Timing of head movements is consistent with energy minimization in walking ungulates

Abstract

Many ungulates show a conspicuous nodding motion of the head when walking. Until now, the functional significance of this behaviour remained unclear. Combining in vivo kinematics of quadrupedal mammals with a computer model, we show that the timing of vertical displacements of the head and neck is consistent with minimizing energy expenditure for carrying these body parts in an inverted pendulum walking gait. Varying the timing of head movements in the model resulted in increased metabolic cost estimate for carrying the head and neck of up to 63%. Oscillations of the head–neck unit result in weight force oscillations transmitted to the forelimbs. Advantageous timing increases the load in single support phases, in which redirecting the trajectory of the centre of mass (COM) is thought to be energetically inexpensive. During double support, in which—according to collision mechanics—directional changes of the impulse of the COM are expensive, the observed timing decreases the load. Because the head and neck comprise approximately 10% of body mass, the effect shown here should also affect the animals' overall energy expenditure. This mechanism, working analogously in high-tech backpacks for energy-saving load carriage, is widespread in ungulates, and provides insight into how animals economize locomotion.

1. Background

The majority of larger mammals use the quadruped walking gait as the main form of locomotion for travelling large distances [1–5]. Running gaits are used in bursts and for high-speed locomotion, e.g. for hunting and escaping, so an animal's life depends largely on the effective outputs of this behaviour. Nevertheless, it is the walking gait in which on average the most energy in absolute terms is consumed over an entire day [4,6,7]. As locomotion is an important energy-consuming factor for most mammals [4,6–8], the main selective pressure shaping the walking gait used in slow locomotion should be energy efficiency rather than high performance.

Many large mammals, such as horses, exhibit a conspicuous nodding of the head when walking. The head and neck of mammals form a highly mobile cantilever on the trunk that constitutes a substantial part of the animal's whole body mass. To date, biomechanical studies on the locomotion of quadrupedal mammals have focused almost exclusively on movements of the limbs and the trunk. The potentially energy-consuming effects of head movements relative to the trunk remain for the most part unexplored in large mammals, but have been shown to be significant in birds [9]. The few existing studies on head movements that accompany the quadrupedal mammalian walking gait [10–14], confirm that the head's kinematics are decoupled from the movements of the trunk. It was therefore proposed that the head movements compensated for fluctuations of the trunk in height and velocity, and stabilized optical and vestibular perception [12–14]. However, available data for head-nodding behaviour in horses suggest that it exceeds the vertical motion of the withers [11,14,15]. Thus, the motion of the head relative to the trunk might increase the overall amplitude of cranial oscillation and appears to contradict the hypothesis that this specific movement has a sensory compensatory function. Moreover, when affected by unilateral forelimb lameness, horses show a reduction in the amplitude of vertical head movement during the stance phase of the lame leg and an increase during the stance phase of the sound leg [16]. This implies that head movement influences forelimb loading, but does not explain the functional significance of head nodding for the sound animal. As the head–neck unit makes up around 10% of a horse's body mass [17,18], and can be expected to constitute a similar proportion in other hoofed mammals, it is clear that the movements of this cantilever have a relevant mechanical influence on walking mechanics and energetics.

Over the last decade, collision mechanics have successfully been used to explain experimental observations on the basic mechanics and energetics of walking gaits [19–25]. Although studies have mainly focused on human bipedal walking, collision mechanics have also been applied to the quadrupedal walking gait of larger mammals [26–28]. For the walking gait, it has been demonstrated that during the single support phases the trajectory of the body's centre of mass (COM) closely resembles that of an inverse pendulum. This passively conserves mechanical energy and only very little active mechanical work is necessary [21]. By contrast, according to collision mechanics, the step-to-step transitions during the double support phases require that the impulse of the body's COM is actively redirected. This inevitably involves positive work from the muscles of the trailing limb accelerating the body in a forward–upward direction and negative work from the muscles of the leading limb decelerating the body in a backward–upward direction [21]. During walking, single support phases therefore are considered to be energetically inexpensive and double support phases are considered to be energetically expensive. Considering the possible mechanical effects of head movements on the trunk and legs, it is intriguing that specialized, high-tech backpacks have managed to significantly reduce the energetic costs of transport in humans by decoupling the movement of the load from that of the carrier's body [29,30]. The carried mass is supported in suspension, thus enabling it to swing vertically out of phase to the vertical oscillations of the carrier's body mass. As a result, the fluctuations of the weight force exerted from the load to the carrier's body are phase shifted, so that the carrier's legs are relieved during the expensive double support phases and loaded during the inexpensive single support phases. The collisional energy losses and therefore the overall costs of locomotion are reduced.

We hypothesized that the movements of the head–neck unit of larger mammals fulfil a similar function. To test this, we began with a video analysis of the movements of eight horses during walking. The data were then used as input parameters for a simple computer model of a horse's fore-quarters and head–neck unit. With the model, we assessed mechanical work and estimated energetic effects of the vertically directed forces acting on the shoulder resulting from the observed motions of the head–neck unit relative to the trunk. We subsequently investigated the influence of the timing of these relative vertical movements on the metabolic costs for carrying the head using our model. Finally, in order to facilitate the discussion of the findings in a wider zoological context, the phase relationship of head–neck and thorax movements of 18 additional species of cursorial mammals (ungulates, carnivores and semi-terrestrial primates) were also studied (table 1).

Table 1.

Species sample for kinematic analysis of head and shoulder motion.

species	trivial name	classification	individuals
Bos taurus indicus	zebu	Artiodactyla, Bovidae	3
Connochaetes taurinus	blue wildebeest	Artiodactyla, Bovidae	5
Eudorcas thomsoni	Thomson's gazelle	Artiodactyla, Bovidae	4
Oryx leucoryx	Arabian oryx	Artiodactyla, Bovidae	5
Tragelaphus strepsiceros	greater kudu	Artiodactyla, Bovidae	3
Camelus bactrianus	Bactrian camel	Artiodactyla, Camelidae	4
Giraffa camelopardalis	giraffe	Artiodactyla, Giraffidae	4
Canis lupus familiaris	domestic dog	Carnivora, Canidae	6
Chrysocyon brachyurus	maned wolf	Carnivora, Canidae	4
Acinonyx jubatus	cheetah	Carnivora, Felidae	3
Felis silvestris catus	domestic cat	Carnivora, Felidae	5
Panthera leo	lion	Carnivora, Felidae	4
Crocuta crocuta	spotted hyena	Carnivora, Hyaenidae	4
Equus africanus	African wild ass	Perissodactyla, Equidae	4
Equus ferus caballus	domestic horse	Perissodactyla, Equidae	8
Equus quagga	plains zebra	Perissodactyla, Equidae	6
Erythrocebus patas	patas monkey	Primates, Cercopithecidae	3
Macaca nemestrina	southern pig-tailed macaque	Primates, Cercopithecidae	4
Semnopithecus entellus	grey langur	Primates, Cercopithecidae	3

2. Results

In the walking gait of a horse, the head and withers showed two complete oscillatory cycles of sinusoidal vertical movements per stride within the sagittal plane. The range of absolute vertical motion was higher for the head than for the withers in all individuals. The mean range of vertical cranial oscillation (mean ± 1 s.d.: 9.1 ± 3.4 cm) exceeded thoracic oscillation (3.2 ± 1.0 cm). Thus, vertical displacement and peak accelerations of the eyes and the vestibular system exceeded that of the thorax by 2.9 and 2.5 times, respectively (figure 1).

Figure 1.

Mean (±1 s.d.) vertical accelerations (a) and displacements (b) of head (red) and withers (blue) of eight warmblood horses during one stride at normal walk. Mean displacements (p = 0.0001) and accelerations (p = 0.0004) differed significantly at the p = 0.001 level. For better comparison, the mean vertical anatomical positions of different sized horses were fitted. Horizontal bars indicate substrate contact of the right (grey) and left (black) forelimb; vertical lines indicate changes from phases of double forelimb support to single forelimb support. Note that the stride consists of two steps and thus vertical movements of the body axis are bi-phasic.

The head and withers oscillated vertically with a distinct phase shift of 25.0 ± 2.5% (mean ± 1 s.d.) of the stride cycle duration, leading to an out-of-phase vertical movement of these body parts. In a symmetrical gait, like the walk, the body axis undergoes two vertical oscillations (one per step) during each stride. Therefore, a phase shift between the head and the withers of 0 and 50% of stride duration equals a vertical in-phase movement, whereas a phase shift of 25 and 75% indicates a vertical out-of-phase movement (figure 1). Carried by the inverse pendulum movements of the forelegs during walking, the equine thorax reached its lowest position along with the strongest upwards directed acceleration during the single support phase of the forelegs (figure 1a,b). By contrast, being out-of-phase with the thorax, the head reached its highest position along with the maximum downward directed acceleration during the double support phase. So, as the head and neck were accelerated downward with a certain proportion of gravitational acceleration during the double support phase, the forelegs only had to provide the support necessary to resist the remaining proportion of the gravitational acceleration. Swinging the head with 4.2 m s⁻² (43% of gravitational acceleration) downwards during the double support phase lowered the weight force of the head during this fraction of time by 43% compared to the force on the head in a static position. As the COM of the combined head–neck unit is closer to the pivot of the oscillating cantilever than the head marker, its vertical range of motion (as well as the acceleration) was determined to be just 79% of that of the head marker, but this does not affect the phase shift relative to the thorax and still lowered the weight force of the complete head–neck unit during double support up to 34%.

Accordingly, our simple model of a horse's fore-quarters showed a distinct correlation between energy expenditure and the phase relation of vertical motion between the head and the point in which the trunk is connected to the forelimbs, i.e. the attachment area of the serratus ventralis muscle at the scapula (trunk–forelimb suspension point (SP) hereafter). Moving the head–neck unit with a considerable phase shift relative to the thorax reduced the energetic effort of the whole model in comparison to in-phase movements (figure 2). In the model, the optimum energetic phase relation was reached at a shift of 25.25%, which agrees with the observed phase shift in horses within the experimental error. Deviating from this optimal phase shift resulted in an increase of the energy expenditure share for bearing the load of the unit of head and neck by up to 63.15%.

Figure 2.

Relative metabolic energy (ΔE_m) costs of locomotion depending on the phase shift between the vertical oscillations of the head–neck unit and the trunk–forelimb SP calculated by the simple horse model. (Online version in colour.)

Comparing the timing of head nodding relative to thorax movements in an additional 18 species, we discovered that the timing observed in horses is not uncommon among larger quadruped mammals (figure 3). Taking into account the relative neck lengths of all 19 mammalian species studied, a relationship between these kinematic and anatomical features can be asserted. Species with a neck–trunk ratio below 0.3 did not regularly exhibit appreciable phase shifts of head and thorax motion. Those mammals examined with ratios between 0.3 and 0.4 (dogs and maned wolves) showed a speed-dependent change in head movement. During slower walking cycles, using step frequencies of less than 1.3 Hz, both species moved the head in phase with the trunk. When exhibiting step frequencies more than 1.3 Hz, the rhythm of the vertical head movements shifted to about 20–25% in phase. Without exception, quadrupeds with intermediate neck lengths (ratio between 0.4 and 0.53) moved the head with similar phase shifts to the horse. However, only one of the four species with even longer necks (ratio between 0.55 and 1.35) exhibited phase shifts comparable to that of the horse.

Figure 3.

Phase shift of the vertical oscillations of head and withers (black squares; mean ± 1 s.d.) and the corresponding length ratio of the neck and trunk (white bars; mean ± 1 s.d.) of 19 species of quadrupedal mammals. The horizontal red line indicates the energetically optimum phase shift calculated by the horse model. For two species (maned wolf and domestic dog) data were separated between slower and faster walking trials, because of a speed-dependent change of the phase shift in these species. All other sampled species did not show a speed-dependent phase shift. The vertical lines separate groups of species with mainly observed phase shifts (see text). (Online version in colour.)

3. Discussion

We examined the head movement relative to the trunk in 19 species of cursorial quadruped mammals in order to evaluate the energetic effects of the timing of conspicuous head-nodding behaviour during walking. There are few comparable studies on head kinematics of the quadruped mammalian walking gait, but our findings are consistent with existing data for the head and thorax movement of horses [2,5,6,10]. In horses, the basic kinematic pattern of the head and withers moving vertically out-of-phase appears to be typical in a majority of hoofed mammals, and is even shared by some Carnivora. It was previously hypothesized that these movements stabilized axial elements in horses, thus enabling the nervous system to maintain spatial orientation [5]. However, we found that the observed movement patterns increased the vertical displacements and accelerations of the head, and hence the optical and vestibular organs in comparison to a hypothetical motion of the head rigidly fixed to the trunk, by far. This calls into question the suggestion that the stabilization of a spatial reference frame is a primary function of this kinematic feature.

As an alternative explanation for this conspicuous behaviour, we demonstrate how the specific timing of head movement relative to the thorax is consistent with a way of minimizing energy expenditure. In horses, characteristic vertical motions of head and neck result in a reduced weight force of these body parts transmitted to the forelegs during the double support phases. The weight force is equivalently increased during the single support phases of the forelegs. A similar phase shift in weight fluctuation constitutes the core premise of the energy-saving weight bearing in high-tech backpacks [29,30]. As our model used the dimensions, inertial properties and kinematics of a horse's fore-quarters as input parameters, it directly assesses the mechanical effect of head movements only in horses. However, the highly reductionist model more generally allows qualitative predictions on the basic energetic effect of head movements for relatively large mammalian quadrupeds with similar relative head–neck masses and kinematics. For species exhibiting a comparable timing between head-nodding and thorax movements, we hypothesize a similar mechanical effect.

Comparing the kinematics of the collision-reducing backpack and the observed head movements of quadruped mammals, as exemplified by the horse, the most important difference can be found in the amplitude of displacement of the mass that is moving out of phase with the proximal joints of the supporting limbs. In the backpack, the oscillating load moves vertically with a relative amplitude to the carrier's back that is similar to the absolute amplitude of the walking human's trunk [29]. Therefore, the phase shift of the load's vertical motion results in a strong reduction in the oscillatory amplitude of the load. By contrast, the head movements of the walking horses showed an absolute vertical amplitude that far exceeded that of the withers. It has been shown that head–neck movement, unlike that of the swinging load within the backpack, is not performed exclusively passively, but rather involves muscular activity [11]. With varying phase shift between the vertical oscillations of head–neck unit and thorax, we expect both for the work to be performed by the limbs, and also for the energetic cost of sustaining the head–neck unit's relative movements to change. Generally, to achieve a net energy-saving effect, the changes in energy expenditure for sustaining deviating head–neck movement relative to the trunk must not overcompensate for the energetic disadvantages of bearing the load at a different phase shift. Thus, the observed timing likely reflects an optimization of the combination of both factors.

In this regard, the anatomical and biomechanical studies of Gellman and colleagues on the cervical spines of horses are of particular interest [10,11,31]. The authors showed that the elastic nuchal ligament stores and recovers up to 59% of the energy needed to sustain head movement during walking (i.e. at least 41% need to be actively performed) [11]. Therefore, this elastin-rich structure that dorsally spans from the occiput to the thoracic spinous processes could play a vital role in sustaining the collision-reducing motion of the head and neck at low cost, optimizing or even enabling the net energy-saving effect. Could the presence (or the absence) of a nuchal ligament be a potential explanation for the occurrence of out-of-phase or in-phase movements of the head and trunk in different species? The fact that this ligament is inherent to ungulate and canid species in general [32–34], but is absent in felid species and non-human primates, is in line with our results and seems to support this notion. Hyaenas, however (like other feloid carnivores), do not have nuchal ligaments [35,36] but still exhibit pronounced head movements—albeit with a lower mean phase shift relative to the trunk than ungulates and fast walking canids (figure 3). We can postulate, therefore, that while this ligament probably helps to make the collision-reducing head movements more efficient, it does not seem to be crucial to this kinematic feature.

In vivo electromyography of the cervical musculature of walking horses showed that the splenius muscle exhibits bilateral activity during the single support stance of each forelimb [37–39], that is when head and neck reach the lowest position. It would seem that the activity pattern of this main dorsal muscle of the cervical spine is enough to induce the observed vertical motion of the head–neck unit. To decelerate and reaccelerate the falling cantilever, it was proposed that isometric contractions could be used, thus avoiding any need to shorten muscle fibres [31]. This means that the vertical oscillations of the head and neck can be maintained with highly economic muscle activity—a strategy that is complemented by, but not entirely dependent on, the presence of a nuchal ligament.

Another anatomical feature that must be taken into account is the length of the head–neck cantilever that acts as an oscillating pendulum. To sustain an oscillatory motion with low effort, the oscillating frequency has to be kept close to the natural frequency of the swinging system. As the natural frequency of an ideal pendulum (with frictionless pivot and massless rod) is inversely proportional to the square root of the length of the pendulum, a longer neck should determine a lower natural frequency of the vertical movements of head and neck—and vice versa for a shorter neck. Figure 3 shows that in our sample the length of the cervical spine is related to the timing of head movement. The majority of quadrupedal mammals with relatively long necks, like most ungulates and the spotted hyaena, use out-of-phase movements of the head-and-neck cantilever, which appear to reduce collisional energy loss. The natural frequency of an oscillation is also influenced by material properties, and various studies have shown that the adjustable stiffness of muscle–tendon complexes may modulate the natural frequency of body parts that exhibit spring-like mechanics [40–43]. Therefore, the optimal stiffness to maximize elastic energy storage can be tuned to match a range of step frequencies. While we here focus on neck length, other morphological properties like neck stiffness and neck–head unit inertial properties might further influence the oscillating pendulum and should be addressed in future studies.

In our study, the quadrupeds with relatively short necks, like felids and monkeys, show no phase-shifted movements of the head–neck unit during walking, but keep the cervico-thoracic joint straight and move the head in phase with the thorax. One possible reason for this result is that the natural frequency of the oscillations of a short head–neck unit is too high to be tuned to match the step frequency, and the motion would be too costly to sustain. Given that the modulation of the natural frequency works by muscle contractions stiffening the spring, the adjustable range of modulation mainly extends towards increasing this frequency, not reducing it. One observation that supports this notion is the step frequency-related change in head movements within the two species of canids examined (dogs and maned wolves). In our sample, only these two species displayed a speed-dependent phase shift. As both these species have intermediate neck lengths, the step frequency during a slower walking gait might be too low to meet the natural frequency of the head–neck cantilever, just as in the short-necked species. When walking speed increases, so does the step frequency: this increased frequency may be closer to the natural oscillatory frequency of the cantilever and, thus, relative vertical head movements could be accomplished with much less effort.

For the relatively long-necked ungulates and the hyaena, the length of the neck allows them to adjust the natural frequency to cover the range of usual walking step frequencies at relatively low cost, and thus out-of-phase head motions are a habitual kinematic feature. So, why then do three species of relatively long-necked ungulates (giraffe, Thomson's gazelle and Bactrian camel) show head movements with very little phase shift relative to the vertical movements of the thorax? Firstly, as short necks can determine natural frequencies that are too high, long necks might have low oscillatory frequencies—possibly too low to fall into the range of adequate step frequencies of walking. This seems a particularly plausible explanation, as all three species of ungulates that do not exhibit the timing characteristic for horses fall within the range of the longest necks measured.

Secondly, it must be considered that the cantilever of the head and neck moves like a pendulum around a pivot located at the intersection between the cervical and thoracic spine. Even when the neck is in a perfectly horizontal orientation, the movement of the cantilever always exhibits a slight longitudinal component. The closer the orientation of the neck gets to the vertical axis, the larger the longitudinal component of the motion becomes, and—accordingly—the smaller the vertical component. In fact, the three hoofed species that exhibit the smallest vertical amplitudes of head movements in this study carry the cervical spine relatively straight upwards. A pendulum motion of the neck around the cervico-thoracic joint would therefore lead to a relatively small vertical displacement of the head, but to a large longitudinal motion. Within the three species of ungulates that showed a low phase shift of the vertical oscillations of head and withers, large longitudinal out-of-phase movements of these body parts were qualitatively observed. By exerting forward- and backward-directed accelerations onto the trunk, these movements of the head–neck cantilever—when correctly timed—could attenuate the fluctuations of kinetic energy of the trunk during walk and therefore, again, reduce collisional energy loss at forefoot step-to-step transition. However, the possible collision-reducing effect of longitudinal out-of-phase movements of body compartments has not yet been evaluated in vivo or by computer modelling. We do not, therefore, discuss this potential kinematic adjustment to economize collisional energy loss here, but it could be addressed in future studies along with other adjustments of the phase relationships between different body parts. Nevertheless, we can state that an orientation of the cervical spine that favours longitudinal movements of the head–neck cantilever over vertical movements is a plausible explanation for the observed difference in kinematics of the three ungulate species with diverging patterns of head–neck movement.

The study presented here uses a simple model to demonstrate that the timing of head–neck unit movements relative to the trunk is a mechanism with the potential to considerably reduce the metabolic energy costs of locomotion by reducing collisional energy losses. The predicted optimum phase shift of the head–neck unit relative to the trunk is closely matched by many mammalian species, especially ungulates. In walking horses, the observed head movements relative to the trunk cannot be sustained completely passively and are thus connected to costs [11]. We expect that as a result of the observed timing of vertical head–neck movement, the involved costs for actively sustaining these head movements and the costs for bearing the unit's load are optimized simultaneously to minimize overall costs. The presence or the absence of a highly elastic nuchal ligament, the relative length of the neck determining the natural frequency of the oscillating cantilever, and the orientation of the neck appear to influence the effectiveness of this kinematic adjustment to minimize collisional energy losses. This study indicates—for the first time—that this energy-saving mechanism, already implemented in engineered mechanical carrying aids, also evolved in animals. The fact that the energy-saving effect of this motion could be confirmed for two mechanical structures as different as the mass in a backpack and the head–neck cantilever of a horse and other larger mammals, suggests this mechanism is an expedient way of bearing a load while walking.

In summary, by demonstrating that this mechanism is present in animals, we provide new evidence for the kinematic optimization of natural locomotion. It is possible that vertical head movements are not the only way by which animals use the timing of moving body parts to minimize energy expenditure related to collision effects. The mechanism also has the potential to be used in the field of bio-informed robotics to design more effective walking machines as well as in the field of paleobiology to infer locomotor characteristics in extinct mammalian species, e.g. from the presence of osteological correlates of a nuchal ligament.

4. Methods

Additional information on the methods is provided in the electronic supplementary material and all custom code for data processing and modelling is made available. For the kinematic analysis (figure 4), white markers were placed on the occiput, between the forehead and muzzle, the scapula and withers of eight adult warmblood horses (six geldings and two mares; age: 5–16 years, mean 10.4 ± 3.7 years; height at withers: 150–169 cm, mean: 161.4 ± 6.0 cm). During trials, the horses were led by a loosely held cord through a calibrated space to cover a range of walking speeds. None of the other animals were marked. They were filmed either following their owners (dogs) or walking freely (cats), or in zoological gardens (Zoologischer Garten Berlin and Tiergarten Friedrichsfelde, both Germany; all other species) while walking at their preferred speed on flat ground.

Figure 4.

Modelling the energetic effect of conspicuous nodding of the head–neck unit relative to the thorax in a horse. (a) Position of the withers, head, rostral and scapula markers for the kinematic analysis of amplitude and phase relationship of the modelled trunk–forelimb SP (see text) and COM of the head–neck unit. (b) Illustration of the reductionist model during the double support phase. The model is driven by kinematic data of a reference individual and, in the case of the vertically directed movement (y_hn(t)) of the head–neck unit, also by imposing phase-shifted kinematics. The latter is used to calculate the vertically directed force at the SP. We here considered the opposing force of equal magnitude to maintain equilibrium at the SP (F_y). This force, in combination with the vertically as well as horizontally directed movement of the SP (Δx_sp(t), y_sp(t)), is subsequently used to estimate the energetic effects of the timing of the head–neck unit's vertically directed movement relative to the trunk (see text). We modelled progression as if on a treadmill, i.e. averaged over an entire stride the position of the SP remains stationary. Horizontally directed movement of the head–neck unit (x_hn(t)) and corresponding horizontally directed force (F_x) and moment at the SP (M_z) are not considered in the model.

The animals were filmed using a Sharp ViewCam Z VL-Z3 at a frequency of 50 fps perpendicular to the animals' line of movement. For each individual, 10 full and steady-state movement cycles, covering a range of walking speeds, were selected for data analysis. With the APAS software (v. 9.3, 2003, Ariel Performance Analysis System, Ariel Dynamics, San Diego, CA, USA) the motion of the head and withers of the horses within the sagittal plane was measured by identifying the x- and y-positions of the markers. From these data, the linear vertical displacement and acceleration of the markers were calculated. For the animals without markers, the positions of the occiput and the dorsal spinal processes of the thoracic vertebrae were digitized by hand frame by frame. Spatial calibration was undertaken only for the horses. For all other animals, we calculated the phase relation of the vertical displacements of the head and withers. All motion data were filtered using the integrated second-order, low-pass Butterworth filter of the APAS Software with a cut-off frequency of 5 Hz. To statistically test for differences between observed timing in head and thorax vertical movements we first confirmed the Gaussian distribution using the Shapiro–Wilks test. This allowed us to use Student's t-test for parametric data of paired samples. The significance level was set at p = 0.001.

To determine the ratio of cervical spine length to trunk length of the 18 additionally analysed species, still images were taken from the films of each animal at touchdown of a forefoot. The length between the occiput and the cervico-thoracic joint, as well as between the cervico-thoracic joint and the head of the femur, was measured directly (to determine the ratio no calibration of the lengths was necessary) using the software ImageJ [44]. The positions of the joints were estimated using anatomical drawings of corresponding or closely related species.

Based on the anatomical characteristics of a warmblood horse, in particular with regard to the masses and COM positions of the head and neck, the position of the trunk–forelimb SP and the kinematic restrictions to be imposed upon the forelimbs and the head–neck cantilever, a simple model was built. This was done with the aim of estimating the energetic consequences of the vertically directed forces in the shoulder that result from vertical movements of the head–neck unit relative to the thorax (figure 4). To achieve this, we first recorded the actual kinematics of the head–neck unit and the thorax of a reference individual (warmblood gelding Leroy). In addition, we systematically varied the phase shift between the vertical motions of the shoulder and the head–neck unit in the model. Subsequently, based on the inertial properties of the combined unit of the head and neck and the observed and modelled kinematics, we determined the vertically directed force resulting in the shoulder through an inverse dynamics approach. Hereafter, distribution of the opposing force upon the supporting limbs was identified according to instantaneous orientations of the respective extremities. Next, we calculated the contribution of these forces to the necessary mechanical work to be done by the forelimbs. Finally, in order to estimate the influence of the timing of head movements relative to the trunk on the energy expenditure of carrying the head–neck unit, we used the published empiric relationship between mechanical work and metabolic energy cost [45] to quantify the effect (refer to the electronic supplementary material for a detailed characterization of the model).

In this first approach to study the energetic effects of relative movements of body parts some limitations were accepted. The model does not consider horizontally directed oscillations of the head–neck unit, which were found to be minimal (less than 0.3 cm), nor the corresponding forces in the trunk–forelimb SP. Also, the moment that is necessary to sustain the movement of the head–neck unit relative to the thorax is not considered in the model. It has previously been shown that much, but not all, of this movement can be sustained passively [11].

The head and neck mass of a horse were scaled to the reference individual in accordance with the linear regression formulae available in the literature [18], and then used to determine the corresponding parameters in the model. Motion capture data of the same individual during unrestrained walking were used to drive the model. Combining these methods, inertial and kinematic parameters of the model were identified (table 2).

Table 2.

Parameters of the simple model.

length of stridea	lS	1.813 m
duration of stride	ΔtS	1.328 s
mean velocity	vm	1.365 m s−1
mass of body	mb	620 kg
mass of head–neck unit	mhn	57.7 kg

^aA stride is defined to last from touchdown of a limb to the subsequent touchdown of the same limb.

An inverse dynamics approach was used to derive the mechanical work (W) necessary to bear the load of the head–neck unit with respect to the observed head-nodding behaviour. We discriminated between positive and negative mechanical work (W⁺ and W⁻, respectively). As both forms of mechanical work independently result in the consumption of metabolic energy, they would partially cancel each other out over the course of a full stride cycle if they were not treated separately [22]. The two contributing types of work were thus defined as:

Here, F and v are the vectors of force and velocity, respectively. F can be determined from inverse dynamics and was approximated based on the kinematic analysis of the reference individual as described in the electronic supplementary material. The measurement of oxygen consumption of uphill and downhill walking individuals had previously been used to establish a relationship between metabolic energy costs and mechanical work [45]. Based on this, we estimated the energetic costs (E_m) of bearing the load of the head–neck unit using the following equation:

The head–neck unit is modelled as a point mass (m_hn) fixed to a massless bar, which adequately represents the unit's total weight and its location of the COM. In mammals, the vertically directed forces induced by the head–neck movement relative to the limbs are transmitted to the limbs via the serratus ventralis muscle that connects the trunk with the medial fossae and borders of the scapulae of both forelimbs [46]. We modelled this point of trunk–forelimb suspension (SP) as a frictionless hinge joint and used motion capture data of the scapula marker. Finally, both forelimbs were modelled as actuating struts of variable length transmitting the forces induced at the SP to the points of ground contact. Flexion or extension of these elements at any given moment was determined by the instantaneous positions of the SP and ground contact of the respective limb.

Data accessibility

The datasets supporting this article as well as the source code for custom MATLAB and Maple analyses have been uploaded as part of the supplementary material and to the Dryad archive under http://dx.doi.org/10.5061/dryad.rf5bj [47].

Authors' contributions

D.M.L. conceived the study, carried out the kinematic analysis and drafted the manuscript; F.M. carried out the modelling and participated in drafting the manuscript; K.K. initially helped with conceiving the study and details of the model. J.A.N. also contributed to the design of the study and drafted the manuscript. All authors interpreted the results, and participated in the in-depth revisions of previous versions of the paper, and gave final approval for publication.

Competing interests

We have no competing interests.

Funding

The study was carried out using budgetary funds of the respective institutions and received funding from the German Research Council (DFG-EXC 1027 Image Knowledge Gestaltung. An interdisciplinary laboratory).