Speeding Up Social Waves. Propagation Mechanisms of Shimmering in Giant Honeybees

Shimmering is a defence behaviour in giant honeybees (Apis dorsata), whereby bees on the nest surface flip their abdomen upwards in a Mexican wave-like process. However, information spreads faster than can be ascribed to bucket bridging, which is the transfer of information from one individual to an adjacent one. We identified a saltatoric process that speeds up shimmering by the generation of daughter waves, which subsequently merge with the parental wave, producing a new wave front. Motion patterns of individual “focus” bees (n = 10,894) and their shimmering-active neighbours (n = 459,558) were measured with high-resolution video recording and stereoscopic imaging. Three types of shimmering-active surface bees were distinguished by their communication status, termed “agents”: “Bucket-bridging” agents comprised 74.98% of all agents, affected 88.17% of their neighbours, and transferred information at a velocity of v = 0.317±0.015 m/s. “Chain-tail” agents comprised 9.20% of the agents, were activated by 6.35% of their neighbours, but did not motivate others to participate in the wave. “Generator agents” comprised 15.82% of agents, showed abdominal flipping before the arrival of the main wave front, and initiated daughter waves. They affected 6.75% of their neighbourhood and speeded up the compound shimmering process compared to bucket bridging alone by 41.5% to v = 0.514±0.019 m/s. The main direction of shimmering was reinforced by 35.82% of agents, whereas the contribution of the complementing agents was fuzzy. We discuss that the saltatoric process could enable the bees to instantly recruit larger cohorts to participate in shimmering and to respond rapidly to changes in flight direction of preying wasps. A third, non-exclusive explanation is that at a distance of up to three metres from the nest the acceleration of shimmering could notably contribute to the startle response in mammals and birds.

Some of the mechanisms of shimmering have been clarified [6][7][8][9][11][12][13][14], but it is not fully understood how the waves propagate [7][8][9]. Shimmering has been described as a Mexican-wave-like process [15], following the principles of bucket-bridging [7,9] to transfer information along a chain of agents, such as passing a bucket of water from one person to another to extinguish a fire in the old days. The bucket-bridging hypothesis of shimmering [7,9,12] predicts that surface bees are affected one-by-one continuously and linearly along a propagation line according to three principles: the first principle is that propagation is directed, whereby a surface bee is stimulated to participate in shimmering by her neighbour bees in those angular sectors where shimmering cohorts show maximal activity [9]. The second principle of bucket bridging is that shimmering activity proceeds steadily from one agent to the next in a linear fashion [7,9]. The third principle is that information is transmitted in parallel chains of agents (for a summary of abbreviations and definitions, see Glossary S1). Theoretically, if information generated at a certain spot propagates in parallel queues at exactly the same velocity, the frontline of the wave would represent a single row of shimmering bees. Any deviation from this parallel propagation will broaden the wave front. The directivity in propagation based on these three principles was addressed in the directed-trigger hypothesis of bucket bridging [9].
Although shimmering is a collective, synchronized and cascading behaviour, each agent is seemingly free to decide whether to participate in shimmering or not. Furthermore, agents that do participate flip their abdomens at variable strengths and angles of up to 120u. The starting point of shimmering depends on the position of the threatening cue [11][12], from where it spreads into all directions, similar to mechanical analogues of water or sound waves [16][17]. However, shimmering does have different ways to propagate [9,[11][12]. Apart from bucket-bridging [7][8][9], shimmering waves appear to ''jump'' from specific sites to others [11][12][13], typically in increments of ten to fifteen surface bees, in a saltatoric process.
Here, we investigated the contribution of both, bucket-bridging [9] and saltatoric processes, to shimmering. The main characteristics of wave propagation were determined on the single bee level regarding recruitment, velocity and directionality, and three functional types of information transfer were distinguished. We found that the saltatoric process is associated with the generation of subordinate or daughter waves. By merging of daughter and parental waves, shimmering waves are speeded up by a factor of two to five, likely reinforcing the anti-predatory impact of shimmering [6].

Experimental conditions
Site and recording. The shimmering behaviour of giant honeybees was studied under field conditions during two expeditions to Nepal. The recording setup (see below) was established at three sites on the campus of the Tribhuvan University in Rampur (February 2009: nest A) and at the border of the Chitwan National park (February 2009: nest B [Fig. 1A]; November 2010: nest C). Two synchronized cameras were used to record black-and-white images with a resolution of 2,35261,728 pixels (px), whereby one pixel covered approximately 0.30 mm in real-world coordinates. Therefore, the characteristic abdomen width of 6 mm of a giant honeybee was imaged by roughly 20 px. The cameras captured 60 frames per s (fps), resolving the abdomen-flipping phase of an individual bee of 200 ms within 12 frames (for further details see [6][7][8][9][12][13]). Nests were also filmed with a high-definition video camera (Panasonic HVX 200) at 50 fps and a resolution of 1,2806720 px from distances between 1.5-10 m, whereby the camera angle always covered the whole nest. Nest B was used for an automated in-depth analysis of the data recorded for approximately 11,000 focus bees and 460,000 neighbour bees (see below). The validity of the results from nest B was confirmed by manual analysis of selected aspects from other nests (see Results). This confirmation, together with our previous experience with many other nests [5][6][7][8][9][11][12][13] made us confident that the parameters used to describe shimmering behaviour previously and in this paper do not represent colony-or nest-specific traits, but can be considered as representative for the behaviour of Apis dorsata generally.
Dummy wasp stimulation of bees. For eliciting shimmering waves, colonies were stimulated with a dummy wasp fixed to a cable car device [7][8][9]13] at the sun-exposed side of the nests. A dummy with white, yellow and black stripes was made of Styrofoam (LxWxH: 40615615 mm) and suspended from a thin thread. Close to the mouth zone, the dummy was swung horizontally at an angle of 90u perpendicularly to the nest. The movement of the dummy was computer-controlled at variable velocities (0.1-0.5 m/s) and directions (towards and away from the nest). For more intense stimulation the dummy was connected to a 1.5 m long stick and moved manually (see Movie S1).

Characterization of shimmering
Identification of agents. In a giant honeybee nest, the colony members are arranged on both sides of the central comb in several layers, whereby several functional regions can be discerned, such as the mouth zone [10], the attachment zone, the rim zone and the quiescent zone [6][7][8][9][11][12][13]. Shimmering behaviour is mainly seen in surface bees in the quiescent zone ( [7][8][9]12]; Movies S1, S2, S3, S4, S5, S6), flipping their abdomens upwards at an angle of up to 120u. In each frame, such agents were identified individually by stereoscopic imaging [7,9] using the coordinates of their thoraces, measured as x-(horizontal directions), y-(vertical directions) and z-(directions towards and away from the comb) positions at resolutions of fractions of a millimetre. More than 600 agents were continuously tracked in successive frames throughout multiple shimmering processes. A total of over 50 episodes of 2 min duration, each comprising 2 waves, were recorded and analysed under defined stimulation protocols [7,9]. ; grey open squares refer to the boundaries of the images of the panels B and C; the four yellow points mark the corners of the image part in Movies S1, S3 and S5. In the mouth zone the surface bees are randomly oriented (which is visible at the right side and below the white curve); peripheral to the mouth zone the surface bees are mostly vertically oriented, with their heads upwards and abdomens downwards. (B-C) Differential images (see image analysis in Methods, [7]) from high resolution video recordings (fps = 60 Hz) of nest part specified in the panel A (real-world measures of panels: B/43634 cm; C/ 14610 cm); large capital numbers show the panel count, small numbers give the relative count of frames during the propagation of the shimmering process which took place from left to right ( f L t R) in the image. The rectangle in panel B indicates the boundaries of panel C. The time difference between the images of B 1 and B 2 was 317 ms [; 19 ff]. Pseudo colours (panels B 1,2 and C 1 ) visualize 8-bit D lum values of motion strength pixelwise in the rainbow palette with blue = 0, red = max D lum (see scale in B 1 and Methods). (C 2 ) The same image as in panel C 1 but in black-and-white and inverted for better discrimination of the abdomens; at the crossing points of the both marker lines an agent bee was selected which was moving solitarily ahead the shimmering front which was proceeding from the left to right; the schematic on the right gives the angular sectors of the near neighbourhood of this focus bee (with the directional categories; r Nh ,40 mm). doi:10.1371/journal.pone.0086315.g001 Definition of spreading directions. The bee curtain of giant honeybee nests displays a polar topology where the individual bees are arranged with their heads upwards and their abdomens downwards ( [7,9]; Fig. 1 Assessment of motion strength. The image-analysis software Image-pro plus (Media Cybernetics) was used to measure the parameter M xy [7,9] that quantifies the motion of an individual agent bee. M xy refers to the luminance changes (D lum~lum ½f i {lum ½f i{1 ) in a 60660 px zone around the thorax of the selected agent bee and was assessed in differential images (e.g. Fig. 1B,C 1 ; [6][7][8][9][12][13]15]) by pixel-wise subtractions of data from one frame (f i{1 ) to the consecutive one (f i ) at intervals of D t ff~D t ½f i{1 ,f i ~16:67ms (fps = 60 Hz). The value M xy includes mainly positional changes in horizontal (x-) and vertical (y-) directions, as well as movements of head, abdomen and extremities, such as legs, antennae or wings. Bees can move actively during shimmering [9][10][11][12][13][14], flickering [12][13] or locomotion (i.e., moving around on the nest surface or penetrating into or emerging from the subsurface layers of the nest). Bees can also be shifted passively when affected by the movements of their immediate neighbours [7]. Values for quiescent conditions were M xy~0 :3|10 4 px per Dt ff , and for massive shimmering activity they were M xy~1 ::10|10 4 px per Dt ff . Values for M xy were normalised as rel M xy relative to the maxima in each recording [7,9], because they depend on conditions of video recording such as lighting and zoom mode used, and on image analysis tools such as subtraction, filtering and segmentation.
Determination of time zero. When the wave front arrived at an agent bee, weak deflections in rel M xy values of the agent bee were assessed [7,9] even before she started to flip her abdomen (Fig. 2). Such small motions of an agent bee were caused by the shimmering-active neighbours. During abdominal flipping, the rel M xy value sharply rises and peaks within 40 ms. In addition, the arrival of the wave can be traced by this parameter some frames before the abdominal flipping, which caused a slight increase in the rel M xy value (for details, see Figs. 2,3A and [7]). This transient increase allowed identifying a shimmering incident defined as the flipping of the abdomen of an individual agent bee participating in the shimmering wave. A shimmering incident was only considered if the agent exceeded the threshold (th) value of DM th~1 0 px for at least five successive frames in the differential images (Equation 1). This criterion served to supress noise.
For each shimmering incident (Figs. 2,3A) the following parameters were assessed: (a) the start time zero (t 0 ) was defined one frame before the sharp rise of rel M xy detected by the trigger algorithm at the time t th when the motion strength exceeded the threshold value according to t th~t ½f th and t 0~t ½f th {1.
(b) The motion strength of the shimmering incident of a selected agent was defined by the maximal rel M xy value measured within 10 frames after t 0 . These strength values were graded in eight steps proportional to the rel M xy values (with motion strength categories c Mxy [ f1::8g). Hereby, the assignment of c Mxy~1 was particularly important, as this minimum level of active shimmering motion of a selected agent bee had to be clearly distinguished from unambiguous (passive) sub-threshold motions (cf. Fig. 2A 2 ). Only active, supra-threshold movements with c Mxy §1 were considered. In contrast to the agent-related rel M xy values the pseudo colours of differential images (Figs. 1-2,4-5) visualize the relative motion in the images, displayed over six frames (from f 74 to f 79;100 ms) from the focus bee (whose thoracic position was marked with a white cross throughout the frames) to one of her neighbours (whose thoracic position was marked with a white cross in a white circle); differential images represent luminance [D lum] patterns according to the rainbow scale displayed in panel A 1 (see Fig. 1 and Methods), the grey open rectangles signify the extension of the receiver agent, the numbers at the bottom of the rectangles refer to the relative activation level of the shimmering incident (whereby the pixel-by-pixel luminance differences were scaled on the rainbow palette with rel M xy~0 :0: strength by the luminance differences pixel by pixel, scaled with rel M xy~0 :0::1:0 on the rainbow palette. Focus bees and their neighbours. The participation in shimmering greatly varies from agent to agent and ranges from passive motions ( Fig. 2A 2 ) caused by the approaching wave front to active participation with abdominal flipping at variable strengths (Fig. 3A). To support the automated evaluation and sorting of the various types of shimmering incidents, the concept of the focus bee was introduced. For every focus bee, two sets of neighbour bees (nb) were defined, which were positioned in the near neighbourhood (,40 mm; Fig. 1C 2 ) and in the far neighbourhood (.40 mm, ,100 mm). The numbers were distinguished according to whether these neighbour agents had participated in shimmering before (N pre nb ) the wave front arrived at the focus bee, or afterwards (N post nb ). Thus, an agent bee was treated either as a focus bee, or as a neighbour bee.
In the in-depth study, focus-bee or neighbour-bee status was assigned to agents regarding all four main wave directions (dir WAV~f R t L: N fb = 2,823; N nb = 138,190 [for .40 mm, ,100 mm]/29,248  N nb = 105,356/13,211; f T t B: N fb = 1,577; N nb = 56,595/9,539; all wave directions: N fb = 10,894; N nb = 459,558/96,937, where fb and nb stand for focus bee and neighbour bee, respectively). For most results shown in this paper, waves in the direction dir WAV~f R t L were taken as representative of all four data sets.
Trigger neighbours. The time point t fb 0 of a shimmering incident of a focus bee was used to define the potential trigger neighbour bee for a given shimmering incident (Fig. 3) by considering two criteria: First, the trigger neighbour must have already participated in the same wave before the focus bee; however, this participation was only considered if this neighbour was active within the time window of 88 ms (25 ff) prior to t fb 0 . Second, the agent defined as trigger neighbour had to be positioned closest of all other candidates to the focus bee and in her near neighbourhood. This criterion excluded those bees from analysis that flipped their abdomen outside the near neighbourhood (in many cases such bees generated daughter waves; Figs. 1-2, 4-5). The trigger direction of a focus bee (a fb TRIG ) was defined by the angle of the direction of her trigger neighbour, measured from the position of the focus bee, categorized according to the eight sectors of neighbourhood (dir Nh [f1::8g; Fig. 1C 2 ) in which the trigger neighbour was positioned.   Benchmarking of bucket-bridging. Bucket bridging was characterised for manually selected agents by the assessment of the information transfer between two neighbouring bees ( Fig. 2A), and over a chain of adjacent agents (Fig. 2B). The thorax-tothorax distances were calculated in mm. A motion detection method was used that utilized differential images (see above) considering the following criteria [9]: (a) Chains of agent pairs were selected (Fig. 2), in which emitter bees (which flipped their abdomen first) could clearly be distinguished from receiver bees (which followed the action of the emitter bee) throughout a continuous and linear sequence of actions. (b) Emitters and receivers participated in the same chain (see Introduction and [9]). (c) Bees were excluded from data assessment if they did not show any interaction, if they were activated synchronously (instead of sequentially, i.e. they did not participate in the same chain) and if they showed abdominal flipping with a delay larger than 167 ms (;10 ff). (d) In differential images, the temporal information transfer was estimated using the abdominal flipping of the emitter and the receiver bee. The time

Ethics Statement
The research expedition to Chitwan, Nepal, entitled ''Study on the behaviour of the giant honeybees: Observations and recording of behaviours at the nesting site'' was supported by the Rector of the Centre for International Relations of the Tribhuvan University (Kathmandu, Nepal).

Bucket-bridging
An example of bucket bridging is shown in differential images in Fig. 2A 1-4 (ff 74-79 corresponding to 100 ms). The emitter bee showed maximum activity in f 76. The silhouette of the receiver bee was already visible in f 76, when it was slightly moved passively by the advancing wave front. In f 77, the shape of the right forewing of the receiver bee appeared, indicating active participation in shimmering. In f 79 her abdomen started to be lifted actively, followed by a massive motion of the whole body accompanied by a beat of both wings. An

Saltatoric processes
Saltatoric wave propagation involves bees that are more than 80 mm away from the approaching wave front, and lift their abdomens typically 30-50 ms earlier than other participants in their own near neighbourhood. These bees generate daughter waves and are termed generator agents. In Figs. 4,5 (compare Movies S1, S2, S3, S4, S5, S6), one of these generator agents is marked by a red point on the thorax. During the abdominal lifting of this agent (ff 02-06, Fig. 5), the parental wave (pw) moved from left to right in the image. Using the position of the wave front in two subsequent frames the velocity of the parental wave was found to be v pw = 0.239 m/s, which is in the same order of magnitude as calculated for bucket bridging in the same nest ( Fig. 2A).
In f 09 of Fig. 4 the daughter wave is visible as a small circular batch, and in f 12 this daughter wave started to merge with the parental wave. The frontline of the parental wave ''jumped'' from the left to the right side of the daughter wave (from f 10, immediately before the daughter wave merged with the parental wave, to f 11, just after merging) within 16.67 ms or even less. The frontline of the merged waves advanced over a distance of 160 mm, which corresponds to an acceleration of the shimmering velocity (v sh ) to a value of at least 0.960 m/s by saltatoric information transfer. This is roughly three times faster than bucket-bridging alone. Fig. 5 demonstrates the saltatoric information transfer in more detail for the same focus bee. This bee started abdominal flipping in f 02, and one of her neighbours joined 70 ms later (marked by a yellow arrow, first visible in ff 04-05). Subsequently, other neighbours followed, producing a daughter wave. The parental wave on the left side of the image continued to spread to the right, while the daughter wave spread into all directions by bucket bridging. Finally, 170 ms after the start of abdominal flipping of the generator bee (in f 10), the daughter wave merged on its left side with the parental wave, while the frontline of the merged waves ''jumped'' to the right side of the former daughter wave.

Propagation velocity of shimmering waves
A shimmering wave lasts up to 800 ms (Fig. 3B) and includes the ascending phase of 200-300 ms, in which the number of synchronously shimmering surface bees increases to the maximum, the climax phase, when a maximum number of agents were simultaneously active (lasting approximately 200 ms), and the descending phase (300-400 ms), in which the number of synchronously shimmering surface bees decreases ( [6,12] and Fig. 3B). When measured from the start to the end of a wave by the detection of the positions of the wave fronts in differential images (Figs. [1][2][4][5], v sh was calculated to be 0.36760.020 m/s (n = 10 waves, see Movies S1, S2, S3), similarly slow as found for bucket bridging.
Alternatively, when measured during the climax phase, v sh was 0.51460.019 m/s, significantly (P,0.01, Student's test) faster than the benchmark value for bucket bridging (v bb = 0.325 m/s). The v sh value for the climax phase was calculated for the wave direction dir WAV~f R t L regarding nest B by selecting pairs of agents that were positioned horizontally (at an angle of a WAV = 7.9565.40u; n = 41 waves; definition of angle of wave direction in Fig. 1C 2 ) at a distance of 112.9064.08 mm between those emitters and receivers that were subsequently affected by the same wave fronts within ten frames or more (226.0668.21 ms) in a straight line (cf. Fig. 2B).
This v sh value of 0.51460.019 was taken as a benchmark value for the mixed propagation mode (bucket bridging and saltatoric processing). Hence, v sh during the climax phase is faster than during the start or the end phases, likely due to the higher probability of the occurrence of saltatoric events.

Communication statuses of agents
The contributions of individual agents to the spreading of shimmering waves, i.e. to bucket bridging or saltatoric propagation were assessed by automated techniques using the algorithms described below. The ''communication status'' of a focus bee was characterized spatially and temporally by considering her neighbours in the near neighbourhood as a reference region (,40 mm, see schematic in Fig. 6), and the pre-stroke and post-stroke intervals of 6167 ms (;610 ff) as a reference time window, whereby the abdomen flipping commenced at the time t 0 . Shimmering waves were assessed regarding four wave directions (Table S1, [7]), and for in-depth analysis of communication status the wave direction .|610 ff|). Thick horizontal lines (black, orange, red) symbolize that abdomen flipping is still going on. (B 1-3 ) The panels Pre t 0 and Post t 0 display the near neighbourhood of a focus bee which is marked black when quiescent and marked red when she flips her abdomen. The numbers and colour codes of shimmering-active neighbours are the same as in the panels A 1-3 ; arrows of the central panels (regarding the time t 0 ) explain the directions of information transfer: status I (bucket-bridging) agents in the pre-stroke (Pre t 0 ) phase receive mechanical information (full orange arrows) predominantly from the side from where the wave came, in the post-½t 0 -stroke (Post t 0 ) phase information is transmitted predominantly to the side where the wave is spreading to (open red arrows); status II (chain-tail) agents receive and emit mechanical information but fail to recruit other neighbours as transmitters; status III (generator) agents utilize predominantly visual information from the threatening cues, but not from shimmering-active agents in their near neighbourhood. They are leaders in emitting mechanical information and generate parental or daughter waves. doi:10.1371/journal.pone.0086315.g006 f R t L (N fb = 4,025; N nb = 29,248 [for ,40 mm]) was selected as representative for all waves.
Three communication statuses (I-III) of focus bees were distinguished. Status I agents are driven by shimmering-active neighbours in the pre-stroke phase, positioned at their ipsi-directional side (from where the wave came), passing information on to their contradirectional side (to where the wave is continuing) in the post-stroke phase (Fig. 6A 1 ,B 1 ). This strategy of propagating information conforms to bucket-bridging [7,9], and these bees are termed bucketbridging agents. They comprised the majority of surface bees and were represented by 54.33% of agents at a reference window of 62 ff (whereby ''22 ff'' defines the time window in the pre-stroke phase and ''+2 ff'' the time window in the post-stroke phase). The number of bucket-bridging agents increased (Fig. 7A 1-2 ,B 1 ) with the reference time window (610 ff) up to a value of 74.9862.83%. This increase was due to the definition of status II agents (Fig. 6A 2 ,B 2 : see below), which may turn into bucket-bridging agents with increasing length of the reference window (Fig. 7). The neighbours of the bucket-bridging agents were similarly large in number in the pre-stroke and post-stroke phases (Fig. 7C 1 , Table S2:  Table S1). This data symmetry applies to all phases of a shimmering wave (ascending, climax or descending phase; Fig. 3B and [6]). Furthermore, the selected wave direction dir WAV~f R t L (red symbols in Fig. 8) was found to be representative of all four wave directions investigated (black symbols in Fig. 8), which shows that shimmering is invariant regarding the direction of wave propagation (cf. [9]).
Status II agents flip their abdomens triggered by their shimmering neighbours, (Fig. 6A 2 ,B 2 ), but thereafter their near neighbourhood becomes quiescent. These bees terminate the information transfer in their chains and are, henceforth termed chain-tail agents. The relative numbers of chain-tail agents increased from rel N fb = 9.2060.94% (at 210 ff, corresponding to 166.67 ms, where rel N stands for relative number in per cent) to rel N fb = 28.4663.14% (at 22 ff;33.33 ms; Fig. 7B 2 ) and had N nb = 7.2860.57 shimmering-active neighbours at 210 ff (Fig. 7C 3 , Table S1).
In contrast to chain-tail agents, Status III agents (Fig. 6A 3 ,B 3 ) are not triggered by their neighbours [,40 mm] in a reference time window in the pre-stroke phase of 210 ff, but flip their abdomens before their neighbours do so. They generate parental or daughter waves (as demonstrated in Figs. 1,4-5 and Movies S1, S2, S3, S4, S5, S6) and are termed generator agents. In contrast to status-I and -II agents, generator agents occurred at numbers which were roughly independent of the length of the post-stroke phase (at +3 ff;50 ms: rel N fb = 17.6562.69%; at +10 ff: rel N fb = 15.8262.61%; Fig. 7B 3 , Table S1) and activated N nb = 6.3660.45 neighbours at +10 ff (Fig. 7C 4 ; Table S1), which corresponds to rel N fb = 6.75% of the shimmering-active neighbours of all surface bees.
In Fig. 8B,C chain-tail and generator agents were sorted according to the number of shimmering-active neighbours in the pre-stroke (210 ff) and post-stroke (+10 ff) phases. These data distributions allow discerning agents with less than five shimmering-active neighbours (subgroup 1), and those agents with more than five active neighbours (subgroup 2). In both agent types, subgroup 1 was larger, while subgroup 2 converged to zero with increasing numbers of active neighbours (Table S1). Hence, only few shimmering-active neighbours (,5) suffice to trigger both agent types.

Acceleration of shimmering by saltatoric propagation
The propagation of shimmering is based on complex synchronous and cascaded recruitment of surface bees, and was simplified in a mathematical model (see lookup tables in Fig. 9) that considers both the bucket bridging and the saltatoric propagation mode under climax conditions. This model allows to assess the impact of both processes on the v sh of the compound wave considering three parameters (where bb stands for bucket bridging and sp for the saltatoric propagation): (a) v bb , this is the v sh under solely bucketbridging conditions (Fig. 9A), where v bb was varied in the model from 0.10 to 0.45 m/s (the benchmark value is 0.317 m/s, see above); (b) v sp , this is the v sh characterised by the saltatoric propagation, which was set to v sp = 0.800 m/s (Fig. 9A-C); this velocity was slightly below the value of v sp = 0.960 m/s that was found for a single saltatoric jolt of the wave front in the example in Fig. 4 (note that v sp is only of theoretical relevance because a real shimmering wave never spreads saltatorically alone and that bucket bridging is the dominant propagation mode of shimmering); (c) the distance d E{R between the emitters and the receivers (E2R) in a chain of bucket-bridging agents; according to the conditions in the nest d E{R was varied from 20 to 45 mm (Fig. 9C,D). These three parameters were used to describe two wave properties that result from the combination of bucket-bridging and saltatoric propagation: First, the factor k sp (Equations 3a,b) by which shimmering is speeded up by the saltatoric process (Fig. 9A,B). ) and the complementary 58.5% for k bb (Fig. 9B). Second, I dec was considered, the time interval within which an agent ''decides'' whether or not to participate in shimmering ( Fig. 9C; conforming with I dec~dE{R =v sh ).
The values of I dec calculated for the distance of d E{R = 25 mm (which is characteristic for the side-to-side distance between bees at the surface of the bee curtain) ranged from 78.86 ms for mere bucket-bridging (v bb = 0.317 m/s) to 48.33 ms (Fig. 9C)

Directional control in shimmering
Theoretically, the simplest way of wave propagation is by spreading energy along a straight line by directed and nonstochastic processes. In shimmering, information is transferred by bridging information along chains of surface bees (in agreement with the directed-trigger hypothesis [7,9]), where both, the focus bees and their trigger neighbours, are aligned perpendicular to the extension of the wave front. Although directedness in propagation has adaptive importance for shimmering [6,9,[11][12][13], only less than 5% of agent bees (e.g., under f R t L) contribute to wave propagation in the main direction of a wave [9]. The present study confirms this apparent conundrum. An in-depth analysis was conducted ( Fig. 10) with the data from one selected wave propagation direction ( f R t L) to test if the data confirms the directed-trigger hypothesis [7,9]. PEAK and SINK concepts. The directed-trigger hypothesis [7,9] predicts for every focus bee that their shimmering-active neighbours are distributed in angular sectors (a Nh ) of the near neighbourhood according to Equation 4.
where P Nh is the probability of the occurrence of shimmeringactive neighbours ( rel N nb ) of a focus bee at the respective angle a Nh (for definition, see Fig. 10A).
The directed-trigger hypothesis postulates that in the pre-stroke phase of a shimmering incident the most shimmering-active neighbours ( rel N nb ) of focus bees are at the ipsi-directional side (where the wave is propagating from: see red arrows in Fig. 10A), and the least neighbours are present at the contra-directional side (where the wave is propagating to). For the pre-stroke phase, this hypothetical distribution of neighbours is described by Equations 5a,b and displayed in Fig. 10C 1 ,E 1 , where blue curves show the maximal rates of shimmering neighbours rel N nb at the ipsi-directional side defined by a Nh :a WAV~0 0 (under dir WAV~f R t L). The distribution pattern shown in these curves is referred to as PEAK distribution pattern. In the post-stroke phase, the hypothetical distribution of shimmering-active neighbours conforms to the SINK distribution pattern, with a maximum occurrence of rel N nb at the contradirectional side, and a minimum occurrence of rel N nb at the ipsidirectional side for a dirWAV~0 0 (see Equations 6a,b; green curves in Fig. 10 D 1 ,F 1 ). Matching the empirical data with the hypothetical PEAK and SINK patterns The empirical rel N nb data were plotted against a Nh and the resulting distributions compared with the hypothetical PEAK and SINK distribution patterns. rel N nb data of all three agent types (bucket bridging, chain-tail and generator agents) were tested for correspondence with the hypothetical PEAK and SINK distribution patterns by cross-correlation according to Equation 7.
Here, the neighbourhood angle a Nh (for definition, see Figs. 1C 2 ,10A) was altered by the deviation angle b in steps of 45u (with 2180u#b#+180u) to identify the best match between the empirical data (displayed by full circles and vertical lines) and the PEAK and SINK curves. The hypothetical PEAK and SINK curves were normalized using the empirical differences between the rel min N nb and rel max N nb values for each agent type as reference (the normalized forms are indicated as {P Nh } The sums of the square differences between the rel N nb values with the {P Nh } values were calculated for every angular step of neighbourhood [a Nh zb] of the focus bees. The total deviation (DEV) over the total range of neighbourhood (dir Nh [f1::8g) between the empirical values ( rel N nb ) and the normalized PEAK and SINK values were determined by Equations 8a,b. Their reciprocal values constitute the goodness of the fit PEAK G and SINK G (Equations 9a,b) which can be tested using the x 2 -test.
Furthermore, the probabilities PEAK P m and SINK P m for the match between the empirical data and the hypothetical PEAK and SINK distribution patterns was calculated according to Equations 10a,b, explaining the coincidence of the agents' directional properties with the directed-trigger hypothesis.  Table S1). (C 1 -F 1 ) Relative numbers of shimmering-active neighbours (ordinate: rel N nb with rel max N nb~1 :0 per communication status) of focus bees in dependence of the neighbourhood angle (abscissa: a Nh ) and communication status (status I: C,D; status II: E, status III: F). Closed circles, arithmetic means; vertical bars, SEMs (n = 7 data sets of f R t L) give the neighbours that were shimmering-active prior to (yellow: panels C 1 , E 1 ) and after (red: D 1 , F 1 ) the t 0 of the abdominal flipping of the focus bees. Fitting functions (blue: PEAK distribution pattern; green: SINK pattern) give the occurrence of shimmering-active neighbour bees (P Nh ½a Nh as predicted by the directed-trigger hypothesis (see . The angular positions (a Nh zb), at which the DEV values exhibit a minimum (Fig. 10C 2 -F 2 ), represent the maximal goodness (max G) of the fit (Fig. 10C 1 -F 1 ). The resulting angular mismatch (MM) between the empirical data and the hypothetical PEAK and SINK distribution patterns was estimated by Equation 11, defined as the absolute deviation between the angle a Ã Nh (the angle with the minimum value of PEAK or SINK patterns, see Lastly, the level E dirWAV by which the different agent types contributed to the propagation of the main wave direction was estimated according to Equation 12. where rel N nb is the occurrence of shimmering-active neighbours of a focus bee in her near neighbourhood; max{min rel D N nb is the percentage of active neighbours in defined a Nh angles between maximal and minimal occurrences of rel N nb ; P m is the probability for the match between the empirical data and the hypothetical PEAK or SINK distribution patterns (Equations 10a,b); f R is the reinforcement factor with f R~z 1 when the main wave direction is reinforced and f R~{ 1 when the main wave direction is restrained.
Matching bucket-bridging agents. The bucket-bridging agents conformed to the directed-trigger hypothesis under pre-stroke and post-stroke conditions (Fig. 10 C 1 -D 1 ) in different ways: the shimmering-active neighbours in the pre-stroke phase matched the PEAK distribution pattern four times more likely  [4][5]9) and bucket bridging (panel A 2 ; Figs. 2,9). The lengths of the black arrows on the right side symbolize the propagation velocities of the partial processes (saltatoric processes and bucket bridging) and of shimmering as the mixed form of propagation. (B) Communication statuses of focus bees categorized in the pre-stroke and post-½t 0 -stroke phase by the recruitment level of their neighbours (nb), exemplified for the main wave direction dir WAV~f R t L; B 1 , the occurrence of the three types of agents (cf. Figs. 6-7); B 2 , schematics of the angular rel N pre nb and rel N post nb distributions of shimmering-active neighbours (PEAK and SINK distribution pattern, cf. Fig. 10) addressing the match with the directed-trigger hypothesis; B 3 , the impact of the three communication statuses on the control of directionality of the shimmering wave (cf. Fig. 10). doi:10.1371/journal.pone.0086315.g011 ( PEAK P G = 0.0061, x 2 -test) than the SINK distribution pattern ( Fig. 10 C 1 max{min rel DN nb = 0.3697; PEAK P m = 0.9461; f R~z 1). Conversely, in the post-stroke phase the shimmering-active neighbours matched the SINK distribution pattern rather than the PEAK distribution pattern (Fig. 10 D 1-2 , Table S2: PEAK DEV = 19.20%; SINK DEV = 4.57%; PEAK G = 5.21; SINK G = 21.87, PEAK P G = 0.0014, x 2 -test; SINK H MM = 0u). The factors calculated from equation 8 were rel N nb = 0.4101; max{min rel DN nb = 0.4073; SINK P m = 0.9543; f R~z 1. The total estimate of the influence of bucket-bridging agents to contribute to the wave propagation in the main shimmering direction was 32.44%, by adding up the E fRtL value in the pre-stroke phase (16.50%) and that in the post-stroke phase (15.94%; Fig. 11B, Table S2).
Matching chain-tail agents. The angular distribution of shimmering-active neighbours of the chain-tail agents agreed stronger with the PEAK than the SINK distribution pattern ( PEAK DEV = 9.  Table S3), showing a small mismatch with the PEAK-pattern ( Fig. 10E: ). This result shows that chain-tail agents can be identified by their match with the PEAK distribution pattern in the pre-stroke phase, and as they are the last agents in a chain they are the components that terminate the chain. Their contribution to shimmering (E fRtL = 22.20%) can be considered as constraining the wave propagation, which is noted by the negative value of f R ( rel N nb = 0.0556; max{min rel DN nb = 0.4781; PEAK P m = 0.9077; f R~{ 1; Fig. 11B, Table S3).
Matching generator agents. In their post-stroke phase, generator agents activate their neighbours. Their distribution pattern agreed twofold more with the SINK than the PEAK pattern ( SINK DEV = 8.79%; SINK G = 11.38; PEAK DEV = 15.61%; PEAK G = 6.40; Table S4) Table S4).
Stigmergy [34][35] (for summary and a broader definition see [36]) is another concept that may also help understanding the interaction between individual agents in shimmering. Stigmergic coordination stimulates behavioural responses through traces left in the environment. A shimmering wave affects all layers of the bee curtain mechanically [5,[7][8]12] within tens of milliseconds (which can be considered as traces), thereby stimulating quiescent agent bees [7,9] to join the wave.
For the analysis of shimmering, concentric zones of neighbourhood were implemented around each focus bee, similar to the Lagrangian swarm models [25,[32][33]. This concept allows the assessment of topological properties (cf. [37]) of the neighbours, such as the coordinates of the thoraces, and the strength and time of participation in the wave. Around 70% of shimmering-active bees (Fig. 7A 1-2 , Status I) act in a bucket bridging-like manner [9] while 88% of their immediate neighbours are also active (in the pre-stroke and post-stroke phases of shimmering incidents: Fig. 7A 3 , Status I). These agents contribute to the wave propagation in response to this mechanical commotion which mainly occurred in their ipsidirectional neighbourhood (where the wave came from). Their action also includes the release of Nasonov pheromone [38] that motivates defensive cohorts to stay rather than fly off and attack the enemy. They also stimulate their neighbours at their contradirectional side (towards which the wave is propagating) to join the abdominal flipping (Fig. 7C 1,2 , Status I).
On the other hand, a minority of about 15% (Fig. 7A 1 , Status III) of shimmering-active bees contribute to saltatoric wave propagation, and stimulate 6% (Fig. 7A 3 , Status III) of their neighbours to flip the abdomens. These generator agents start abdominal flipping before any bucket-bridging activity can be detected in their near neighbourhood, independent of the length of the discrimination window of ,10 ff (Fig. 7B 3 ). Generator agents are not prompted by their neighbours and therefore, have a lead role in decisionmaking by processing the visual cues of the threats, such as wasps in front of the nest, generating daughter waves. A daughter wave can merge with the steadily proceeding parental wave, forming a new wave front distally to the former parental wave front. Thus, the resulting saltatoric wave propagation is based on visual control which adds two new properties to the shimmering process [6]: it speeds up the wave front by a factor of up to three and allows the wave front to change direction rapidly with the potential that shimmering may even ''follow'' a threatening cue topologically on the nest (see Movies in [6,39]).

Goals of social waves
Social waves are hallmarks of animal aggregations. Besides giant honeybees, which display shimmering waves in response to wasps, birds and mammals [4][5][6]12], social waves have also been described for avian flocks [24,29,37,[40][41][42][43][44][45], for the huddles of Emperor penguins [30], and for aggregations of humans in football stadiums [15]. A main question concerning social waves is whether they are advantageous for the aggregation, having received evolutionary function and conveying fitness benefits, or alternatively, whether they are simply a mere epiphenomenon of reactivity [43]. In the following, this question is discussed comparing the ephemeral rolling patterns in starlings and the shimmering waves in giant honeybees.
Waves in starling flocks. Starlings form flocks especially before they roost, to effectively maintain cohesion of the group, strongly supporting survival [37,[41][42]. Under predation they display spectacular ephemeral rolling patterns. Individual starlings may be kept informed in the flock about the progress and the state of such an ongoing ''wave'' by continuous inspection from the momentary vantage point. In reality, this information is restricted to a topological group of 6-7 neighbours [37]. Although an individual starling has some room to modify its decision to participate in the concerted flight manoeuvre, it cannot decline to conjointly move together with its neighbours. This constraint is less due to the potential danger of being eaten by the predator [29] but more due to the danger of collision [37,[41][42]. These rolling patterns display trains of pulses of optical density that propagate across the flock and are produced within the swarm body, mostly without affecting the swarm surface. The authors explain [29] that such pulses are formed in proximity to the bird of prey, mostly laterally, and are propagated typically away from it. These patterns may lead to confusion [6,46] of the predator and encounter dilution [47] but the cause for their evolution is only poorly understood.
Shimmering waves. The goals of shimmering [5][6][7][8][9][10][11][12][13] in giant honeybees differ from the rolling waves of starling flocks [29,37,[41][42] in the following ways: Shimmering waves are produced by stationary agents (Movies S1, S2, S3, S4, S5, S6). Most of the colony members, including those in subsurface layers or at the opposite comb side, continually receive information about the shimmering status mechanoceptically [7][8]12]. Surface bees also release Nasonov pheromone [38], motivating others to participate in the wave. Shimmering giant honeybees may or may not join the wave, and if they join, they can determine when to start their contribution and at which strength [7,9]. Both, starlings and giant honeybees have the potential to respond rapidly to changing visual cues, and in both cases directivity seems to be strongly controlled.
In giant honeybees, the directivity of bucket bridging was described previously [8]. In the present work, we show that the majority of shimmering-active bees (63.18%) do not contribute to the wave propagation in the main direction (Fig. 11B 3 ). Of the other 36.82% of the shimmering-active bees which contribute to the directional control, 88.09% are bucket-bridging agents, 5.98% are chain-tail agents and 5.92% are generator agents (Table S2, S3, S4). Generator agents are less affected by the oncoming wave than the two other agent types (Fig. 7B 3 ,10F), contributing less, and fuzzily, to wave direction control. This fuzziness of generator agents is important, because it enables rapid changes in wave direction in response to rapidly moving cues (cf. [39]).

Why do giant honeybees speed up shimmering waves?
Taken together, the aspects discussed above appear to contribute to the confusion and repellence of predators [6,12,39], but they cannot explain why saltatoric propagation has evolved. Three possible explanations for this conundrum are discussed below: saltatoric propagation could reinforce (a) the recruitment of shimmering agents, (b) the repelling effect on predating wasps by directed visual patterns and (c) the bottom-up attention in vertebrate predators.
Saltatoric propagation reinforces recruitment. More agents are recruited when shimmering is accelerated above the ''base'' speed of bucket bridging (v bb = 0.317 m/s) by saltatoric propagation (v sp = 0.517 m/s) than during bucket bridging alone (Figs. 1-2, [3][4]9). This increased recruitment is due to the generation of daughter waves, causing rapid, exponential growth of the visual pattern with a concomitantly greater repelling effect, which likely benefits the giant honeybee colony. This explanation is in agreement with previous findings [6] that more bees contribute to shimmering the faster and the nearer to the bee nest a preying wasp flies.

Saltatoric propagation increases the repelling effect on
predating wasps by directed visual patterns. Giant honeybees have the capability to align the direction of their shimmering waves with the flight path of a preying wasp [6,39]. In this preypredator interaction both, the bees and the wasp emit signals and both respond to stimuli of their counterparts, whereby the honeybees can change the direction of their shimmering waves faster than the wasps can turn [39]. This asymmetry between bee and wasp is based upon the greater speed of wave propagation and the fuzziness in directional control by saltatoric propagation.
Saltatoric propagation may reinforce bottom-up attention in vertebrates. Giant honeybees display shimmering in response to wasps, but also to birds and mammals that approach the nest within distances of about 3 m (own observations). Shimmering produces repetitively moving circular areas, typically with a diameter of 20 cm or more. Mammals and birds may perceive these patterns as supernormal cues that represent moving, headlike structures [48], outwitting their perceptual systems. Such cues may release a startle response [48][49] in vertebrates based on bottom up (BU) attention [50][51][52][53]. Bottom up attention depends upon the properties of a sensory stimulus to capture full attention, such as a bright spot of colour, an area of sharp contrast, or a rapidly moving pattern, such as typically involved in shimmering [1][2][3][4][5][6][7][8][9][10][11][12][13].
In higher vertebrates, the retinal fovea is the main sensory interface for attention retrieval [50][51]. When a shimmering wave is imaged by the fovea under covert conditions (with fixated eyes), BU attention is likely stronger the greater the image of the shimmering wave is, until the image fully covers the fovea. The fovea receptors are most densely packed in the central 1-2u, and the maximal trace length of a moving object that crosses a central projection at the main acuity region of the fovea is less than 10u [50][51]. The trace length of a shimmering wave on a vertebrate fovea in the 200 ms of the ascending phase (Fig. 3B, [6]) will cover an angle of 10u, provided that circular patterns of 20 cm or more are produced and viewed from a distance of 1 m from the nest. In the climax phase of a shimmering wave (Fig. 3B), when the patterns move at a speed of v sh = 0.514 m/s (Fig. 9A) they will be viewed with a foveal path length of 8.154u, whereas mere bucket-bridging propagation (v bb = 0.325 m/s) will affect only a path length of 5.400u.

Summary and Conclusions
Characteristics and benchmark data of two propagation mechanisms (Fig. 11A) of shimmering, bucket bridging and the saltatoric processes were investigated. Combining both mechanisms speeds up the shimmering waves from 0.317 m/s (bucket bridging only) to 0.517 m/s, which shows that the saltatoric component, also associated with the generation of daughter waves, increases the overall propagation speed of shimmering by 40%. Three categories of shimmering-active surface bees were identified regarding their communication status: the bucket-bridging, chain-tail and generator agents. These agents comprise a characteristic proportion (Fig. 11B 1 ) and a characteristic recruitment status with respect to their neighbours (Fig. 11B 2 ), and they contribute differently to the directivity of the main shimmering wave (Fig. 11B 3 ). Summarizing, the wave-like shimmering process in giant honeybees displays adaptive complexity, particularly regarding the generation and propagation of information, and is an impressive example of swarm intelligence [19][20][21]. Shimmering conforms to rules of bucket-bridging such as linearity, continuity and graduality [9] and involves an additional saltatoric strategy to speed up signal transmission, but also to provide a high level of fuzziness (Fig. 11B 3 ) which may enable giant honeybee colonies to respond to rapidly changing threats [39].

Supporting Information
Glossary S1 List of important terms, definitions and abbreviations used in this paper. (DOCX) Movie S1 In this episode two successive waves were generated by a dummy wasp that was moved from left to right (in the image).
The episode started at f 1, but the playback session refers only to ff 550-650 (corresponding to 1,700 ms). The dummy wasp was moved 20 cm in front of the nest (shown at the left bottom corner of the image). The part of the nest displayed in the film corresponded to the area marked by the four yellow spots in Fig. 1A and comprised about 1,250 bees of the surface layer; the thorax of the focus bee selected for this film was marked in red. Left panel: black-and-white images, inverted to enhance the contrast of the abdomens; right panel: the same view but displayed as differential image with the motion strength in pseudo colours (with relative motion strength scaled from blue = 0.0 to red = 1.0; see rainbow scale on the right bottom side). Original recording speed: fps = 60 Hz; playback speed: fps = 60 Hz. (AVI) Movie S2 The same episode as in Movie S1, but zoomed for about 60 bees; original recording speed: fps = 60 Hz; playback speed: fps = 60 Hz.