Dynamics of quadrupedal locomotion of monkeys implications for central control

Abstract

We characterized the three-dimensional kinematics and dynamics of quadrupedal gait of young adult rhesus and cynomolgus monkeys while they walked with diagonal and lateral gaits at 0.4–1.0 m/s on a treadmill. Rigid bodies on the wrist, ankle, and back were monitored by an optical motion detection system (Optotrak). Kinematic data could be normalized using characteristic stride length, reducing variance due to different gait styles, to emphasize common characteristics of swing and stance parameters among animals. Mean swing phase durations fell as walking speed increased, but the swing phase durations increased at each walking velocity as a linear function of increases in amplitude, thereby following a main sequence relationship. The phase plane trajectories of the swing phases, i.e., plots of the relation of the rising and falling limb velocity to limb position in the sagittal (X–Z) plane, had unique dynamic characteristics. Trajectories were separable at each walking velocity and increases in swing amplitude were linearly related to peak swing velocities, thus comprising main sequences. We infer that the swing phase dynamics are set by central neural mechanisms at the onset of the swing phases according to the intended amplitude, which in turn is based on the walking velocity in the stance phases. From the many dynamic similarities between swing phases and rapid eye movements, we further suggest that the swing phases may be generated by neural mechanisms similar to those that produce saccades and quick phases of nystagmus from a signal related to sensed or desired walking velocity.

Introduction

Locomotion is a periodic pattern that linearly accelerates the body forward so that it follows a planned trajectory (Muybridge 1887; Grillner 1975; Grillner and Wallen 1985; Grillner and Dubuc 1988; Whelan 1996; Orlovsky et al. 1999; Pearson and Gordon 2000). Previous studies of mammalian quadrupedal locomotion have largely focused on cats (Grillner 1975; English and Lennard 1982; Cruse and Warnecke 1992; Gorska et al. 1993a, b; Jankowska and Edgley 1993; Howland et al. 1995; Carlson-Kuhta et al. 1998; Smith et al. 1998), dogs (Vilensky 1987; Korvick et al. 1994; Griffin et al. 2004) and horses (Hoyt et al. 2000; Wickler et al. 2000) as exemplars of quadrupedal gait. Locomotion studies in the cat have determined the basic outline of central organization (Wetzel and Stuart 1976). These studies have shown that locomotion can be elicited by stimulation of both subthalamic and mensencephaslic locomotor regions (SLR and MLR; Grillner 1975; Shik and Orlovsky 1976), which activate central pattern generator neurons in the spinal cord to produce the muscle activation patterns for periodic limb motion (Shik et al. 1966; Grillner 1975; Grillner et al. 1976; Perret and Cabelguen 1976; Shik and Orlovsky 1976; Stein 1978; Grillner and Wallen 1985; Patla et al. 1985; Ivanenko et al. 2004, 2005). Rubro-spinal and reticulo-spinal neurons are active during flexion, and vestibulo-spinal neurons are active during the extension phase of the step cycle (Orlovsky and Feldman 1972; Grillner 1975). It is possible to produce locomotor patterns in the central pattern generator without peripheral input, but peripheral feedback can also interact with the pattern generator to modify its output (Grillner 1975).

Over the last two decades, there has been a growing interest in the gait of subhuman primates (Hildebrand 1967; Vilensky and Larson 1989; Solomon and Cohen 1992a, b; Demes et al. 1994; Mori et al. 1996; Dunbar and Badam 1998; Schmitt 2003; D’Aout et al. 2004; Courtine et al. 2005a, b). In contrast to other quadrupeds, many subhuman primates are able to switch easily from quadrupedal to bipedal locomotion (Dunbar et al. 1986; Hirasaki and Matano 1996; Aerts et al. 2000; Mori et al. 2001; Hirasaki et al. 2004). It has been questioned whether the central organization of the locomotor system is similar in subhuman primates and humans to that in cats, dogs, and horses (Courtine et al. 2005b; Vilensky and O’Connor 1998), but although the quantity of supraspinal control may vary in different species, the central pattern generator in subhuman primates and humans is likely to be similar to that of other mammals (Pearson and Gordon 2000). Furthermore, although there are differences in the bipedal walking of monkeys and humans, such as leg length, configuration of the feet, extension of the legs, and tilt of the body on the legs (Alexander 2004; Hirasaki et al. 2004; Nakajima et al. 2004), it is also likely that aspects of neural control determined from quadrupedal and/or bipedal locomotion of monkeys can be applied to human locomotion.

Characteristics of quadrupedal gait of sub-human primates are different from those of cats, dogs, and horses (Muybridge 1887; Hildebrand 1967; Orlovsky and Feldman 1972; Grillner 1975; Wetzel and Stuart 1976). Monkeys walk with more protracted forelimbs (Demes et al. 1994; Larson et al. 2000; Schmidt 2005), have a larger excursion of both the fore- and hindlimbs (Vilensky 1987; Schmitt 1999, 2003), have smaller vertical oscillations of the body, and a long contact time (Vilensky 1987; Schmitt 1999, 2003). Sub-human primates generally engage in a diagonal sequence in which the footfall of each hindlimb follows that of the ipsilateral forelimb, and the hindlimb moves together with the contralateral forelimb in one half of the gait cycle in a diagonal couplet, whereas other mammals utilize a lateral sequence in which the hindlimb footfall leads the ipsilateral forelimb with either a diagonal or a lateral couplet (Muybridge 1887; Vilensky and Larson 1989; Cartmill et al. 2002; Courtine et al. 2005a, b). Monkeys can also walk with lateral sequences, however, particularly at a younger age (Vilensky and Larson 1989; Dunbar and Badam 1998; Courtine et al. 2005a; Shapiro and Raichlen 2005). Differences between the locomotion of subhuman primates and other quadrupeds have been postulated to be the result of increased use of the forelimbs for grasping or for other precise actions during arboreal locomotion by monkeys, rather than being due to a primary association with bipedal ambulation (Dunbar and Badam 1998; Schmitt 2003). Consistent with this, the posterior weight shift is seen as a dynamic strategy to reduce the load on the forelimbs, making them more accessible to prehension (Reynolds 1985a, b; Demes et al. 1991, 1994; Polk et al. 2000; Schmitt 2003, 2005).

Quantification of quadrupedal locomotion was facilitated with the use of cinematic recordings to characterize angular joint movements in the sagittal plane (Muybridge 1887; Hildebrand 1967). Video recordings with markers placed on individual body parts increased accuracy, and fundamental findings emerged about gait kinematics (Vilensky and Larson 1989; Courtine et al. 2005a, b). One insight has been that the stance and swing phases have significantly different characteristics (Brown 1911; Grillner 1975; Mori et al. 1996; Courtine et al. 2005a; Orlovsky et al. 1999 for review). During the stance phases on a treadmill, the velocity of limb movement is fixed at the velocity of the moving belt, whereas the speed of the swing phases varies (Grillner 1975; Mori et al. 1996; Courtine et al. 2005a). Moreover, there are large decreases in stance durations with increases in walking speed that are not reflected in changes in swing phase durations (Mori et al. 1996; Courtine et al. 2005a). The development of motion detection systems that can accurately track rigid bodies in three-dimensional space has made it possible to measure the angular and linear limb positions, velocities and accelerations at high sample rates. From this, it is possible to determine the dynamic characteristics of the swing phases, including their phase plane characteristics, i.e., the relation between position and velocity. The timing and amplitude of the swing and stance phases have been well studied, but no studies have analyzed velocities and accelerations in three dimensions using phase plane plots to help determine the central commands that generate the activity responsible for locomotion. This was one aim of this study.

In preliminary experiments, we noted that the velocity characteristics of the swing phases in the forward (X-axis) direction bore a striking similarity to the velocity characteristics of rapid eye movements, i.e., of saccades and quick phases of nystagmus. By analogy to models that closely simulate the slow and quick phases of nystagmus (Raphan et al. 1977; Robinson 1977; Chun and Robinson 1978; Raphan and Cohen 1996, 1981, 2002; Kushiro et al. 2002), we hypothesized that the two phases of locomotion are governed by neural mechanisms that are driven by a signal related to walking velocity. If the analogy to the oculomotor system were to be proven correct, walking velocity would not only set the velocity of the stance phases, but also would drive the dynamics of the swing phases and be responsible for timing and switching between the two. Such techniques have been valuable in understanding the organization of the central vestibulo-oculomotor system (Raphan and Cohen 2002, for review).

The purpose of this study, then, was first to determine the amplitude, velocity, and acceleration, and the timing relationships of stance and swing phases of young adult cynomolgus and rhesus monkeys in three dimensions as a function of walking velocity to establish their basic characteristics. Second, we intended to examine the dynamic characteristics of the swing phases and determine if there were relations between limb amplitude and velocity in their phase plane trajectories as a function of walking velocity. A main sequence, which was originally used to describe the aging of stars, has been fruitfully applied to characterize the relationship between duration, peak velocity, and magnitude of saccades, which also have unique phase plane trajectories (Stark 1971; Bahill et al. 1975). We postulated that the swing phases would have characteristic main sequences and phase plane trajectories, similar to those of the quick phases of nystagmus. If these dynamic relationships had the same characteristics, it would imply that similar pulsatile networks in the spinal cord provide suboptimal control of the limbs during the swing phases as during quick phases and saccades (Stark 1971). From this, it would be possible to compare aspects of neural control of the central neural organization of the oculomotor and locomotor systems. It would also support the idea that in the locomotor as in the oculomotor system, a signal related to velocity, here walking velocity, emanating from cortical and subcortical neural networks, as well as from feedback from joint and muscle proprioceptors in the feet and legs, is likely to be a critical signal driving both the stance and swing phases.

Methods

Two young adult rhesus monkeys (Macacca mulatta, Rh426 and Rh488) and two young adult cynomolgus monkeys (Macacca fasicularis, Cy091 and Cy101) of approximately the same size, weight, and age were used in this study (Table 1). The experiments conformed to the Guide for the Care and Use of Laboratory Animals and were approved by the Institutional Animal Care and Use Committee of the Mount Sinai School of Medicine.

Table 1.

Animal ages, sizes, and weights

Monkey no.	Age (years)	Body length (cm)	Forelimb length (cm)	Hindlimb length (cm)	Weight (kg)
Cy091	5	26.7	29.2	26.7	2.5
Cy101	4	30.5	29.2	27.9	3.8
Rh426	3	29.9	31.8	30.5	3.6
Rh488	5	30.5	30.5	29.2	3.8
Mean ± SD		29.4 ± 1.8	30.2 ± 1.2	28.6 ± 1.6	3.4 ± 0.6

Body measurements were taken when the animals were lightly anesthetized. The length of the body was measured from the base of the skull to the rump, the hindlimb from the greater trochanter to the plantar surface of the dorsiflexed hindpaw, the forelimb from the acromion to the ventral surface of the flexed forepaw

Training

The monkeys were first accustomed to treadmill walking, and then were acclimated to a specially designed, light cloth body suit that held the LEDs while walking for several weeks. Walking was qualitatively similar at the beginning and at end of the training period except that the walking became more regular with experience. The treadmill (Key Co., NY, USA) was 1.2 m long and 0.45 m wide (Fig. 1). A loose chain attached to a light collar (Primate Products, Inc.) around the neck restrained the animals on the treadmill, but they were otherwise unrestricted. The monkeys faced a clear plastic screen and were intermittently rewarded with sugar pellets placed in a cup in front of them. Each animal walked about 60 min a day, 3 days a week during training. The monkeys were tested at ten speeds between 0.4 and 1.0 m/s. They would not walk slower than about 0.4 m/s in a regular fashion, and galloped or bounded when pressed to walk faster than about 1.0 m/s. Slow and fast treadmill speeds were interspersed to prevent tiring. Gaze was unconstrained and the monkeys were free to look where they wished. After the monkeys had attained a steady gait, about 90 s of data were collected at each treadmill velocity.

Fig. 1.

Coordinate frame for quadrupedal walking. The monkey is shown in a diagonal gait. Limb movements were recorded with reference to the spatial coordinates, X, Y, Z, which were determined by LED markers affixed on the wall. Positive directions of rotation follow the right-hand rule. The origin of the coordinate frame was the lower left LED on the wall (X-, Y-, Z- 0), which was 32.5 cm above the treadmill surface. Abbreviations: RFL right forelimb, RHL right hindlimb, LHL Left hindlimb. Three rigid bodies with LED markers were attached to the RFL above the wrist, the RHL above the ankle, and the right side of the chest to measure both linear and angular displacement of the limbs and body. A single marker on the rigid body on the LHL measured its linear displacement. The lower left LED on each of the rigid bodies was the point of measurement for linear movement, while all four LEDs were used to compute angular movements

Coordinate frame

To establish the spatial coordinate frame, a 30.5 · 30.5 cm² square containing four LEDs 25 cm apart was placed on the wall along the boundary of the treadmill (Fig. 1). The lower left LED was the zero coordinate. The Z-axis was parallel to the spatial vertical (positive upward), the positive X-axis was parallel to the direction of forward walking and orthogonal to the Z-axis, and the Y-axis was orthogonal to the X–Z plane, positive to the monkey’s left. The plane of the treadmill belt on which the animal walked was 32.5 cm below the origin of the spatial coordinate frame (Z₀=−32.5 cm) and was parallel to the X–Y plane. The zero on the ordinates in the position graphs of Figs. 4, 5c, and 6 is the zero coordinate of the spatial coordinate frame, not the treadmill surface. The markers on the right wrist (right forelimb, RFL) and on the left and right ankles (left and right hindlimbs, LHL and RHL, respectively) defined the positions of these limbs in space. The directions of positive rotation (roll, pitch, and yaw) of the limbs in the spatial coordinate frame were around the X-, Y- and Z-axis and followed a right-hand rule (Fig. 1).

Fig. 4.

Three-dimensional limb trajectories in spatial coordinates. The XZ-plane (a b, e, f) was viewed from the right side of the animal, while the XY-plane (c, d, g, h) was viewed from the top of the animal. About 15 stride cycles were used to form each sequence, with the thick lines representing the averaged trajectories. The circles of progressively varying gray scales show the differences of average body and limb positions as time progressed in a complete stride cycle. a XZ-plane at 0.53 m/s for Cy091; b XZ-plane at 0.58 m/s for Rh426; c XY-plane at 0.53 m/s for Cy091; d XY-plane at 0.58 m/s for Rh426; e XZ-plane at 0.85 m/s for Cy091; f XZ-plane at 0.85 m/s for Rh426; g XY-plane at 0.85 m/s for Cy091; h XY-plane at 0.85 m/s for Rh426

Fig. 5.

Differences between lateral and diagonal gaits in the rhesus monkey Rh426. The ordinates are in the spatial coordinates of the experiment in cm. The abscissa indicates time (s). In b–e, zero time is at the onset of the swing phases. a The animal switched from a lateral (left) to a diagonal (right) gait at the vertical line. The top trace shows the fore-aft (X-axis) positions of the right wrist; the bottom trace shows similar X-axis positions of the right ankle. The line on top of each of these traces is a line fit to the average forward X-position during the swing phase (Ante X), and the bottom line shows the average posterior position (Post X). The average positions are shown by the numbers to either side of the lines. Values are given at the left side of a for lateral walking and on the right side for diagonal walking. b–e Differences in displacement (b–d) and pitch (e) of the right hindlimb (RHL, solid line), right forelimb (RFL, dashed line), and body (dotted line) between lateral and diagonal gaits for X-axis (b), Z-axis (c), Y-axis (d), and pitch (e)

Fig. 6.

Three-dimensional kinematics for the ankle and wrist in Cy091 walking at 0.53 m/s. The thick lines are the averages of 13 cycles. a–i X-axis displacements (a, g), velocities (b, h) and accelerations (c, i) of the ankle (a–c) and wrist (g–i). d–l Z-axis displacements (d, j), velocities (e, k) and accelerations (F, l) of the ankle (d–f) and wrist (j–l). m–u Y-axis displacements (m, s), velocities (n, t) and accelerations (o, u) of the ankle (m–o) and wrist (g–i). p–r and v–x are pitch movements in a similar format. See text for further details

Measurement of body and limb movements

Movements of the right forelimb, the right and left hindlimbs, and the right posterior chest wall (body) were detected by tracking the position of 5 mm infrared LEDs with a optically-based motion detection system (Optotrak 3020, Northern Digital Inc., Canada). A specially designed, light cloth suit anchored the rigid body on the chest. The LEDs were embedded in rigid bodies that were fixed with flexible Velcro bands to the right forelimb just above the wrist and to the right hindlimb just above the ankle (Fig. 1, RFL, RHL). One LED was attached to the left hindlimb (LHL) to track its linear translation. The left wrist was also tracked in one animal.

The rigid bodies were small Plexiglas plates, each with four embedded LEDs. Four LEDs were necessary to establish the angular and linear position of the rigid bodies in space. The arm and leg rigid bodies were 5.1 x 3.8 cm² with the longer side along the length of the limb segment. The trunk rigid body LEDs were placed at the corners of a quadrilateral to fit the contour of the animal’s trunk. The rostral-to-caudal distance between the top LEDs was 10.2 cm, and the distance between the bottom LEDs was 10.7 cm. The ventral–dorsal distance between the LEDs on the wrist and ankle was 5 cm and the anterior–posterior width was 1.8 cm. The origin of each of the rigid bodies was the lower left LED. This LED was tracked in the computation of linear movements, while all four LEDs on the rigid bodies were tracked for the computation of angular movements.

Thin wires connected the LEDs to an external strobe unit fixed above the monkey. This unit pulsed the LEDs. A high-resolution, three-camera sensor sensed activation of the LEDs, and a central control unit and computer recorded the information about the LED positions. The sampling frequency was 90 Hz, which was sufficient to establish the 3D angular and linear positions of the rigid bodies on the trunk and limbs, and to capture the full spectrum of the body and limb movements. With the cameras placed 4 m from the monkey, the Optotrak created a 2 m space in all dimensions between 3 and 5 m from the camera in which the accuracy of measurement of horizontal and vertical translations was 0.3 mm (Hirasaki et al. 1999). In depth the accuracy was 0.45 mm, with a resolution better than 0.1 mm for all axes (Manufacturer’s specification). The angular accuracy of the device for rotation around any axis was approximately 0.1° at this distance from the sensor.

Translation and rotation of the rigid bodies in space were computed during post-processing of the raw position data. Simultaneous digital video was recorded while the animals walked on the treadmill. One camera viewed the animals from the side, and a second camera from the back. The video footage was stored on separate channels in a computer, and images of the animals were displayed in synchrony with the parameters of walking on a computer monitor. Having synchronous video proved invaluable for identifying sections of data for analysis when the animals were actively walking with a stable gait.

The sampled data were converted into digital files, and a 3D reconstruction was done by the Optotrak software. A five point average window filtered the data. Approximately 10~20 cycles of gait were used at each walking velocity for each condition to characterize the stride frequencies and stride lengths, together with other relevant kinematic measures of the limbs. If the animals’ position on the treadmill did not vary, as determined from the X-axis coordinates of the rigid bodies on the trunk and limbs and from the synchronous video, their walking velocity was the treadmill velocity. Main sequence graphs were used in this study to show the relationship between amplitude and maximum velocity of the swing phases in the X- (forward) direction, while phase plane graphs were used to demonstrate the relationship between the position and velocity of the movements during the swing phases.

Measurement of eye movements

Eye movements were recorded in two cynomolgus monkeys with perilimbal scleral search coils that were implanted on the left eye under anesthesia and sterile surgical conditions (Robinson 1963; Judge et al. 1980; Yakushin et al. 2000). During eye movement recordings, the monkey sat with its head fixed in a primate chair in a multi-axis vestibular and optokinetic stimulator (Yakushin et al. 2000, 2003). Optokinetic nystagmus (OKN) was induced in Cy357 by rotation of the visual surround, which had alternating 10° black and white vertical stripes, at 60, 90 and 120°/s. Per- and post-rotatory vestibular nystagmus was induced in Cy014 by constant velocity rotation in darkness at 60, 90 and 120°/s. Since the amplitude of the quick phases of nystagmus are related to the velocity of stimulation (Cohen et al. 1965), it was possible to obtain quick phases with amplitudes that increased with increases in stimulus and slow phase eye velocity during OKN. We utilized ≈40 beats of nystagmus for analysis of OKN at each velocity. Since the velocity of the vestibular nystagmus induced by constant velocity rotation declines in darkness (Raphan et al. 1979), only the first 9 s of the response was used to calculate the phase plane plots as shown in Fig. 9d, while the velocity of the slow phases was close to the velocity of the stimulus. This gave ≈15 beats of nystagmus at each rotational velocity for analysis during the rotation in darkness. Voltages related to eye position were recorded with amplifiers having a bandpass of DC to 40 Hz. Eye position voltages were digitized at 500 Hz with 16-bit resolution, smoothed, and digitally differentiated with an algorithm corresponding to a filter with a 3 dB cutoff above 40 Hz. Eye movements were calibrated by rotating the animals in light at 30°/s. Under these conditions, it was assumed that the horizontal gain was unity (Raphan et al. 1979; Yakushin et al. 1995).

Fig. 9.

Characteristics of rapid eye movements of two young adult cynomolgus monkeys during optokinetic nystagmus (OKN) (a–c, e; Cy357) and per- and post-rotatory vestibular nystagmus (d; Cy014). Both OKN and vestibular nystagmus were induced at three velocities, 60, 90 and 120°/s. a Horizontal eye positions (top trace) and eye velocities (bottom trace) during OKN in response to a stimulus velocity to the animal’s right of 90°/s. The eyes moved to the right (down) across the mid-position during the slow phases (zero line) and were reset to the left during the quick phases. In the differentiated record (bottom trace), slow phase velocities to the right are below and quick phase velocities to the left are above the zero velocity line. b Peak velocities of OKN quick phases (ordinate) vs. quick phase amplitudes (abscissa). c Phase plane plots (eye position vs. eye velocity) for 40 quick phases of OKN at each of the three stimulus velocities. d Phase plane plots for 15 quick phases of vestibular nystagmus at the same three stimulus velocities. e Amplitude/duration relationship of quick phases of OKN. The data obtained at each of the three stimulus velocities were grouped to generate the plots in b and e

General gait characteristics and definitions

Monkeys walked with either a diagonal sequence with diagonal couplets, or with a lateral sequence with lateral couplets. In this paper we will use the term, ‘diagonal gait’ for the first sequence and ‘lateral gait’ for the second. During diagonal gait, the hindlimb was in phase with the contralateral forelimb. In this small series, the cynomolgus monkeys generally walked with a diagonal gait, while the rhesus generally walked with a lateral gait, but each species could and did switch between the two (see Fig. 5a). In both species, the hindlimbs were placed medial to the forelimbs. The movement of the right ankle, which faced the Optotrak cameras, was taken as the origin of the parameter definitions. Stride cycles were composed of a swing phase followed by a stance phase. The swing phases were defined by the time when the limb moved forward in space, while the stance phases occurred when the same limb moved backward in space with the foot planted on the treadmill. Stride duration was composed of a swing and stance duration. Stride length was defined as the distance between the successive placements of a single hindlimb relative to the platform (belt) of the treadmill and calculated as the spatial distance traveled by the right hindlimb in one stride cycle less the negative distance traveled by the treadmill, which was calculated from the product of treadmill speed and the stride duration. Stride frequency was the inverse of the stride duration. If we assume that the animals were not limping and that the strides were equivalent on the two sides, then step length was half the stride length and step frequency was double the stride frequency. For diagonal gait, latency was the time from the onset of the swing phase of one hindlimb to the start of the swing phase of the contralateral forelimb. The latency during lateral walking was the time between the onset of the swing phase of one hindlimb and that of the ipsilateral forelimb.

Statistical analyses

Average values are reported in the form of means ± 1 standard deviation. Statistical comparisons of pairs of data were made with a Student’s t test. To determine variance of amplitude and durations around a mean value, we utilized a principal component analysis in two dimensions to explore the nature of this variance based on the Karhunen–Loeve transform (Kunin 2005). In this analysis, the r-value, which represents a measure of the goodness of fit, is the ratio between the variance along the principal component divided by the sum of the variance along the principal component plus that along the orthogonal axis. Values of 0.5 indicate equal variance along each axis, and values of 1.0 indicate that all of the values fell along the principal axis. A linear regression was also utilized in some cases.

Results

Kinematics of movement along the X-axis

The kinematics of these young adult animals from two species were typical for macaque monkeys (Vilensky and Larson 1989; Mori et al. 1996; Courtine et al. 2005a, b) and did not depend on whether the monkeys walked with a diagonal or lateral gait. Stride length increased as walking speed increased, going from about 30 to 55 cm in three of the four animals with a slight saturation at 0.8 m/s and above (Fig. 2a, open circles, Cy091; open triangles, Rh426; filled triangles, Rh488). One animal had substantially longer stride lengths, extending from about 50 to 70 cm (Fig. 2a, filled circles, Cy101). Fore- and hindlimb stride frequencies were the same and also increased with walking speed, ranging between 1.2 and 1.9 Hz for the three monkeys (Fig. 2b). Because the stride lengths of Cy101 were longer, its stride frequencies were lower, ranging from 0.8 to 1.2 Hz (Fig. 2b, filled circles). In all four animals, the product of stride length and stride frequency approximated the walking speed.

Fig. 2.

Kinematic (a–d) and normalized (e–h) relationships as a function of walking velocity for the four monkeys in this study. The abscissae in a–d are in m/s, and in e–h are dimensionless numbers due to the normalization. Filled and open circles are data from cynomolgus and filled and open triangles are data from rhesus monkeys. a Stride length; b Stride frequency; c, Swing duration; d Stance duration; e Normalized stride length L_strideⁿ; f Normalized stride frequency F_strideⁿ; g Normalized swing duration T_swingⁿ; h Normalized stance duration Tⁿ_stance. Dashed lines are linear least square error fits to the data

At lower walking velocities, the durations of the stance phase were substantially longer than the swing phase (Fig. 2c, d). As walking speed increased, however, stance durations decreased faster than swing durations. As a result, the percentage of the stride cycle occupied by the stance phases fell from 62% at 0.45 m/s, to 58% at 0.67 m/s, and 53% at 0.89 m/s. At 0.89 m/s when the durations of the swing and stance phases were approximately the same, the double stance phase was close to being eliminated, and for treadmill velocities above this, the monkeys would begin to bound or gallop.

Stride lengths were approximately equal in the fore- and hindlimbs in each monkey at each speed and on average, ranged from ≈39 cm at 0.45 m/s to ≈56 cm at 0.89 m/s. Swing and stance durations were also essentially the same for the fore- and hindlimbs, decreasing from ≈0.33 to ≈0.29 s and ≈0.54 to ≈0.35 s, respectively, for walking velocities from 0.45 to 0.89 m/s. Latencies of forelimb and hindlimb forward movement were examined for both diagonal and lateral walking. The ankles led the relevant wrists at all walking velocities with latencies that generally fell within 50–100 ms with no relation between latency and walking speed. Thus, as walking velocities increased and stride durations decreased, the temporal coupling between displacements of the hind and forelimbs was modified so that the latencies between them remained constant. This is consistent with the requirement that the gait be adjusted to follow the speed of the treadmill.

The standard deviations of the swing phase durations were between 20 and 30% of the mean durations at each walking velocity for individual animals. In view of the declining or flat slope of the relationship between swing durations and walking velocity (Fig. 2c), and the increase in swing phase amplitudes as walking velocity increased, it might be predicted that there would also be a decrement in duration for larger swing phases around each mean swing duration. This was not the case. Rather, the variations in duration of the swing phases increased approximately linearly as a function of the swing amplitude (Fig. 3a, c). We utilized a principal component analysis in two dimensions based on the Karhunen–Loeve transform to explore the nature of this variance (Kunin 2005). This demonstrated that the increase in duration along the principal axis of variation increased as a function of amplitude at each walking velocity (Fig. 3a, c). With one exception, the r-values all fell between 0.66 and 0.85, showing that the variance along the principal axis was more than double that along the orthogonal axis. These linearly increasing relationships between duration and amplitude were present in both cynomolgus and rhesus monkeys and for both forelimbs and hindlimbs, despite the decline in the means (Fig. 3b, d). Thus, the local increase in swing durations and amplitudes at individual walking velocities was a general property of the swing phases. These data imply that a signal related to walking velocity was important in conditioning the swing phase timing and dynamics.

Fig. 3.

Relationships between swing durations (ordinates) and swing amplitudes (abscissae) while walking at velocities of 0.45 m/s (left; a, c), 0.71 m/s (middle; a, c), and 0.85 m/s (right; a, c) for cynomolgus monkey Cy101 (a) and rhesus monkey Rh426 (c). The durations increased with amplitude at each velocity. b, d The mean durations decreased, however. The slopes of the relationships, which tended to be shallower at higher walking velocities are shown above each graph with the r-value. The latter is the ratio of the variance along the principal axis divided by the sum of variances along both the principal axis and the orthogonal axis (see text for details)

Normalization of kinematic parameters along the X-axis

Variations in stride length and frequency among animals in Fig. 2a, b could be attributed to the unique walking styles of the individual monkeys and were not related to the size of the animals (Table 1). For example, Cy101 walked with a ‘swimming motion’, holding its body lower and extending its forelimbs more than the other animals. Its stride lengths were larger, and its swing and stance durations were longer. Human gait has been successfully normalized to compare walking parameters among subjects of different sizes using leg length (Wagenaar and Beek 1992). We performed a structurally similar normalization to create dimensionless measures for stride length, frequency, swing duration, and stance duration by employing a characteristic length (L_c) for individual animals based on their average stride length over the whole range of velocities (see Appendix). The values for the four monkeys were 41.5 ± 9.4 cm (Cy091), 62.5 ± 6.4 cm (Cy101), 50.1 ± 8.4 cm (Rh426) and 46.5 ± 6.0 cm (Rh488).

When fit with a linear regression, the R² coefficient for stride length (Fig. 2a, e), stride frequency (Fig. 2b, f), swing duration (Fig. 2c, g), and stance duration (Fig. 2d, h) was larger for the normalized (Fig. 2e–h) than the original data (Fig. 2a–d), demonstrating a more cohesive relationship (P = 0.002). The estimated error variances for each of the four parameters, when normalized relative to the mean square value of the data set, were smaller for the normalized than for the original data (P = 0.0026). Thus, with this normalization, relationships between walking velocity and stride length, stride frequency, swing duration, and stance duration had less variance between subjects than the original data. There was also less variance in the normalized data than when the various parameters were normalized using leg length (L) (P = 0.0025 for R² and P = 0.0037 for error variance). The ratio of the slopes of the declines for the swing and stance phase durations (−0.41 ± 0.21 and 0.12 ± 0.21, respectively) was still high (4.5) after normalization, indicating that it is possible to derive a single intrinsic attribute that provided convergence for the kinematic characteristics of quadrupedal walking of individual monkeys, as for the kinematic parameters of human bipedal walking (Wagenaar and Beek 1992).

Three-dimensional kinematics

Although the kinematics were similar for the wrist and ankle along the X-axis, there was considerable difference between wrist and ankle movements in the vertical dimension. Figure 4 shows comparative movements of the right ankle, the right wrist, and the right side of the posterior chest (body) of Cy091 and Rh426, viewed from the right side in the X–Z plane (Fig. 4a, b, e, f) and from the top in the X–Y plane (Fig. 4c, d, g, h). The cynomolgus walked with a diagonal and the rhesus with a lateral gait. The strides of the right ankle started at the white circles and ended with the black circles with increasing shades of gray marking the progression of the right ankle through the stride cycle. These markers are also shown for the right wrist and body, synched to the swing/stance cycle of the right ankle. The clear circles show the positions of each of these structures at the beginning of the hindlimb swing phase, and the circles get increasingly darker as the hindlimb progressed through the swing and stance phases.

The walking patterns of the cynomolgus monkey were similar at both low (0.53 m/s, Fig. 4a, c) and higher walking velocities (0.85 m/s, Fig. 4e, g). The hindlimb moved medial to the forelimb (Fig. 4c, g). The right wrist (RFL) was lifted slightly (≈4 cm) and thrust forward during the swing phase at both walking speeds with little upward (Z-axis) movement (Fig. 4a, e). The upward movement of the ankle (RHL) was substantially larger (≈9 cm), probably due to both the dorsiflexion of the foot and the upward movement of the leg (Fig. 4a, e). The right side of the body translated up and back during the stance phase and down and forward during the swing phase in synchrony with the right forelimb.

The walking patterns of the rhesus using a lateral gait were somewhat different. In Cy091, the right wrist and ankle were close to 180° out of phase (Fig. 4a, c, e, g), while they moved in synchrony in Rh426 (Fig. 4b, d, f, h). There was less vertical movement of the ankle and more of the wrist in the rhesus (Fig. 4b, f) than in the cynomolgus monkey (Fig. 4a, e). Additionally, the body was held higher and was more stable in space (Fig. 4b, f) than in the cynomolgus monkey (Fig. 4a, e).

Although the rhesus monkeys walked predominantly with a lateral gait, they could easily switch between lateral and diagonal gaits. In the example shown in Fig. 5a, Rh426 transitioned from lateral to diagonal gait at the vertical line while walking at 0.85 m/s simply by extending the duration and amplitude of the hindlimb stance phase (bottom trace). Thus, the right hindlimb did not start the swing phase until the right forelimb was about to finish the swing phase. This enabled the subsequent movements of the hindlimb to be out of phase with the ipsilateral forelimb.

The mean anterior position of the wrist was approximately the same during lateral as well as during diagonal walking (104 vs. 103 cm), but the anterior position of the ankle fell back by 9 cm from 85 to 76 cm during diagonal walking, extending the animal’s stride. The predominant increase in the stride was in the extension of the forelimb. The forward amplitude of the wrist movement increased from 21 to 26 cm while the ankle movement remained at 26 cm. The longer forward movement of the wrist along the X-axis is also shown in the averaged traces of Fig. 5b (dashed line). The rigid body on the trunk moved back in space relative to the forward excursion of the wrist during diagonal walking (Fig. 5b, dotted line), as compared to its position during lateral walking, and it was slightly lower (1 cm) on average during the diagonal than the lateral gait (Fig. 5c, dotted line). The Y- and Z-axis movements of the wrist, ankle, and body (Fig. 5c, d) were the same during the lateral and diagonal gaits except for a change in the phase of the hindlimbs (dashed lines). Along the Y-axis, the body moved in the opposite direction against the forelimb during lateral walking but in the same direction as the forelimb during diagonal walking (Fig. 5d, dotted lines). The pitch angles were also similar between the two walking gaits except for the phase difference (Fig. 5e).

Thus, the difference between lateral and diagonal gaits in three dimensions in this example embodied many of the differences that have been described previously between mammals with lateral and diagonal gaits, namely that the forelimbs are more protracted during diagonal gait (Larson et al. 2000; Schmidt 2005), there is a larger excursion of the forelimbs (Larson et al. 2001) and lower vertical position of the body (Vilensky and Larson 1989; Schmitt 1999).

Temporal characteristics of ankle and wrist movement

The dynamic characteristics of the stance and swing phases can be seen in the relations between position, velocity, and acceleration while walking at a median velocity of 0.53 m/s (Fig. 6). The swing phases begin at the origin of the graphs for both the ankle and wrist, and the start of the stance phases is indicated by the vertical dotted line so that the temporal components of the variables can be compared. The forward movements of both the ankle and the wrist followed sigmoidal trajectories during the swing phases (Fig. 6a, g), and the swing phase velocities rose monotonically, reaching a peak value of 1.2 m/s close to the middle of the movement (Fig. 6b, h). The velocities then fell to zero at the onset of the stance phases, which was also the time of peak negative acceleration (Fig. 6c, i; vertical dotted lines). The cycle-averaged forward (positive) accelerations were ≈10 m/s² (≈1g) at the onset of the swing phases, and the average decelerations were −23 and −25 m/s² (≈2.3 and 2.5g) at paw placement at the onset of the stance phases. During the stance phases, the ankle and wrist moved backwards at a constant velocity of ≈0.5 m/s (Fig. 6b, h), the treadmill velocity, with no acceleration (Fig. 6c, i).

The predominant movements of the ankles and wrists were in the X–Z plane with rotation about the pitch axis. There was relatively little lateral (Y-axis) movement of either the ankle or the wrist in the swing or stance phases (Fig. 6m, s), consistent with the measurements in Fig. 4c, d, g, h. The upward (Z-axis) movements were larger in the ankle (Fig. 6d) than in the wrist (Fig. 6j). The ankle had begun moving upwards late in the previous stance phase, reaching a peak amplitude for a brief period at the beginning of the swing phase with an upward velocity of 0.5 m/s (Fig. 6e), and an acceleration of 4.5 m/s² (Fig. 6f) The ankle then moved down in the middle of the swing phase with a larger velocity (−0.7 m/s) and acceleration (−15 m/s²). The large downward velocity and deceleration of the ankle was not present in the wrist (Fig. 6k, l). During the stance phase, the Z-axis velocities were small and there was no acceleration until the end of the stance phase, when the ankle lifted from the treadmill surface to start the next swing phase (Fig. 6e, f). Pitch movements of the ankle (Fig. 6p, q, r) and wrist (Fig. 6v, w, x) followed very similar patterns as the Z-axis movement. The ankle and wrist were flexed 70 and 65°, respectively, at the end of the previous stance phase and extended during the swing phase. The upward pitch of the ankle and wrist during the swing phase was rapid (650 and 600°/s) with decelerations close to 10⁴°/s². The acceleration in pitch at the onset of the stance phase was also large (≈10⁴°/s²). During the stance phase, the ankle and wrist each rotated at a constant speed of about 200°/s, with no acceleration. Thus, the characteristics of the movements of the wrist movements were close to those of the ankle in the fore-aft direction, and there was less vertical excursion of the wrist (Fig. 6j), less vertical velocity (Fig. 6k), and less vertical acceleration (Fig. 6l) than for the ankle.

The mean amplitudes, peak velocities, and accelerations along the X-, Y-, and Z-axes of the ankle (Fig. 7a–c) and wrist (Fig. 7d–f) were compared at three treadmill speeds (0.45, 0.67, and 0.89 m/s) using data from the four animals during the swing phases. Fore-aft (X-axis) displacements of the ankle (Fig. 7a, light gray bars) and wrist (Fig. 7d) were significantly larger than the vertical (Z-axis, dark gray bars; P = 0.004) and lateral (Y-axis, medium gray bars, P = 0.003) displacements in that order. Since Y-axis displacements (medium gray bars) were small in both the ankle (Fig. 7a) and the wrist (Fig. 7d), their velocities and accelerations are not shown and are not considered further. X-axis velocities and accelerations and decelerations (Fig. 7b, c, light gray bars) were also larger than Z-axis velocities and accelerations (Fig. 7b, c, e, f). Displacements along the X-axis were similar in the ankle (light gray bars, Fig. 7a) and wrist (Fig. 7d), as were their velocities (Fig. 7b, e) and accelerations (Fig. 7c, f). The X-axis decelerations were larger in the ankle than in the wrist (Fig. 7c, f). The parameters of both ankle and wrist movement, i.e., displacements, velocities, and accelerations/decelerations, increased with increases in walking speed (Fig. 7a–f), and peak decelerations were larger than accelerations in both the ankle and the wrist during the swing phases.

Fig. 7.

Peak swing phase displacements (a, d), velocities (b, e) and accelerations (c, f) for both the ankle (a–c) and wrist (d–f). The bar graphs are the average of four monkeys walking at slow (0.45 m/s), medium (0.67 m/s) and high (0.89 m/s) velocities. The error bars (±1 SD) are shown on the top. a X-, Y- and Z-axis swing amplitudes for the ankle. b X- and Z-axis peak swing phase velocities for the ankle. c X- and Z-axis peak swing phase accelerations for the ankle. d–f Same as a–c except for the wrist

Peak Z-axis displacements, velocities, accelerations, and decelerations tended to increase with walking velocity at all speeds in the ankle (Fig. 7a–c, dark gray bars), but not in the wrists (Fig. 7d–f). Consistent with the increases in Z-axis movement of the ankles with walking speed, the vertical velocities (Fig. 7b) and the vertical accelerations and decelerations of the ankles also increased with walking velocity (Fig. 7c). In both the ankle and wrist, the peak downward (negative) velocities and decelerations were larger than the upward (positive) velocities and accelerations, although all of these parameters were smaller than those along the X-axis (Fig. 7b, c, e, f). None of the vertical (Z-axis) measures of the wrist were consistently dependent on walking velocity. Thus, the fore-aft and vertical changes in position, velocity, and acceleration of the ankle were linked to walking velocity, as were the fore-aft movement of the forelimb. The Z-axis decelerations and accelerations in the wrist (Fig. 7f, dark gray bars) were smaller than in the ankle (Fig. 7c), probably related to the different joint structure of the fore- and hindlimbs. Since we did not have a direct measure of the movements of the toe or heel, it is not possible to compare whether paw clearance was different in the fore and hindlimbs. From observation of the video frames taken simultaneously, there was not a striking difference between foot and forepaw clearance, however.

Dynamic characteristics of the swing phases

The phase plane plots of X-axis velocity followed characteristic patterns as a function of X-axis position during the swing phases (Fig. 8a–d). The trajectories had a rapid rise in velocity at the onset of the swing phase, a plateau in the central portions, and a rapid fall at the end of the swing phases. The trajectories were unique in that the amplitude/velocity curves were separate at each walking velocity, and expanded as a function of walking velocity. There were minor differences in the phase plane trajectories for the wrist and ankle. Ankle velocities rose more slowly, and the peak velocity of the ankle was closer to the end of the swing phase (Fig. 8a, b). For the wrists, there was a long period of constant velocity during the middle of the movement, and the peak velocities occurred earlier in the movements (Fig. 8c, d). Regardless, curves with larger swing amplitudes were associated with higher walking speeds and had higher peak swing velocities.

Fig. 8.

a–d Phase plane plots (limb position vs. limb velocity) for swing phase during locomotion at three different walking velocities. a, b Swing velocity of the ankle vs. position for Cy091 (a) and for Rh426 (b). c, d Swing velocity vs. position of the wrist for Cy091 (c) and for Rh425 (d). e, f Peak swing velocity vs. swing amplitude of the ankle for Cy091 (R² = 0.71; e) and for Rh426 (R² = 0.54, f). g, h Peak swing velocity vs. swing amplitude of the wrist for Cy091 (R² = 0.91; g) and for Rh426 (R² = 0.76; h). Slopes were calculated from linear regression. Data from e–h were taken from all velocities of walking

The relationships between the X-axis swing amplitude and the peak X-axis swing velocity were linear, increasing as the swing amplitudes became larger, and forming a main sequence (Fig. 8e–h). Although qualitatively similar for both the ankle (Fig. 8e, f) and the wrist (Fig. 8g, h) in both species, slopes of the relationships were steeper for the cynomolgus monkey walking with a diagonal gait than for the rhesus monkey walking with a lateral gait (6.7 and 7.8 vs. 3.3 and 4.0, respectively, Fig. 8e–h). There was also less variance in the main sequence plots for the wrists than for the ankles across all four monkeys. This was confirmed by the R² statistics between amplitude and peak swing velocity, which were significantly larger for the wrist than for the ankle (0.91, 0.55, 0.76, 0.77 vs. 0.71, 0.39, 0.54, 0.62, for the four animals, respectively; P < 0.001; paired student’s t test). Despite minor differences, the close relationship of the swing phase dynamics to walking velocity was consistent with the earlier finding in Fig. 3 that the variation around the mean swing phase durations was a function of walking velocity.

Discussion

In this study, we characterized the kinematics and dynamics of the swing and stance phases of monkey quadrupedal locomotion in three dimensions while the animals walked with both diagonal and lateral gaits. As shown in Fig. 5a, the monkey switched easily between the two gaits simply by delaying the next swing phase in the hindlimbs. Consistent with the findings of others, the diagonal gait was associated with an increased length of the forepaw step (Demes et al. 1994; Larson et al. 2000; Schmidt 2005), a longer stride length, and a lower position of the body (Vilensky 1987; Schmitt 1999, 2003). Motions of the wrist and ankle were mainly confined to the X–Z plane, and there was little variation along the Y-axis. As in previous studies, stride length and stride frequency increased with walking velocity, swing phase durations were shorter than the stance phase durations, and increases in walking velocity caused decrements in average stance and swing phase durations that were greater in the stance phases (Mori et al. 1996; Courtine et al. 2005a, b). Using average stride length as a characteristic length (L_c), it was possible to normalize and consolidate the kinematic data from the two species.

Comparison of velocities and accelerations of the wrist and ankle

The X-axis position changes, peak velocities, and peak decelerations at the onset of the stance phases were approximately equal at the ankle and wrist (Compare Fig. 6a–c, g–i), but the Z-axis position changes, peak velocities, and peak decelerations were considerably greater in the ankle than in the wrist (Compare Fig. 6d–f, j–l). The peak decelerations of the hindlimb occurred in the middle of the swing phases, reflecting the onset of the incipient downward thrust of the leg.

Shortly thereafter, there was a peak in downward velocity as the shank moved toward the treadmill surface (Fig. 6e), and the swing phase concluded with the paw being placed on the treadmill (Fig. 6d). The deceleration was three times greater in the ankle than in the wrist (−15 vs. −5 m/s, Fig. 6f, l), and the downward velocity was twice as large in the ankle as in the wrist (−0.7 vs. −0.35 m/s; Fig. 6e, k). Although movements of the wrist and ankle do not give information about the flexion and extension of the foot or hand during the locomotion, they can give critical information about the timing and magnitude of the accelerations and velocities of these distal points on the fore- and hindlimbs during the swing and stance phases. These results are consistent with those from EMG recordings (Courtine et al. 2005a) and from force plate measurements (Reynolds 1985a, b; Polk 2002, 2004; Schmitt 2003), supporting the postulate that quadrupedal locomotion in the monkey is indeed ‘back-driving and front-steering’ (Kimura 1985; Demes et al. 1994; Mori et al. 1996) in contrast to the ‘front-driving’ behavior of other quadrupeds. Such a ‘back-driving’ gait would be a reasonable precursor to bipedal locomotion, which is entirely back-driving with arm swing as a stabilizing mechanism.

Normalization of kinematic parameters

To consolidate the data across different gait strategies and between the two species, the kinematic values of gait along the X-axis were normalized using average stride length for the four monkeys in this series. This normalization reduced the variances of the kinematic parameters of the locomotion across both the rhesus and the cynomolgus monkeys, regardless of whether there was a lateral or diagonal gait. The normalization of walking speed in this study, $V^n=V/\sqrt{g·L_c}$, was similar to the Froude number, $Fr=V^2/(g·L)$, which is a dimensionless quantity representing a ratio of kinetic energy to gravitational potential energy for pendular motions. The Froude number has been used to characterize the dynamic similarity of gait patterns for bipedal and quadrupedal walking across species with a wide range of sizes (Alexander and Jayes 1983; Alexander 1989, 2004; Zatsiorky et al. 1994; Preuschoft 2004). However, the Froude number depends on a measure of leg length (L) rather than stride length (L_c) to normalize the gait parameters. Neither limb nor body length could be used as the length characteristic for the monkeys in our study, since they were approximately the same for the four animals (Table 1) and therefore would not limit the variance across different monkeys. Rather, the characteristic stride length, which represented a preferred pace for a particular animal, brought the data together across monkeys. This indicates that the variations in kinematics were caused by the different walking styles rather than by leg or body sizes, and implies that if all animals were to walk with the same preferred stride length, the composition of swing and stance durations would be exactly the same, and the kinematics would be a uniform function of walking velocity. The finding that it was possible to consolidate the data and form precise relationships between swing and stance durations, positions, and velocities among monkeys suggests that there is an internal representation of stride length that governs the gait of these quadrupeds, which is likely to contain a measure of compliance (Schmitt 1999, 2003; Larney and Larson 2004), which can be different among monkeys. These insights could be important in formulating system models of the control of quadrupedal gait in monkeys that can be compared to experimental data.

Signals driving the stance and swing phases

During the stance phases, since the movement of the limbs was largely dependent on the movement of the treadmill surface, there was no significant acceleration. Thus, the velocity and acceleration characteristics of the stance phases were governed by the velocity of the treadmill, and the predominant signal driving the locomotion was likely a proprioceptive velocity feedback signal from the limbs related to treadmill velocity. There must have also been a cognitive component during treadmill walking arising in the visual and vestibular systems, as the monkeys used ‘sensed’ walking velocity to guide them in their attempt to maintain a constant position in space. During overground walking, the more natural situation, the stance phases generate the forward velocity of the body in space related to ‘desired’ walking velocity, which entails a heavier cognitive component, although sensed walking velocity may also be important to maintain a steady gait when attention is directed elsewhere. Thus, in both conditions, the signal driving the limbs in the stance phases is related to walking velocity with a cognitive component. For convenience, we have termed these components ‘sensed’ and ‘desired walking velocity’.

Less obvious is the nature of the signal driving the swing phases. The phase plane trajectories of velocity versus position of the forward motion of the wrist and ankle during the swing phases (Fig. 8a–d) followed a unique sequence at each walking velocity from the onset of the movement (Fig. 8a–d), and the peak swing velocities were linearly related to walking velocity (Fig. 8e–h). The separation of the trajectories of the position/velocity relationship at each walking speed indicates that the dynamics of the swing phases had been determined at or before the beginning of the movement. From this, we postulate that the swing phases are also driven by a signal related to sensed or desired walking velocity, derived from information present during the stance phases, which sets the state of both the stance and swing phases and is responsible for timing and switching between the two. Consistent with this was the earlier finding that increases in the frequency of stimulation of the mesencephalic locomotor region in the decerebrate cat successively produced walking, running, and galloping (Shik et al. 1966), a direct indication that an increase in firing frequencies in descending pathways causes an increase in the velocity of locomotion.

Additional information about the nature of the central organization driving the swing and stance phases came from the analysis of the dynamics of the swing phases. In both the raw and normalized data the mean durations of the swing phases declined as walking velocity and amplitudes increased, but a new finding was that the duration of the swing phases increased with increases in amplitude at any single walking velocity. Thus, the swing durations and amplitudes had stochastic behavior in which variation in duration increased with variations in swing amplitude, although the means were lawfully associated with a decline in duration as walking velocity increased. This finding shows that the variations in swing duration and amplitude are governed by a main sequence relationship at any single walking velocity, which we infer to be one of the basic characteristics of the dynamical system that produces the swing phases. It was further possible to establish other characteristics of the central neural organizations driving the swing phases. In particular, the instantaneous changes in position and velocity were closely linked to walking speed. Consequently, the phase plane plots were characteristic at each walking velocity, and the circular trajectories were separable at each walking speed. Thus, not only was there a main sequence relationship between amplitude and maximum velocity at each walking velocity, but there was also an overall linear or main sequence relationship between amplitude and peak velocity that governed the swing phase dynamics. From this, we conclude that the swing phases are produced by pulsatile signals consistent with a suboptimal control neural strategy (Stark 1971), that the stance and swing phases are driven by independent neural mechanisms, and that both use signals that are based on and timed by ‘desired’ and/or ‘sensed’ walking velocity.

From the concentric circles of the phase plane plots we can conclude that the longer stride during the stance phase for higher walking velocity sets the initial condition for higher velocity swing and a close to undamped pendular motion. A neural feedback control alters the pendular frequency of the swing, giving rise to the appropriate swing amplitude and timing. This has been modeled for bipedal locomotion as a non-linear switching between stance and swing phases dependent on load together with a central feedback control law, which modifies the pendular frequency (Osaki et al. 2006a). While there must be some hierarchical central control to coordinate the fore- and hindlimbs, based on the phase plane and main sequence plots, the timing and switching that is accomplished for each limb should be similar to that for bipedal locomotion (Osaki et al. 2006a). We have recently studied human bipedal walking at a given velocity by imposing different frequencies of stepping. This shows that frequency of stepping and walking velocity can be independently controlled (Osaki et al. 2006b). However, because of the usual monotonic relationship between stepping frequency and walking velocity during natural locomotion (Osaki et al. 2006b), the gait dynamics can be related to walking velocity alone.

Comparison of locomotor patterns to nystagmus

The observation that the X-axis velocities of the swing phases along the X-axis on the treadmill had a striking resemblance to the velocity characteristics of rapid eye movements, i.e., saccades and quick phases of nystagmus, prompted a more detailed comparison of their dynamic characteristics. The dynamic characteristics of rapid eye movements are well known, but typical examples are presented here to facilitate comparison. During optokinetic nystagmus (OKN) that excites the visual system or during rotation in darkness that activates the vestibular system, the slow phases are linear as the eyes accurately follow the angular velocity of the stimulus (Fig. 9a, bottom trace; Cohen et al. 1977; Raphan et al. 1979). This is similar to the constant velocity stance phases that follow the velocity of the treadmill (Fig. 6b, h). At the conclusion of the slow phases, the quick phases reset the eyes with an impulsive sigmoidal movement (Fig. 9a, top trace; Cohen et al. 1977; Raphan et al. 1979), similar to that of the limbs in the swing phases (Fig. 6a, g). The frequency of nystagmus induced by vestibular and optokinetic stimulation generally ranges from 1.5 to 3–4 Hz (Cohen et al. 1977), which is close to the step frequencies, as derived from Fig. 2b (twice the stride frequency) despite the differences in mass and structure between the eyes and the legs. During each quick phase, eye velocity increases and decreases rapidly (Fig. 9a, bottom trace), as does the velocity of the swing phases (Fig. 6b, h). The peak velocities and amplitudes of the quick phases are linearly related (Fig. 9b), similar to the amplitude/velocity relationships of the swing phases along the X-axis (Fig. 8e–h). Thus, both have main sequence relationships, i.e., a linear relation between amplitude and peak velocity (Bahill et al. 1975), and the phase plane trajectories of eye velocity as a function of change in eye position are characteristic for each stimulus velocity for OKN (Fig. 9c) or for vestibular nystagmus (Fig. 9d). This is similar to the phase plane characteristics of the swing phases (Fig. 8a–d), and is based on the finding that the amplitudes of the quick phases or of the swing phases increase with either eye or walking velocity. Thus, the characteristics of either the swing phases or the quick phases of different amplitudes are dependent on stimulus or walking velocity. One obvious difference is that the quick phase durations increase as a function of increases in amplitude of the movements (Fig. 9e), whereas the mean durations of swing phases either decrease or are constant with increases in walking velocity (Figs. 2c, g, 3b, d) and swing amplitudes (Fig. 3b, d). As we have shown, however, the variations about the mean at any single walking velocity, i.e., the principal component durations of the swing phases also increase as a function of swing phase amplitude (Fig. 3a, c). Thus, there are many similarities between the dynamic characteristics of nystagmus, and the swing and stance phases of locomotion.

There are obvious differences in mass, bone, muscle, joints, and rotation axes between the legs and eyes that must be reflected in differences in the neural organizations that are driving the leg and eye muscles. Nevertheless, some important aspects of oculomotor function could also be utilized in locomotor control. In particular, a neural integrator in the brainstem transforms pulses from the visual and vestibular systems into positional signals that move the eyes during saccades and quick phases of nystagmus, with the amplitude of saccades being dependent on the number of pulses in the preceding bursts (Robinson 1971; Henn and Cohen 1972, 1973, 1976). This neural mechanism also integrates activity related to head velocity from the vestibular system to generate the eye position changes during slow phases of nystagmus (Robinson 1976) (see Raphan and Cohen 1981, 2002 for review).

Patla et al. (1985) proposed a model that simulated a pattern generator for the limbs, extending previous formulations of Brown (1911), Shik and Orlovsky (1976), and Miller and Scott (1977). The model utilized a constant input, which they related to walking velocity to excite a pattern generator that was modeled as a relaxation oscillator. In it, the basic functions of the oscillator were summed to form the muscle extensor and flexor patterns. Our study showing the relation of the phase plane trajectories and main sequence plots of the forward limb movement to walking velocity supports such a model. Similarly, in analogy with oculomotor control during nystagmus (Raphan 1976; Chun and Robinson 1978), we suggest that the basic functions that generate the muscle patterns in response to the walking velocity command in burst neurons in the spinal cord (Grillner and Dubuc 1988; Ivanenko et al. 2006) must be mathematically integrated to position the limbs. Whether this integration takes place centrally in the spinal cord or is due to the inertia of the muscles is not known. If these speculations are correct, however, they could have important consequences for identification of the coding in descending central pathways to the central pattern generator in the spinal cord and in identification of different neural groups in the CNS that control the limbs in the swing and stance phases.

Appendix

Kinematic data were normalized relative to the average stride length (characteristic length). This characteristic length was denoted as $L_c$, the stride frequency as $F_{\text{stride}}$, and the swing and stance durations as $T_{\text{swing}}$ and $T_{\text{stance}}$, respectively. From this, with the acceleration of gravity $g$, the normalized stride length $L_{\text{stride}}^n$, stride frequency $F_{\text{stride}}^n$, and swing/stance durations ($T_{\text{swing}}^n$ and $T_{\text{stance}}^n$) were represented as

$$L_{\text{stride}}^n=\frac{L_{\text{stride}}}{L_c}$$ (1)

$$F_{\text{stride}}^n=F_{\text{stride}}·\sqrt{\frac{L_c}{g}}$$ (2)

$$T_{\text{swing}}^n=T_{\text{swing}}·\sqrt{\frac{g}{L_c}}$$ (3)

$$T_{\text{stance}}^n=T_{\text{stance}}·\sqrt{\frac{g}{L_c}}$$ (4)

Similarly the original walking speed, V, was converted into a normalized walking speed $V^n$ and represented as

$$V^n=\frac{V}{\sqrt{g·L_c}}$$ (5)