# Data-driven stochastic modelling of zebrafish locomotion

## Abstract

In this work, we develop a data-driven modelling framework to reproduce the locomotion of fish in a confined environment. Specifically, we highlight the primary characteristics of the motion of individual zebrafish *(Danio rerio)*, and study how these can be suitably encapsulated within a mathematical framework utilising a limited number of calibrated model parameters. Using data captured from individual zebrafish via automated visual tracking, we develop a model using stochastic differential equations and describe fish as a self propelled particle moving in a plane. Based on recent experimental evidence of the importance of speed regulation in social behaviour, we extend stochastic models of fish locomotion by introducing experimentally-derived processes describing dynamic speed regulation. Salient metrics are defined which are then used to calibrate key parameters of coupled stochastic differential equations, describing both speed and angular speed of *swimming* fish. The effects of external constraints are also included, based on experimentally observed responses. Understanding the spontaneous dynamics of zebrafish using a bottom-up, purely data-driven approach is expected to yield a modelling framework for quantitative investigation of individual behaviour in the presence of various external constraints or biological assays.

### Keywords

Fish locomotion Computational biology Stochastic models Zebrafish Ornstein–Uhlenbeck### Mathematics Subject Classification

37N25 46N60 97M60 60H10 91B70 37B70## 1 Introduction

The coordinated motion of groups of animals, and in particular of fish shoals, has been widely studied both from experimental and theoretical perspectives (Partridge 1982; Huth and Wissel 1992; Camazine et al. 2001; Couzin et al. 2002; Krause and Ruxton 2002; Buhl et al. 2006; Kolpas et al. 2007; Pillot et al. 2011; Moussaïd et al. 2011; Gautrais et al. 2012; Vicsek and Zafeiris 2012). Numerous authors have proposed a wide variety of theoretical models, for example the canonical Vicsek-type models (Vicsek et al. 1995) along with those by Czirók et al. (1997, 1999) which describe mobile agents as identical self-propelled particles with heading directions updated via the integration of noisy local-neighbourhood interaction rules. More elaborate models of collective motion have also been proposed which may account for repulsive and attractive forces between fish (or other animals), for example those by Aoki (1982), Reynolds (1987), Huth and Wissel (1992), Couzin et al. (2002), Grégoire and Chaté (2004), Chaté et al. (2008), Kolpas et al. (2013). Within all of these models, the dynamics of the group behaviour are dissected into individual rules from which complex coordinated motions emerge. It is critical therefore to establish predictive and tractable models for the behavioural response of isolated individuals, upon which to study and construct models of sociality.

The degree to which individual behaviour modulates group dynamics, and correspondingly, how interactions with conspecifics affects individual response, can be tested with a modelling cycle driven by precise experimental data. The recent work of Gautrais et al. serves as an important example of this process. Specifically, a data-driven model of spontaneous fish movement was first derived by Gautrais et al. (2009). Then, using a bottom-up methodology, a model of group motion from data gathered at the level of the individual was developed by Gautrais et al. (2012). Unlike many other models of collective motion, this approach enables all model parameters to be estimated directly from experimental data. Based on evidence that the fish considered (*Kulia Mugil*) are best described in terms of their turning speed and its autocorrelation, Gautrais et al. have developed a model referred to as a ‘Persistent Turning Walker’ (PTW). This model is based on an Ornstein–Uhlenbeck (O-U) stochastic differential equation (SDE) governing the turning speed of an agent with a fixed forward speed. In addition, the effects of environmental confinement were considered, providing a versatile methodology for incorporating fixed boundaries, obstacles and other fish, within the same model framework.

Recent experimental studies, including those by Krause and Ward (2005), Herbert-Read and Perna (2011), Katz et al. (2011), Berdahl et al. (2013), Herbert-Read et al. (2013), show that the speed response of fish play an important role in fish interaction. For example, a comprehensive study by Katz et al. (2011) reveals the subtle modulation of turning and speeding responses of groups of golden shiners in relation to their conspecifics. Important conclusions from this work include the observation that speed regulation may be a dominant component of interaction, where subsequent alignment between neighbouring fish emerges from the interplay between attraction and repulsion. With respect to this latter conclusion, some models including that of Strömbom (2011), have demonstrated the characteristic hallmarks of collective motion with a rich diversity of dynamics such as swarming, and circular and directed milling, emerging solely from inertial local attraction between individuals. The study by Katz et al. also examined the importance of higher order interactions, namely a non-trivial 3-body component which may contradict the pervasive assumption of models which exclusively integrate pairwise interactions. Supporting experimental work by Herbert-Read and Perna (2011) also suggested the absence of an empirically justifiable alignment rule for schooling mosquitofish, suggesting that group polarisation is an emergent property. In this study, speed regulation was again found to be a key reaction mechanism due to group interactions, especially repulsion from close neighbours, with clearly defined zones of interaction.

Most commonly, models of schooling consider fish as agents with fixed forward speed (Couzin et al. 2002; Gautrais et al. 2009, 2012), and thus prevent us from exploring the role of speed regulation in collective dynamics. Existing models of collective motion which do consider variable speed agents (Reynolds 1987; Huth and Wissel 1994; Toner and Tu 1998; D‘Orsogna et al. 2006; Strefler et al. 2008; Ebeling and Schimansky-Geier 2008; Abaid and Porfiri 2010; Strömbom 2011; Mishra et al. 2012) either describe self-propelled particles as a continuum dynamical system, or rather consider the effects of noise on the absolute velocity. Thus far however, none of these approaches have been fully validated against experimental data in terms of their description of speed modulation.

The primary aim of this paper is to extend the approach by Gautrais et al. (2009) to develop a data-driven modelling framework describing the individual locomotion of zebrafish. Selected here primarily for their strong propensity to form social groups (Miller and Gerlai 2008; Saverino and Gerlai 2008; Miller and Gerlai 2012), laboratory studies with zebrafish also benefit from their short intergenerational time and comparatively high reproductive rate. Yielding extensive genomic homologues with both humans and rodents, zebrafish have emerged as one of the predominant species for neurobiological, developmental and behavioural studies (Gerlai 2003; Kuo and Eliasmith 2005; Miklósi and Andrew 2006; Lawrence 2007; Kalueff et al. 2014). In this work, we find that modelling zebrafish motion requires an additional, experimentally calibrated process governing the variation of swimming *speed*. In light of recent studies such as those by Katz et al. (2011), Herbert-Read and Perna (2011), Berdahl et al. (2013), indicating that speed regulation is a key response of similar fish to external stimuli, this latter modification represents a shift away from many canonical models, which prescribe a constant speed, and provides the foundations for a novel modelling approach for studying zebrafish social behaviour.

The modelling process addressed in this work employs a bottom-up approach, using data analysed from experimentally observed zebrafish trajectories, primarily in terms of position/velocity time-series data, to inform an empirical model of individual swimming locomotion. Based on the direct analysis of experimental zebrafish trajectory data obtained via automated computer vision techniques at the Dynamical Systems Laboratory (New York University Polytechnic School of Engineering, NY, USA), we clarify whether the stochastic PTW models of spontaneous fish motion, developed by Gautrais et al. (2009), can be applied to suitably describe the locomotion of zebrafish. We wish to emphasise that the modelling framework presented in this paper can provide the foundation for future extensions which capture group level dynamics of zebrafish shoals and their interaction with semi-autonomous artificial stimuli (Aureli et al. 2012; Kopman et al. 2012; Aureli et al. 2010; Aureli and Porfiri 2010).

## 2 Materials and methods

### 2.1 Ethics statement

The experimental data for this analysis was provided by the Dynamic Systems Laboratory, New York University Polytechnic School of Engineering, NY, USA. Trajectory data for isolated fish analysed in this study are derived from source data published in the recent work of Butail et al. (2014) (‘No Robot’ control condition). All experiments were conducted following the protocols AWOC-2012-101 and AWOC-2013-103 approved by the Animal Welfare Oversight Committee of the New York University Polytechnic School of Engineering.

### 2.2 Animals and environment

Wild-type zebrafish *(Danio rerio)* were used in all experiments, acquired from an online aquarium (LiveAquaria.com, Rhinelander, Wisconsin, USA). Subjects age was between 6 to 8 months, inferred from their average body length (BL) of approximately 3 cm. Fish were kept in 37.8 l (10 US gallon) holding tanks with a maximum of 20 individuals in each. A photoperiod of 12 h light / 12 h dark was sustained prior to experimentation as per Cahill (2002). Water temperature in the holding tanks was maintained at \(27 \pm 1~^\circ \)C with a pH of 7.2. Fish were fed daily at 7 pm with commercial flake food (Hagen Corp./Nutrafin Max, USA). Experiments were started after a 10 days acclimatization period.

### 2.3 Apparatus

The setup and apparatus for this study is described by Butail et al. (2014). Experimental subjects were monitored in a \(120\times 120\times 20\) cm tank, supported on an aluminium frame, with a water depth of 10 cm (see Fig.1). The side length of the tank was thus approximately equal to 40 BL. The surface of the tank was covered with a white contact paper to enhance the contrast for automated tracking. Video recording was accomplished using a Microsoft LifeCam (USB interfaced) camera mounted 150 cm above the water surface, providing a single overhead video feed at a resolution of \(640 \times 480\) pixels at 5 fps. At this resolution, the location of a fish was represented by approximately 20–50 pixels of each frame. Illumination was provided by diffused overhead lighting from four 25 W fluorescent tubes (All-Glass Aquarium, preheat aquarium lamp, UK). Video image analysis and real-time multi-target tracking was achieved using software developed in MATLAB (R2011a, Mathworks), sampled at 5 Hz on a 2.5 GHz dual-core Intel desktop computer with 3 Gb RAM (detailed description of tracking presented by Butail et al. (2013).

### 2.4 Experimental procedure

A total of ten experimental observations were used for this investigation, each tracking the free-swimming trajectories of a different, experimentally naive, individual randomly selected from the population. Fish were removed from the holding tank using a small hand net and released into the experimental tank. Each observation was preceded with ten minutes of habituation time allowing the fish to swim freely and acclimatise to the novel environment of the experimental tank, as reported by Wong et al. (2010). The motion of each fish was recorded for 5 min (300 s), sampled at 5 Hz producing 1,500 position and velocity data samples per individual. Automated tracking was performed on video frame data in real-time, with minor adjustments made after each observation to repair missing trajectory points.

### 2.5 Data extraction and pre-processing

*freezing*or

*thrashing*near obstacles (boundaries) as defined by Bass and Gerlai (2008). To obtain suitable data representing swimming behaviour, we used a simplified version of a method described by Kopman et al. (2012), pre-processing the ten raw observation data sets (denoted \(F_1\ldots F_{10}\)) to extract data segments (60 s) of equal duration in the following way:

- 1.
Raw (speed) data was initially smoothed with a moving average window of 3 samples, then segmented such that contiguous segments are isolated when fish is moving with a speed above the threshold \(u_\text {min} = 1\) BL s\(^{-1}\). The original (unsmoothed) data is subsequently used for the steps that follow.

- 2.
If the duration between consecutive segments was less than a time threshold \(\tau _s = 2\) s, the two segments were joined so that fish were regarded as

*not*swimming only if the duration of the speed dropping below threshold, \(u_\text {min}\), exceeded \(\tau _s\). - 3.
Resulting data segments were subdivided into intervals of equal length \(\tau _l\) representing a continuous time-series of swimming data from an individual fish. Segments shorter than \(\tau _l\) were discarded such that we obtain continues data segments of equal duration.

### 2.6 Numerical implementation

Following the calculation of both \(f_W\) and \(f_c\) at a given time step \(t\) (discussed in §4), the updated values of \(U_t\) and \(\varOmega _t\) were found using the method in (2), with values of \(\varOmega _t\) restricted in the range \(\pm 15\,\hbox {rad\,s}^{-1}\) in accordance to the observed maximum angular speed.

## 3 Experiments

*(thigmotaxis)*(e.g. \(F_6, F_7, F_{10}\)) along with extended periods of freezing or thrashing behaviour (e.g. \(F_2, F_8, F_9\)).

### 3.1 Swimming trajectory analysis

Using the data segmentation process described earlier, we isolated 28 segments of swimming data, each 1 min in duration, representing 8 out of the 10 raw observations, where \(F_4\) and \(F_8\) failed to produce data which met all filtering criteria for swimming. The pre-processing method was also found to eliminate periods of excessive thrashing, characterised by large amplitude fluctuations in \(\omega _t\). Speed and turning speed time series data \(u_t\) for each segment, labelled consecutively from \(S_1\) to \(S_{28}\), are shown in-situ with the corresponding raw data in supplementary Figs. S1 and S2.

*left*, negative to the

*right*. Maximum and minimum values of turning speed were found to be \(-14.73\) and \(14.28\,\hbox {rad\,s}^{-1}\) respectively, suggesting a (global) absolute maximum turning speed \(\hbox {max}(|\omega |) \approx 15\,\hbox {rad\,s}^{-1}\). Maximum turning speeds were found to be close to the upper limit detectable between consecutive samples at frequency \(f_s\), where \(\omega _{\mathrm {max}} = \pi f_s = 5\pi \approx 15.71\,\hbox {rad\,s}^{-1}\). Such high speed turns however are observed with very low frequency across filtered swimming segment data, with turns faster than \(5\,\hbox {rad\,s}^{-1}\) accounting for less than 1% of all samples. Isolated swimming trajectory portraits (Fig. 3) display the variety of different characteristic behaviours described earlier. In particular, we observed strong wall-following behaviour which leads to an individual bias of the turning speed in the direction of rotation around the walls.

The distribution of instantaneous speed \(u_t\) was found to be approximately normal with a natural truncation at \(u = 0\) cm s\(^{-1}\). Individually parameterised Gaussian density functions therefore yield a good approximation to the distributions of \(u_t\) (Fig. 4a). Analysis of \(\omega _t\) (Fig. 4b) similarly indicated that a normal probability density function provides reasonable approximations to experimental data, with a mean close to zero. In general however, the distributions of \(\omega _t\) were found to be more sharply peaked than a Gaussian, with heavy tails due to a low proportion of extreme values of turning, both left (\(\omega \gg 0\)) and right (\(\omega \ll 0\)), resulting in larger estimates of the standard deviation and flattening of the corresponding Gaussian probability distribution function (pdf). As such, a normal distribution is found to be appropriate when the sample standard deviation \(\hat{\sigma }_\omega < 1.5\,\hbox {rad\,s}^{-1}\). Above this value, a normal distribution fails to capture both the sharp peak around the mean, and the finite probability of rapid changes in heading. In the model we describe later in this section, the speed process \(u_t\) is assumed to be a stationary Gaussian process, whilst the turning speed \(\omega _t\) is assumed to be a Gaussian process with varying variance. Additionally, these processes are coupled such that we recover both the observed cross-correlation and a correction to the distribution of the turning speed.

We considered the associated correlation time \(\tau \) for the ACFs, where \(\tau = -\varDelta t/\hbox {ln}(r_1)\) was used to parameterise an exponential function \(\hbox {ACF}_{est} = \exp (-t/\tau )\) estimating the autocorrelation decay envelope. Across the majority of segments, a exponential approximation provides a good estimate for both \(\hbox {ACF}_u\) and \(\hbox {ACF}_\omega \) (example shown for segment \(S_9\) in Fig. 5). The average autocorrelation half-life (\(\tau \,\hbox {ln}\,2\)) for \(u_t\) and \(\omega _t\) across segments, were found to be approximately 1.37 and 0.28 s, respectively.

## 4 Modelling

*Kulia mugil*(BL \(\approx 20\) cm) modelled by Gautrais et al. (2009), the variance in swimming speed for smaller zebrafish (BL \(\approx 3\) cm) is large. Many factors influence the range and fluctuations of swimming speed, with drag being the primary physical component, scaling with the square of the wetted surface area. In the presence of viscous drag, with a flow regime dependent on the specific Reynolds number, different aquatic species have evolved a range of swimming styles as described by Sfakiotakis et al. (1999). Specifically for zebrafish, their small size and tail morphology results in a burst-and-coast mode of locomotion, which has been found to be more efficient than a continuous swimming style, as discussed by Weihs (1972), Muller et al. (2000).

*differentials*of position and heading respectively.

In order to recover the observed correlation between the magnitude of \(u_t\) and variance of \(\omega _t\) (e.g. Fig 4c), we introduce the function \(f_c = f_c(U_t,\sigma _\omega ,\sigma _0,\mu _u)\), which couples the two processes such that the variance of the random fluctuations of turning speed \(\varOmega _t\) depends on the speed \(U_t\). Wall (boundary) avoidance is achieved by incorporating a second function \(f_W = f_W(\phi _W,d_W)\) in (8b), which models the tendency of fish to avoid collisions with the tank walls, where \(d_W\) and \(\phi _W\) are the distance and angle of projected collision with a boundary, given the velocity at a given time step. The features encapsulated by these two additional functions \(f_W\) and \(f_c\), including the estimation of all model parameters, are described in what follows.

### 4.1 Selection of wall avoidance function \(f_W\)

*distance*\(d_W\) (Gautrais et al. 2009), or

*time*\(t_W\) (Gautrais et al. 2012) with which a projected collision with the boundary would occur given the current position and velocity of the fish. To quantify this effect, we calculate the distribution of a ‘wall-corrected’ value of the turning speed \(\omega _c\) which is positive when the direction of a turn is away from the collision boundary and vice versa, such that

*increases*the projected distance (or time) to collision. Parameters \(A\) and \(B\) control the strength and decay of the repulsion \(f_W\) and are estimated from experimental data as described in the following section (Parameter estimation). A similar analysis on the effect of wall boundaries on speed regulation suggested that speed is only marginally influenced by the wall distance (Fig. S5, S6). For this reason we opted not to include a functional dependence on either \(d_W\) or \(t_W\) for the speed process in (8a) as we have for turning speed.

As the repulsive turning effect supplied by \(f_W\) does not implicitly prevent trajectories from crossing the simulated boundaries, we also include an additional *hard*-boundary condition. Our simple strategy is to model wall encountering events as fully inelastic collisions, such that the speed \(U_t\) of a random walker passing through a boundary at time \(t\) is set to zero, leaving its position unchanged from the previous time step, so that \(\mathbf{x}_t \leftarrow \mathbf{x}_{t-\varDelta t}\).

In order to replicate experimental conditions, simulated trajectories were modelled in a bounded, rectangular arena. However, a finite, rectangular simulation arena presents discontinuous boundaries at each corner which must be smoothed to prevent competing repulsion by perpendicular walls near the vertices from creating singularities for point-like random walkers. By rounding the edges of the simulation arena with quarter-circles of radius \(R_c\), we avoid unrealistic turning behaviour at the corners of the tank, where a value of \(R_c=10\) cm was found to sufficiently reduce undesirable artefacts in these regions.

### 4.2 Selection of coupling function \(f_c\)

- 1.When \(U_t\) approaches zero, the function returns the upper bound, say \(\sigma _0\) on the variance of the turning speed (to be estimated from the experimental observations), or equivalently$$\begin{aligned} \lim _{U_t \rightarrow 0}f_c = \sigma _0,\quad \sigma _0 > \sigma _\omega \end{aligned}$$(11)
- 2.
The function approaches zero as \(U_t\) goes to infinity.

- 3.
The function returns a value dependent on the variance \(\sigma _\omega \) of the turning speed when \(U_t\) is equal to the average speed \(\mu _u\), which will be estimated so as to better capture the observed experimental distribution (see Fig. 6).

As demonstrated in Fig 6b, this function allows to recover a distribution of \(U_t\) and \(\varOmega _t\) which are highly comparable with experimental data.

### 4.3 Parameter estimation

Mean parameter values for each fish \(F_1\ldots F_{10}\), calculated from 28 isolated data segments.

\({F}_{1}\) | \({F}_{2}\) | \({F}_{3}\) | \({F}_{4}\) | \({F}_{5}\) | \({F}_{6}\) | \({F}_{7}\) | \({F}_{8}\) | \({F}_{9}\) | \({F}_{10}\) | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|

# of segments | 4 | 3 | 4 | 0 | 4 | 4 | 4 | 0 | 1 | 4 |
| |

\(u(t)\) | \(\mu _u\) | 7.555 | 8.899 | 14.406 | N/A | 12.841 | 16.135 | 20.964 | N/A | 11.719 | 16.642 |
02 |

\(\theta _u\) | 0.592 | 0.661 | 0.298 | N/A | 0.426 | 0.633 | 0.771 | N/A | 0.717 | 0.741 |
59 | |

\(\sigma _u\) | 3.350 | 4.447 | 3.981 | N/A | 4.086 | 4.118 | 4.810 | N/A | 4.651 | 4.689 |
21 | |

\(\omega (t)\) | \(\mu _\omega \) | 0.028 | 0.280 | -0.183 | N/A | -0.187 | 0.206 | -0.384 | N/A | \(-0.221\) | 0.221 | \(-\) 02 |

\(\theta _\omega \) | 3.077 | 2.858 | 1.971 | N/A | 2.996 | 3.347 | 2.175 | N/A | 3.456 | 2.606 |
74 | |

\(\sigma _\omega \) | 3.651 | 4.676 | 2.304 | N/A | 3.649 | 2.355 | 1.599 | N/A | 2.914 | 2.128 |
85 |

*turning*) effect of the boundary on turning speed as a function of \(d_W\). An example of the two-step interpolation for segment \(S_{27}\) is shown in Fig. 8a. By considering only segments with which a reasonable fit could be obtained \([S_2, S_3, S_4, S_{6-11}, S_{13}, S_{16}, S_{17}, S_{19}, S_{20}, S_{22}, S_{23}, S_{25-28}]\), we found average parameter values \(A=2.25\pm 0.70\,\hbox {rad\,s}^{-1}\) and \(B = -0.11\pm 0.04\) cm. For completeness, we also calculated parameters values for \(A\) and \(B\) for a time-to-collision dependence \(t_W\) (see Fig. 8b), averaging over segments \([S_4,S_7,S_{8-11},S_{13},S_{16},S_{18-28}]\) to give \(A = 2.25 \pm 0.62\,\hbox {rad\,s}^{-1}\) and \(B = -1.68\pm 0.53\) s. From our analysis we found no compelling evidence supporting a stronger functional dependence of turning speed on either \(d_W\) or \(t_W\) thus we proceeded with a wall-avoidance function dependent solely on projected collision distance \(d_W\) and the collision angle \(\phi _W\). Graphical depictions of the dependence of \(\omega _c\) and \(u_t\) on projected collision distance \(d_W\) and time \(t_W\) for all segment data can be found in supplementary Figs. S3–S6.

Heuristically, we found that a magnified value of \(A\) was required to produce turning behaviour comparable to experimental observations. After many realisations of random walkers with various calibrations, we chose to increase the above value of \(A\) by a factor of 3 such that we simulate all random walkers with \(A = 6.75\,\hbox {rad\,s}^{-1}\) and \(B = -0.11\) cm, calculating \(f_W(d_W,\phi _W)\) from Eq. (10). This discrepancy results either from interpolating \(f_W\) with an insufficient number of data points close to the boundaries, or the compensation required to account for oversimplification of the wall avoidance model.

### 4.4 Model consistency

Global simulation parameters

Parameter description | Symbol | Unit | Value |
---|---|---|---|

Simulation time step | \(\varDelta t\) | s | 0.2 |

Simulated tank (square) side length | \(L\) | cm | 120 |

Rounded edge circle radius | \(r_c\) | cm | 10 |

Maximum turning speed variance | \(\sigma _0\) | \(\hbox {rad\,s}^{-1}\) | 12 |

Max. turning speed (cut-off) | n/a | \(\hbox {rad\,s}^{-1}\) | 15 |

Wall avoidance function amplitude | \(A\) | \(\hbox {rad\,s}^{-1}\) | 6.75 |

Wall avoidance function decay | \(B\) | cm | –0.11 |

Data simulated across a range of sample generation frequencies, 1000–5 Hz (\(\varDelta t\) = 0.001–0.2 s) using identical stochastic processes \(dW_t\) and \(dZ_t\), indicated that trajectories and their underlying statistics (distributions, ACFs etc.) were sufficiently robust to increasing values of \(\varDelta t\) over three orders of magnitude (see Fig. S7). Using a value \(\varDelta t = 0.2\) s was therefore found to provide a good compromise between numerical accuracy and computational efficiency^{1}, with a corresponding sample generation rate of 5 Hz matching that of the experimental acquisition frequency.

To support the inclusion of the coupling function \(f_c\) in our proposed model, we simulated comparable trajectories in the absence of coupling (fixed \(\sigma _\omega \)). Simulated realisations for representative experimental segments \(S_3\) and \(S_{17}\) are shown in Fig. S14, both with and without the coupling between \(U_t\) and turning speed variance \(\sigma _\omega \). In the uncoupled trials, we clearly fail to capture a reasonable estimate for the joint distribution of speed and turning speed (see column B in Fig. S14). Without \(f_c\) to restrict the turning speed variance at high speeds, we find a more normal spread of \(\varOmega _t\) which fails to capture the sharp peaks of the experimental distributions (column D). The one-way coupling between two processes should have no effect on the distribution and autocorrelation of speed data (columns B and E), however we also note that we do not find significant effects on the turning speed ACF (column F). Importantly, we find that trajectories produced by the coupled model appear to be qualitatively more consistent with experimental segment trajectories (column A). We note a higher propensity to enter longer lasting/long path length spiralling when the process are uncoupled, features which are reduced by the coupling as large turning speed variance (increased range in either direction) is reduced at high speeds—and therefore only available to the random walker at lower speeds where less distance will be covered during such a turn. We also note that decoupling the processes reduces the propensity to elicit wall-following behaviour when calibrated on segments exhibiting these phenomena (again due to the increase range of turning speeds when decoupled or conversely because, when coupled, the turning speed distribution is more sharply peaked around zero).

Plots for all segments (coupled model), comparing a single random walker realisation to experimental source data, can be found in the supplementary information (Figs. S8–S13). We also refer again to plots depicting the dependence of speed and wall-corrected turning speed on projected distance and time to boundary collision in supplementary Figs. S3–S6.

Further tests of model consistency are provided by comparing eight random walker trajectories, simulated using the averaged parameter values for individual fish \(F_1\), \(F_2\), \(F_3\), \(F_5\), \(F_6\), \(F_7\), \(F_9\) and \(F_{10}\) found in Table 1, to composite zebrafish trajectories from the corresponding experimental segments \(S_1\ldots S_{28}\). Single random walker realisations, calibrated for each fish are shown in Fig. 10, simulated for a time \(T=60n_s\) where \(n_s\) is the number of segments isolated for each fish. We observe that broadly similar qualitative turning characteristics of each zebrafish are recovered, including the propensity for wall-following behaviour. From these simulations, we find that the model is able to effectively extract and reproduce trajectory data which closely approximates the swimming motion, and subtleties of individual fish, and also how the underlying statistics may be used to predict a form of ‘passive’ thigmotactic-like behaviour^{2}. Specifically, the approximate ratio \(\sigma _\omega / \theta _\omega \) is found to provide a good predictor of the observed thigmotactic-like behaviour that is well captured by the model. In order of increasing ratio, fish \(F_6\), \(F_7\) and \(F_{10}\) exhibit the most consistent wall-following behaviour, with values of \(\sigma _\omega / \theta _\omega < 1\). Consequently, fish which are found to spend a larger fraction of time away from the walls, for example \(F_2\), \(F_5\), \(F_1\), in order of *decreasing* ratio, are found to have \(\sigma _\omega / \theta _\omega > 1\).

## 5 Conclusions

A model of spontaneous zebrafish motion has been presented which captures the approximate distribution of speed and angular speed of swimming fish, accounting for both the autocorrelation and interdependence of these processes. Analysis of simulated trajectories suggests that our model describes many of the salient features of zebrafish locomotion, including the emergence of a thigmotactic-like (wall-following) behaviour when model parameters are calibrated on fish exhibiting similar patterns of motion. Specifically we find that this ‘passive’ wall-following behaviour results from a model in which only repulsion from the wall is present. The novel feature of this model, extending the ‘Persistent Turning Walker’ model due to Gautrais et al., is to capture the intrinsic speed variation of zebrafish and other small fish.

Importantly, by allowing speed to vary in our model, further progress can be made in the development of group models which can address the most recent experimental findings for similar fish species. We refer specifically to the findings of Katz et al. (2011) and Herbert-Read and Perna (2011), which report that speed regulation is the primary response governing the interaction between conspecifics and their environment.

Further development of these models, informed directly from experimental data, represents a significant departure from some canonical approaches where fish are modelled with constant speed and conspecific interactions result in changes only to their heading direction, or angular speed. Direct calibration of the model to experimentally observed fish trajectories results in a purely data-driven model and provides the necessary foundations for the future objective of understanding modelling the dynamics of multi-fish shoals. The results of the model are encouraging and provide a solid basis for future investigations into fish social response.

## Footnotes

- 1.
Random walk trajectories with a duration of 60 s are computed in approximately 0.2 s in the current implementation with \(\varDelta t = 0.2\,\hbox {s}\)

- 2.We denote ‘passive’ thigmotactic-like behaviour as occurrences of wall-following which is not driven by explicit modelling of psychological effects, for example in seeking protections from predators

## Notes

### Acknowledgments

We also gratefully acknowledge the contributions of Fabrizio Ladu and Sachit Butail at the Dynamical Systems Laboratory, New York University Polytechnic School of Engineering, for providing experimental data and visual tracking software used in our analysis. MdB would like to thank the Dynamical Systems Laboratory for hosting him during the preparation of this manuscript and to acknowledge support from the Network of Excellence MASTRI Material e Strutture Intelligenti (POR Campania FSE 2007/2013).

## Supplementary material

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