2.1 Data

The behavioral data used in the present article comes from a publication that examined the kinematic of free exploration in zebrafish larvae [24]. The experimental design (Fig 1a) enables recording the trajectories of multiple freely swimming larvae aged 5-7 days at temperatures of 18°C, 22°C, 26°C, 30°C, and 33°C. At each temperature, the trajectories of multiple fish are combined into a single dataset, and a set of kinematic parameters is extracted at each bout n, such as the angular change in heading direction, the time elapsed since the previous bout and the traveled distance (see Material and methods Sect 4.1). Water temperature was found to systematically impact the statistics of navigation, leading to qualitatively different trajectories as illustrated in Fig 1b. As the temperature increases, trajectories tend to become more winding and erratic. We have also re-analyzed a second dataset of long-trajectories for 18 fish tracked individually for over two hours at 26°C, in order to assess inter-individual variability (see Material and methods Sect 4.1).

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Fig 1. Behavioral and neuronal datasets:

(a) Overview of the experimental setup: Zebrafish larvae are free to move in a tank that is kept at a desired constant temperature by a Peltier module. An imaging system records images of the fish from above at a rate of 25 frames per second. The upper right panel provides a close-up view of a larva in a raw image. Adapted from [24]. (b) Example trajectories of zebrafish larvae in 2D space at various temperatures. Each point represents a swim bout, with the color indicating the corresponding re-orientation angle defined in panel c. The trajectories’ starting points are denoted by black arrows. (c) Convention used for the reorientation angle () between two consecutive swim bouts (n and ). (d) Distribution of re-orientation angles () for each ambient temperature. The grayed-out area corresponds to the re-orientation angles classified as forward bouts by a thresholds at . (e) Diagram of the Anterior Rhombencephalic Turning Region (ARTR) in larval zebrafish. Adapted from [29]. (f) Example ARTR activity at 22°C. Top : Raster plot of neurons located in the left and right ARTR (blue and red respectively). Bottom : Mean activity m_L and m_R of neurons in the left and right ARTR. (g) Mean activities of the ARTR for all recordings in the dataset. The blue contour line represents 90% of the joint distribution.

https://doi.org/10.1371/journal.pcbi.1013762.g001

The neural data comes from another publication in which the spontaneous activity of the Anterior Rhombencephalic Turning Region (ARTR) [29] (Fig 1e) was recorded from 5-7 days old immobilized larvae expressing the calcium indicator GCaMP6f, using light-sheet functional imaging. Several neural recordings (3-10) for each one of the five temperatures (from 18°C to 33°C (Fig 1b)) were analyzed. The fluorescence signal of each neuron was further deconvolved to estimate an approximate spike train (see Material and methods Sect 4.2).

2.2 Modeling of behavior

2.2.1 Markov models.

The distribution of reorientation angles after each bout, shown in Fig 1d, appears to be trimodal, suggesting a classification of the bouts in 3 types: forward (F), left-turn (L) and right-turn (R). In practice, this categorization is generally carried out by thresholding the distribution of re-orientation angles. Denoting the state of swim bout n by s_n we have:

(1)

The use of the same threshold (in absolute value) to detect left and right turns relies on the hypothesis that zebrafish larvae, as a group, have no preferred direction (a.k.a. non-handedness). As the exact value of has minimal qualitative impact on the results of the Markov Chains, we adopt the same value as in [24]; notice that is the same across all temperatures to avoid introducing ad hoc, temperature-dependent biases. An example of the classification of states along a swimming trajectory is presented in Fig 2b.

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Fig 2. 3-state Markov Chain and hidden Markov models - Stronger persistence emerges from better labeling:

(a) Diagram of the 3-state Hidden Markov Model (HMM) with normal emissions for Forward bouts, and gamma emissions for Turning bouts. Example emission distributions were taken at 26°C. (b) Example trajectory at 22°C. Each point represents a swim bout, with the left color for the threshold labeling (Markov Chain model), and the right color for the HMM labeling using the Viterbi algorithm. Top: 2D trajectory. Bottom: reorientation angle for this trajectory, with the threshold as a dashed line. (c) Probability of observing a streak of consecutive forward bouts (black) or same-direction turning bouts (pink), for MC (circles) and HMM (triangles), at 22°C. Inset: Exponential decay characteristic length (, solid lines), and theoretical persistence length computed from the transition matrix (, dashed lines). (d) Ratio of persistence length (observed vs. no-memory null model) vs. temperature, for Forward (s = F, black) and turning (, pink) bouts. (e) stubbornness factor at q = 0 intermediary Forward bouts, . (f) stubbornness factor at q = 1 intermediary Forward bouts, . (e-f) Shaded bands represent the estimated errors from aggregated fish data (see Materials and methods 4.5).

https://doi.org/10.1371/journal.pcbi.1013762.g002

Once the bout types are identified, we define a dynamical model for the trajectories using a three-state Markov Chain (MC). Informally, the sequence of states (associated with the 3 different bout types) is described by the probabilistic automaton in S3 Fig a. In this model, after each bout n, a new state s_n+1 is drawn randomly, conditioned only on s_n (and not on previous states). The transition probabilities between states, , are estimated by counting the numbers of occurrences of the transitions along the trajectories:

(2)

with .

The top right eigenvector of the transition matrix gives access to the stationary probabilities P(s) of the 3 states. As a self-consistency check, we confirmed that this stationary distribution matches the empirical fraction of threshold-labeled states measured in the same dataset, with a maximum absolute difference <0.003 for every bout type and temperature.

2.2.2 Hidden Markov model.

We then turn to an alternative categorization method, where states are inferred rather than a priori assigned. To do so, we consider a three-state Hidden Markov Model (HMM), see Fig 2a. Unlike MC, HMM makes a clear distinction between the observations (here the reorientation angles treated as ‘symbols’) and the states of the system (here s_n, which are not directly accessible from the knowledge of , in contradiction with the key assumption underlying MC). The HMM is defined by:

• The transition probabilities between the hidden states.
• The emission probabilities, , relate the observations to the hidden states s. For the forward state, we choose normally distributed reorientation angle emission distributions, centered in zero: . For turn states, we use Gamma distributed reorientation angles, with a positive or negative sign according to whether the state is Left or Right: and , constraining and . These specific functional forms (Normal and Gamma) were chosen empirically because they adequately capture the experimental turn angle distributions across all temperatures (see Material and methods Sect 4.3 for details about the validation of these emission distributions).
• A probability distribution for the initial state at the beginning of a trajectory.

We consider a symmetric HMM, where left-right directions are equivalent. This is done in practice by enforcing the following constraints over the transition probabilities,

and for the emission distributions. We have checked that relaxing these constraints on the transition matrix leads to equivalent results, see S9 Fig and S10 Fig, while maintaining them leads to faster training convergence.

We train both types of HMM models using Expectation Maximization (Baum-Welch algorithm) aggregating all trajectories at a given temperature in the first dataset, and for each individual fish in the long-trajectory data. We employ a customized Julia [30,31] implementation (available at https://github.com/ZebrafishHMM2023/ZebrafishHMM2023.jl).

Results show that the inferred parameters for the unconstrained and symmetric models are very similar, see S9 Fig. This is expected since our experimental setup is compatible with left-right symmetry. In the following, we therefore employ symmetric HMM only, since imposing symmetry from the beginning results in faster and more robust training of the HMM. This in turn ensures that steady state bout probability is left-right symmetric ().

2.3 State classification and behavioral persistence

2.3.3 Consistency of the MC and HMM descriptions of behavior.

Taken together, the results above suggest that the Hidden Markov Model better captures persistence in reorientation by labeling bouts with small reorientation angles based on context. This leads to a more flexible and thus stable classification than the thresholding method. However, given the absence of a ground truth, it remains unclear whether the labeling produced by the Hidden Markov Models is more accurate than the one produced by the standard threshold-based approaches.

One way to address this question is to examine to what extent each of these methods are self-consistent, i.e. guarantees that the inferred labeled sequences are truly markovian such that the bout type at a given time only depends on the type of the preceding bout. It has been previously noted that the thresholding methods lead to significant non-markovianity. In particular, in a transition with , the two turning bouts tend to have the same orientation (). This means that the memory of orientation T₁ is maintained during the forward bout, in violation of the Markovian assumption. This observation led to propose a 4-state Markov system comprising two independent Markov chains, independently controlling the forward-turn bout transitions, and directional left-right bout transitions (see S4 Fig b for a diagram of this 4-state model) [22,24].

Given that our 3-state Hidden Markov Model (HMM) re-labels numerous Forward bouts as Turn bouts, we ask whether this new classification might alleviate this non-Markovianity issue, such that the ad hoc 4-state model might no longer be needed. We thus propose a new test of Markovian violation specifically designed for our use case, that we apply to both the HMM and MC models.

We introduce the stubbornness factor f_q to empirically assess the tendency of larvae to retain their orientation after a sequence of q intermediary forward bouts (S4 Fig b, Materials and methods Sect 4.5):

(3)

with and .

Owing to the loss of orientational memory after a forward bout, a non-handed 3-state Markovian model should have f_q = 1 for (Materials and methods Sect 4.6). On the other hand, f_q = 0 is a measurement of directional persistence during uninterrupted sequences of turning bouts.

We found that most of the memory effects captured by the HMM occur at q = 0, and that the stubbornness reaches for , suggesting that the HMM-inferred bout sequences are quasi-Markovian. In comparison, and for lower temperatures, the thresholded MC classification displays lower persistence at q = 0 but higher stubbornness at q = 1 as seen in Fig 2e–2f (and less significantly at q = 2, see S4 Fig d). This suggests that the thresholded labeling leads to Markov violation primarily due to the mislabeling of turn bouts as forward bouts during turning streaks, as anticipated in the previous section and illustrated in Fig 2b. As this stubbornness is mostly significant at q = 1, we expect that most mislabelings are one-off errors.

In summary, previous works using an ad hoc threshold to classify bouts had dismissed 3-state Markov models because the resulting sequences were non-markovian. We found that by using an unsupervised method to simultaneously label the data and infer a Markov Model, we could unveil previously underestimated memory effects in zebrafish reorientation statistics. These results suggest that the HMM labeling is more markovian, making it a more coherent model than MC. This is probably due to a reclassification of forward bouts as turns during sequences of small reorientations.

2.4 Behavioral phenotyping from long individual fish trajectories

As HMM provides an unbiased quantification of the behavior, we now ask whether the approach is accurate enough to detect behavioral differences between specimen (inter-individual variability) and whether it can enable the unambiguous identification of each animal.

In the preceding sections, the dataset used to infer the models comprised trajectories from multiple fish, as the different individuals swimming together during a given assay could not be distinguished. To address the question of individuality, we used additional experiments reported in [24], in which individual fish were tracked at 26°C (see Materials and methods 4.1). A total of 18 fish were recorded for over 2 hours. We first trained HMMs on such long trajectories (the “global HMM”), by enforcing left-right symmetry on the transition rates as for the aggregated fish data. For comparison, we show in S10 Fig the results when we trained HMMs by relaxing the left-right symmetry on the transition matrix. The resulting transition matrices are close to be symmetric, even if slight deviations may be found for some fish.

We then split the 2h-long recorded sequence of each individual fish into smaller periods (chunks) of minutes each, and trained an HMM on each of these chunks (see diagram in Fig 3a–3b).

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Fig 3. Fish identification from long trajectories:

(a) Dataset : 18 fish recorded individually for 2-hour sessions. Each session is split into 10 chunks (mean = 9.5 0.5 trajectories per chunk). (b) Example trajectories for fish 1. (c) HMM parameters inferred from all the trajectories of a fish (referred to as global) vs. inferred per-chunk. Only four HMM parameters are shown for clarity (see S5 Fig for all parameters). Each dot represents a fish, and the error bars correspond to the standard error of the mean. Points labeled in red correspond to fish misidentified in panel d. (d) Confusion matrix of the relative likelihood of data coming from fish i and HMM trained on fish j. The fish identity most likely according to each model is indicated with a star (black : correctly identified fish, red : misidentified). (e) Average number of correctly identified fish when a fraction f of the test data is used for identification. Shaded band : standard deviation across 100 trials. In each trial, the trajectories of each fish were randomly split into train and test sets (50%).

https://doi.org/10.1371/journal.pcbi.1013762.g003

For each fish, the parameters of these HMMs exhibit significant variability (as shown by the vertical error bars in Fig 3c). This variability between the different chunks reflects both intra-individual (temporal) variability and, to a lesser extent, inference uncertainty due to the limited sampling of the HMM (see S5 Fig). We stress that small deviations from symmetry (e.g. by relaxing left–right symmetry, see S10 Fig) would make individual fish even easier to recognize. However, since chunk trajectories are shorter and thus more sensitive to symmetry-breaking fluctuations, imposing left–right symmetry also reduces susceptibility to such errors. Fig 3c compares selected parameters of the global HMM for each fish, against the average parameters over several HMMs trained on the chunk trajectories (see S5 Fig for all parameters). There is a clear trend between the global HMM and the average behavior of the chunk HMMs. Therefore, although a fish exhibits variability during a long sequence of bouts, the variability between distinct fish is larger.

These results suggest that the HMM models can be used to distinguish different fish from observations of their bout sequences. To test this hypothesis, we split the trajectories of each fish into a training and a withheld test set. After training the HMM on the train set for a particular fish, we computed the likelihood of all fish trajectories in the test set, and compared them. For 14 out of the 18 fish, the test set that yield the maximum likelihood rightly identifies the fish used to train the HMM (Fig 3d). This finding suggests that the HMM captures behavioral parameters which are distinctive enough to discriminate between different fish. Given the large variability exhibited by a single fish, one expects this discriminative ability to increase with the duration of the training sequences. To quantify this, we further evaluated the likelihoods of subsets of the test fish trajectories, and recorded the number of times that the maximum likelihood HMM corresponded to the correct fish (Fig 3e). Even when withholding 80% of the sequence, we were able to correctly identify 10 out of the 18 fish. These results suggest that individual fish exhibit variable but distinctive behavior which can be captured by the HMM.

2.5 Modeling of neural data

The selection of turning bouts orientation in zebrafish is known to be controlled by a small bilaterally distributed circuit in the anterior hindbrain, called Anterior Rhombencephalic Turning Region (ARTR). This circuit displays self-sustained alternating activity between its left- and right-lateral sub-population, with a period of the order of tens of seconds (Fig 1e). The animal tends to execute left turns when the left ARTR is active while the right ARTR is inactive (and vice versa for right turns) [20].

In contrast, no specific circuit has yet been identified for the selection of turn vs forward bouts. The hypothesis that two distinct circuits are involved in bout-type selection is consistent with the 4 states Markovian model of navigation, in which two independent Markov chains drive the two selection processes. However, the 3-state Markovian model supported by the HMM analysis suggests that the same circuit (ARTR) could drive the selection of all 3 bout-types.

In order to test this hypothesis, we re-analyzed the ARTR recordings reported in [29] using a 3-state HMM (Fig 4a). We posit an independent neural model for the activity of the N recorded neurons, yielding, for each state, the emission probability:

(4)

where is a neuronal configuration, s is the hidden state, and is the local field representing the effective excitability of neuron i in state s. The model thus includes parameters , associated to each neuron and each hidden states. Notice that for the neural HMM, the non-handedness of the behavioral HMM is not enforced. We also performed comparisons of neural HMM with different numbers of hidden states in a cross-validation test, and found that including more than 3 hidden states results in marginal improvement, see S8 Fig.

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Fig 4. 3-state Hidden Markov Model (HMM) describes ARTR neuronal statistics:

(a) Diagram of the 3-state Hidden Markov Model (HMM) with emissions described as independent models of the ARTR neuronal population, see Eq (4). Distributions of fields are shown for all fish for neurons in the left (blue) and right (red) ARTR. (b) Example ARTR activity (see Fig 1f) classified by the 3-state HMM. Solid lines represent the mean activity of neurons in the left (m_L, blue) and right (m_R, red) ARTR. (c) HMM classification in space. Dots : neuronal configurations from the example recording in panel b. Solid lines : 90% of the distributions for all recordings combined. (d) Distributions of per state (all recordings combined). (e-f) Empirical vs. HMM-generated neuronal statistics (all recordings combined). (e) Mean activity of neuron i. (f) Covariance of neurons i and j on opposite sides (left plot) and on the same sides (right plot) of the ARTR.

https://doi.org/10.1371/journal.pcbi.1013762.g004

The distribution of fields for the 3 hidden states, shown in Fig 4a, are used to assign labels to the three states (see Materials and methods 4.7). Consistent with our current understanding of the ARTR function for turn selection, the state with large values of the fields on the left and smaller values of the fields on the contralateral side is labeled "left" (and vice versa for "right"). The third state exhibits similar distributions of fields for neurons on the left and right side of the ARTR, and is labeled forward in analogy with behavior. The ARTR activity is thus modeled as a sequence of left-right-forward states.

With this classification, the forward state corresponds to a low mean neuronal activity of both the left (m_L) and right (m_R) sides of the ARTR, while turning states are associated with large activity on the ipsilateral side of the ARTR (left state : , right state : , see Fig 4b–4d).

This model accurately captures the mean activity of each neuron (Fig 4e), as well as the pairwise correlations between contralateral neurons. However, ipsilateral pairwise correlations are not as well reproduced, showing lower covariance in the generated data (Fig 4f). This mismatch presumably comes from the fact that the activities of neurons within a state are uncorrelated in our emission probabilities, while recurrent interactions in the ARTR circuit produce correlations. These would be better modeled with emission probabilities including effective interactions between neurons [29].

2.6 Comparison of behavior and neuronal HMMs

In the preceding sections, we demonstrated that both the reorientation behavior and the neuronal activity of the Anterior Rhombencephalic Turning Region (ARTR) can be effectively modeled using three-state Hidden Markov Models (HMMs). However, it remains unclear whether the three states identified in the Behavioral HMM (B-HMM) directly correspond to those inferred in the Neuronal HMM (N-HMM).

Unfortunately, there is currently no publicly available dataset offering simultaneous recordings of freely swimming larvae kinematics and neuronal activity, which would enable direct comparison of B-HMM and N-HMM states for individual bouts. Current research addressing this question largely relies on experimental paradigms where larvae are either paralyzed with electrophysiological recording of motor nerve signals (fictive swimming preparations) [20,32,33], or head-embedded with a free-moving tail (head tethered preparations) [34–37]. In fictively swimming preparations, whilst the classification of left-vs-right bouts is feasible based on the asymmetric nature of the motor command, such experiments are not, to the best of our knowledge, capable of discriminating forward-vs-turning bouts [20]. On the other hand, head tethered preparations allow forward-left-right bout classification [34,36], but typically rely on visual stimuli to elicit behavior [34–37] as the spontaneous sequence of bouts is strongly disrupted in comparison with freely swimming contexts [38]. The disruption of behavioral and neuronal dynamics caused by animal immobilization is a general problem in behavioral neuroscience, as illustrated by studies in C. elegans where the direct comparison of neuronal dynamics between freely-moving and immobilized worms was quantified [39].

We hereafter propose to circumvent these experimental challenges by comparing the statistical structures of the reorientation sequences inferred from the two datasets presented in Sects 2.1 and 2.5. The transition probabilities obtained from B-HMM and N-HMM at all recorded temperatures are shown in Fig 5b. Comparison of these transition rates require to first correct them for differences in sampling rates. Indeed, neural transition rates are computed from neuronal recordings performed at ∼6 Hz (depending on the dataset, see Materials and methods 4.2), while for behavior, the sequences are divided into swim bouts triggered at an average rate of ∼1 Hz depending on the temperature.

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Fig 5. Behavior vs. Neuronal temporal structure:

(a) Sojourn-time distribution for forward (left) and turn states (right) : behavior (black), neuronal before (orange) and after (magenta) temporal re-scaling. A single re-scaling factor is used for forward and turning states, for all temperatures, and recordings. (b) Behavior vs. neuronal state-transition probabilities (for all state pairs and all temperatures), before (left) and after (right) temporal re-scaling. Each dot represents a single transition probability at a given temperature. For neuronal state-transition, we show the mean and standard error over all recordings. (c) Diagram showing two possible transition trajectories between left and right states in ARTR space. Transitions through the forward state are more probable (see panel d). (d) Distributions of for behavior (black) and neuronal data before (orange) and after (magenta) temporal re-scaling (all temperatures combined). These distributions are shown as standard box plots (median, quartiles, and outliers beyond the inter-quartile range from the median).

https://doi.org/10.1371/journal.pcbi.1013762.g005

To bridge the gap between neuronal and behavioral datasets, one needs to estimate how the behavior is subsampled from the neuronal activity. Previous observations have shown that internal dynamics, and precisely ARTR pseudo-period, tends to be slower in head-tethered assays [29], as well as experiments comparing free and fictive swimming which showed that the fictive swimming frequency is reduced in paralyzed animals [20]. To verify this, we computed the distribution of sojourn times of all three states in both B-HMM and N-HMM, where is the duration of a sequence of k consecutive states s observed at times . We found the neuronal sojourn times to be significantly longer than the behavioral sojourn times (Fig 5a). The optimal temporal scaling factor f_N/B for which the distribution of neuronal sojourn times matches the distribution of behavioral sojourn times (see Materials and methods 4.8) was . Interestingly, this value appears to be consistent with findings from [20], which reported the mean interbout interval for fictive swimming to be 0.41 times slower than for freely swimming.

Using this temporal re-scaling factor, we find that the transition probabilities for behavior and ARTR models are similar (, see Fig 5b), indicating that the behavioral and neuronal state sequences share similar underlying structures. This is remarkable as the number and meaning of the neuronal internal states were not a priori fixed, but entirely assigned by N-HMM after training.

This result supports our hypothesis that the ARTR not only governs the selection between rightward and leftward turning bouts, but also controls the bout-type selection, forward vs turn. To test this claim further, we analyzed in more detail the statistics of trajectories in the bout space inferred from the ARTR dynamics and from behavioral data. We specifically examined the bout sequences leading to a change in orientation, such as transitions from L to R and vice-versa. Such orientational switches can be either direct, e.g. , or may include an intermediate forward bout, (Fig 5c). Using the ARTR signal, we found that the second path is strongly favored as evidenced by the the fact that . A comparable value of this ratio is observed in the behavioral data (Fig 5d), indicating that fish indeed tend to execute a forward bout when changing orientation. This statistical bias would be difficult to understand under the standard model that posits the existence of independent neural circuits governing orientation and bout-type selection, respectively. In contrast, in our model, it emerges naturally from the phase space structure of the ARTR dynamics as shown in Fig 4c and Fig 5c. The L-Shaped distribution of constrains the Left-to-right (or Right-to-Left) trajectories to pass through a symmetrical, low activity state, thus favoring intermediate forward bouts.

2.7 Generation of synthetic behavior with the neural model

So far we have compared neuronal and behavioral data by examining only the short-scale statistical structures of the HMM-inferred state sequences. We now use the N-HMM model to identify state sequences from activity recordings, then generate long synthetic trajectories and compare their statistics with those of freely swimming fish. This approach allows us to assess whether the N-HMM, when combined with appropriate scaling and behavioral parameters, can reproduce the complex statistical properties of exploration at various temperatures.

2.7.1 Generation of trajectories from neural recordings.

Consider a neural activity recording. Following Sect 2.6, we hypothesize that the internal states obtained by parsing the activity recording with the Neural HMM (N-HMM) match the behavioral internal states, after proper temporal rescaling. Therefore, we expect that it should be possible to generate artificial swim trajectories from these neural states.

We report in Fig 6a, 6b, 6c the pipeline to generate 2D swim trajectories from neural recordings of individual fish. This is done by first identifying neural states from recordings using the Viterbi algorithm. We then generate a sequence of discrete bouts by sampling the behavioral distribution of inter-bout intervals , rescaled by the scaling factor (see S7 Fig a for example sequences). This simulates a stochastic bout-initiation process with the correct temporal characteristics. The bout internal state b_n is given by the neural state identified at time .

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Fig 6. Generative ability of HMM models and trajectory reconstruction from neural HMMs:

(a,b,c) Pipeline to convert neuronal activity to swim trajectory. (a) ARTR activity is first converted to the most likely sequence of forward/left/right hidden states using the Viterbi algorithm on the Neuronal Hidden Markov Models (N-HMM). (Example empirical ARTR activity at 26°C). (b) Time is then re-scaled using the scaling factor identified in Fig 5, and sampled based on the empirical distribution of inter-bout intervals . Bout-angle sequences are then sampled based on the Behavioral Hidden Markov Models (B-HMM). (c) Swim trajectories are constructed for each bout sequence by sampling the bout distances d_n from the empirical distribution. An example trajectory generated from the empirical neuronal activity at 26°C is shown. Point color corresponds to bout type (left, right, forward), and point size corresponds to inter-bout interval. (d,b,e) Pipeline to convert N-HMM-generated state sequences to swim trajectories. (d) The N-HMM is first sampled to generate a sequence of forward/left/right internal states. The sequence is then temporally rescaled and sampled as before (see panel b). (e) Swim trajectories are constructed as before (see panel c). An example N-HMM-generated trajectory at 26°C is shown. (f) Distributions of reorientation angles , inter-bout intervals , and bout distances d_n; for the aggregated multiple-fish trajectories (gray), trajectories generated from B-HMM (black), and generated from N-HMM (blue); at 22°C. (g) Mean Square Reorientation (MSR) accumulated after q bouts for aggregated multiple-fish trajectories (normal : grey, shuffled : red dashed), single-fish long trajectories (green), and trajectories generated from N-HMM (blue). For both long and N-HMM-generated trajectories, we show the mean and standard deviation over all individuals (solid line and band). (see S7 Fig for individual trajectories and all temperatures) (h) MSR at q = 10 bouts, with mean (horizontal bars) and standard deviation (vertical bars).

https://doi.org/10.1371/journal.pcbi.1013762.g006

We then sample, for every n, the traveled distance d_n from the experimental distribution. We ignore the potential dependence of and d_n to bout type. Lastly, we sample the emission probability associated to the Behavioral HMM (B-HMM) inferred from all fish data to get a realization of the reorientation angle .

We reconstruct the coordinates of the virtual fish after k bouts, , through

(5)

where is the orientation angle of the fish at bout n, constructed as the cumulative sum of re-orientation angles at previous bouts.

In Fig 6c, we show a representative synthetic trajectory generated from neural data with the above procedure (see S7 Fig b for more examples).

2.7.2 Generation of synthetic 2D trajectories from neural model.

We can further push the N-HMM models associated to individual fish to generate de novo synthetic 2D trajectories, not associated to a specific neural recording. To this aim we first generate synthetic temporal sequences of neural states (Fig 6d) with the N-HMM. We then use the distributions of inter-bout intervals, traveled distances, and reorientation angles to sample synthetic trajectories as explained above. An example trajectory is shown in Fig 6e.

The similarity with real trajectories is quantified by the comparison of the distributions of bout angles, inter-bout duration intervals and traveled distances. As shown in Fig 6f the distribution of bout angles, inter-bout duration intervals and traveled distances is in very good agreement with the ones observed in the behavioral data.

We further characterize the trajectories using the Mean Square Reorientation (MSR) after q bouts:

(6)

where the average is taken across all bouts n and all trajectories.

Fig 6g shows the values of obtained from N-HMM-generated trajectories at different temperatures (see S7 Fig d for the remaining temperatures), as well as the MSR directly obtained from multiple-fish trajectories and long single-fish trajectories (only at C).

We first notice that long individual fish trajectories at C display large variability in values, compatible with the presence of fish-to-fish variability. This variability is washed out for the multiple-fish dataset (since individual trajectories are combined) providing an averaged for each temperature. Interestingly, the MSR for the long sequences of individual animals significantly differ from the MSR obtained from multiple-fish trajectories. This could be due to differences in experimental conditions, and in particular the effects of collective vs. isolated navigation [24].

N-HMM-generated trajectories have a MSR distribution compatible and encompassing the behavioral data in their variability. Such large variability is expected from the large fluctuations in neural brain states. Some trajectories generated with N-HMM show anomalously large angular persistence (see S7 Fig a), which may correspond to brain states where the Anterior Rhombencephalic Turning Region (ARTR) displays no left-right alternating behavior. This is expected, as the N-HMMs were established from spontaneous activity recordings of immobilized fish which were not constrained to swim-like behaviors. In Fig 6h, we summarized and compared the results for the Mean Square Reorientation after 10 bouts, (see S7 Fig e for all temperatures). We found, consistently across temperatures, that the MSR of behavioral data are comprised within the one-standard-deviation confidence interval of N-HMM-generated trajectories.

As we show in S1 Text, the MSR of HMM-generated trajectories can be decomposed as the sum of a purely diffusive contribution, associated to the variance of bout angles, and of terms arising from time-correlations in bout type selection along a trajectory (see Eq (12)). The increase of bout-angle variance with temperature is sufficient to explain the increasing trend of the mean MSR with temperature observed in Fig 6h (see S7 Fig f–g).

4.1 Behavioral datasets

The behavioral dataset used in the present study is derived from [24], and can be accessed directly at https://doi.org/10.5061/dryad.3r2280ggw. This dataset comprises spontaneous swimming trajectories of 5 to 7 dpf zebrafish larvae, collected at controlled bath temperatures of 18°C, 22°C, 26°C, 30°C, and 33°C. A camera was used to continuously record the swimming behavior of the fish within an arena of 100454.5mm³ for 30 minutes at 25 frames/second. To eliminate border effects, a Region of Interest (ROI) was defined at a distance of 5mm from the arena’s walls. Fish that swam outside the defined tracking ROI were considered lost, and a new trajectory was initiated upon their re-entry into the ROI. The identity of the fish is thus lost each time it exits the ROI. Therefore, the dataset contains a varying number of fish trajectories, ranging from 532 to 1513 trajectories across the different temperatures (mean = 1148). Individual trajectories were tracked offline using the open-source FastTrack software [52], and were then discretized into sequences of swimming bouts.

Each trajectory consists of a sequence of swim bouts, spanning from 9 to 748 bouts per trajectory (mean=60, distributions shown in S1 Fig a). From this extensive dataset, we primarily utilized the re-orientation angles, defined as the difference between the heading direction at bout n + 1 and the heading direction at bout n:

(7)

(a graphical illustration of this definition can be found in Fig 1c). This parameter encapsulates the angular change between consecutive bouts, providing insight into the fish’s ability to modify its orientation during swimming.

We also used the interbout interval representing the elapsed time between 2 consecutive bouts, and the traveled distance .

On top of these multi-fish trajectories, we used in Sects 2.4 and 2.7 a second dataset from [24] consisting in single-fish recordings. For this dataset, each fish (N = 18) is placed alone in the arena at 26°C, and is recorded for 2 hours. With this experimental paradigm, the identity of the fish is conserved across trajectories, even when the fish leaves and re-enters the ROI.

4.2 Neuronal datasets

The neuronal dataset used in the present study is derived from [29], and can be accessed directly at https://gin.g-node.org/Debregeas/ZF_ARTR_thermo. This dataset contains 32 one-photon Light-Sheet Microscopy recordings of spontaneous brain activity, for 13 zebrafish larvae (5 to 7 dpf) at 18°C, 22°C, 26°C, 30°C, and 33°C. It focuses on neurons from the Anterior Rhombencephalic Turning Region (ARTR), with neurons (mean 307, std 119), recorded during (mean 23, std 4 min) at ∼6 Hz (mean 5.9, std 2.1 Hz). The approximate binarized spike trains of segmented neurons were inferred from fluorescence signals using a previously described deconvolution algorithm [53].

4.3 Emission of reorientation angles in the Hidden Markov model

To validate the hypothesis that the re-orientation angles can be modeled using normal and gamma distributions, we compared the distribution of the data with a Gaussian Mixture Model (GMM) and a Gaussian & Gamma Mixture Model:

where , and w_F, w_L, and w_R denote the weights for forward, left, and right states, respectively.

Using Quantile-Quantile (QQ) plots, we show that this last mixture model accurately reproduces the observed distribution of in the data, and is much better than a GMM, especially in the tails of the distributions (S2 Fig a). We also confirmed that, once trained, the emission distributions do indeed match the observed reorientation distributions (S2 Fig b–c).

4.4 Hidden Markov model training

Given a data set of trajectories (neuronal or behavioral), the Hidden Markov models were trained using the standard Baum-Welch algorithm [54]. We train for a maximum of 500 expectation-maximization iterations, stopping earlier if the gain in the total log-likelihood of all the data in one iteration becomes less than a small threshold, that we fix at 10⁻⁶. In practice we find that all HMMs trained in this study converge before reaching 500 iterations.

The transition matrix parameters are initialized at random, respecting left-right symmetry in the case of the behavioral models.

4.5 Stubbornness factor

The stubbornness factor f_q is a measurement of the animal’s preference towards turning in the same direction over changing direction, after q intermediary forward bouts (S4 Fig c), as defined in Eq (3).

It can be computed from a sequence of classified bouts b_n by first identifying and counting the q-plets where and where :

(8)

and then computing their ratio:

(9)

In practice, this ratio has a physical interpretation only for long sequences of bouts where N₌ >>1 and . As the trajectories in our dataset can be quite short (S1 Fig a), we compute f_q from all trajectories at a specific temperature, increasing the chance of observing a high number of stubborn (N₌) and non-stubborn () trajectories.

By considering that the probability of a given q-plet is stubborn follows a binomial distribution ( and with ), we can evaluate the uncertainty in stubbornness as:

(10)

It is to be noted that these uncertainties are conservative estimates, as there exits a bias inherent to the dataset. Indeed, a very stubborn fish will tend to stay longer within the Region Of Interest (ROI) of the camera, leading to longer trajectories and therefore weighing more on the final result. Hence, it is unclear whether a stubbornness factor is truly significant (as suggested by the estimated error bars on S4 Fig d).

Furthermore, as the stubbornness factor is computed from all trajectories (and thus all fish) at a particular temperature, it represents an average behavior rather than an individual fish.

4.6 Stubbornness factor and 3-state Markov Chain

The stubbornness factor can be defined directly from the transition matrix:

For q = 0, calculations are simple:

(11)

For , the stubbornness factor is defined from the transition matrix as:

with S_L,q the probability of a trajectory which starts and ends with a left bout, W_L,q the probability of a trajectory which starts with a left bout and ends with a right bout, and S_R,q W_R,q their symmetrical opposites.

For a 3-state model, the forward-forward bout probability cancels out, giving:

and with our non-handedness hypothesis: , , and , yielding:

(12)

4.7 Labeling of states in the neuronal Hidden Markov Model

The internal states of the Hidden Markov Models (HMMs) trained from neuronal activity are not a priory assigned to the Left, Right and Forward labels, and must therefore be re-ordered post-training.

We expect a certain symmetry in the system, where neurons in the left side of the ARTR will be more active during a Left state (and vice versa). Hence, we can use the excitability of neuron i in each internal state s, as defined in the emission distribution of the HMM (see Eq 4). We define the lateralized excitability:

(13)

where is the standard logistic function, and and are the sets of neurons located respectively in the left and right side of the ARTR. We thus label the HMM states such that

(14)

with F, L, and R the Forward, Left and Right internal states.

4.8 Temporal re-scaling

To find the temporal re-scaling factor f_N/B between behavioral and neuronal models, we first compute the distributions of sojourn times for all states in both behavioral and neuronal Hidden Markov Models.

We then find the optimal re-scaling factor f_N/B for which the combined distributions and are as close to each other as possible :

(15)

where Q(D) is the quantiles of a distribution D, and is the Root Mean Squared Error between vectors x and y (see S6 Figa).

For Markov chains, the transition matrix represents the probability of transitioning in one step from state s to state . The transition probability in steps is then the matrix power P^k.

In order to apply the temporal re-scaling f_N/B between behavioral and neuronal models, we can thus compute the re-scaled transition matrix :

(16)

where P_n is the transition matrix inferred from neuronal data recorded at a frequency .

4.9 Mean square reorientation

To characterize the orientational diffusivity of the trajectories, we use the Mean Square Reorientation (MSR) accumulated after q bouts, as defined in Eq (6) [22].

For infinitely large datasets with no left-right bias, we expect a centered distribution of reorientation angles . However, this is not the case, particularly for the neuronal dataset where experimental limitations can induce strong biases. In particular, two of those limitations are due to the one-sided illumination in our Light Sheet Fluorescence Microscope [55]. First, due to scattering, the illumination beam is not uniform left-right across the brain, which can induce biases in the detection of neurons and their activity. Second, the non-uniform perception of light by the zebrafish larvae can elicit a phototaxis response, which is known to bias the activity of the Anterior Rhombencephalic Turning Region (ARTR) [43].

Since a non-zero bias can result in a distortion of the MSR (see S1 Text), the MSR is computed from instead of .