Two modes in the absolute velocity statistics in cautious walks of laboratory rodents

Abstract

We have analyzed a large number of rodent tracks in open field tests to elucidate the statistics of their velocities. We found that the probability distribution of the absolute velocity of individual rodents can be approximated by a superposition of two Rayleigh distributions, with distinct characteristic velocities, $v_1$ and $v_2$, with $v_1<v_2$; this is in contrast to the single Rayleigh distribution for the speed of a Brownian particle executing 2D random motion. We propose that the part of the distribution near the larger speed, $v_2$, characterizes rodents’ progressions in space, while the part near $v_1$ describes other types of motion, such as lingering and body micromovements. We observed that the animals switched randomly between these two modes. Since the existence of the modes is observed both in preweaned, blind pups and in older animals, it cannot be ascribed to foraging, but rather reflects risk assessment and proactive inhibition. We called such motion “cautious walks.” Statistical analysis of the data further revealed a biphasic decline in the absolute velocity autocorrelation function, with two characteristic times, ${\tau }_s<{\tau }_l$, where ${\tau }_s$ characterizes the width of absolute velocity peaks, and ${\tau }_l$ is associated with the timing of the switches between progression and lingering. To describe the motion, we propose a stochastic model whose 2D Langevin-like equation has a damping coefficient that switches between two values, representing mode switching in rodent locomotion. A mechanistic analogy of rodent cautious walks is a Brownian particle that randomly switches its mass between a lighter weight and a heavier weight. Techniques developed here may be applicable for locomotion studies in a wide variety of contexts, as long as tracking data of sufficient resolution are available.

Significance

This study combines experimental measurements and modeling to show that rodent velocities follow a polymodal Rayleigh distribution, in contrast to the Brownian motion’s single mode. Distinct velocities of cautious walks reflect innate behavioral modes: lingering (i.e., risk assessment/micromovements) and rapid progression, randomly switching both in preweaned rodent pups and adult animals. A biphasic autocorrelation decline reveals two timescales: one for pulse-like structure of movement and the other one for slower mode switching. The statistical methodology advances analysis of locomotion data archives, offering tools for translational neuroscience. Furthermore, our stochastic approach can be used to study animal models of human motor disorders, emotional regulations of motor patterns, the degree of variation among animals in the same age group, and also determine which parameters experience change as a result of development.

Introduction

Laboratory rodents are among the most frequently used subjects of experimental neuroscience and, in particular, the studies of motor and exploratory behavior (1,2,3,4,5,6). Quantifying rodent motor patterns is important for characterizing motor behaviors across diverse experimental scenarios, including human disease models, studies involving gene or circuit disruptions, investigations of post-induced brain or spinal cord injuries, and assessments of drug responses (7,8,9,10,11). Motor assays, such as motion tracking, offer a cost-effective and labor-efficient method to quantify different aspects of locomotor performance (5,12,13,14).

A number of statistical models of animal locomotion have been proposed. The motion of simpler species, such as invertebrates, has been successfully described by models based on the Brownian motion (see, e.g., the reviews in (15,16)). Evolutionary cephalization facilitated goal-directed locomotive strategies (17,18). A search for a randomly distributed resources might include a Brownian strategy in mammalian and avian species (19,20,21,22), or its modifications such as correlated random walks (23,24,25). Models that contain Lévy flights/walks have been widely used to describe animal movement in the context of foraging (see, e.g., (26,27,28) and reviews (25,29)). Many useful methods for studying velocity distributions and correlations have been developed in the field of microbiology (30,31,32,33).

In the mammalian locomotion models, animals are often assumed to move to find a rare reward. In a natural habitat, however, foraging is not necessarily the main driving force behind locomotion, as small animals are under continuous pressure to assess the risk of becoming prey (34). Exercising caution is crucial for small animals to increase their chances of survival in environments where they face constant threats from predators. Evolution has shaped their behaviors and adaptations to minimize the risks associated with being preyed upon, which includes camouflage, sheltering, and nocturnal behavior.

Caution is also a natural feature of small animals’ spontaneous walks in a new environment. Animals evaluate potential environmental resources and dangers during stops, which usually punctuate locomotion (35) and are often used for reorientation. Reorientation behavior is characterized by many types of 3D posture changes (2), whose intensity and repertoire depends on the animal locomotive activity and anxiety level (5,36,37). In the case of danger, a stop may turn into a complete arrest (“freezing”) (13,38), or it could be followed by a spontaneous flight to escape predation (39).

In this paper we used laboratory rodent (rat and mouse) experiments and mathematical modeling to create a statistical model of rodent walks, which incorporates deceleration to reflect caution, as a factor that modulates the statistics of the animals’ speed. To dissociate reward-related foraging behavior from risk-avoiding environmental exploration, we analyzed short-term (of the order of minutes) tracking data from a large number of experimental observations of rodent walks in an open field. Our test subjects included adult rodents in a regular satiated state (i.e., not deprived of food/water before the tests), as well as preweaned rodent pups of different ages, before the age of solid food ingestion. Therefore, in our experiments we assume that food search is not the leading motivation to move in a new environment. The tests were conducted by placing a single animal in an empty two-dimensional (2D) arena, to observe spontaneous (i.e., not provoked by external events) locomotion in a stimulus-free space. We refer to this type of motion as “cautious walks.” To observe various stages of locomotion and spatial navigation, we included animals of different ages, to see if locomotor pattern exhibit any age-related features.

In this setting, we expect that the animal movement behavior is largely independent of the environment, but rather determined by the animal’s internal brain dynamics, at its different levels of motor control. Many authors have modeled the trajectories of this type of motion by using random walks (see, e.g., (15,35)). The question becomes: To what extent are these walks random and uncorrelated? The first step in answering this question is to reduce the system’s complexity (1,5,40) and to analyze the speed, rather than the trajectories or specific behavioral motifs (36) of the animals, by representing the whole rodent as a single body centroid. This is what we focus on in this paper. We find that the statistics of individual instantaneous velocity of animals deviates from the classical Brownian particle in two ways: 1) the absolute velocity distributions for individual animals are not well described by a single Rayleigh distribution, but instead seem to contain two Rayleigh components with distinct characteristic velocities, and 2) the autocorrelation function of the absolute velocity is characterized by two distinct correlations times, corresponding to short and long correlations, ${\tau }_s<{\tau }_l$.

To describe these statistics, we created a model, where an individual animal accelerates according to a stationary pulse Poisson process. It is assumed to be subject to random pulses of acceleration, whose correlation time corresponds to the smaller value, ${\tau }_s$. On the other hand, deceleration is modeled as a consequence of “friction,” or “viscosity,” whose attenuation constant takes two values. The larger of the values corresponds to the slowest mode of movements, which includes the so-called lingering (i.e., drifting, sniffing, rearings, lateral head scans, and grooming), while the smaller attenuation constant corresponds to the mode of “quasilinear raids” in pups (41), or short progressions in adult rodents (42). The random switching between these two modes yields the second, larger, correlation time, ${\tau }_l$. The probability distribution of the absolute value of the velocity is given by the Rayleigh function for each of the two modes, while, overall, the probability distribution has the form of a 2-component Rayleigh function.

The results of our paper also contribute to the study of the constructive role of noise in complex system dynamics (43,44,45,46,47). It is generally assumed that the neural control signals are subject to noise interference, where the noise’s variability escalates as the magnitude of the control signal increases (48), although it is not fully understood how neural activity in biological control networks shapes variability in motor output (49). The noise “warms up” the system and helps the animal start moving. It also makes the rodent movement unpredictable and difficult for a predator to detect and catch. Another constructive role of this randomness is preparation for search behavior of adult animals, which has been shown to be partly random.

This model is an attempt to create a statistical description of rodent movements that is closely related to the experimental observations. Using this model, we propose causal relations between the different components of velocity control loops. Furthermore, it can be used to study animal models of human motor disorders, emotional regulations of motor patterns, the degree of variation among animals in the same age group, and also determine which parameters experience change as a result of development.

Materials and methods

Animals and track recording

The animals were housed in standard plastic cages, on a 12-h light/dark cycle, with food and water available ad libitum. All the rat pups included were of an outbred Wistar strain; adult rats were outbred Wistar (the in-house archive) or Long-Evans (the open source video archive, see below) males. The mice were of an outbred ICR strain. The numbers of animals in every age group are given in the results.

The rat dams were separated from their cage mates since the last days of pregnancies. Mouse mothers were kept with the litter, the mate and, optionally, one more female cage mate. The rodent litters were intact until the day of experiment. On the postnatal days 11 (mice, M11), 13 (rat, R13), 15 (rat, R15), or 17 (rat, R17) the dams were separated from their pups. The rodent nestlings were allowed to calm in their huddles for about 30 min. Then, the pups were gently grasped individually from the huddle periphery, and transported to the experimental chamber next door. Each pup was released in the center of an open field arena (black chamber $59\times 59$ ${\text{cm}}^2$, the same as reported in (41)), with a camera set 1 m above. The pup’s placement was a start for video tracking which lasted for 120 s $n=336$, where $\mathcal{N}$ is the number of pups in the sample, 300 s $\left(n=11\right)$, or 600 s $\left(n=14\right)$. The arena was cleaned after each individual session. All the pups were tested once, weighed after the experiments and returned to their mothers immediately. Rat and mouse litters of 6–14 pups were enrolled. Body weights of the rodent pups are reported in Table 1.

Table 1.

The best-fit parameters of function (Eq. 8) for each animal group.

Parameter	M11	R13	R15	R17	R adults
$b$	0.785 (0.782, 0.788)	0.80 (0.792, 0.803)	0.70 (0.693, 0.704)	0.78 (0.771, 0.798)	0.90 (0.895, 0.902)
${\tau }_s$ (s)	0.19 (0.188, 0.201)	0.36 (0.349, 0.373)	0.41 (0.395, 0.415)	0.21 (0.200, 0.225)	0.60 (0.588, 0.610)
${\tau }_l$ (s)	15.41 (15.09, 15.74)	11.81 (11.42, 12.20)	5.54 (5.43, 5.66)	3.33 (3.09, 3.57)	17.60 (16.75, 18.45)
Weights (g)	7.02 $\pm$ 1.1	22.7 $\pm$ 4.0	31.3 $\pm$ 4.6	36.4 $\pm$ 7.2	250–400

The average body weight of the animals in each group is also provided ($\pm$standard deviation). The numbers in parentheses show the 95% confidence interval.

We used mouse and rat pups of a comparable developmental stage (M11 and R13; i.e., 2–3 days before eye opening), and two older cohorts (R15 and R17). The second rat pup cohort (R15) had started to open their eyelids, but had not yet acquired visual sensing (50), while the oldest pup group (R17) fully possessed the visual sensory inflow (50). Therefore, the adult animals (R adult) and 17-day-old pups (R17) were regular sighted rats, studied under normal daylight conditions, with fully available visual cues.

Video data of the adult rat group (R adult) consisted of 16 individual recordings made in a square ($120\times 120$ ${\text{cm}}^2$) “open field” arena, according to the standard procedure: a rat was released at the arena center, with its back to the experimenter, and this was a start for video tracking. The adult rats were males weighing 250–400 g and aged 4–5 months. Additionally, an open video archive of 20 Long-Evans rats performing an open field test ($100\times 100$ ${\text{cm}}^2$ arena) was downloaded from example materials (51), and tracked as described below. The tracking data were pooled for analysis. The open field tests lasted 150 s $\left(n=20\right)$ or 300 s $\left(n=16\right)$ for adult rats (R adult).

Movement tracking was done offline. A tracker software (ToxTrack, the algorithm ToxID (52)) determined the body position by calculating the white mass center against the black background each 40 ms. In brief, animals were detected as bright moving objects using a threshold intensity value. The obtained objects or blobs were filtered by size to remove false-positive events. Thus, the multidimensional mixture of animals’ posturing and behavioral motifs (see, e.g., (2,36)) were reduced to a single point movement.

To confirm the robustness of our findings and to exclude the possibility of error related to the specific tracking system, an additional subset of video data recorded in 19, 13-day-old rat pups, was analyzed by two independent tracking software. ToxTrack was used as described above, and also DeepLabCut (53) was used to track the positions of the body parts for each animal: the head, trunk, and base of the tail. The results, which are consistent with our main findings, are shown in Appendix 3.

The experiments were carried out in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals; the experimental protocol was approved by the Institutional Animal Care Committee. All efforts were made to minimize animal discomfort.

Trajectories

We digitized the motion of the rodent groups M11, R13, R15, R17, and R adult. Typical trajectories are shown in Fig. 1, three for different types of animals, including a close-up (the bottom row).

Figure 1.

Examples of rodent trajectories. Three typical trajectories are presented for each animal group. The bottom row (below the dashed line) shows a close-up of a trajectory. The frames show distance in cm.

We note the presence of a large degree of variability in trajectory patterns, with some immature animals remaining within a few ${\text{cm}}^2$ of their starting position, while others traversed the arena several times, and yet others spent time within a small area as well as engaging in longer runs. The blind walks of the youngest rat pups (R13) have been recently characterized as a superposition of localized walks and quasilinear runs (41). The trajectories of older animals obtained here were typical for this type of tests, and included open center investigation, thigmotaxis (i.e., movements along walls), and localized walks during lingering.

Most of the studies in the literature focus on displacement, because foraging efficiency depends directly on it. In this paper, we address the relative lack of research on speeds, since, much as in mechanics, speed is the cause of displacement. Accordingly, information about speed provides a direct window into the neurological processes that prompt rodents to start and stop moving. Having verified that the rodents’ movement is isotropic, that is, the direction angle of their velocity is uniformly distributed over $[0,2\pi )$, we built the empirical distributions of the velocity projections. It was confirmed that they match the distributions obtained by converting the speed-magnitude distribution from polar to Cartesian coordinates. We omit these plots for brevity; instead, we include example trajectories that visually demonstrate the isotropic nature of the movement.

In all cases, visually, blind walk trajectories (M11, R13, R15) resemble those of Brownian motion, and support the suggestion that the motion is nearly isotropic.

Velocity data collection

The data were obtained as coordinates of each animal at 0.04 s intervals. Denote the obtained 2D coordinates of the $i$-th animal as ${\mathit{r}}_k^{\left(i\right)}$, where $k=1,\dots ,N$ are the consecutive moments of time, $N$ is the number of measurements in the time series, and $i=1,\dots ,n$, where $n$ is the number of pups in the sample. The absolute value of the instantaneous velocity (or speed) at time $k\Delta t$ is numerically estimated as

$$v_k^{\left(i\right)}=\frac{\left|{\mathit{r}}_{k+1}^{\left(i\right)}-{\mathit{r}}_k^{\left(i\right)}\right|}{\Delta t}\text{,}$$ (1)

where $\Delta t$ is the duration of time between two measurements (0.04 s). To gather the statistics of this quantity, all the data from the tracks were taken into account.

To create numerical probability distributions of speed values for individual animals, we specified the bin size for capturing the experimentally observed speed values. This quantity was chosen to represent typical values for the majority of animals in the age group. We used bin sizes 0.25, 0.25, 0.5, 0.5, and 2.0 cm/s for the groups M11, R13, R15, R17, and R adult, respectively.

The autocorrelation function of the absolute velocity (Eq. 1) was calculated as follows. Define two sample means of the absolute velocity:

$${\overline{v}}_1^{\left(i\right)}\left(j\right)=\frac{1}{N-j}\sum _{k=1}^{N-j}{v}_k^{\left(i\right)},0\le j\le N-1\text{,}$$

$${\overline{v}}_2^{\left(i\right)}\left(j\right)=\frac{1}{N-j}\sum _{k=1}^{N-j}{v}_{k+j}^{\left(i\right)}\text{,}$$

where again, the index $i$ denotes the animal, index $k$ the time point, and $j$ is time shift. These quantities reflect the mean behavior of the different parts of the sample. Then, we define (see, e.g., (54)) the autocorrelation function as

$$C^{\left(i\right)}\left(j\right)=\frac{1}{N-j}\sum _{k=1}^{N-j}{v}_k^{\left(i\right)}{v}_{k+j}^{\left(i\right)}-{\overline{v}}_1^{\left(i\right)}\left(j\right){\overline{v}}_2^{\left(i\right)}\left(j\right)\text{.}$$

The normalized autocorrelation function (also called the autocorrelation coefficient) for pup $i$ is then given by

$$K^{\left(i\right)}\left(j\right)=\frac{C^{\left(i\right)}\left(j\right)}{C^{\left(i\right)}\left(0\right)}\text{,}$$

where the quantity $C^{\left(i\right)}\left(0\right)$ has the meaning of a sample variance. The average (over the rat pups) autocorrelation function for delay $\tau =j\Delta t$ is calculated as

$$K_v\left(\tau \right)=\frac{1}{n}\sum _{i=1}^n{K}^{\left(i\right)}\left(j\right)\text{.}$$ (2)

In what follows, we analyze individual animals’ speed distributions to uncover the existence of two modes of motion. When it comes to the autocorrelation functions, we have to turn to an ensemble of animals, because there are not enough data in the individual animal tracks to construct an interpretable autocorrelation function. Inevitably this leads to smoothing out individual differences, but nonetheless, as will be shown below, one clearly observes the existence of two timescales in the autocorrelation function. This is likely a consequence of the sufficient difference between the characteristic times, such that individual differences cannot erase the signature of this phenomenon in the group data.

Stochastic modeling

We use stochastic differential equations, which in general can be written as

$$\dot{v}=a\left(v\right)+b\left(v\right)\xi \left(t\right)\text{,}$$ (3)

where $a\left(v\right)$ and $b\left(v\right)$ are functions of $v$, and $\xi \left(t\right)$ is a stationary white Gaussian noise with a zero mean $⟨\xi \left(t\right)⟩=0$ and a correlation function $⟨\xi \left(t\right)\xi \left(t+\tau \right)⟩=2D\delta \left(\tau \right)$.

The stationary solution of the corresponding Fokker-Planck equation for the absolute velocity distribution in Stratonovich interpretation (55,56,57) is

$$w_{st}\left(v\right)=\frac{C}{\left|b\left(v\right)\right|}\exp \left[\frac{1}{D}\stackrel{v}{\int }\frac{a\left(v^{\text{'}}\right)d\left(v^{\text{'}}\right)}{b^2\left(v^{\text{'}}\right)}\right]\text{,}$$ (4)

where $C$ is a normalization constant.

The traditional way of applying this formalism is to write a stochastic differential equation first and then obtain a solution as a result. In this paper, on the contrary, we use the stationary distribution obtained from experimental data and then find the corresponding equation to construct the mechanical model of specific random walks.

Results

We investigated free walks of rodents in an open field. We digitized the motion of five groups of animals: 13-day-old $\left(n=150\right)$ rat pups (R13), 15-day-old $\left(n=159\right)$ rat pups (R15), 17-day-old $\left(n=24\right)$ rat pups (R17), adult $\left(n=36\right)$ rats (R adult), and 11-day-old $\left(n=33\right)$ mouse pups (M11). The young animals were of both sexes, since all pups in the litters were included. A part of the R13 dataset $\left(n=51\right)$ was analyzed in a recently published manuscript (41). Adult rats were males, 4–5 months old, with body weights of approximately 250–400 g at the moment of recording. Typical trajectories are shown in Fig. 1, including a close-up (bottom row). The digitized tracks have length between 120 and 600 s for all the animals.

The distribution of the absolute value of the instantaneous velocity: The existence of two characteristic velocities

Fig. 2 shows graphs of the absolute value of the instantaneous velocity of animals, where (a) shows motion over the interval of 100 s, while (b) zooms into a shorter interval of motion (5 s). We observe the rugged appearance of these graphs with sharp accelerations and decelerations, suggesting the presence of short-time correlations. Despite a high degree of heterogeneity among the animals with respect to the speed of motion, some patterns can be discerned. For example, we can see in Fig. 2 b that the average width of the high-speed peaks is less than a second. Zooming out to (a), we further see that there are intervals of high peaks and intervals of low peaks of a typical duration of about a few seconds. Fig. 3 a illustrates some of these patterns further by plotting points of the trajectory in yellow if the absolute velocity was smaller than 1.5 cm/s, and in blue otherwise. We can see that the motion consists of an intermittent sequence of faster and slower regions. The threshold value is arbitrary. Using a different threshold value for the speed cutoff would make the yellow parts longer or shorter, but would not change the overall picture. Later in the paper we present a more rigorous way to quantify the timescales.

Figure 2.

Typical graphs of the absolute value of the instantaneous velocity (Eq. 1), plotted against time, for the four groups of animals. Please note the different range in the plots for adult rats. (a) Plotted over 100 s (note the different vertical scale for adult rats), (b) a close-up over the interval of 5 s.

Figure 3.

Absolute velocity variation patterns. (a) An example of a typical trajectory of a 13-day-old rat pup, where the yellow segments correspond to speeds $<1.5$ cm/s, while blue segments to speeds $\ge 1.5$ cm/s. (b) The two modes corresponding to strong and weak decelerations as a dichotomous random process. ${\vartheta }_{1i}$ are the times of localized walks during lingerings and ${\vartheta }_{2i}$ the times of progressions.

In Fig. 3 b we present a typical graph of the absolute velocity as a function of time. There are intervals during which the peak heights remain approximately constant, with alternating stretches of high- and low-amplitude impulses. By comparing these speed profiles with direct observations of the animals, we infer that the high-amplitude peaks correspond to relatively quick progressions, whereas the low-amplitude peaks reflect shuffling, turning, and sniffing the immediate surroundings. We denote the time intervals of slow motion as ${\vartheta }_{11}$, ${\vartheta }_{12}$, ${\vartheta }_{13}$, … and the intervals of fast motion as ${\vartheta }_{21}$, ${\vartheta }_{22}$, ${\vartheta }_{23}$, ….

Examples of the statistics of the absolute velocity are presented in Fig. 4, where we show several typical frequency versus speed curves for animals from different groups. Such frequency versus speed curves for individual animals were fitted to several functions. First they were fitted to the function proportional to the Rayleigh distribution,

$$f_1\left(v\right)=\frac{cv}{v_1^2}{e}^{\frac{-v^2}{2v_1^2}}\text{,}$$ (5)

where $c$ is a normalization constant. The integral of the frequency is equal to the number of experimental measurements. Then we fitted them to a weighted sum of two Rayleigh distributions with characteristic velocities, $v_1<v_2$,

$$f_2\left(v\right)=\frac{{\mu }_{1}v}{v_1^2}{e}^{\frac{-v^2}{2v_1^2}}+\frac{{\mu }_{2}v}{v_2^2}{e}^{\frac{-v^2}{2v_2^2}}\text{,}$$ (6)

where ${\mu }_1$ and ${\mu }_2$ are the relative weights of slow and fast movements. The details of the fitting procedure are described in Appendix 1.

Figure 4.

Sample individual animal graphs of the frequency versus absolute velocity for four groups of animals are presented by dots (with the bin size determined for each animal as described in Appendix 1). These data are fitted by Eq. 7 (yellow curves) and Eq. 6 (blue curves). The three graphs represent three typical examples of individual animals for each group.

It was apparent that, in the great majority of the cases, the 2-component Rayleigh distribution (function Eq. 6) provided a better fit. The superiority of model Eq. 6 with respect to model Eq. 5 was further demonstrated by using the Akaike information criterion (AIC) (Fig. 5 a). The lower value of this criterion corresponds to the stronger modeling choice. Our calculations show that, despite having two additional parameters, function Eq. 6 is a stronger choice than function Eq. 5. Because of the two contributions with different characteristic velocities, the shape of this function better reflects a somewhat prolonged “shoulder” of the speed distributions of individual animals.

Figure 5.

Model selection for the absolute velocity distributions: (a) comparing a 1-component Rayleigh distribution (Eq. 5) and a 2-component Rayleigh distribution (Eq. 6), (b) comparing the Gamma distribution (Eq. 7) and a 2-component Rayleigh distribution (Eq. 6). Fitting was performed for all the animals individually. Shown are the proportions of animals better modeled with each of the models according the AIC.

In addition to the functions Eqs. 5 and 6, we also tested a different two-parametric function,

$$f_0\left(v\right)=\frac{\mu v}{v_1^2}{e}^{-\frac{v}{v_1}}\text{,}$$ (7)

which is a Gamma distribution with the shape parameter $k=2$. The motivation behind this choice is to use a function that is zero at $v=0$ and features a slower decay compared with the Rayleigh distribution. As before, this function’s AIC was compared with that for the 2-component Rayleigh distribution. Again, the 2-component distribution (function Eq. 6) presented a better fit for the majority of cases (Fig. 5 b), but here some interesting age-dependent patterns were observed. Based on the data, we see a trend wherein, as the animals get older, their absolute velocity distributions tend to be described better by a multiscale Rayleigh distribution, with a sharp improvement reached by eye-sighted animals (R17 and R adult vs. M11, R13, and R15). Presumably, the mature visual input helps in reorientation trials and thus in a better splitting of the modes. In Appendix 1 we also describe fitting the speed data with multicomponent Rayleigh functions, where the 3- and 4-component functions involve two and four additional parameters, respectively. It turns out that, especially for adult rats, the multicomponent Rayleigh functions provide even better fits, because these animals explore a much wider range of speeds and utilize multiple “gears” (42). In this work, however, we focus on cautious walks in a novel environment, and therefore restrict our modeling to a 2-component Rayleigh distribution.

Examples of fitting are presented in Fig. 4, where three instances of individual animals’ absolute velocity distribution are plotted together with the best fitting function Gamma distribution Eq. 7 (yellow curves) and a weighted sum of two Rayleigh distributions Eq. 6 (blue curves). The two terms in function Eq. 6 can be interpreted as two components of motion, progressions and lingerings.

Using the 2-component Rayleigh function Eq. 6, we considered the dependence of $v_2$ on $v_1$ (see Fig. 6). The dependence is well described by a linear function with the proportionality coefficient between about 3 and 5 for all the animal groups.

Figure 6.

Best fitting characteristic speed values, $v_1$ and $v_2$. Each dot corresponds to an individual animal. The best fits with the linear function, $v_2=a{v}_1$ (a single-parametric fitting) are shown as straight lines, with 95% confidence intervals indicated by dashed lines. The values of the proportionality coefficient $a$ are given by $a=4.00\left(3.42,4.60\right)$ for M11, and $a=3.05\left(2.85,3.26\right),3.45\left(3.27,3.64\right),4.00\left(3.55,4.46\right),4.27\left(3.74,4.81\right)$ for R13, R15, R17, and R adults, respectively (listed are the the best fit values and the 95% confidence intervals in parentheses). The difference between the slopes for the youngest (R13) and the oldest (R17 or R adults) rat groups is significant with $p<{10}^{-2}$.

Patterns of age progression in the context of the two characteristic velocities

Fig. 7 shows the means of the two characteristic speed values for each group of animals. There is a clear age progression for the values of $v_1$ and $v_2$, which significantly differ from one another (see Appendix 2).

Figure 7.

Characteristics of motion for different groups of animals. (a) The means (over the animals in each group, dots), and standard errors of the means (vertical bars), for the best fitting values for $v_1$ (yellow symbols) and $v_2$ (blue symbols), for each group of animals. All the means for $v_2$ are different among the groups ($p<0.05$ by t-test). (b) The values of ${\mu }_1/\left({\mu }_1+{\mu }_2\right)$, that is, the relative contribution of small motions, for all animal groups.

Interestingly, the relative weight of slow movements, ${\mu }_1/\left({\mu }_1+{\mu }_2\right)$ seems to be stable during ontogeny and between the two rodent species: the lingerings constitute as much as a half of registered movements (Fig. 7 b). The functional blindness of the previsual pups (M11, R13, and R15) also does not affect the locomotor ratio studied. During the free walks inside the empty test arenas, possible navigation efforts of the animals appear to influence trajectories, but not the studied velocity dynamics and statistics.

Two modes in the absolute velocity autocorrelation function

We have calculated the normalized absolute velocity autocorrelation function, or autocorrelation coefficient (see Eq. 2). Fig. 8 presents the results, where the blue lines (experimental data) show a distinct biphasic decline. These curves were fitted with the function

$$g\left(\tau \right)=b{e}^{-\tau /{\tau }_s}+\left(1-b\right)e^{-\tau /{\tau }_l}\text{.}$$ (8)

Figure 8.

The absolute velocity autocorrelation function calculated as described in Eq. 2 for each animal group. Top row: ensemble means (dark lines) and standard deviations (lighter shading) of the autocorrelation functions, plotted on a linear scale. Bottom row: ensemble means of the autocorrelation functions (blue lines) together with the best fitting function (Eq. 8) plotted on a logarithmic scale. The confidence intervals of fitting are shown in red dashed lines (hard to see, because of the narrowness of those intervals).

The black lines in Fig. 8 represent the best fit results for function Eq. 8. The values of $b$, ${\tau }_s$, and ${\tau }_l$ thus obtained are summarized in Table 1.

A note on the terminology. In the literature there are two somewhat contradictory definitions of the correlation time. The first one (call it “experimental”) comes from the method of the approximation of the correlation functions by one or more exponents of the form $e^{-t/{\tau }_{\exp }}$. In this case, ${\tau }_{\exp }$ corresponds to an $e$-fold decrease in correlation function, but after this time correlations still exist and cannot be neglected. The other definition of the correlation time (call it “theoretical”) comes from the theory of random processes and corresponds to the time, ${\tau }_{\text{theor}}$, after which correlations can be neglected; the process can be considered a Markov process if the time step $\Delta t>{\tau }_{\text{theor}}$. In this paper we use the first, experimental, definition of the correlation time.

The existence of two modes in the absolute velocity autocorrelation function inspired a model that is described next. In the model, the two correlation times are understood as corresponding to two mechanisms of speed regulation: acceleration and deceleration.

Stochastic modeling of rodent walks

We have formulated a stochastic model that describes the motion of rodents, which is based on the observations reported above. If the rodent motion was completely random (which means being subject to uncorrelated acceleration pulses, such as random uncorrelated pushes along both axes in a 2D space for Brownian particles, or random directional choices), then its velocity projections would be distributed normally, and the absolute velocity would be described by the Rayleigh distribution (55). This is a direct consequence of the Langevin equation with a constant noise intensity and a constant attenuation coefficient. In this case, the absolute velocity autocorrelation function would be exponential with the characteristic time inversely proportional to the attenuation coefficient. This, however, is not what we observed.

Indeed, our data analysis demonstrated that the absolute velocity is distributed according to a 2-component Rayleigh function with two characteristic velocities; a deviation from the Rayleigh distribution indicates the presence of a correlation in acceleration (58,59). Furthermore, the absolute velocity autocorrelation functions for the rodents in our experiments are characterized by not just one, but two decay times.

To proceed with the mathematical model, we further note that the time dependence of the absolute value of the instantaneous velocity presents as a sequence of peaks, whose width is similar to the smaller of the two correlation times (${\tau }_s$ in Table 1) (see, e.g., Fig. 3 b). Moreover, we observe the existence of two types of regions in the time series, those with relatively low peaks and those with relatively high peaks. The duration of those regions varies, but is similar, roughly, to the larger of the two correlation times (${\tau }_l$ in Table 1). The envelope of the speed versus time graph resembles a dichotomous process, with a sequential alternation of higher and lower horizontal regions (Fig. 3 b). Based on the idea of the previously reported “gears” that were observed in the adult rodent locomotion (42), we propose the existence of two distinct modes in our experimental system, which correspond to (on average) faster and slower movements. The slower mode corresponds to lingering (and includes postural movements, rearings, grooming, lateral head scans, etc.) and the faster mode to short walks and runs in adults, and to the quasilinear raids in pups. An animal enters one of the modes and moves accordingly until it switches to the other mode, and so on.

Finally, we note that, while model parameters change as a function of age (as shown in Fig. 7), the dynamics within each age group is assumed stationary. This is a reasonable approximation, since there is a large timescale separation between the observed dynamics and the slower maturation dynamics that effectively change the model parameters.

To construct a stochastic differential equation for the absolute velocity and to obtain probability distribution function Eq. 6, it is convenient to split the change of the absolute velocity into acceleration and deceleration,

$$\dot{v}=-{\xi }^{-}\left(t\right)+{\xi }^{+}\left(t\right)\text{,}$$ (9)

and construct a model for each of these two processes.

Deceleration

In the absence of environmental cues, the switching between the slower and the faster modes is modeled as transitions in a two-level system (60), or a dichotomous random process (55). We assume that the transitions between the two movement modalities are temporally uncorrelated. This idea is consistent with the findings of, e.g., (61).

Our two-mode description is analogous to intermittent search strategies, where (quoting from (62)): “it is assumed that the searcher randomly switches from phase 1 (2) to phase 2 (1) with a fixed rate per unit time, ${\lambda }_1$ $\left({\lambda }_2\right)$, that is, with no temporal memory. It leads to exponentially distributed phase durations, in agreement with numerous experimental studies (63,64,65,66), the mean duration of phase $i$ being $T_i=1/{\lambda }_i$ .” The random switching model is typical for animal locomotion, despite the fact that the internal mechanisms of such switching have not yet been understood.

Let ${\vartheta }_{1i}$ be the times of slower movements during lingerings and ${\vartheta }_{2i}$ the times of faster walks (see Fig. 3 b). These two modes correspond to strong and weak decelerations, respectively. For example, in Fig. 3 b, we observe faster motion during the first second, that is, ${\vartheta }_{21}\approx 1$ s, and then slower motion for about 1 s (${\vartheta }_{11}\approx 1$ s), then ${\vartheta }_{22}\approx 0.5$ s, ${\vartheta }_{12}\approx 4$ s, and so on.

To describe this behavior, we assume that deceleration depends linearly on the speed, but the proportionality coefficient is described by a dichotomous process (or telegraph process) with constants ${\gamma }_1>{\gamma }_2$. Deceleration, being dependent on the speed, experiences fluctuations within each mode. These fast oscillations are accounted for in our model by the short correlation time. Long-term correlations related to mode switching are incorporated in deceleration. Since the speed oscillates rapidly during the time interval between the switchings, each mode is characterized by its own deceleration, ${\xi }_1^{-}$ and ${\xi }_2^{-}$. These are related to the damping coefficients through ${\xi }_i^{-}={\gamma }_i{⟨v⟩}_i$. To summarize, we have two separate timescales in the system, the shorter one is related to the changes in acceleration (see below), and the longer one to the changes in deceleration.

In the absence of evidence otherwise, transitions from one mode to another are assumed to be random and uncorrelated. The assumption that transitions among the two movement modalities are temporally uncorrelated is a first approximation, and additional correlations can be discovered in the future. In our model, ${\vartheta }_{1i}$ and ${\vartheta }_{2i}$ are independent exponentially distributed random variables (see also the assumption in (21)); we denote their averages as $⟨{\vartheta }_{1i}⟩=T_1$ and $⟨{\vartheta }_{2i}⟩=T_2$. The ratio of these times is determined by the ratio of relative weights of slow and fast movements in experimental distribution Eq. 6:

$$\frac{T_1}{T_2}=\frac{{\mu }_1}{{\mu }_2}\text{.}$$ (10)

The correlation function of the deceleration is a correlation function of the Markovian process in a two-level (${\xi }_1^{-}$ and ${\xi }_2^{-}$) system (60),

$$K_{{\xi }^{-}}\left(\tau \right)=\frac{{\tau }_l^2{\left({\xi }_1^{-}-{\xi }_2^{-}\right)}^2}{T_1{T}_2}\exp \left(-\frac{\left|\tau \right|}{\tau l}\right)\text{,}$$ (11)

where the correlation time of deceleration (${\tau }_l$ in Table 1) is

$${\tau }_l=\frac{T_1{T}_2}{T_1+T_2},T_i={\tau }_l{\mu }_i\left(\frac{1}{{\mu }_1}+\frac{1}{{\mu }_2}\right)\text{.}$$ (12)

We used Eq. 10 here.

Acceleration

Next we turn to the description of acceleration. Examining the time series of absolute velocity, we observe pronounced upward peaks of similar shape and width across different behavioral modes, and across individuals. We do not observe any periodicity in the pulse train. In the absence of periodicity, we use the model of pulse train with Poisson statistics. We propose that locomotion of animals in a homogeneous environment corresponds to Poisson statistics of acceleration impulses:

$${\xi }^{+}\left(t\right)=\sum _{p}G\left(t-t_p\right)\text{,}$$ (13)

where $t_p$ are random start times of the pulses and

$$G\left(t\right)={\xi }_0^{+}\exp \left(-t/{\tau }_s\right)$$

is the shape of the pulses, where ${\xi }_0^{+}$ (cm/${\mathrm{s}}^2$) is an amplitude of the acceleration and ${\tau }_s$ is a characteristic width of each pulse.

As demonstrated below, this shape corresponds to an exponential autocorrelation function of acceleration with the characteristic time given by ${\tau }_s$ from Table 1. The pulses do not correlate with one another (an assumption valid in the absence of environmental cues). The correlation of acceleration occurs due to the correlation of the pulse with itself: the time lapses between two successive pulses, or waiting times, ${\Delta }_p=t_p-t_{p-1}$, are distributed exponentially with the mean $⟨\Delta ⟩=T$. The width of the pulses ${\tau }_s$ is assumed to be small compared with the long correlation time: ${\tau }_s\ll {\tau }_l$. The average of the pulse process is $⟨{\xi }^{+}⟩={\xi }_0^{+}{\tau }_s/T$.

The pulse process ${\xi }^{+}\left(t\right)$ can be represented using a sequence of $\delta$-shape pulses or a point process (67,68)

$$\eta \left(t\right)=\sum _p\delta \left(t-t_p\right)\text{,}$$ (14)

such that

$${\xi }^{+}\left(t\right)=\int G\left(t-t^{\prime }\right)\eta \left(t^{\prime }\right)d{t}^{\prime }\text{.}$$ (15)

The spectral density of the process $\xi {\left(t\right)}^{+}$ can be expressed as follows (69,70,71):

$$S_{{\xi }^{+}}\left(\omega \right)=S_{\eta }\left(\omega \right){\left|G\left(\omega \right)\right|}^2\text{,}$$ (16)

where $G\left(\omega \right)$ is the Fourier transform of the shape function $G\left(t\right)$ and

$${\left|G\left(\omega \right)\right|}^2=\frac{{\left({\xi }_0^{+}\right)}^2}{{\omega }^2+1/{\tau }_s^2}\text{,}$$

and the spectral density of the sequence of $\delta$-shape pulses is

$$S_{\eta }\left(\omega \right)=\frac{1}{2\pi T}\text{.}$$

Substituting these expressions into Eq. 16 and making the Fourier transform in accordance with the Wiener-Khinchin theorem (72,73) we obtain the correlation function of the acceleration

$$K_{{\xi }^{+}}\left(\tau \right)=\frac{{\left({\xi }_0^{+}\right)}^2{\tau }_s}{2T}\exp \left(-\frac{\left|\tau \right|}{{\tau }_s}\right)\text{.}$$ (17)

Exponential acceleration correlation is often used to describe animal movement (74).

Correlation function of the absolute velocity

We assume that the acceleration and deceleration are independent of each other, therefore the correlation function of the acceleration-deceleration is a weighted sum of the two functions, Eqs. 11 and 17,

$$K_{\xi }\left(\tau \right)=\frac{{\left({\xi }_0^{+}\right)}^2{\tau }_s}{2T}\exp \left(-\frac{\left|\tau \right|}{{\tau }_s}\right)+\frac{{\tau }_l^2{\left({\xi }_1^{-}-{\xi }_2^{-}\right)}^2}{T_1{T}_2}\exp \left(-\frac{\left|\tau \right|}{{\tau }_l}\right)\text{,}$$

which is a sum of two exponents. The correlation function of the absolute velocity can be estimated using the following equation (70,71):

$$K_{\xi }\left(\tau \right)=-\frac{{\partial }^2}{\partial {\tau }^2}{K}_v\left(\tau \right)\text{.}$$

These transformations, related to the calculation of derivatives or integration, do not change the structure of the function consisting of exponents. As a result, the correlation function of the absolute velocity also contains two exponents:

$$K_v\left(\tau \right)=\frac{{\left({\xi }_0^{+}\right)}^2{\tau }_s^3}{2T}\exp \left(-\frac{\left|\tau \right|}{{\tau }_s}\right)+\frac{{\tau }_l^4{\left({\xi }_1^{-}-{\xi }_2^{-}\right)}^2}{T_1{T}_2}\exp \left(-\frac{\left|\tau \right|}{{\tau }_l}\right)\text{.}$$

Comparing the ratio of the two prefactors in this theoretical correlation function with the ratio in the experimental one given by Eq. 8, we find:

$$\frac{2T{\tau }_l^4{\left({\xi }_1^{-}-{\xi }_2^{-}\right)}^2}{T_1{T}_2{\left({\xi }_0^{+}\right)}^2{\tau }_s^3}=\frac{1-b}{b}\text{.}$$ (18)

Langevin equation

The conventional method of studying dynamics of a fluctuating parameter is to write a stochastic differential equation in accordance with the law of motion, and to obtain the probability distribution as a result. Our situation is reversed. We have a PDF deduced from experimental observations Eq. 6, and want to reconstruct the underlying law of motion. Each of the two Rayleigh distributions can be obtained from the standard Langevin equation for a 2D Brownian motion with a damping coefficient $\gamma$ and the noise intensity

$$D=\frac{1}{2}\underset{-\infty }{\overset{\infty }{\int }}{K}_{{\xi }^{+}}\left(\tau \right)d\tau =\frac{{\left({\xi }_0^{+}\right)}^2{\tau }_s^2}{2T}\text{,}$$ (19)

where the correlation function of the short-correlated noise ${\xi }^{+}$ is given by Eq. 17.

We have two Rayleigh distributions corresponding to the two modes. We suggest that, for each of the two modes, the noise is the same, but the damping coefficient is different and given by ${\gamma }_i$, where $i=1,2$ is an indicator of the mode. This coefficient changes as a dichotomous process. Substituting acceleration and deceleration into Eq. 9, we obtain the following Langevin equation corresponding to the PDF given in Eq. 6:

$$\dot{v}=-{\gamma }_{i}v+\frac{\left(n-1\right)D}{v}+{\tilde{\xi }}^{+}\left(t\right)\text{.}$$ (20)

Here, the second term on the right-hand side of the equation comes from an $n$-dimensional Wiener process, or a Bessel process. In our case, $n=2$. The random process ${\tilde{\xi }}^{+}={\xi }^{+}-\frac{{\xi }_0^{+}{\tau }_s}{T}$ has a zero average and can be considered a white noise, since ${\tau }_s\ll {\tau }_l$. We added a constant value $\frac{{\xi }_0^{+}{\tau }_s}{T}$ to the deceleration, to compensate the average acceleration. The deceleration also has a part proportional to the absolute velocity, $-{\gamma }_{i}v$.

To illustrate the behavior of Eq. 20, we performed numerical simulations (please see Fig. 9 for a typical realization). The value of $\gamma$ switches from slow to fast and back to slow as a dichotomous noise. Solutions of Eq. 20 are consistent with the observed speed versus time graphs for the rodents.

Figure 9.

Numerical solution of Eq. 20 with $D=7$ is shown with a black solid line. The underlying telegraph process for the values of $\gamma$ is depicted with a blue dashed line, where the value for the slow mode is ${\gamma }_1=7$ and that for the fast mode is ${\gamma }_2=1$.

The stationary PDF corresponding to this equation is a weighted sum of two Rayleigh distributions. Comparing it with Eq. 6 we have:

$$w\left(v\right)=\frac{v}{D\left({\mu }_1+{\mu }_2\right)}\left({\mu }_1{\gamma }_1{e}^{\frac{-{\gamma }_1{v}^2}{2D}}+{\mu }_2{\gamma }_2{e}^{\frac{-{\gamma }_2{v}^2}{2D}}\right)\text{,}$$ (21)

where $D/{\gamma }_i=v_i^2$ and $i=1,2$ is an indicator of the mode.

The origin of noise

Next we comment on the nature of “noise” in our system. The random terms in our model do not represent an “external” source of noise, which might come, e.g., from the inaccuracies of measurement. Here, we describe a random process, which drives important aspects of rodent walks, and which is an essential feature of brain dynamics, seen in many contexts, ranging from single spiking neurons to collective EEG phenomena. This inherent randomness of rodent walks has been reported previously in the literature (see, e.g., (21,24,61,74)).

Analogies with mechanical systems

The rodent’s speed distribution consists of two components, each of them a Rayleigh distribution. In other words, each of the modes resembles the behavior of a Brownian particle for 2D motion. This gives us the opportunity to describe the motion using Langevin-like Eq. 20. As a result, the rodent can be modeled as a Brownian particle with the mass and size, which are constant but different for each mode.

Consider a Brownian particle, whose mass and size randomly vary in time. The mass randomly switches between two values $M_1>M_2$, where $M_1$ corresponds to the slow motion, and $M_2$ describes the fast movement. This system is equivalent to Brownian motion with fluctuating diffusivity (see (75) and the references therein). The short correlation time ${\tau }_s$ is the time of a single collision with one of the surrounding particles. The large correlation time ${\tau }_l$ is the average time the particle retains a given mass and size before switching to the other mass and size. The friction magnitude and the resulting deceleration are determined by the mass and size.

The probability distribution of the masses is as follows:

$$P\left(M_i\right)=\frac{T_i}{T_1+T_2},i=1,2\text{.}$$ (22)

The kinetic energy distribution $E$ does not depend on the mass distribution and it is determined only by the Boltzmann constant and the temperature, $k_{B}T$,

$$w\left(E\right)=\frac{1}{k_{B}T}\exp \left(-\frac{E}{k_{B}T}\right)\text{.}$$ (23)

As a result, to obtain their joint distribution, we multiply them, then transform this distribution to the PDF for $v$ and $M_i$ and get

$$w\left(v,M_i\right)=\frac{P\left(M_i\right)v}{k_{B}T}\exp \left(-\frac{M_i{v}^2}{2k_{B}T}\right)\text{.}$$ (24)

After averaging over $M_i$, we obtain a mixture of two Rayleigh distributions (Eq. 6), where $v_i=\sqrt{k_{B}T/M_i}$ and $P\left(M_i\right)={\mu }_i/\left({\mu }_1+{\mu }_2\right)$. The masses ratio of the two-mode particle should be $M_1/M_2={\left(v_2/v_1\right)}^2$. The PDF of the absolute velocity and autocorrelation function are the same as those observed for a rodent.

One way in which a particle can randomly change its mass can be illustrated using two dipoles moving in ideal gas (76). The model has two quasistable states minimizing the Helmholtz free energy. Consider the intermittent process for the velocity of one of two dipoles moving in ideal gas (see Fig. 10). There is one mode, when the two dipoles are connected and move together with a relatively small speed, and another mode, when they move independently and faster on average. The first type of motion minimizes the Helmholtz free energy by the negative energy of interaction. The second one corresponds to a larger entropy.

Figure 10.

The model of a dipole moving in ideal gas (focusing on the motion of the dipole marked by the arrow): (a) the dipoles are connected corresponding to the slow mode and (b) they move independently corresponding to the fast mode.

Discussion

Locomotor tests in laboratory rodents are part of experimental routine worldwide. One of the basic ways to observe animals’ behavior is to track their movements (1,5,6,53), or even to mimic it by creating a virtual rodent (49). Since motion tracking software became available, animal locomotion has been digitized routinely. Mainly, it is done by calculating the instant body centroid coordinates against the background. This approach provides trajectories with a relatively high resolution. Also, more sophisticated deep learning-based algorithms (e.g., DeepPoseKeet, DeepLabCut, etc.) enable tracking of user-specified body parts, adding precise information on micromovements of virtual body axes (reviews in (1,53,77)). In this study we represented the rodent motion as the velocity of a single point, thus reducing the complexity of the problem (see, e.g., (40,74,78) for a related discussion of dimensionality reduction in animal motion studies). We have digitized and analyzed a large number of rodent tracks in open field tests, to elucidate the dynamics of the slowest locomotor grade, which is pronounced in a novel environment. We called this type of motion “cautious walks,” reflecting the idea that caution is a natural reason for frequent stopping and deceleration during environmental exploration. Short (e.g., of the subsecond range) behavioral motifs such as sniffing, lingering, or rearing (1,2) comprise an essential part of the slowest locomotor grade, being a complex mixture of small progressions and orientation trials, environmental estimation, etc.

We found that the probability distribution of the absolute instantaneous velocity of rodents can be approximated by a superposition of two Rayleigh distributions, which are characterized by distinct characteristic velocities $v_1$ and $v_2$, with $v_1<v_2$. This is in contrast to the velocity of a Brownian particle executing random motion, which obeys a single Rayleigh distribution. The two contributions come with approximately equal weights. We propose that the part of the distribution associated with the larger of the two velocities characterizes rodents’ progressions in space, while the part of the distribution near the smaller speed describes a range of other types of motions, such as lingering with closing steps (79,80) and body micromovements (lateral head scans, torso switches, etc.) (37). Importantly, in the context of the present study, the robustness of the obtained results was confirmed by a triple tracking of the head, the trunk, and the base of tail for individual pups (see Appendix 3). Since all these three points yielded the same qualitative results, we expect that any representative point located between these three will possess a similar speed distribution.

Our finding are consistent with those reported in (42), where the motion of adult rats was decomposed into four separate “gears,” with the lowest gear corresponding to “stopping” and “staying in place.” Similar results were also reported for adult mice (81). While it is difficult to provide a quantitative comparison (due to differences in the animals’ age, the arena size, and the duration of the experiments), it appears that our two modes may correspond to subtle distinctions of movements within the first gear of (42). Another difference between our approach and that of (42) is that we kept all speed values in our statistics, while the lowest speeds were not considered by (42). More detailed approaches, such as those used in (2,36,49,53,77), can shed further light onto the exact nature of these behaviors by tracking rodent posture data and identifying short-time behavioral motifs, such as grooming, sniffing, head bobbing, etc. The subseconds ethological “syllables” are also important for pharmacobehavioral studies in laboratory rodents, to describe stereotypical patterns or to catch anxiety signs.

Information on both larger-scale motions (locomotion) and smaller movements is likely contained in many laboratory studies of movements that use motion tracking (82,83) (the most common frontal view used to monitor test animals picks up small movements of the head and the torso). Therefore, by necessity, the obtained signal contains different types of movement, such as forward progressions, lingering episodes, and vertical activity.

Since in our experiments the rodent pups (below the age of solid food ingestion) demonstrated velocity patterns similar to those in adult animals (Figs. 4, 6, and 8), we could not ascribe the stops or lingering to putative foraging attempts. As the decelerated episodes were characterized by the lateral head scans (37) and used for reorientation (35), we hypothesize that they reflect the process of risk assessment, necessary for prey animals in the wild. One might attribute the animals’ deceleration (i.e., action-stopping) behavior to the so-called proactive inhibition (84,85,86), a condition where inhibition is strongly facilitated.

Apart from the existence of two characteristic velocities, statistical analysis of the data also revealed a biphasic decline in the absolute velocity autocorrelation function, with two characteristic times, ${\tau }_s<{\tau }_l$. The time ${\tau }_s$ characterizes the width of the absolute velocity peaks, and ${\tau }_l$ is associated with the timing of the switches between progression and lingering. A central observation is that the animals slow down to reevaluate the environment, with its potential danger and rewards (these stops might further develop into fear- or startle-induced behavioral arrests (38), or present as just a punctuation in the ongoing locomotor activity (35)). These patterns inspired a stochastic model that is compatible with the observations of the rodent cautious walks. To describe this, we assumed the existence of two interfering processes: in addition to the impulses to move arriving at random times, continuous deceleration is taken into account. This leads to the pulse structure of the dependence of speed on time (see Fig. 2).

Our model can be contrasted with the “race model” where, during task discrimination trials, the execution and inhibition of an action are often presented as two concurrent motivations (reviewed in (87,88)). This race model approach implies a competition of “stop” and “go” processes, and the winner determines the final action. Instead, our theory suggests a summation of two opposite effects acting upon the locomotor output, which results in acceleration and deceleration.

It is further interesting to compare our findings with those of (89), where quasiperiodic limb movements lead to sinusoidal correlations that can last up to 1 s timescales. The main difference is that, in our work, we are not studying animal runs with some purpose/intention, which happen in a chosen direction and are characterized by periodic steps. Instead, we are focusing on cautious, open field walks with many random stops, lingering, and rotations on the spot. While periodicity is observed during running episodes in (89), in our system it is destroyed by the randomness of lingering episodes.

It is essential that the statistical analysis of animal speed distribution is performed on an individual basis. Absolute velocity distributions were constructed and analyzed separately for individual animals. Pooling animals together and analyzing the ensemble behavior produce different statistical results, which are the subject of future work.

Having identified two different modes of motion with different characteristic velocities, we can analyze some of the implications of this finding. A surprising observation was that the two characteristic velocities, $v_1$ and $v_2$, while quite variable among individual animals, are roughly proportional to each other. Our study provides an estimate for the ratio between $v_2$ and $v_1$ of about 2–5. This means that the energy cost for regular locomotion is an order of magnitude higher than what is needed for the lingering and micromovements. Presumably, this estimate will be shifted by neuromuscular/neurodegenerative disorders, drugs affecting the vigilance state, etc. Further studies are needed to pursue these ideas.

The proportionality of $v_1$ and $v_2$ might also point to the idea of slow local movements as a precursor of major locomotor acts (79,80). An analogy may be drawn between these patterns and other cases involving smaller and larger motions, such as movements of athletes before start of run trials (“warming-up”), or motor behavior of patients recovering from lesion-induced akinesia (80), as well as comatose or deeply anesthetized patients; in such cases, minor motor acts appear before gross locomotor activity. The same is known for the ontogenetic flow of motor skills: local motions of limbs precede its ability for large-scale movements (79).

Both characteristic velocities, $v_1$ and $v_2$, have been found to increase with age. This can be naturally attributed to a growing muscular power, as the simplest explanation. It is interesting that the ratio between the two velocities, $a=v_2/v_1$, also grows (albeit relatively slowly) with maturation.

In about one-third of the young blind rats (R13, R15), the absolute velocity distributions appeared to be blended, with poorly discriminated peaks for $v_1$ and $v_2$ (Fig. 5). For older animals, we observe a clearer separation between the two models of motion. The separation of locomotor modes in more mature animals might therefore serve as a developmental milestone, marking the activation of a stress-response system (90), which initiates escape reactions.

It is instructive to consider our results in the broader context of the work that was done on the free moving behavior of other model organisms, such as invertebrates (nematodes (40,91,92) and flies (93)), and also mammals (94,95,96), where high-resolution tracking of positions and postures have become commonplace (see also reviews in (1,5,78)). Both similarities and differences exist between the velocity patterns we report in rodents, and those found in other systems. When comparing with the motion of predators, it appears that dogs (94) and cheetahs (95) are characterized by a relatively compact speed distribution, such that the animals typically move at the speed that is close to its average. In contrast to these examples, the rodents studied in this paper seem to avoid moving at the average speed (due to their 2-component speed distribution). On the other hand, two-humped absolute velocity distributions have been reported in different contexts. In (93), the locomotor behavior of Drosophila melanogaster was recorded by using a video tracking system, and two behavioral components were found by using frequency versus speed graphs: a near-zero speed segment and a peak in the distribution at a nonzero speed. It was suggested that the flies, who typically move in a “staccato” manner with bouts of fast walking episodes interspersed among brief stops, had a preferred walking speed. In (96), the movement of wolves was captured by using GPS collars, revealing a bimodal distribution of the animals’ speed, with the lower speed corresponding to resting and feeding, and the higher speeds to traveling movements, which often were associated with the wolves moving along human-made “linear features” such as roads, trails, pipelines, etc. Although these systems are different in nature from our experiments, it is interesting to see evidence of speed bimodality in other contexts.

Extension of this work may consider further complexities of motion, including additional sources of correlation in the noise. Speed measurements at much higher temporal frequencies might reveal a more nuanced picture of the dynamics, possibly requiring colored noise (see, e.g., (97)). While we do not propose a detailed model of switching modes at the moment, we can suggest, as one of possible hypotheses, that a model of two metastable states perturbed by external white noise can be used. In this model, the noise is the reason for the transition between the two metastable states. In this case, the mode-switching probabilities depend on the noise intensity. This dependence is not trivial and can include a phenomenon similar to noise-enhanced stability (98,99). The noise properties can also be different; it can be a pulse train with regulated periodicity (100,101) or Lévy noise (102). Alternatively, models of posture dynamics have indicated the possibility of deterministic chaos underlying the emergent stochasticity (see, e.g., (91,103)). This study describes the simplest model that approximates the current measurements; future work may provide deeper insights into the nature of the random pulses that govern the motion.

Techniques developed here may be applicable for locomotion studies in a wider range of contexts. For example, by analyzing data archives accumulated for various drug-affected and/or pathological conditions, one may gain new insights into the mechanisms of a potentiation of gross locomotor acts by micromovements, in a variety of physiological states. Such an empirical tool to estimate the action of inhibitory control over body micromovements could be particularly important for modeling developmental disorders (104,105,106), ADHD, (107,108,109), Parkinson disease (110,111,112), etc.

Conclusion

We developed a simple, interpretable model of rodent locomotion that identifies two behavioral modes, each characterized by a distinct timescale derived from experimental correlation decay. The model quantitatively reproduces features of the data, including 1) the two empirical timescales, mean peak width, and mean mode duration, which closely match the two model parameters and 2) the shape of the speed distribution, which is well fit by a mixture of two Rayleigh distributions. While this model is necessarily minimal due to limited available data and the absence of existing models for rodent pup behavior, it captures key dynamic features of the observed trajectories and provides a foundation for future quantitative comparisons and model refinement.

This study reveals the existence of distinct locomotor modes in rodents exploring novel environments: slow micromovements or reorientations and faster spatial progressions, described by superimposed Rayleigh distributions with velocities $v_1$ and $v_2$, with the ratio $v_2/v_1$ lying between 2 and 5. A stochastic model incorporating intermittent acceleration-deceleration pulses explains a biphasic velocity autocorrelation function, reflecting rapid micromovements and slower state transitions. The proportional relationship between $v_1$ and $v_2$ suggests lingering as a precursor to larger movements, with the modes’ separation sharpening during maturation. Bimodal speed distributions, contrasting with Brownian unimodal patterns, highlight adaptive exploration-risk balance. These findings provide a framework to study motor and motivational control in neurodevelopmental disorders and other pathologies, bridging behavior and pathophysiology.

Acknowledgments

The authors thank Prof. Vladimir V. Raevsky and Dr. Marina L. Pigareva for their valuable scientific discussion of ontogeny of laboratory rodents.

Author contributions

I.S.M., V.S., N.L.K., and O.A.C. conceptualized the study. O.A.C., V.S.S., and N.L.K. developed the theoretical approach. N.L.K. visualized the results. A.A.R., O.S.I., and I.S.M. performed the animal experiments. A.A.R. and F.S.S. preprocessed the experimental data. All authors wrote the paper.

Declaration of interests

The authors declare no competing interests.

Editor: Elena Koslover.

Appendix 1: The fitting procedure

Bin-based versus maximum likelihood fitting methods

We have attempted two different methods of fitting the probability distribution of the rodent speed. Both methods have advantages and disadvantages.

• 1)Bin-based fitting: we split the experimentally obtained data into bins and fit the resulting points by the standard method of minimization of the squared error. This method, by its nature, contains an extra parameter, which is the bin size. The dependence on the bin size, however, is informative. It helps to estimate the characteristic scale of a random variable. Below, we show how the results depend on the choice of the bin size.
• 2)The maximum likelihood method: we minimize the logarithm of the likelihood of the observed data given the distribution function. This method amplifies the contribution of the tail of the distribution, because of the operation of taking the log. The latter operation is the standard way of handling the extremely small numbers that appear when calculating the likelihood. This results in exaggerating the weight of large speeds, for which the data are unreliable due to sparsity and noise. It also shifts the focus to the distribution tail from the region of our interest in this work, which is cautious walks of the rodents.

For the bin-based method, we were guided by the Terrell-Scott rule (113,114), which suggests to use at least $k={\left(2n\right)}^{1/3}$ bins, where $n$ is the number of data points. In the main text we used the bin size that was calculated individually for each animal, $i$, and given by $\frac{V_{99\%}^i}{{\left(2n\right)}^{1/3}}$, where $V_{99\%}^i$ is the $99\%$ quintile of the speed values for animal $i$. Below, we refer to this procedure as individualized bin size fitting. Figs. 4, 5, 6, and 7 use individualized bin size fitting.

To check the sensitivity of the estimated parameters to the bin size, for each of the animal group, we denoted

$$M=⟨\frac{V_{99\%}^i}{{\left(2n\right)}^{1/3}}⟩\text{,}$$

where the triangular brackets denote the ensemble average. $M$ is the mean of the maximum bin size according to the Terrell-Scott rule. We repeated the fitting procedure (using the double Rayleigh distribution, $f_2\left(v\right)$) for bin sizes $M/2,M/4,M/8$, and $M/16$, as well as $2M$ and $4M$, for each animal group. Resulting best fitting values for the characteristic speed, $v_1$ and $v_2$, were compared. Fig. 11 shows how the $v_2$ vs. $v_1$ graph changes as the bin size changes. For reference, the black dashed line shows the slope of the data obtained by individualized bin size fitting. We can see that, for all the bin sizes, the dots align along a similar linear dependency, but for older animals (and especially for R17) increasing the bin size leads to an increase in the estimated characteristic speed values.

Figure 11.

A scatterplot of the characteristic speed values, $v_1$ and $v_2$, obtained for each group of animals by fitting with different bin sizes. Here, blue symbols correspond to bin size $M/16$, yellow to $M/8$, green to $M/4$, and red to $M/2$. The dashed black line is the slope of the scatterplot obtained from individualized bin size fitting.

Fig. 12 shows a summary of this information. For each animal group and for each of the bin sizes, we plot the slope of the $v_2$ vs. $v_1$ graph; the slopes obtained from individualized bin size fitting are given by horizontal dashed lines. We can see that, except for R17, the slopes are more or less independent of the bin size. For R17, the largest of the bin sizes shows a significant deviation in the slope.

Figure 12.

The slopes of the $v_2$ vs. $v_1$ graphs obtained by fitting the data in Fig. 11 with the function $v_2=a{v}_1$ for each animal group. Here, blue symbols correspond to bin size $M/16$, yellow to $M/8$, green to $M/4$, and red to $M/2$. The dashed black line is the slope of the scatterplot obtained from individualized bin size fitting.

Fig. 13 shows the ensemble average of the characteristic speed $v_1$ (left) and $v_2$ (right) for each animal group, for each bin size. Again, the means are largely independent of the bin size except for the case of R17.

Figure 13.

The ensemble means of the characteristic speeds $v_1$ (left) and $v_2$ (right) for each group of animals, obtained by fitting the data using bin size $M/16$ (blue), $M/8$ (orange), $M/4$ (green), and $M/2$ (red).

Fig. 14 shows an example of the fitting procedure using a 2-component Rayleigh distribution, where the bin size increases by a factor of 2 for each consecutive panel. The resulting best fitting functions are shown by blue lines. They are also plotted together in Fig. 15 a. We can see that in this example, all the fits are very similar except the two that were performed with bin sizes larger than the maximum value $M$ obtained by applying the Terrell-Scott rule. Although there was individual variation, we generally saw that the fitted values were consistent across a range of bin sizes from $M/16$ to $M/2$ or $M$ (see also Fig. 15 b, which presents an example from an adult rat). The only exception was the group R17, where the best fitted parameters depended more strongly on the bin size, as was evident in Figs. 11, 12, and 13.

Figure 14.

Examples of fitting an individual animal speed data in group R13 with function $f_2\left(v\right)$ using bin sizes, $M/16,M/8,\dots ,4M$. The histograms are drawn with the bin size indicated in each graph, and the best fitting 2-component Rayleigh distribution is shown by the blue line.

Figure 15.

Fitting the distribution by using the bin method, with bin sizes ranging from $M/16$ to $4M$. (a) An individual animal from group R13; the same best fitted functions are shown as in Fig. 14. (b) Results for an individual animal from group Radult.

Next, we comment on the maximum likelihood method. The challenge in maximum likelihood estimation, when dealing with highly asymmetric distributions, is that the log-likelihood transformation emphasizes the tail of the distribution. The logarithm compresses smaller likelihood values (corresponding to data in the tail) into a narrower range, effectively giving them relatively more weight in the estimation process. As a result, when we apply this methodology to the animal speed data and use the 2-component Rayleigh distribution function, the resulting second characteristic speed $\left(v_2\right)$ often becomes shifted to incorporate the tail end of the distribution. The effect is similar to using a bin-based fitting method after applying the log transformation. Fig. 16 shows several examples of this behavior. Here, the blue and the green lines show the best fitting distribution obtained by the maximum likelihood method and by the bin-based method where the log transform was applied, respectively. The dashed lines represent the bin-based method without taking the log (this method was used in the main body of the paper). We can see that the two methods that implement the log transformation are better at describing the tail (high speeds) but tend to “miss” the shoulder of the distribution. In other words, they are suboptimal in describing the “cautions walks,” which are the focus of this study. They give extra weight to the faster movements, which are closer to running and galloping, and for which the collected data are sparse and less reliable.

Figure 16.

Examples of fits performed by different methods. Blue is the maximum likelihood method, dashed purple is the bin-based method, and green is bin-based fitting after the log transformation.

There are several ways of dealing with this problem within the maximum likelihood method methodology, including the weighted likelihood method, truncated likelihood method, or using more parameters in the fitting function. All of these involve extra assumptions, and were rejected in favor of the (nonlog-transformed) bin-based method, which was shown to remain robust in a wide range of bin size choices, as described above.

Two versus more modes in the speed distribution

In addition to a 2-component Rayleigh distribution, Eq. 6, we have also experimented with using other multiple-component Rayleigh distribution,

$$f_n\left(v\right)=\sum _{i=1}^n\frac{{\mu }_{i}v}{v_i^2}{e}^{\frac{-v^2}{2v_i^2}}\text{,}$$ (25)

where $n=2,3,4$, etc., $v_1<v_2<\dots$, and all ${\mu }_i>0$. In particular, the $n=3$-component model contains two extra parameters compared with model (6), and the $n=4$-component model contains four extra parameters. Typical examples of fitting with model (25) are presented in Fig. 17, where the best fits with $n=2,3$, and $4$ are presented in the top row together with the data, and the best fitting distribution structure is shown in the next two rows. It was found that, in many cases, and especially for the Radult group of animals, the $n=3$-component Rayleigh distribution overall provides an even better fit than function (6) $\left(n=2\right)$. This is consistent with adult animals utilizing a much wider range of speeds, such that only part of their locomotive behavior can be classified as “cautious walks.” Higher speeds at the tail end of the distribution are better described by more complex models.

Figure 17.

Fitting the speed distribution by using the bin method with the number of Rayleigh components $n=2,3,4$ (Eq. 25). The results are illustrated by using four animals in the R adults group (bin size 0.5 cm/s). For each animal, the experimental speed probability distribution is plotted in the top row (blue dots), together with the best fits by the functions in (25) with $n=2$ (blue), $n=3$ (yellow), and $n=4$ (green). The structure of the obtained distribution for each $n$ is shown in the next two rows. The bar graph in the middle row shows the relative contribution of each of the modes, and the values in the bottom row are the best fitting characteristic speeds.

Fig. 18 shows how the characteristic speed values obtained by the $n=2$ and the $n=3$ method compared with each other. In (a), the animals in Radult group are listed along the $x$ axis, and the corresponding values of the three characteristic speeds obtained by the $n=3$ distribution are shown as the blue, yellow, and green points, respectively. For comparison, the two characteristic speed values obtained by the $n=2$ distribution are marked by black points. As expected, using three as opposed to two components expands the difference between the smallest and the largest characteristic speed. The $n=2$ distribution places the two speeds between the three $n=3$ values. The dependence of $v_2$ on $v_1$ (Fig. 18 b) is similar to that presented in the main text (see Fig. 6), except that the values of $v_1$ and $v_2$ are somewhat lower if fitting with the 3-component distribution.

Figure 18.

Comparison of the best fits obtained by the 2-component and 3-component Rayleigh function (Eq. 25), for R adults. (a) The best fitting characteristic speeds for each animal. The blue, yellow, and green points show the values for $v_1,v_2,v_3$ for each animal using the $n=3$ distribution. The black points are the two characteristic speeds obtained from the $n=2$ distribution. (b) The $v_2$ vs. $v_1$ graph using the $n=3$ distribution.

With an increased number of parameters in the fitted models, two issues have come up that are noteworthy. 1) It becomes increasingly difficult to find relevant fits due to the issue of parameter nonidentifiability. The usual searching methods with $n=4$ often return values of $v_4$ that are extremely high, and also the values of intermediate mode contributions (e.g., ${\mu }_3$) extremely small. For example, for animal D in Fig. 17, $v_4\approx 50$, and the relative contribution of the third mode is less than $1\%$. Similarly, fitting the $n=4$ distribution for animal $A$ we found the characteristic speeds $v_2\approx v_3$. (2) It would intuitively make sense if these 4-component models were rejected in favor of a similarly fitting 3-component model. This was, however, not the case, and both AIC and BIC (Bayesian Information Criterion) values were lower for the $n=4$ models compared with the $n=3$ models in all the four cases in Fig. 17. This is because the “cost” value assigned to introducing additional parameters in this problem is an order of magnitude smaller than the contribution of the maximum likelihood.

While distributions with a higher number of components may be a better model to describe the whole spectrum of velocities, for the cautious walks studied here, we chose to use a 2-component distribution. The mathematical simplicity of the 2-component model is a necessity at this stage, because the data collected so far would not support further complexities of the mathematical description.

Appendix 2: Significance of the speed difference for different groups of animals

Fig. 19 shows the best fitted parameters, $v_1$ and $v_2$ (model Eq. 6) for all the animal groups. To determine whether the mean values of the two characteristic velocities, $v_1$ and $v_2$, were different between different ages, we performed the t-test to compare the group mean of $v_1$ for R13 with the group mean of $v_1$ for $R15$ and the group mean of $v_1$ for R15 with the group mean of $v_1$ for adult rats, to find that both comparisons yielded a significant difference with $p<5\times {10}^{-4}$. Similarly, we compared the mean values of $v_2$ among different groups, and again the t-test showed significant differences for all comparisons with $p<2\times {10}^{-4}$.

Figure 19.

Best fitting characteristic speed values, $v_1$ (yellow) and $v_2$ (blue), presented as histograms for each animal group. Individualized bin size fitting is used.

Appendix 3: Additional analyses

Sampling time

The data used in this paper contained tracking the animals’ position at regular time intervals of $\Delta t=0.04$ s. To see if any of our results are an artifact of this particular choice of time interval, we have repeated all the calculations based on sampling that was three times less frequent ($\Delta t=3\times 0.04$ s $=0.12$ s). Fig. 20 shows that the results did not change noticeably compared with those reported, and definitely no qualitative changes were observed.

Figure 20.

Analysis of the data with a different sampling time, $\Delta t=3\times 0.04$ s. (a) A typical frequency versus absolute velocity graph (blue dots) fitted with a 2-component Rayleigh distribution (blue line) and the Gamma distribution (yellow). (b) The number of animals for which the 2-component Rayleigh and Gamma distributions were a better fit by AIC (blue and yellow, respectively), for different animal groups. (c) The characteristic speed values, $v_1$ and $v_2$, obtained by fitting the 2-component Rayleigh distribution, for different animal groups (the means and standard errors are shown). (d) A comparison of the autocorrelation function, $K_{\nu }\left(\tau \right)$, which was calculated based on position measurements at $\Delta t=0.04$ s intervals (blue) and $\Delta t=0.12$ s intervals (yellow).

Tracking algorithm

Putative artifacts of tracking could be a source of external noise in the experimental data. For this research, we used a freely available tracker, ToxTrack (52), which has been widely used in a variety of other publications worldwide. However, to make sure that the resulting noise does not affect the conclusions of our study, we have performed additional measurements with a limited sample $\left(N=19\right)$, with double tracking by ToxTrack (52) and DeepLabCut (53). These are two independently developed tracking systems based on two different algorithms to find the regions of interest. The results obtained were robust: the two different tracking approaches yielded very similar outcomes.

In Fig. 21 we summarize the findings of the study where we used DeepLabCut tracking and compare the tracking results for the 3 body parts of 19, 13-day-old rat pups. In Fig. 21 a we show typical single-animal frequency versus absolute velocity plots for the head (pink), trunk (purple), and the base of tail (black) of the animals. We can see that, on average, the head speed is the highest and the base-of-tail speed is the lowest. In Fig. 21 b we show the same data fitted with the three functions, a 2-component Rayleigh (blue), a single Rayleigh (yellow), and Gamma (blue). This procedure was repeated for all the animals, and the AIC obtained for all the functions. It turned out that, consistent with the rest of this study, the 2-component Rayleigh function was the best fit for the majority of the animals, for all the three body parts. Finally, the clear linear correlation between the two characteristic speeds, $v_1$ and $v_2$, obtained from the best fitting 2-component Rayleigh functions, can be seen in Fig. 21 d. The slopes for the three body parts are similar to each other and are also similar to those reported in the paper.

Figure 21.

Results of the study tracking different body parts. (a) A representative frequency versus absolute velocity graph for one of the 40 animals: head (pink), trunk (purple), base of tail (black). (b) For the same animal, the data are fitted with three functions, a 2-component Rayleigh (blue), a single Rayleigh (yellow), and Gamma (blue). (c) The number of animals for which the 2-component Rayleigh (blue) and the Gamma function (yellow) provided a better fit (by AIC); the single Rayleigh distribution was never the best fit. (d) The pairs of the two best fitting characteristic speeds, $\left(v_1,v_2\right)$, for all the animals, fitted with the function $v_2=a{v}_1$; the best slope, $a$, is given for each body part, together with the 95% confidence intervals.

We conclude that results obtained with DeepLabCut tracking are qualitatively and quantitatively similar to those obtained with ToxTrack, and therefore the errors/noise inherent in ToxTrack do not influence our study’s findings.

Comparison with the Lévy walks model

Lévy walks are another model often used to describe animal movement. An important distinction of Lévy walks is the associated speed distribution characterized by the presence of heavy tails (such as a power-law distribution for high speeds) (102,115,116). The animals in our experiments, and especially groups M11, R13, and R15, do not often move with speeds much higher than the average, and their speed probability distributions are not consistent with a power-law description (see e.g., Fig. 22, where the distributions for several typical animals from the R13 group are plotted on a log-log scale). We can see that the speed probability distributions in our system decay faster than a power law. One of the reasons for this behavior is that, in our open field experiments, the animals are not engaged in foraging behavior.

Figure 22.

Several typical speed distributions plotted on a log-log scale (animals from the R13 group).

The distribution of the dwell times

We are assuming that the dwell times in each mode are distributed exponentially based on the exponential shape of the correlation function. This is, however, difficult to determine directly because we are lacking a criterion of mode switching, which could be justified uniquely. To create such a criterion one would need to incorporate data on the motion of different body parts (which goes beyond our current approach and beyond the data we have). There is a related quantity that we considered here. Fig. 23 shows several typical instances of the “slow motion” time distribution, where we simply recorded the amounts of continuous time where the animal moved with speed less than 1.5 cm/s. These distributions do appear exponential. (But we note that this is not the same as a dwell time in a given mode, because a mode is characterized by a typical speed, but may contain speeds that are much above and much below that.) Because we do not have a good criterion of determining which mode the animal is in at each moment of time, we cannot make any statements about the age dependence. We of course could determine the age dependence of the mean time in distributions of the type shown in Fig. 23, but this would show a trivial dependence where older animals spend less time moving slower than a threshold.

Figure 23.

Several typical distributions of time spent continuously moving with speed less than 1.5 cm/s (animals from the R13 group).