Spontaneous and Information-induced Bursting Activities in Honeybee Hives

doi:10.21203/rs.3.rs-2118394/v1

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Spontaneous and Information-induced Bursting Activities in Honeybee Hives

https://doi.org/10.21203/rs.3.rs-2118394/v1

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published 07 Jul, 2023

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In this study, we used a cohort population of approximately 1000 adult worker honeybees and a single queen bee in a single layer of an artificial hive to analyze how honeybee hives self-organize (i.e., how social role differentiation progresses). The temporal self-organization of the beehive was analyzed based on data obtained by tracking all the bees for a week using a “bcode” attached to their backs. We found that global synchronization of individual activities was produced, which was linked to foraging behavior and dancing. Moreover, individuals labeled as initiators of the synchronous activity have their roles determined on a weekly scale. This study demonstrates that individual bees behave in a binary manner, either quiet or excited, and their populations engage in spontaneous or informed collective bursts of activity. These collective bursts might evolve a social structure in the beehive through the opening and closing of the hive entrance.

It is well known that populations of agents, whether animate or inanimate, sometimes exhibit complex self-organization and emergence phenomena. The memory and learning capability of agents synthesized under the appropriate conditions support the dynamic structure of the macroscopic population.

An example of such self-organization in social organisms is known as "social entrainment/synchronization"[1]. Synchronization of grouping behavior is essential for the daily survival of a group, as this practice enables the achievement of common goals, such as breeding, defense against predators, collective hunting, and gathering as a means of energy conservation. Groups of starlings and sardines are well-known examples of groups that function as super-individuals[2, 3]. Grouping reduces the susceptibility to predator attacks. Drosophila increases the probability of mating by synchronizing its circadian rhythm with that of the group[4, 5, 6]. Meanwhile, honeybees perform the so-called waggle dance to announce the locations of feeding areas and to synchronize the global clock within a hive to transfer information[7].

In honeybees, numerous studies have been published regarding temporal synchronization—with respect to circadian rhythms in particular—of groups within the hive. Southwick et al. (1987) suggested that bees use direct contact or vibrations, rather than volatile substances, to synchronize activity levels among individuals[8]. Korst et al. suggested that trophallaxis is a major form of communication mechanism among honeybees[9].

A recent study suggests that direct contact, for example, via contact pheromones or tactile communication, is not necessary for the synchronization of circadian rhythm in the hive[10]. Bloch et al. proposed the following four hypotheses: 1. Entrainment through zeitgebers, 2. Social entrainment via direct contact, 3. Entrainment through environmental cycles driven by forager activity, and 4. Self-organized social synchronization. These hypotheses were considered the possible entrainment mechanisms of circadian rhythm in honey bee colonies by summarizing these previous studies[1].

In the past decade, machine learning methods have greatly advanced tracking techniques, allowing for more detailed analysis of individual behaviors. To illustrate, by identifying all individuals in a nest and tracking them with high spatiotemporal resolution, more detailed interaction structures between individuals can be revealed [11–14]. Furthermore, tracking whole individuals can be used to obtain detailed activity time series at individual and population levels. This consequently enabled us to determine which individual (micro) characteristics contribute to self-organization at the population level.

In the present study, we report the global synchronous activity (bursts) discovered in a population of Apis mellifera bees, and the relationship between this global activty and individual activities. This collective burst is an extremely robust phenomenon observed under all five different conditions in artificial beehives consisting of approximately 1000 honeybees (fig. S1).

In this study, we used the entire tracking data, which were acquired by tracking all honeybees individually by placing an artificial hive in one planar layer (a normal honeybee hive has its planes in layers) with a population consisting of approximately 1000 adult worker bees and a single queen bee (a single cohort population). Initially, the entrance door of the hive was closed and the bees were not allowed to go outside. When the entrance door was opened, the figure-eight dance (waggle dancing) emerged in this artificial hive, but when it was closed, the waggle dance was not observed.

Later when the entrance door was opened, the bees started to go out and eventually began showing the waggle dance. In response to this, bursts of activity began to appear. Accordingly, the spontaneous bursts without dancing gradually disappeared.

When we observed this burst at the level of each individual, it was obvious that the waggle dance and its observer were causing the burst. The discovery of spontaneous and informed (from foragers) collective bursts enabled us to deduce the linkage between individual and collective views of the beehive organization.

Entire tracking of bee behaviors

Gernat et al., a collaborative research team, developed an automated, high-throughput individual monitoring system for honeybee hives [11]. Briefly, they used one cohort hive consisting of approximately 1000 adult worker bees aged 1 day and one unrelated, naturally mated queen bee. As in other studies that have tracked individual social insects [12–14], they successfully tracked the location and orientation of individual bees every second by placing a 2DQR barcode (bCode) on the thorax of each bee (fig. S2). In July 2012 and July 2013, five trials were conducted, with each trial lasting approximately 1 week. The hive was located in a dark, quiet place, and its glass window was changed daily in the morning and at night to ensure a high detection rate of individual bees night and day throughout the experiment. The hive was connected to the outside by a door, which was closed for a first few days and then opened. Then, the worker bees could exit the hive and start exploring the outside and foraging. The distribution of time intervals for communication between individual bees followed a power law on scales shorter than 100 s, which Gernat et al. termed “bursty” and reported in their study [11]. We used the tracking data to analyze different types of bursting behavior, namely, global synchronized activities of much longer time scales.

All individual bees were identified by the bCode and tracked over 7 days. The obtained data were analyzed by measuring the “physical quantities,” such as the kinetic energy, location, whether the bee was inside or outside the hive, whether it was performing a waggle dance, and whether they located close to the entrance. Moreover, by completely utilizing a machine learning technique, the components of the burst were analyzed microscopically. Results showed that each bee could be functionally identified as forager(F), waggle dancer(W), dance follower(DF), initiator of the burst(P), and hive bee(H). As there were no larvae or pupae in this artificial nest, no caretaking role existed. Importantly, these functional labels were not necessarily mutually exclusive, so that the same individual could be identified as F, W, or P. In addition, the labels were not permanent and could vary over time. For instance, the role of a single bee transits among F, W, DF, and P.

Finally, the time in the hive was punctuated by circadian rhythms and bursts. The circadian rhythm is a 24-h cycle, whereas the bursts are intermittent; hence, the rhythms are not similar. Bursts occur only during the daytime; they may occur when feeding information is brought from the outside, so in this sense, bursts are the carriers of information from the outside. In this study, we confirm this fact of bursts and analyze the transition of individual labels by focusing on the temporal development of the nest that proceeds with the rhythm of bursts.

Characteristics of bursts

The activity of an individual bee (i) can be measured using its kinetic energy (K), i.e., a square of velocity. The hive activity level ${K}_{G}\left(t\right)$at time t is defined by averaging the individual kinetic energies. By tracking the individual bee behavior, we defined and measured the activity of the individual bee (i) using the kinetic energy as follows:

$${K}_{i}\left(t\right)=\varDelta \hspace{0.17em}{x}^{2}+\varDelta \hspace{0.17em}{y}^{2}\hspace{0.25em},$$

where $\varDelta \hspace{0.17em}x$ and $\varDelta \hspace{0.17em}y$ denote the respective displacements of the x and y coordinates of each bee (i) per second. The overall hive activity level ${K}_{G}\left(t\right)$ was defined by averaging over the individual kinetic energy as follows:

$${K}_{G}\left(t\right)=\frac{1}{n}{\sum }_{i=1}^{n}{K}_{i}\left(t\right),$$

which quantifies the global activities of the hive. When the bursts are synchronized, ${K}_{G}\left(t\right)$ should increase. In fact, ${K}_{G}\left(t\right)$ was found to increase rapidly in time and then relax, which we identified as a bursting phenomenon (Fig. 1A, B). For the mechanical detection of bursts, we applied a commonly used algorithm known as Kleinberg’s burst detection method [15]. An advantage of this method is that bursts are systematically identified based on a variable baseline of activity. The number of bursts detected in each trial was as follows: 71 in Trial 1201; 48, Trial 1202; 58, Trial 1203; 69, Trial 1301; and 66, Trial 1302. During the bursting phase, we observed that the excited honeybees moved around in the hive and frequently collided with other bees.

Initially, we closed the door of the hive, and after 2 days, we opened the door to allow the bees to go outside. However, the bursts occurred spontaneously irrespective of the opening and closing of the door. We termed this activity as a spontaneous burst. After the door is opened, a burst accompanied by a waggle dance will occur. If we assume that opening any door will bring information from the outside to the hive, we term the bursts as information-induced bursts. We also use the term “information-induced” as the waggle dancing that is known to be related to foraging (i.e. reporting the information of food sources).

These two bursts cannot be distinguished by the apparent burst alone, but they can be distinguished by the individual bees that make them up. There are also other bursts caused by artificial disturbances (i.e., caused by the act of cleaning the glass of the hive case). The features of the artificial bursts are sudden increases in ${K}_{G}\left(t\right)$, which occurs as a result of external stimuli so that the artificial bursts have a relatively more steep increae than that of the other bursts.

FKDO measures

In addition to the abovementioned activity level (K), we identified individuals that were absent from the hive for a certain amount of time (O) and individuals that performed waggle dance (D)(Fig. 1C). By calculating the cross-correlation coefficients between these, K, D, and O time series, we confirmed that the dance event preceded both the burst event (maximum coefficient = 0.69, lag = 1 min) and the “(nest) leaving” event (maximum coefficient = 0.43, lag = 13 min). Burst was preceded by O (maximum coefficient = 0.69, lag = 3 min) (Fig. 1D). These data confirmed that the return of the bees from the outside and the waggle dancing of the bees (if present) occurred before the onset of the burst.

Foraging (F) and outing (O) were not identical. We identified foragers (F) using two criteria. First, if a bee returned from outside 10 min before the onset of the burst, we labeled it as F. The 10-min criterion was determined from the probability distribution of the time interval between the time the bee returned from outside and the time the burst started. The second criterion was to exclude “orientation flights,” which are exploratory activities (16,17). We defined the first day of foraging by ignoring the first six flights outside the door (18). The flight activity after this day was defined as the foraging activity.

Classifying Pioneer Bees

As we approximately determined the temporal ordering of K, D, and O, we investigated each burst with respect to the activities of the individual constituents. We used a technique known as nonnegative matrix factorization (NMF) to dissect bursts[19, 20](Fig. 2). This technique does not decompose bursts into periodic modes like the Fourier transform, but it expresses a given matrix from each burst by a multiplication of any two matrices. As this expression is not unique, we selected a best rank order of matrix to minimize the Kullback–Leibler (KL) divergence of the original and reexpressed the matrix.

To explain NMF in more detail, the matrix from each burst expresses the activity of all the bees by its bCode (n) at time (t), and the matrix elements are the kinetic energy K(n,t) of the bees. The matrix K(n,t) is expressed by the product of two new matrices, W and M, and the group (m) of bees with high activity at each time is obtained from K(n,t) = W(n,m)M(m,t). When M(m,t) is plotted along the time axis, we can identify the individual groups with higher activity in the burst period. The contribution of an individual to each group is represented by W(n,m). The optimal size m is determined by the KL divergence (see Methods section for details). Here, we name “pioneer bees” (P), which is a particularly active group at the very beginning of each burst. P is an additional label to identify the role of the bee. Various individuals are involved in each burst, but it is necessary to investigate whether P is randomly determined for each burst or if there is a rule for determining P.

It is important to examine the biological meaning of P, because P is just a label derived from NMF. However, as discussed subsequently, F will become P with a high probability. After 3 days of opening the door, that is, after the burst was accompanied by a waggle dance, the percentage of P that were also F was 40.1% ± 3.89% (Fig. 3A). Moreover, we found that almost all (89.6%) bees that exhibited the waggle dance will become P. For instance, in Trial 1201, the dancing behavior gradually emerged at burst 50, and inversely approximately 10% of P came from W (Fig. 3B).

We found that dance followers (DFs) were the candidates for P. DFs are literally bees that observe dancing bees. DFs have been known as agents collecting information about food locations by watching the dance (21,22,23). Here, we defined a DF as an individual that is close enough to the dancer to turn its face toward the dancer. The percentage of Ps that were also DFs after the third day of the door opening was 35.0% ± 4.31% on average (Fig. 3C).

To summarize, the honeybee hive responded to the opening of the door, and when the door was opened, the bees went out to forage, and half of the returning bees became P that initiated the burst. A few days after the door was opened, dancers and dance followers emerged, and most bursts were accompanied by dancers and dance followers. Approximately 90% of the dancers and dance followers were included in the member of P. These trends were also observed in all five trials (Fig. 3D). In addition, by calculating the mean frequency of dancing and dance following for bees that became P at least once and for bees that did not become P at all during the experimental period, we found that bees that experienced P tended to perform more frequently both waggle dances and dance follows than bees that did not become P (Fig. 4A, 4B).

We further distinguished “forager pioneer bees” (FP bees) that foraged and returned to become P and “non-forager pioneer bees” (NFP bees) that were P but did not leave a nest. FP and NFP bees spatially spread differently in the hive (Fig. S12). Both FP and NFP bees came near the door, but at the end of the post-phase (at around day 7), FP bees were still near the entrance, whereas NFP bees moved to the middle of the nest. In terms of information spread, we argue that FP bees brought information from the outside into the nest, but NFP bees were the nonforaging bees that spread information throughout the hive.

Maturation of the beehive

A mature hive indicates that individual bees are tied by the entire hive (downward causation)[24], and according to E.O. Wilson et al., for example, it would imply role differentiation[2]. We hypothesize here that the burst phenomenon promotes/regulates role differentiation. We analyzed the difference between the bursts in the earlier stage and the bursts in the later stage (7 days after setup) from the perspective of the role of P.

First, each burst was indexed according to the individual members in P (Fig. 5). Then, using this index (pioneer bee binary matrix), the distance between bursts was determined, and each burst was projected onto a two-dimensional plane (this method is known as multidimensional scaling [MDS])[25]. As shown in Fig. 6A, comparison of the bursts before the door was opened is represented in blue text, and that after the door was opened is represented in green to yellow text; they formed clusters at different locations in the MDS space.

The bursts were moving over time on the MDS plane, indicating that the members of P on the first day after the door was opened and P after 7 days were different. The pioneer bees were not randomly selected, and its members changed over time. The members of P on day 7 appeared to be fixed, suggesting that they were converging at this point. Another discovery was that spontaneous bursts that did not involve waggle dance did not show a fixed P. However, P was fixed only when accompanied by a waggle dance (Fig. 6B).

In this study, we used the tracking data of approximately 1000 honeybees (× 5 trials) to investigate the large-scale bursting behavior of honeybee hives, which is a social synchronization on the locomotion activities of honeybees. In conclusion, we believe that hive growth can be measured by observing the dynamics and evolution of bursts.

The overall burst is self-organizing. When the door is open, the bees can go out to forage and perform a waggle dance when they return, and if any information is brought back, a global burst occurs. When the door was closed, there were no dancers, but a global burst still occurred, which is a spontaneous burst. A distinction was made between exogenous bursts of foraging information accompanied by dancing and endogenous spontaneous bursts without dancing. The third type of burst is the one that became fixed over the course of a week or so.

After opening the hive, dancer pioneer bees and their followers were found to subsequently trigger global bursts. Individuals that returned to the nest after performing a foraging act performed a dance, and half of them were initiators of the global burst. In this case, we speculated that information was being transmitted from the outside to the nest members.

Regarding the mechanism that causes the global spontaneous burst, when a bee in an excited state comes into physical contact with a bee in a quiescent state, the excited state is propagated. Therefore, burst detection was performed on the basis of the kinetic energy time series (K_i) of each individual, and the burst period was defined as the active state and the other periods as the inactive state. The active state was approximately 10 times faster than the inactive state. Based on the results of this calculation, we simulated a simple agent model to confirm our hypothesis of the activity transfer among agents (see Supplementary Information for details).

By choosing the parameters, we could quantitatively reproduce the global burst curve (the slope of the amount of burst against time) obtained from the experimental data. We thus proposed that the number of physical collisions triggers the spontaneous global bursts. In other words, information introduced from the external world is transferred through physical contact between bees.

The hive must have a mechanism to initiate bursts spontaneously in the first place. The spontaneous burst was also demonstrated in the abovementioned model. The model assumes that the individual bee is in two states, viz., excited and calm. By assuming that the number of collisions causes a transition between the two states, the bursting behavior was replicated. Spontaneous bursts appear as a universal property of excitatory media, suggesting that the beehive is also an excitatory medium and uses its properties (as does the brain) to transmit information.

Data

All data were provided by G. Robinson and his group from the University of Illinois at Urbana-Champaign; their detailed data collection methods are reported in their paper (11). To summarize briefly, they used a single-cohort hive consisting of approximately 1000 adult worker bees aged 1 day and one unrelated, naturally mated queen. This type of colony is commonly used in experiments analyzing the division of labor to control the effect of age and, instead, focus on the behavioral development of worker bees as a function of their genetics and social interactions (26).

Similar to other approaches for tracking insects [12–14], their method used custom 2D QR-barcode devices, known as bCode. These devices were attached to the thoraxes of individual bees and provided sequences of digital images that enabled reliable identification and tracking of every individual in the hive. The digital images were converted into data with coordinates (x, y) and the orientation of each bee every second, with x ranging from 0 to 6576 and y ranging from 0 to 4384. The hive was installed vertically in a dark room with a consistent temperature (approximately 35°C) and humidity, and it was connected to the outside through a tunnel, whose entrance was closed until 2 or 3 days after the start of the observation and then opened to allow the workers to exit the hive and begin foraging (Fig. S1). The bees produced sufficient honey to feed the entire hive for the duration of the experiment, with sufficient bee nutrition for 2 days. Some bees could not be tracked because they died and because of sublations of bCode. The researchers removed the bees that could not be tracked from the tracking data before opening the entrance. The software could identify the bees in 94% of all cases (determined by a manual analysis of 60 images), with an error rate of 1.4% (determined by a manual analysis of 5000 detected barcodes). There were five separate trials (five different hives) of the experiments over July 2012–2013 (Table S1).

Detection Of A “forager”

After opening the entrance, bees often went outside. It is well known that bees go outside for “foraging.” Although the point at which a bee becomes a forager varies, under normal colony conditions, forager bees are workers with an age of more than 3 weeks, and they shift to perform outside tasks, such as collection of water, nectar, pollen, and resin (27,28).

Because the time duration of our experiment was shorter than 3 weeks, the experimental duration and bees’ age were almost similar, so that there were no “real foragers” in our experiment. We considered bees that went outside as “forager candidates”; however, they could become real foragers after 3 weeks. In this study, we identified a bee that returned from outside within 10 min before each burst as a “forager candidate.” Moreover, to focus on foraging activities, we had to omit “orientation flights” (16,17) that occurred several days before the foraging behavior. To omit orientation flights, the forager candidate’s first day of foraging was defined as the first day on which it has at least six reads of going outside and > 25% of which occurred before 12:00 (based on personal observations, Gene Robinson’s group was aware that in their locality, most orientation flights occur in the afternoon). This criterion for the detection of the orientation flight is similar to that in the previous study by Gene Robinson’s group (18). Any flight activity on or after this day was defined as foraging activity.

Detection Of “waggle Dance”

By performing a “waggle dance,” foragers can share directions and the distance to patches of flowers and water sources with other hive mates (21,29,30). The dancing bees waggle back and forth as they move forward in a straight line and then circle around to repeat the dance. We developed an algorithm for detecting a waggle dance in which the rotation of the bee was used as an indicator of the waggle dance.

The orientation of a bee at time t, where ${{n}_{x}}_{t}$ and ${n}_{{y}_{t}}$ denote the respective $x$ and $y$ components of the head direction of each bee, is as follows:

$${n}_{t}={(n}_{{x}_{t}},{n}_{{y}_{t}})$$

If ${n}_{t}\cdot {n}_{t+1}<0$, then the orientation of the bee changed to > 90°. As the first condition was to determine whether a bee was dancing, they used the following parameter: in 5 consecutive seconds, there should be at least four negative scalar products, which implies that, in 5 consecutive seconds, the potential dancer had to change its orientation at least four times to > 90°.

As the second criterion, if ${n}_{t}\times {n}_{t+1} >0$, then the bee made a right turn. When the orientation of the bee alternated between left turn and right turn over 5 consecutive seconds, it was identified as a “potential dancer.” We identified a bee that met both conditions as a “dancer.” To confirm the success rate of the detection, we attempted to validate the results by watching the videos of the detected dances and by considering the features of the “dancer” (e.g., circling, have followers, and trophallaxis) that were related to dance. We found that, in our tested list of 500 dances, the success rate of the detected dances was 75–82%. In Fig. 5, to examine the relationship between a burst and its most recent dance event, bees that were dancing within 10 minutes before each burst are considered dancers for that burst.

Detection Of “dance Followers”

Dance followers receive information about the feeding area from dancers (21,22). Their existence was observed by von Frisch in 1967 (20). The dance follower tracks the motion of a dancer by keeping its head facing. In this study, we defined a dance follower as a bee that is within 600 pixels (approximately the size of a bee) of the detected dancer when the dancer is dancing and where the direction vector from the follower to the dancer and the vector of the follower’s head direction are always within 90°.

Detection Of A “burst”

We also required a criterion to detect the “bursting” phase. Kleinberg’s burst detection algorithm was used to detect bursting events (15). This algorithm assumes that the intervals of the events occur independently according to the following exponential distribution:

$$f\left(x\right)=\stackrel{-}{\lambda }{e}^{-\stackrel{-}{\lambda }x}$$

Here,$\stackrel{-}{\lambda }$ is the mean frequency, defined as $N/T$, where N is the total number of events over the time series and T is the total length of the time series. In addition, $x$ is the interval between consecutive events. Bursts were detected by comparing the expected frequency with the actual event frequency observed within the specific time window.

We defined the burst level at every time point $t$ of individual events. The burst level was expressed as $bl\left(t\right)$. When $bl\left(t\right)$ was ≥ 1, we considered it as a burst. When the local event frequency at time $t$, shown as $\lambda$, exceeded a certain threshold, the burst level was updated. When ${\lambda }_{t}$ exceeded $\stackrel{-}{\lambda }{s}^{1}$, the burst level $bl\left(t\right)$ became 1 from 0. In the same manner, if ${\lambda }_{t}$ exceeded $\stackrel{-}{\lambda }{s}^{2}$, $bl\left(t\right)$ became 2 from 1, and so on. We used $s=2$, so the burst level would increase by 1 when the frequency doubled the previous one.

In case we had followed this process naively, the algorithm would have detected a large number of bursts, when the actual frequency of events fluctuated around the boundary between two burst levels, 0 and 1. To alleviate this, we introduced another parameter $\gamma$ into the algorithm. We used $\gamma =1$ (see reference [15] for details). We defined the “burst period” as a period when the burst level was ≥ 1.

This algorithm originally required timestamps of the sequence of events in question. However, our data are provided as a time series ${K}_{G}$. Here, we consider ${K}_{G}$ to be a frequency-based sequence, and we must convert our frequency-based data into interval-based data. We simply calculated the inverse of the frequency as the interval between events. We applied the algorithm after removing the diurnal cycle of ${K}_{G}\left(t\right)$.

In this study, we explored the bursting behavior detected by Kleinberg’s burst detection algorithm. However, there were also small burst-like activities that were not detected by Kleinberg’s algorithm (depending on the parameters of the algorithm). We speculate that there are some reasons why these small bursts do not grow into larger, global bursts. By examining these small bursting behaviors, we may elucidate a more detailed mechanism for the bursting behavior.

Classifying Bee Behavior By Activity And Activity Timing

To examine when and which bees were initializing and committing to organizing a global burst, we used NMF, which has become a popular decomposition algorithm (19,20).

Given a nonnegative matrix $X$, this algorithm finds the nonnegative matrix factors $W$and $H$, where $W\in {R}^{m \times n}$, $H\in {R}^{r \times n}$, and $r$ is the number of components (known as rank), as follows:

$$X \cong WH$$

In practice, $r$ is often chosen such that $r\ll \left(m,n\right)$.

A critical parameter of NMF is the factorization rank $r$ that defines the number of representative bee activities used to approximate the target matrix. The method of setting the value of $r$ is to attempt different values and compute some quality measures and then select the best value. Several approaches are available to decide the optimal value of $r$—for example, Brunet et al. proposed choosing the first value of $r$ for which the cophenetic coefficient starts decreasing (19). Hutchins et al. suggested choosing the first value where the residual sum of the squares curve presents an inflection point (35). We decided to use the approach of Brunet et al. in this study. The advantage of this algorithm is that it can factorize the input matrix without destroying the cluster structure of the original matrix. To estimate $r$, we calculated the cophenetic coefficient, changing the value of $r$ from 2 to 10.

In general, $H$is known as the base (feature) matrix, and $W$ is known as the weight matrix. In the case of our analysis, $X\in {R}^{m \times n}$ implies the ${K}_{i}$ matrix at the bursting phase, as determined by Kleinberg’s algorithm. In addition, $n$ is the number of individual bees, $m$ is the time length of the bursting phase, and the matrix elements are ${K}_{i}$. To extract $X$ from the original ${K}_{i}$ matrix, we explored the timestamps of the peak points for each burst $t{s}_{b}$, where $b$ is the index of the detected bursts. We then defined ${X}_{b}$ as follows:

$${X}_{b}=\left[\begin{array}{c}{k}_{{ts}_{b}-\text{1000,0}} \cdots {k}_{{ts}_{b}+1000,n} \\ ⋮ \ddots ⋮\\ {k}_{{ts}_{b}-\text{1000,0}} \cdots {k}_{{ts}_{b}+1000,n} \end{array}\right]$$

We confirmed that the minimum interval of the peaks of the detected bursts was 2538 s. Moreover, $H$ describes the representative examples of the time series ${K}_{i}$ (the number of examples [base] depends on $r$). The method starts by randomly initializing the matrices $W$ and $H$, which are iteratively updated to minimize a functional divergence. To estimate the matrices $W$ and $H$, as a local minimum, the NMF algorithm solves the following minimization problem, where $D$ is a loss function based on the Kullback–Leibler divergence:

$$min\left\{ W,H \ge 0 \right\}\left[ D\left(X ,WH\right)\right]$$

There are some types of update functions for solving the prior minimization problem iteratively. In this study, we used the “multiplicative update rules.” The initial entries for $W$ and $H$ were drawn from a uniform distribution. The number of iterations of the update function was 500. The error rate converges sufficiently well at around 500 iterations. We executed it 50 times, changing the initial matrices, and we then selected the solution in which the cost function was minimized.

Classifying “pioneer Bees” Using Nmf

NMF decomposes the original matrix into a matrix composed of feature vectors and its weight matrix; hence, it is often used as a soft-clustering algorithm. The cluster member was computed as the index of the dominant basis component for each bee. For instance, if the maximum value of the weight vector $w$ ( $= {W}_{\left(,beeZ\right)}$) of bee Z was ${w}_{\left\{i\left(1\le i \le r\right)\right\}}$, bee Z was classified as the ith base.

We identified that a “pioneer bee” is an individual corresponding to a base vector whose largest amplitude was observed before that of the global burst. If the largest value of the recomposed ${K}_{i}$ of bee Z at burst $i$ was smaller than the mean ${X}_{i}$, we did not identify bee Z as a “pioneer bee.”

Such bees were identified for each observed burst. We created the “pioneer-bee binary matrix” $P{M}_{\left\{i,j\right\}}$, where the columns represent each burst index and the rows represent each individual bee index (Fig. 5). The matrix element was either 0 or 1—if a bee $i$ was categorized as a “pioneer bee” of the burst $j$, then the matrix element $\left(i,j\right)$ was assigned 1; otherwise, it was 0.

Dimension Reduction Of The “pioneer-bee Binary Matrix”

To examine the similarity between the combination of pioneer bees in different bursts, we conducted a dimensional reduction of $P{M}_{\left\{i,j\right\}}$ using nonmetric multidimensional scaling (nMDS) (25).

The input data of nMDS is a distance matrix. In this study, we first created the distance matrix from the “pioneer-bee matrix.” We used the Jaccard index for computing the distance of a binary matrix. Next, we reduced the dimension of the distance matrix using nMDS. In the reduced space, the distance between the points exhibited the behavioral similarity of the pioneer bees in each burst. As such, if the same bees were pioneer bees during different bursts, then these bursts are represented close to each other; otherwise, they are presented further apart. We iterated until the stress (the loss function of nMDS) was smaller than 0.2.

Acknowledgments

The authors would like to thank Dr. Tim Gernat and Prof. Gene E. Robinson for providing the experimental data for the analysis. We would particularly like to express our gratitude to Prof. Yoshi Oono for introducing this massive data-driven analysis and the deep insights into this study. The basic idea for the experiment of individual bee tracking system is attributed to him. Without him, this work would not have been possible.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Spontaneous and Information-induced Bursting Activities in Honeybee Hives

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Abstract

Figures

Introduction

Results

Characteristics of bursts

Classifying Pioneer Bees

Maturation of the beehive

Discussion/conclusions

Methods

Data

Detection Of A “forager”

Detection Of “waggle Dance”

Detection Of “dance Followers”

Detection Of A “burst”

Classifying Bee Behavior By Activity And Activity Timing

Classifying “pioneer Bees” Using Nmf

Dimension Reduction Of The “pioneer-bee Binary Matrix”

Declarations

References

Additional Declarations

Supplementary Files

Status:

Journal Publication

Version 1