Semicircular canals and agility: the influence of size and shape measures

Abstract

The semicircular canals of the inner ear sense angular accelerations and decelerations of the head and enable co-ordination of posture and body movement, as well as visual stability. Differences of agility and spatial sensitivity among species have been linked to interspecific differences in the relative size of the canals, particularly the radius of curvature (R) and the ratio of the canal plane area to streamline length (P/L). Here we investigate the scaling relationships of these two size variables and also out-of-plane torsion in the three semicircular canals (anterior, posterior and lateral), in order to assess which is more closely correlated with body size and locomotor agility. Measurements were computed from 3D landmarks taken from magnetic resonance images of a diverse sample of placental mammals encompassing 16 eutherian orders. Body masses were collected from the literature and an agility score was assigned to each species. The R and P/L of all three semicircular canals were found to have highly significant positive correlations with each other and no statistical difference was found between the slope of 2P/L against R and 1. This indicated that, contrary to initial hypotheses, there is little difference between 2P/L and R as measures of semicircular canal size. A measure of the in-plane circularity of the canal was obtained by dividing 2P/L by R and out-of-plane torsion was measured as angular deviation from a plane of best fit. It was predicted that deviations from in-plane and out-of-plane circularity would increase at small body size due to the constraints of fitting a proportionately larger canal into a smaller petrous bone. However, neither measurement was found to have a significant correlation with body mass, indicating that deviations from circularity (both in-plane and out-of-plane) are not sufficient to alter P/L to an extent that would impact the sensitivity of the canals. 2P/L and R were both shown to be significantly correlated with locomotor agility. The posterior canal was the least correlated with agility, suggesting that it may be generally less closely aligned to the direction of movement than the anterior canal. Of the three canals, the lateral canal was the most highly correlated with agility. In particular, it could be used to distinguish between species that move in a largely 2D environment and those that locomote in 3D space (aerial, arboreal and aquatic species). This complements previous work suggesting that the lateral canal primarily commands navigation, whereas the vertical canals control reflex adjustments. It was also found that 2P/L is substantially better correlated with agility than is R in the lateral canal. This result is intriguing given the above finding that there is no statistical difference between 2P/L and R, and requires further investigation.

Introduction

The semicircular canals of the vertebrate inner ear, located within the petrous part of the temporal bone of the skull, are responsible for sensing angular accelerations and decelerations of the head. The output from the canals supplies information about how an animal is moving within its environment and enables it to co-ordinate posture, body movements and gaze direction during locomotion. Mammals, like most gnathostomes, possess three semicircular canals on each side of the head: two vertical (anterior and posterior canals) and one horizontal (lateral canal). These three semicircular canals show substantial variation in size across the mammals and these variations have been linked to functional differences of locomotion (e.g. Spoor & Zonneveld, 1998; Spoor et al. 2002, 2007). Here we investigate canal size and possibly more bio-physiologically realistic shape measurements in relation to patterns of locomotion as well as covariation with body size. The aim of this study was to see whether the more realistic measurements yield any tangible improvement in reconstructing patterns of locomotion.

Canal size

Semicircular canal size has historically been measured as the radius of curvature of the osseous canal (R). This measurement is distinct from the radius of the membranous duct seen in a cross-sectional slice through the canal (r). R has been measured from fixed, dissected specimens by superimposing a circle of best fit (estimated by eye) over the course of the canal either directly (Lindenlaub et al. 1995; McVean, 1999) or from photographs (Curthoys & Oman, 1986, 1987). More recently, semicircular canals have been imaged using computed tomography, obviating the need for dissection (Spoor & Zonneveld, 1998; Spoor et al. 2002, 2007). Here, the radius of curvature has typically been calculated from the width (w) and height (h) of each canal (R = 0.5[h+w]/2). Fitted circles and calculated R measurements reveal notable size differences between the three semicircular canals. Generally, amongst mammals, the anterior canal is larger than the posterior and lateral canals. This has been demonstrated in rats (Cummins, 1925), mice (Calabrese & Hullar, 2006), chinchillas (Hullar & Williams, 2006), little brown bats (Ramprashad et al. 1980), rhesus and squirrel monkeys (Blanks et al. 1985), as well as guinea pigs, cats and humans (Curthoys et al. 1977). This relationship is also seen in most of the wide range of mammals studied by Ramprashad et al. (1984) and the large number of primates examined by Spoor & Zonneveld (1998) although the anterior and posterior canals are often very close in size and, in some cases, the radius of the posterior canal exceeds that of the anterior canal (e.g. Cercopithecus aethiops and Callithrix jacchus).

As well as variation between the semicircular canals within an individual animal, there is also a great deal of variation in the relative size and shape of each semicircular canal between mammal species. Numerous studies (Jones & Spells, 1963; Spoor & Zonneveld, 1998; Spoor et al. 2007) have found that the canal dimensions increase with body size but with strong negative allometry, i.e. the relative size of the canals tends to decrease as body mass increases. Spoor & Zonneveld (1998) calculated the regression slope of a double logarithmic plot between body mass and mean radius of curvature of all three canals to be around 0.14, well below the value of 0.33 that would indicate an isometric relationship. This result was confirmed by Spoor et al. (2007) with the data adjusted using phylogenetic least squares regression.

The relative size of the semicircular canals has long been assumed to be related to the method or speed of locomotion of an animal. For example, Gray (1907), in his comprehensive study of the vertebrate labyrinth, unequivocally attributed the diminished size of the canals in the sloth to its sluggish movements. This hypothesis was tested by Spoor et al. (2007) who, after controlling for body size, found a significant positive correlation between semicircular canal radius and agility across a wide sample of mammals. It is likely that this correlation is a result of the increased sensitivity to angular acceleration conferred by a larger radius of curvature. A relationship between canal radius and afferent sensitivity has been demonstrated between different canals of the same labyrinth in cats (Blanks et al. 1975; Curthoys et al. 1977) and pigeons (Dickman, 1996), and also between corresponding canals of different individuals in neonatal rats (Curthoys, 1982) and mice (Yang & Hullar, 2007), as well as across different species of mammals (Yang & Hullar, 2007).

Canal shape

Despite its frequent use in studies of the semicircular canals (Jones & Spells, 1963; Curthoys et al. 1977; Spoor et al. 2007), R is limited in that it provides information on canal size but not on canal shape. This is important as deviation from circularity will affect the area enclosed by the canal, a parameter that has been shown to be closely related to canal sensitivity (McVean, 1999). A more useful measure may be 2P/L, referred to as the ‘average radius’ of the canal by Oman et al. (1987), in which L is the streamline length of fluid flow, encompassing the entire membranous duct, ampulla and utricle, and P is the plane area enclosed by L (Curthoys & Oman, 1986, 1987; Lindenlaub et al. 1995). This measure takes shape into account as, for a given streamline length, the greater the deviation from circularity, the smaller the value of P and hence P/L. A perfectly circular canal should have a value of 2P/L that equals R (because P = πR² and L = 2πR). Therefore, the dimensionless variable (2P/L)/R gives a measure of circularity; a perfect circle will have a value of 1 and an ellipse will have a value of <1. Oman et al. (1987) found that mechanical sensitivity was directly related to the ratio P/L. Assuming constant area, the deviations from a circle would yield increases in L, whereas if length is held constant P decreases. In both cases, the mechanical sensitivity of the canals is presumed to decrease as P/L decreases, although for small deviations the effect on P/L, and therefore sensitivity, is minimal (McVean, 1999). Variations of P/L from a perfect circle could indicate a spatial constraint on the plane area that can be accommodated in the petrous bones. This could be a particular problem for small mammals in which the semicircular canals are relatively larger compared with the size of the head (Jones & Spells, 1963; Spoor & Zonneveld, 1998).

The ratio P/L and its derivatives can vary with out-of-plane as well as in-plane deviations from circularity. Out-of-plane deviations can be defined in terms of canal torsion, which refers to twisting of the canal arc out of its principal plane, and not to the sharpness of the curve (as it is in differential geometry). Torsion can allow the canals to pick up accelerations outside its plane of best fit at the expense of in-plane sensitivity (L increases relative to P). This can be particularly important as the maximal response directions of the nerves frequently differ from the anatomical planes of the canals (Rabbitt, 1999). The misalignment between the anatomical plane and maximal response direction has been calculated to be approximately 6° in monkeys (Reisine et al. 1988) and 7° in cats (Estes et al. 1975). Blanks et al. (1985) calculated the planarity of rhesus and squirrel monkey semicircular canals as the mean angular disparity between the extreme points above and below the plane of best fit to the canal, measured relative to the centre of the canal. They found that the posterior and lateral canals were approximately planar but that the anterior canal showed substantial bending. Similar results but with less torsion in the anterior canal were reported for chinchillas (Hullar & Williams, 2006). Spoor & Zonneveld (1998) attempted to quantify canal torsion by measuring the difference in orientation between the superiormost and inferiormost parts of the vertical canals, and the lateralmost part and greatest arc width of the lateral canal. However, they found few discernable trends in the pattern of torsion across their sample of primates.

Hypotheses

The present study seeks to examine whether the use of the measurement radius of curvature (R) in morphological studies of the semicircular canals and agility (e.g. Spoor et al. 2007) is justified or whether a more complex shape variable is more appropriate. The following hypotheses are proposed.

1. It is predicted that the semicircular canals are not perfectly circular and therefore the shape-inclusive variable P/L will differ significantly from the trends for the size variable R. Thus plots of 2P/L against R will have slopes greater or less than 1. This simply tests for equivalence between the two measurements.

2. (i) It is proposed that deviations from in-plane circularity are primarily due to size scaling constraints encountered among smaller species rather than function. This predicts that values of (2P/L)/R are negatively correlated with body mass. (ii) It is proposed that torsion is also a solution to size scaling and is sufficient to significantly alter the mechanical sensitivity of the canal in terms of P/L by increasing L.

3. Finally, it is hypothesized that P/L will be a better predictor of the degree of agility than R, as it is a more complex variable incorporating shape as well as size.

Materials and methods

Sample and imaging

This study was based on data gathered from magnetic resonance imaging, a technique that allows the fluid inside the semicircular canals to be very clearly visualized (Fig. 1). Image data were collected and collated for 120 specimens, representing 58 extant mammalian species from 16 eutherian orders (as defined by Wilson & Reeder, 2005). Specimens were kindly provided by numerous institutions and were imaged on seven different systems (see Table S1 and Acknowledgements for details). Most specimens were preserved in formalin but a number of museum specimens had been stored in alcohol and had to be soaked in distilled water prior to imaging. Great care was taken to minimize the effects of preservation by selecting the best specimens. However, it is unlikely that any deformation would have influenced the morphology of the inner ear deep in the petrous bone. The majority of specimens were imaged with the 4.7-Tesla imaging and spectroscopy unit (Sisco-Varian, Palo Alto, CA, USA) at Queen Mary, University of London with a T2-weighted spin-echo multi-slice sequence (Echo time [TE], 20–50 ms; repetition time [TR], 8000–16 000 ms). Data were zero-filled to between 256 × 256 and 512 × 512 data points, Fourier transformed and exported as raw binary files representing fields of view from 30.8 to 115.2 mm. The slice thickness ranged from 0.24 to 0.90 mm. Similar apparatus and comparable sequences were used to acquire data on the other systems (see Table S1). Data for 13 primate specimens, including all of the modern humans, were provided by James Rilling (Emory University), Dirk Bartz (University of Leipzig) and the NIH Visible Human Project (http://www.nlm.nih.gov). Slices for all 120 specimens were interpolated to form isometric voxels (vertices ranging from 0.05 to 0.9 mm) with the bicubic spline function in ImageJ (W. Rasband, National Institute of Mental Health, Bethesda, MD, USA).

Fig. 1.

Magnetic resonance image of the anterior semicircular canal of a Sunda flying lemur (Galeopterus variegatus). (A) Landmarks defining canal height (A–B) and width (C–D). (B) Landmarks defining the streamline length and canal plane area; streamline length calculated from total of perimeter lines and area calculated from total area of triangles formed by radials and perimeter lines.

The imaging systems were regularly serviced and checked. Testing and checking for geometric distortion is standard for all of the systems used in this study. It is not possible to measure repeatability among the instruments due to the differences of bore size. Nevertheless, it is unlikely that differences of instrumental error would be of any great significance. Moreover, the repeatability of measurements given in similar studies suggests that variability due to measurement precision and accuracy is of little consequence compared with the intraspecific and interspecific biological variability found in such large samples.

Measurements

Four landmarks were recorded on each canal: the apex of the canal arc and the point diametrically opposite it on the margin of the vestibule (A and B in Fig. 1A), and the two landmarks demarcating the canal’s greatest diameter perpendicular to the apical–basal axis (C and D in Fig. 1A). From these, the height and width of each canal were determined, thus allowing the calculation of the radius of curvature (R = 0.5[h+w]/2) (Spoor & Zonneveld, 1998). To determine the streamline length of the semicircular canals, 3D co-ordinate data were taken along the arc of all three canals from the right-hand side of each set of magnetic resonance images using Amira 5.2 (Mercury Systems Inc., Chelmsford, MA, USA). The number of points recorded varied between 8 and 42 depending on the resolution of the image and the relative size of the canal but in all cases they were spaced sufficiently closely to accurately capture the entire circumference of the canal including the membranous duct, ampulla and utricle. The streamline length (L) of each canal was calculated by summing the distances between consecutive data points, including the distance between the first and last landmarks (Fig. 1B). The area (P) of each canal was determined by calculating the lengths of radials running between each point and a centroid point and then summing the areas of triangles constructed between consecutive radials using Heron’s formula:

where a, b and c are the sides of a triangle.

The out-of-plane torsion of each canal was measured as the mean angular deviation of the landmarks from the plane of best fit (calculated using principal components) measured relative to the centre of the canal (Blanks et al. 1985; Hullar & Williams, 2006).

Body mass was used as the measure of size in this study in order to make the results more directly comparable with those of Spoor et al. (2007). Many of the analyses were repeated using brain mass as the size variable as it was thought that this measure might be more relevant to semicircular canal morphology. However, the results were exactly the same as those calculated using body mass, so only these results have been reported. The body masses of primates were taken from Smith & Jungers (1997), Galidia elegans from MacDonald, (2001) and all others from Silva & Downing (1995). All are listed in Table 1. Where the sex was known, mean body mass values for that sex were calculated. Where the sex was unknown, the mean of values marked ‘both’ was calculated. The agility of each species, also given in Table 1, was scored from 1 (low agility) to 6 (highly agile) with reference to Spoor et al. (2007). As only one species fell into category 1 (Linnaeus’s two-toed sloth, Choloepus didactylus), it was decided to combine categories 1 and 2.

Table 1.

Semicircular canal dimensions and torsion, plus body mass and agility of the 58 mammal species under study.

	ASC			PSC			LSC
Species	R	P/L	Torsion	R	P/L	Torsion	R	P/L	Torsion	BM	Agility
Capra hircus	2.93	1.49	5.04	3.14	1.63	6.34	2.39	1.31	4.74	46 750	4
Hemitragus jemlahicus	3.32	1.64	6.38	2.98	1.56	4.83	2.89	1.40	10.30	14 700	4
Ovis aries	2.61	1.33	5.74	2.56	1.38	4.07	2.35	1.27	6.16	33 750	4
Naemorhaedus caudatus	1.80	0.94	4.91	1.83	0.92	8.02	1.64	0.98	5.50	27 000	4
Hydropotes inermis	2.49	1.30	3.97	1.82	1.02	6.88	1.68	0.88	5.11	1285	4
Giraffa camelopardalis	4.23	2.16	8.15	3.44	1.79	6.77	3.41	2.05	5.32	1 190 000	3
Tragulus javanicus	1.77	0.87	3.46	1.87	0.88	5.98	1.37	0.54	2.64	2000	4
Sus scrofa	2.58	1.41	4.34	2.08	1.08	6.37	1.76	1.01	8.58	101 900	4
Vicugna vicugna	3.15	1.57	3.84	2.70	1.44	7.98	2.42	1.33	2.15	45 000	4
Camelus bactrianus	4.32	2.33	7.79	4.16	2.18	4.44	3.55	1.97	7.95	415 000	4
Equus caballus	4.17	2.07	3.90	3.80	1.97	3.12	3.23	1.68	6.33	250 000	4
Equus grevyi	3.19	1.65	2.25	3.10	1.60	2.55	2.55	1.34	2.91	341 000	4
Tapirus terrestris	3.79	1.97	4.80	3.29	1.74	1.66	2.93	1.59	6.02	207 500	4
Tapirus indicus	3.59	1.84	7.79	3.41	1.76	4.37	2.93	1.53	4.19	296 250	4
Galidia elegans	1.84	0.91	4.34	1.66	0.84	3.32	1.32	0.71	2.14	900	4
Felis catus	1.82	0.93	7.13	1.83	0.95	8.43	1.63	0.85	7.41	3713	4
Neovison vison	1.42	0.79	5.17	1.47	0.78	3.75	1.48	0.74	9.52	1045	4
Lutra lutra	1.50	0.81	6.10	1.67	0.80	10.13	1.89	1.04	7.77	9600	5
Halichoerus grypus	4.53	2.45	5.48	4.39	2.12	3.61	3.44	1.87	7.32	233 000	5
Vulpes vulpes	2.26	1.13	3.29	2.33	1.06	7.99	2.56	1.17	4.47	4380	4
Manis tricuspis	1.64	0.80	1.61	1.45	0.75	3.93	1.25	0.67	3.04	2000	3
Pteropus rodricensis	1.40	0.69	4.93	1.17	0.57	4.72	1.03	0.57	6.53	350	6
Epomophorus gambianus	1.29	0.64	5.93	1.28	0.63	8.37	1.08	0.62	5.59	83	6
Molossus molossus	0.95	0.46	2.77	0.72	0.39	4.11	0.49	0.37	4.14	14	6
Scotophilus robustus	1.06	0.51	4.16	0.79	0.41	7.39	0.76	0.46	7.59	83	6
Erinaceus europaeus	1.40	0.73	6.08	1.21	0.53	5.67	1.19	0.67	9.81	780	4
Solenodon paradoxurus	1.75	0.88	2.85	1.61	0.83	4.54	1.39	0.72	4.11	900	4
Pan troglodytes	2.55	1.36	8.17	2.45	1.19	6.52	2.39	1.21	7.32	43 820	4
Pan paniscus	2.59	1.37	10.50	2.17	1.10	10.58	2.41	1.34	5.13	39 100	4
Homo sapiens	3.29	1.65	4.94	2.89	1.50	7.11	2.40	1.26	8.50	58 313	4
Gorilla gorilla	2.85	1.37	8.50	2.54	1.27	3.74	3.05	1.60	7.41	109 790	2
Pongo pygmaeus	3.11	1.48	11.53	2.57	1.50	3.80	2.43	1.18	10.81	35 700	2
Symphalangus syndactylus	2.55	1.24	9.44	2.06	1.17	10.44	2.21	1.18	7.81	11 300	4
Hylobates lar	3.34	1.54	2.28	2.36	1.17	3.94	2.28	1.27	6.65	5340	6
Tarsius bancanus	1.49	0.76	4.34	1.36	0.69	2.16	1.39	0.76	5.21	123	6
Perodicticus potto	2.10	1.11	4.07	1.70	0.80	4.49	1.29	0.69	7.36	1230	2
Loris tardigradus	1.57	0.78	4.75	1.31	0.67	6.41	1.08	0.57	3.95	193	2
Nycticebus coucang	1.93	0.95	6.74	1.44	0.83	11.96	1.18	0.61	17.50	679	2
Lemur catta	2.45	1.18	2.97	1.94	1.05	5.23	1.94	1.01	4.50	2210	4
Galeopterus variegatus	1.80	0.92	5.19	1.69	0.81	3.95	1.37	0.73	7.17	1000	6
Tupaia glis	1.78	0.93	10.28	1.56	0.79	3.21	1.40	0.76	7.93	165	4
Sciurus carolinensis	2.09	1.11	4.70	1.94	0.98	5.11	1.69	0.95	5.01	519	6
Glaucomys volans	1.15	0.67	4.40	1.04	0.54	11.22	1.15	0.62	9.60	62	6
Mus musculus	1.00	0.50	3.70	0.74	0.39	3.62	0.66	0.38	4.76	14	4
Rattus norvegicus	1.27	0.67	3.76	1.03	0.55	4.77	0.91	0.55	3.06	279	4
Cavia porcellus	1.80	0.93	8.49	1.51	0.79	12.50	1.46	0.81	9.77	728	4
Chinchilla lanigera	2.02	1.04	3.80	1.35	0.74	3.00	1.55	0.80	14.31	485	6
Heterocephalus glaber	0.85	0.41	4.10	0.73	0.37	7.89	0.73	0.37	6.71	60	2
Oryctolagus cuniculus	2.65	1.44	5.92	1.92	1.04	7.35	2.23	1.14	8.65	1437	5
Ochotona macrotis	1.73	0.89	3.36	1.46	0.74	5.86	1.25	0.70	6.88	143	5
Choloepus didactylus	2.06	1.09	6.11	1.88	1.02	4.79	1.45	0.72	7.53	4150	2
Cyclopes didactylus	1.59	0.78	3.22	1.23	0.63	9.83	1.06	0.53	7.64	400	3
Chlamyphorus truncatus	0.87	0.41	7.81	0.88	0.44	3.35	0.82	0.41	5.05	130	5
Limnogale mergulus	1.34	0.66	5.66	1.06	0.54	2.25	0.87	0.49	7.23	80	4
Tenrec ecaudatus	1.28	0.65	3.38	1.03	0.55	6.94	1.07	0.56	6.39	832	5
Potamogale velox	2.00	1.07	2.67	1.41	0.66	5.27	1.43	0.73	6.45	660	5
Rhynchocyon cirnei	2.13	1.06	6.73	1.89	0.99	6.78	1.79	0.88	5.34	490	4
Procavia capensis	1.87	1.02	4.82	2.35	1.16	7.21	1.64	0.92	6.78	3140	4

ASC, anterior semicircular canal; BM, body mass (g); L, streamline length (mm); LSC, lateral semicircular canal; P, canal plane area (mm²); PSC, posterior semicircular canal; R, radius of curvature (mm).

Analysis

For simple bivariate comparisons, standard reduced major axis (RMA) regression slopes and product-moment correlations were computed in PAST v1.89 (Hammer et al. 2001). In order to account for phylogenetic information in the measurements, significant statistics were recalculated using phylogenetically independent contrasts (PIC; Felsenstein, 1985; Harvey & Pagel, 1991; Garland et al. 1992). The phylogeny used in this analysis is shown in Fig. 2 and was constructed from the mammalian phylogeny of Bininda-Emonds et al. (2007). Divergence dates of the groups under study were also taken from this source. Independent contrast RMA regressions of canal lengths and canal torsion against body mass and each other were performed using the PDAP module (Midford et al. 2003) of the Mesquite package (Maddison & Maddison, 2009).

Fig. 2.

Phylogeny of mammal species used in this analysis, constructed from Bininda-Emonds et al. (2007). Branch length scale is in millions of years.

Phylogenetically corrected anova (Garland et al. 1993) was employed to investigate whether canal size variables differ significantly between agility groups. PDSIMUL was used to simulate the evolution of size-corrected P/L and R for all three canals on the phylogeny given in Fig. 2. A Brownian motion model of evolution was used and 1000 simulations were performed with seed number 42. Bounds of 0 and 2 were set. anovas were performed on each simulated dataset using pdanova and 95% percentiles of the resulting frequency distribution of F-statistics were determined. An F-statistic calculated using conventional anova falling beyond the 95% percentile was deemed to be statistically significant under phylogenetic adjustment (Garland et al. 1993).

Results

R vs. P/L

The radius of curvature and ratio of area to streamline length of each semicircular canal of each of the species under study are given in Table 1. Where multiple specimens were imaged, mean values are reported. It was hypothesized above that, although the two variables will be very closely correlated, the values of R and 2P/L will differ significantly due to the added shape component in 2P/L. Table 2 and Fig. 3 show that there is indeed a very close correlation between these two variables (r> 0.960, P< 0.001 for all three canals after phylogenetic correction). However, it can be seen that the slopes of the RMA regression lines for the vertical canals, after phylogenetic correction, are not significantly different from 1. Although the slope of the RMA for the lateral canal is significantly > 1, even after phylogenetic correction, it appears that this result is being unduly influenced by one point (see Fig. 3C) that has a much larger value of 2P/L than any of the others. This point represents the node where the giraffe branches off from the cervids and bovids. Removal of this one point decreases the value of the slope from 1.15 to 1.05, which is not significantly different from 1. Therefore, it is implied by these results that there is little difference between R and 2P/L as measures of canal size.

Table 2.

Product-moment correlation coefficients (r) and RMA line fittings (a, slope; b, intercept; 95% CI, 95% confidence interval on the slope) of 2P/L against R for uncorrected and phylogenetically corrected (PIC) data.

Canal	r	P	a	95% CI	b	P (a = 1)
Standard
ASC	0.99	*	1.04	0.98 – 1.10	−0.03	NS
PSC	0.99	*	1.04	0.98 – 1.10	−0.02	NS
LSC	0.98	*	1.07	1.01 – 1.15	−0.01	***
PIC
ASC	0.94	*	1.02	0.94 – 1.10	0.00	NS
PSC	0.88	*	1.03	0.93 – 1.13	0.00	NS
LSC	0.87	*	1.15	1.02 – 1.30	0.00	***

*P< 0.001; ***P< 0.05. Abbreviations as for Table 1.

Fig. 3.

Bivariate plot of independent contrasts of 2P/L against R for the (A) anterior semicircular canal (ASC), (B) posterior semicircular canal (PSC) and (C) lateral semicircular canal (LSC). Arrow marks potentially anomalous data point in the LSC data set.

Size scaling constraints

The second hypothesis given in the Introduction states that, owing to the need to pack a relatively larger canal into a smaller volume in the petrous bone, smaller species will show a greater deviation from circularity, i.e. they will have smaller values of 2P/L compared with R. Table 3 shows the product-moment correlation coefficients of (2P/L)/R against log₁₀ body mass. It can be seen that this variable does not show a significant correlation with body mass in any of the three canals. Therefore, there seems to be no tendency for the semicircular canals of smaller species to become less circular and more elliptical. Analysis of the SD of the circularity measure across the entire sample (Table 3) indicates that there is very little variation in this measure throughout mammals as a whole, which is consistent with the previous result that showed that there is very little difference between R and 2P/L.

Table 3.

Mean values (± SD) of (2P/L)/R and product-moment correlation coefficients (r) of (2P/L)/R against log₁₀ body mass for each semicircular canal for non-phylogenetically corrected data.

Canal	Mean±SD	r	P
ASC	1.02 ± 0.05	0.22	NS
PSC	1.03 ± 0.06	0.06	NS
LSC	1.07 ± 0.09	−0.10	NS

Abbreviations as for Table 1.

Out-of-plane torsion

The degree of out-of-plane torsion of each canal as measured by angular deviation from the plane of best fit is given for each of the species under study in Table 1. Values ranged between 1.6° and 11.5° for the anterior canal, 1.7° and 12.5° for the posterior canal, and 2.1° and 17.5° for the lateral canal, with a minimum torsion of 1.6° being found in the anterior canal of the tree pangolin, Manis tricuspis, and a maximum torsion of 17.5° in the lateral canal of the slow loris, Nycticebus coucang. It was hypothesized that a high degree of out-of-plane torsion would be negatively correlated with 2P/L as a trade-off between maximizing in-plane and out-of-plane sensitivity. Table 4 shows that no significant correlation was found between 2P/L and torsion. Therefore, it was hypothesized that the degree of out-of-plane torsion exhibited by a canal may be influenced by the volume available into which the canal can expand, i.e. a greater degree of torsion may be found in smaller species that have relatively larger semicircular canals in a more constrained space. This was tested by calculating correlation coefficients between torsion and log₁₀ body mass (see Table 5). It can be seen that there is no correlation between out-of-plane torsion and body size for any of the three canals (the correlation seen initially in the anterior canal appears to be artefactual as it is lost under phylogenetic correction). As above, in the analysis of in-plane circularity, examination of the SD of the whole sample (Table 5) indicates that there is very little variation in torsion across placental mammals – mean torsion falls between 5° and 7° and the SD is between 2° and 3° for all three canals. Thus, although all mammals show some torsion in all three canals, the degree of torsion is reasonably small and does not change substantially between species.

Table 4.

Product-moment correlation coefficients (r) of 2P/L against out-of-plane torsion for each semicircular canal for non-phylogenetically corrected data.

Canal	r	P
ASC	0.20	NS
PSC	−0.16	NS
LSC	0.07	NS

Abbreviations as for Table 1.

Table 5.

Mean values (± SD) of out-of-plane torsion, product-moment coefficients (r) and RMA line fittings (a, slope; b, intercept; 95% CI, 95% confidence interval on the slope) of out-of-plane torsion against log₁₀ body mass for each semicircular canal for uncorrected and phylogenetically corrected (PIC) data.

		Standard					PIC
Canal	Mean ± SD	r	P	a	95% CI	b	r	P
ASC	5.32 ± 2.19	0.30	*	3.73	2.90 – 4.70	−0.62	0.22	NS
PSC	5.87 ± 2.57	−0.16	NS
LSC	6.65 ± 2.73	0.07	NS

*P< 0.05. Abbreviations as for Table 1.

Agility

It is hypothesized in this investigation that the increased complexity of the variable P/L will make it a better predictor of locomotor ability than R. Table 6 gives the results of an anova conducted to investigate whether the mean values of size-adjusted P/L and R are statistically different between the agility groups from 2 to 6. P/L and R were corrected for body size by dividing by the cube root of body mass. It can be seen that both P/L and R of all three semicircular canals could be used to distinguish between fast- and slow-moving mammals, even after phylogenetic correction. It can also be seen that the P-value is about five times smaller for P/L than R for the lateral canal, suggesting that this measure correlates substantially better with agility than does R. This is not true for the anterior or posterior canals – P/L and R have approximately the same power to discriminate between agility groups in these canals. In addition, the posterior canal appears to be less strongly correlated (P< 0.01) with agility than the anterior and lateral canals (P< 0.001). Post-hoc Duncan’s tests indicate that both P/L and R can distinguish between the two most agile (5 and 6) and the three least agile (2, 3 and 4) groups but not within these groupings (e.g. there is no statistically significant difference between groups 5 and 6).

Table 6.

Results of anova between agility categories (2–6).

Dependent	F	P	Significant under PIC?
R(ASC)	5.63	0.00075	Yes
R(PSC)	5.10	0.00150	Yes
R(LSC)	5.77	0.00062	Yes
2P/L(ASC)	5.71	0.00067	Yes
2P/L(PSC)	4.82	0.00216	Yes
2P/L(LSC)	7.00	0.00013	Yes

R and 2P/L divided by ³√BM to correct for size. Other abbreviations as for Table 1.

Discussion

The results show that, as hypothesized, the radius of curvature and the ratio of canal plane area to streamline length are highly significantly correlated. This was expected as both R and 2P/L represent some form of the radius of the semicircular canals. However, the prediction that 2P/L would differ greatly from R was firmly rejected. Indeed, it has been shown that the slope of 2P/L against R is not significantly different from 1 for any of the three canals, indicating that R and 2P/L are two ways of expressing the same measure. This is particularly marked in the vertical semicircular canals. The lateral semicircular canal initially appears to show a slope >1 for 2P/L against R but further analysis indicates that this result is heavily influenced by one potentially spurious data point. The original hypothesis was formulated because it was inferred that deviations away from circularity in the semicircular canals would result in a smaller area and thus a smaller value of 2P/L but would not be accounted for in R. However, the findings reported here – that there is little difference between 2P/L and R – suggest that deviations from in-plane circularity are not great enough to produce a significant difference between 2P/L and R. Indeed, McVean (1999) notes that deviations from circularity have to be extremely severe (above 0.8 eccentricity) before any significant reduction of area occurs. Thus, small- and medium-sized deviations from perfect circularity are unlikely to be reflected in the value of 2P/L.

It was hypothesized that 2P/L divided by R would give a dimensionless variable that would represent the degree of circularity of a semicircular canal, as it would indicate the discrepancy between the two input variables. Hypothesis 2 predicted an inverse relationship between (2P/L)/R and body mass as it was hypothesized that smaller species would need to have a greater deviation from circularity in order to fit their relatively larger semicircular canals (Jones & Spells, 1963; Spoor & Zonneveld, 1998) in the petrous part of the temporal bone. However, no correlation was found between this circularity measure and body mass. The correlation between body mass and (2P/L)/R in the anterior canal was found to be almost significant but even this weak relationship is likely to have been further weakened by phylogenetic adjustment. However, it was noted above that, unless the deviations from circularity were extremely pronounced, they would not greatly reduce 2P/L compared with R. Thus, the value of (2P/L)/R would be unlikely to vary much over the entire dataset. This was shown to be the case by the very small SDs seen for this measure in all three canals (Table 3).

Out-of-plane torsion was calculated as the mean angular deviation from the plane of best fit of the canal, measured from the centroid of the canal. It was predicted that torsion helps to optimize overall sensitivity by sacrificing in-plane mechanical sensitivity, as denoted by P/L, in order to shift the best fit plane closer to the plane of maximal sensitivity (Rabbitt, 1999). This predicts that P/L and torsion are negatively correlated. The alternative arguments are that the torsion is not sufficient to influence P/L (e.g. McVean, 1999) or that torsion is due to spatial constraints and will correlate negatively with body size (see hypothesis 2ii). No correlation was found between 2P/L and torsion and nor was a significant correlation between torsion and body mass found in any of the canals after phylogenetic correction. Thus, it was inferred that the degree of torsion found in the mammalian semicircular canals is not sufficient to substantially alter the value of P/L. This conclusion was supported by the low means and SDs calculated for all three canals across the entire data set.

The anova showed that both measures of semicircular canal size (P/L and R) are correlated with locomotor agility in all three canals in mammals. Spoor et al. (2007) found that this was true for the radius of curvature, treating agility as a quantitative measure. In this investigation, agility has been more realistically treated as a qualitative variable with discrete groups, hence the use of anova here instead of least-squares regression. The results indicate that P/L has a greater power to distinguish between fast- and slow-moving species for the lateral canal. This is not the case for the anterior or posterior canal, the latter of which appears to be the least closely correlated with agility. It is difficult to see why the posterior canal should be less closely associated with agility of locomotion than the anterior canal but it may indicate that the posterior canal is positioned such that it is less closely aligned to the usual plane of movement (i.e. directly forwards) than the anterior canal. This would suggest that the vertical canals are not orientated at 45° to the midline (as noted by Simpson & Graf, 1981; Ezure & Graf, 1984; Cox & Jeffery, 2007).

The most intriguing aspect of the results presented here is the outcome that 2P/L and R are very highly correlated, and have an RMA slope not significantly different from 1, yet 2P/L is more powerful than R at distinguishing between agility groups for the lateral canal. This suggests that, although over the entire mammalian sample deviations above and below an RMA slope of 1 cancel each other out, at the individual species level 2P/L is a better predictor of agility (at least in the case of the lateral canal). Investigations requiring a sensitive discrimination among agility types may benefit from using P/L as opposed to R.

Overall, the lateral semicircular canal has been shown to be the only canal in which 2P/L and R are potentially significantly different, although it has been noted that this may be an artefact due to one erroneous data point. Of the three semicircular canals, it is also the most closely associated with the degree of locomotor agility (as also noted by Spoor et al. 2007). The post-hoc Duncan’s tests show that the canal variables in particular distinguish groups 2, 3 and 4 from groups 5 and 6. Mammals that have been classified in groups 5 and 6 are largely aerial, arboreal, aquatic or ricochetal, i.e. they have a strong 3D element to their locomotion. Thus, it appears that the lateral canal is especially important for movement in 3D space. This is consistent with the suggestion of Fitzpatrick et al. (2006) that the lateral canal primarily controls navigation, whereas the vertical canals are concerned with reflex adjustments in response to movement.

Supporting Information

Table S1. Number, sex, source and imaging details of species used in this analysis.

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