior and rests appear as distinct peaks (S9 Fig), the turn-like behavior is not a separate maximum in the probability distribution along MDS dimensions 1 and 2, but rather a wing of the scoot-like maximum. This suggests that turn-like and scoot-like trajectories form more of a continuum of behavioral patterns than discrete behavioral states. Visual inspection of trajectories in postural space (Fig 4E and 4F) supports this view. This broad range in behavior does not appear to be a consequence of variability between fish. Trajectories from individual fish reveal the same behavioral variability within the bouts of the single fish as for the population, and there is no strong clustering of the single-fish trajectories in behavioral space (see S10A Fig).

The behavioral space also reveals differences between age groups. In Fig 4A (left panel) showing MDS dimensions 2 and 3, trajectories from younger (filled symbols) and older (open symbols) larvae occupy separate regions. The distinction with age is even more apparent for cycles 1 and 2 (S8B Fig). Inspection of Fig 4C, 4D, 4E and 4F shows that older fish display more variation in U₃ in cycle 1 in comparison to younger fish; their turn interval (red trajectories) is considerably more bent than that of the younger larvae, indicating a systematic difference in shape. Nevertheless, although fish of different age seem to adopt different postures, they undergo the same temporal sequence of patterns in the behavioral space (Fig 4A).

A simple neuro-kinematic model reproduces scoots and turns in postural space

Having established a robust, parameter-free method for analyzing zebrafish behavior, can we reproduce the observed patterns using a simple model of free swimming? We surmised that a simple neural network may suffice in describing the simple behavioral patterns observed. Constructing a network with a channel controlling the left and the right of the organism, we anticipated that signals of identical amplitude to both channels would produce a symmetric fish spine oscillation (i.e. a scoot) while unequal amplitudes would produce a turn in one direction. To test this idea, we adapted a coarse-grained neuro-kinematic model of xenopus swimming [23] with two input channels.

Our version of the neural network divides the fish backbone into a right and left half, each with 10 equal neural segments (s_j) of neurons interconnected by appropriate synapses (see S11A Fig). The right and left halves each receive an input trigger signal F_osc(s_j, t_i) in the form of a train of sharp pulses whose firing times τ_f, amplitudes a, and segment-to-segment delays d are model parameters (Fig 5). The output of each neural segment along the backbone is convolved with a bi-exponential function representing the temporal response of the neuromuscular junction. The result is the force exerted by the muscles F_m(s_j, t_i) at each segment on the left and right. From this force function, we determined the radius of curvature R(s_j,t_i) = F_m(s_j,t_i)/W(s_j) of the zebrafish backbone at each segment position, where W(s_j) is the fish stiffness along its spine. Finally we calculated the tangential angles Δθ(s_j,t_i) by integrating R^-1(s_j,t_i) = |dθ/ds|. The neuro-kinematic model thus produced simulated time traces of Δθ(s_j,t_i) on which we could perform the same dimensionality reduction analysis detailed above.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Spike train of zebrafish free swimming.

(A) Schematic of neuro-kinematic model depicting the fish backbone divided into ten segments on the right and left sides. (B) Spike train generated by an optimized neural model. The right and left spike trains are shown in red and green, respectively. The height of each spike represents the amplitude a of the stimulus. τ_f is the firing time for each spike in the head segment. The segment-to-segment delay d is the time difference between spikes in adjacent segments of the fish backbone.

https://doi.org/10.1371/journal.pone.0128668.g005

Using a genetic algorithm, we optimized the model parameters τ_f, a, and d and the stiffness function W(s_j) against two experimental data traces (see S1 File and S11B Fig). Fig 5 shows the spike train generated by an optimized neural network model of a turn. In Fig 6A the fish backbone from neural network simulation (red dotted line) is compared with the one from experimental data (cyan). Fig 6B and 6C show that three eigenshapes are again sufficient to describe the neuro-kinematic model’s behavior. In order to simulate real larval free swimming accurately, we found that our model stiffness function W(s_j) had to decrease monotonically from the head toward the tail, indicating greater flexibility towards the tail, then increase again over the last tail segment (Fig 6D). We also varied the model parameters to produce different stereotyped behaviors. We construct the postural space for the simulated fish free swimming from neural network model in Fig 6E. The simple neuro-kinematic model generates turn-like trajectories by increasing the amplitude a (Fig 5) to one side of the organism relative to the other and increasing the segment-to-segment delay d relative to a scoot during the first oscillation cycle (see S12 Fig). Thus, despite the high dimensionality of the parameter space of the model, two are sufficient to generate the observed behavioral patterns. A movie comparing a real swim and a neuro-kinematic model swim is shown in S6 Movie.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Minimal neuro-kinematic simulation of zebrafish free swimming.

(A) Still images from a free swimming zebrafish movie. Each snapshot shows the fish backbone reduced to a thin skeleton fitted to a cubic spline fit (cyan) and obtained from the neural simulation (red), respectively. The neural model was optimized against the experimental data as described in S1 File. (B) Decomposition of simulated swimming traces from neuro-kinematic model into eigenshapes V_k(s_j). The eigenshapes from the neural model of a turn (dashed light red, blue, and green lines) match those from experimental data of a turn (solid red, blue, and green). (C) Bar plot of % weights of each eigenshapes V_k(s_j) (j = 1, 2… 9) (left axis) and cumulative contribution of each eigenshape (right axis, cyan). The first three eigenshapes contribute 98% in total to the variance in Δθ. (D) Plot of head-to-tail zebrafish compliance (1/W(s_j)) obtained after optimization of neuro-kinematic model against experimental data. (E) Simulated neuro-kinematic model trajectory of a turning bout in postural space.

https://doi.org/10.1371/journal.pone.0128668.g006

We also tested the robustness of the model by adding varying levels of white Gaussian noise to the model parameters a, τ_f, and d (S13A and S13C Fig). We found that the free swimming behavior of the neuro-kinematic model was stable against noise as large as 70% and 90% in the amplitude a and delay d. Fluctuations in d simply affected the duration of the cycle. The model was most sensitive to noise in the firing time τ_f. Not surprisingly, noise that significantly altered the timing pattern between the left and right sides of the organism affected the cyclic motion required for free swimming. A behavioral space was constructed to compare the simulated trajectories with varying signal-to-noise ratio. S13D and S13E Fig show how trajectories scattered in behavioral space with increasing noise for a and τ_f, respectively, whereas the trajectories with the highest noise overlapped with those with the lowest noise in parameter d.

Ethics statement

All experimental procedures in this study were approved by the University of Illinois Institutional Animal Care and Use Committee protocol # 13327.

Animals

All experiments were performed on wild-type AB genotype zebrafish (Danio rerio) larvae age 7–10 dpf (days post fertilization). The larvae were raised without food until 6 dpf and were fed food every day (Hatchfry 0, Argent Labs) before taking data. These larvae were obtained from breeding of wild-type zebrafish adults. Zebrafish were maintained in a Z-hab mini system (Aquatic habitats, Beverly, MA) fish facility at 28.5°C on a 14h:10h light:dark cycle. The embryos were obtained from adult fish breeding and were raised at 28.5°C in 10% Hanks solution [8]. The total number of larvae used for experiments were 20: 8 (7–8 dpf) and 12 (9–10 dpf) old. All larvae were freely swimming in petri-dish before starting the experiment.

Imaging setup

We imaged free swimming larvae using a high speed camera (IDT vision N-3) mounted to a stereo microscope (Edmund Optics 6V head + 10X eyepiece). This allowed us to image of the entire 21-mm diameter Petri dish in which the larvae swam. A halogen light source (Edmund Optics MI-150 high-intensity illuminator) was used to illuminate the sample, and its light passed through a series of long-pass filters (780 nm and 830 nm) in order to obtain IR wavelengths (>810 nm). IR light is preferable to visible light since the larva cannot detect it [9]. A circular diffuser (100 mm dia. 120 grit ground glass diffuser, Edmund Optics) was used for uniform, homogeneous illumination. The diffuser and Petri dish were mounted on a custom-built stainless steel holder on a bread board. To ensure that the movement of the larva was restricted to the x-y plane as much as possible, measurements were carried out in water 2 mm in depth.

Behavioral experiment

For each experiment, a single 7–10 dpf fish larva was placed in a 21-mm Petri dish in 10% Hanks solution at room temperature. The larvae were illuminated from the bottom with IR light. We recorded videos of a free swimming larva at 500 fps using a high speed camera (Diagnostic Instruments) and the Motion Studio Software Suite. Each video typically had 4–5 swimming episodes ~250 ms with intermittent pauses. There were a total of 18 videos from 7–8 dpf and 21 from 8–10 dpf old fish. All videos were recorded when larvae were swimming in the center of the Petri dish, away from the dish edges that obscure the fish and making backbone shape extraction difficult. We used the frame difference method [27] with an appropriate threshold to extract swim bouts from each video. A total of 115 movies containing one free swimming bout each were collected for data analysis. All free swim movies were analyzed without selection.

Image segmentation

All the zebrafish free swimming movies were analyzed using the image and video processing toolbox of MATLAB. A stepwise image segmentation algorithm was followed. The images were first preprocessed using a customized background subtraction algorithm. Next, the image was thresholded to 1 bit (black/white) to yield a fish outline. A skeletonization algorithm was applied to find the center of the outline, corresponding to the backbone. A cubic spline was then fitted to n points evenly sampling the fish backbone along the backbone arc length. Each step is explained in S1 File and is shown in S1 Fig.

Eigenshape analysis

Each fish swimming bout was of 180–200 ms duration which accounts for 90–100 frames in total. We took 140 frames in total, including the resting behavior preceding and following the swimming bout of the fish. We analyzed a total of 115 swimming bouts from 20 different fish. We performed singular value decomposition as described above on the matrix Δθ(s_j,t_i) containing m = 16100 total frames from the catenation of all recordings of free swimming fish. To analyze multiple fish trajectories together, the time traces of the amplitudes {U_k(t_i)} (with k = 1, 2 and 3) were scaled and shifted in time to maximize the overlap between trajectories (S1 File). For Fig 4, we determined the “distance” between trajectories from the Euclidean distance d_αβ between pairs α,β of time-aligned spine bend angles Δθ^α,β(s_j,t_i). Based on Eq (2) and the orthonormality of the basis functions that compose Δθ, it follows that (S1 File): (2) where is the normalized time for trajectory α.

Neural network model

We used a modified version of the neuro-kinematic model described in [23], with 10 neurons on each side, instead of 25. A detailed description of the model is in S1 File. Briefly, the model parameters were the firing times τ_f, stimulus amplitudes a, and segment-to-segment delays d. The number of half cycles of tail motion, n_c, was based on our experimental values (6–8 half cycles observed in each swimming bout). The three model parameters were adjusted by a genetic algorithm to minimize the difference between experimental and simulated trajectories Δθ(s_j,t_i). The genetic algorithm used 50 family members with constraints on each variable obtained from eigenshape data in each generation and typically converged after ~50 generations. The step-wise algorithm for optimization is explained in S1 File.