The passive, human calf muscles in relation to standing: the non-linear decrease from short range to long range stiffness

Abstract

During human standing, tonic ankle extensor torque is required to support the centre of mass (CoM) forward of the ankles, and dynamic torque modulation is required to maintain unstable balance. Passive mechanisms contribute to both but the extent is controversial. Some groups have revealed a substantial intrinsic stiffness (65–90%) normalized to load stiffness, ‘mgh’. Others regard their methodology as unsuitable for the low-frequency conditions of quiet standing and believe the passive contribution to be small (10–15%). Here we applied low-frequency ankle rotations to upright subjects who were supported at the waist allowing the leg muscles to be passive and we report normalized stiffness. The passive calf muscles provided: (i) an extensor torque capable of sustaining unstable balance without tonic activity at a mean CoM–ankle angle of 1.6 deg, (ii) a long range stiffness of 13 ± 2% and (iii) a short range (< 0.2 deg) stiffness of 67 ± 8%. Chordal ankle stiffness, derived from the torque versus angle relationship for 7 deg rotations, shows a non-linear decrease (stiffness α rotation^{−0.33±0.04}) from 101 ± 9% to 19 ± 5% for rotations of 0.03–7 deg, respectively. Thus, passive stiffness is well adapted for the continuum of postural and movement activity and has a substantial postural role eliminating the need for continuous muscle activity and increasing the unstable time constant of the human inverted pendulum. Ignoring the non-linear dependence of passive stiffness on sway size could lead to serious misinterpretation of experiments using perturbations and sensory manipulations such as eye closure, sway referencing and altered support surfaces.

During human standing the whole body centre of mass (CoM) is usually maintained several centimetres in front of the coaxial ankle joints (Gatev et al. 1999). A continuous ankle extensor torque is required to prevent forward toppling of the body and dynamic modulation of ankle extensor torque is required to regulate the unstable balance of the body. It is generally recognized that passive properties of joints, ligaments and muscle–tendon units have a tonic postural role in relieving muscle from the fatiguing burden of continuous activity (MacConaill & Basmajian, 1977). Dynamic postural control benefits from a variety of short range passive phenomena including short range stiffness (Rack & Westbury, 1974), thixotropy (Lakie et al. 1984; Campbell & Lakie, 1998), cartilage deformation (MacConaill & Basmajian, 1977), low-speed muscle viscosity (Winters et al. 1988) and the mild, passive friction-like and stiction-like properties of a joint complex (Winters et al. 1988).

The term passive can have two meanings – absence of neural modulation or absence of active contraction. In papers on human standing, passive often means absence of neural modulation and this can apply to active or to inactive muscle. In the experiments of this paper we use the term passive to mean both absence of active contraction and the absence of neural modulation.

Humans usually display continuous calf muscle activity in the act of standing. This observation is implicit in modelling which assumes a continuous, time-invariant, active control system for maintaining upright balance (Peterka, 2002; Maurer & Peterka, 2005b; Masani et al. 2006) but whether or not this continuous muscle activity is necessary is controversial. Electromyography (EMG) has shown that in most subjects soleus at least is active tonically (Basmajian & De Luca, 1985; Horak et al. 1996); however, some observe that during quiet standing, triceps surae activity is not continuous (Gatev et al. 1999) and that individual subjects can have periods of very low muscle activity with minimal modulation (I.D. Loram, C.N. Maganaris & M. Lackie, unpublished observations). Moreover, at least one study has reported that the upright human configuration can be maintained entirely passively (Kelton & Wright, 1949). These authors found periods in which the soleus and tibialis anterior muscles were simultaneously inactive for periods of up to 32 s. The possibility of entirely passive balance depends finely on the position of the CoM in relation to the ankle joint and also on the existence of a passively stable zone within which neither plantarflexor nor dorsiflexor muscle activity is required (Kelton & Wright, 1949). Here we ask: where would the CoM be in relation to the ankle joint if the ankle extensor torque required for balance were entirely passive?

For several decades, there was a body of opinion advocating that the passive (no neural modulation) stiffness of the activated calf muscles was sufficient to ensure dynamic stability of the unstable human ‘inverted pendulum’ in the face of small changes of joint position (Gurfinkel et al. 1974; Nashner, 1976; Winter et al. 1998). In a contemporary review, considerable prominence was given to the passive role of the calf muscles in maintaining upright stance (Horak et al. 1996). More recently, two groups investigated this passive or intrinsic stiffness using small (0.06 and 1 deg, respectively) ankle rotations. They both found the stiffness to be less than 100% of the gravitational toppling torque per unit angle ‘mgh’ and thus insufficient for static stability, and yet large enough to make a substantial contribution to stabilization. The reported values were 91%mgh (Loram & Lakie, 2002b) and 65%mgh (Casadio et al. 2005). The very recent ultrasound demonstration of ‘paradoxical muscle movements’ in human standing has shown beyond doubt that the stiffness of the calf muscles is limited by a tendinous series elastic component (SEC) of less than 100%mgh with 85–90% being the estimated SEC stiffness (Loram et al. 2004, 2005a,b). We, the authors, assumed that the contractile portion of the muscle was considerably stiffer than the SEC.

While we believed the matter was settled, recent publications (Peterka, 2002; Jeka et al. 2004; Loram et al. 2004; Maurer & Peterka, 2005b) and conference communications (Maurer & Peterka, 2005a) have demonstrated another body of opinion which understands the passive (no neural modulation) ankle stiffness to be low at around 10–15%mgh or less. These low values of passive stiffness are supported by fitting transfer functions to human ankle torque responses to continuous support surface perturbations of peak-to-peak amplitudes varying from 0.5 to 8 deg (Peterka, 2002). Furthermore, the ankle stiffness values of Loram & Lakie (2002b) and Casadio et al. (2005) were discounted for the reason that the short-duration individual perturbations given were equivalent to a high frequency (7 Hz; Loram & Lakie, 2002b) and therefore not relevant to the low-frequency ankle rotations in quiet standing (∼0.5 Hz). This argument was strengthened theoretically by using a linear Hill-type model (see following companion paper (Loram et al. 2007)) which predicts that the combined stiffness of the muscle–tendon unit is independent of the size of ankle rotation, but highly dependent on the frequency of ankle rotation. At high frequency, perturbations would not measure the ankle stiffness that is relevant at low frequency.

The existing literature provides some information concerning this controversy. It has been shown that ankle stiffness increases as stretch size decreases (Kearney & Hunter, 1982) although this work did not reveal whether the increase in stiffness was a consequence of changes in sensory modulation (reflexes) or was an intrinsic (non-sensory) property of the ankle joint. Additionally, this work did not study stretches below 1 deg and so did not reveal the stiffness at the small rotations of quiet standing which are approximately 0.05–0.2 deg (Loram & Lakie, 2002b). Also, for dorsiflexion rotations of 1 deg and higher stretching the tibialis anterior muscle it has been shown that the ankle is stiffer for 1 deg ankle rotations than larger ones although the values of stiffness were low for inactive muscle (∼10%mgh) (Sinkjaer et al. 1988). Stretching of cat soleus muscles has revealed that muscle stiffness changes little if at all with frequency and that amplitude of stretch is the most important factor with velocity being secondarily important (Rack & Westbury, 1974). For short stretches, there is a short range stiffness which is considerably higher than the long range stiffness. However, the extensive work of Rack and Westbury also revealed that the range of short range stiffness diminishes when the velocity of stretch is very low (Rack & Westbury, 1974). This is further confirmed in the experiments by Nichols & Houk (1976) on denervated cat soleus muscle. In attempting to extrapolate from animal work to human muscle stretches in quiet standing, it became clear that the velocity of muscle lengthening in quiet standing is very low and it was uncertain whether one would observe a functionally useful short range stiffness or not. Several investigations have confirmed that relaxed human limbs have a disproportionately large stiffness for small movements (Lakie et al. 1984; MacKay et al. 1986; Axelson & Hagbarth, 2001) and this is always attributed to the short range stiffness of muscle (their short range elastic component (Hill, 1968). However, the size of the effect has not generally been quantified but one relevant study (Hufschmidt & Schwaller, 1987) indicates that human passive, short range ankle stiffness has a low value of 10–15%mgh for stretches at 1–40 deg s⁻¹ which are an order of magnitude faster than those encountered in quiet standing.

For passive muscles we have used slow ankle rotations to investigate the short range and long range torque–angle relationship at the ankle joint. In this paper we address the following questions: (i) At what CoM angle from the vertical does passive extensor torque balance the gravitational torque and how does this compare with the angle at which subjects normally stand? (ii) How does the low-frequency ankle stiffness progress from the short range of quiet standing to the longer range of perturbed standing and sensory degradation experiments? (iii) What are the implications of passive short range stiffness for postural balance and the study of standing?

In the companion paper, we report the use of ultrasound measurements to investigate the progressive contribution of contractile and series elastic components to ankle stiffness as stretch size increases.

Methods

Subjects and set-up

Ten healthy subjects, seven women and three men, aged 26 ± 12 years (mean ±s.d.), stood quietly while strapped securely around the hips to a vertical supporting board (Fig. 1A). Subjects stood on two footplates with the centre of their ankles approximately 22 cm apart. Their ankles were positioned to be coaxial with the axis of rotation of the footplates. Series of slow footplate rotations were applied to the subject who was instructed to keep their leg muscles inactive. The subjects gave written informed consent to these simple non-invasive experiments, and the study was approved by the local human ethics committee and conformed to the principles of the Declaration of Helsinki.

Figure 1. Set up and apparatus.

A, apparatus. The subject was maintained in a vertical configuration, and was supported securely against the adjustable back support by a strap around the pelvis. The calf muscles were passive. The subject had audible and visual feedback of their EMG. Subjects' ankles were coaxial with the axis of rotation of the two separate footplates (F). The footplates were connected through horizontally mounted load cells (LC) to a mobile horizontal platform (P) which shared its axis of rotation with the footplates and ankles. The platform was rotated by a servo motor (not shown) which drove it through a precision toothed belt and gearwheel (G). The applied movements were large or small slow, low-frequency ankle rotations. The small rotations were representative of those during quiet standing. We recorded ankle torque from each leg using the load cells, footplate angle using a contactless potentiometer, EMG from the left calf muscles and tibialis anterior. B, joint stiffness model. Ankle torque is generated mechanically in response to footplate rotation (θ) relative to the backboard by a spring, K, and a damper, B, F, representing whole joint ankle stiffness and viscosity-friction.

Apparatus and measurements

Rotation of the footplates was achieved using a custom-built, servo motor geared down using a steel reinforced precision timing belt. The angle of the footplates was recorded using a precision, Hall effect, contact-less potentiometer of resolution 0.002 deg. Ankle torque from each leg was recorded using a horizontally mounted load cell with resolution 0.03 Nm. Having shaved and cleaned the skin, surface EMG (Delsys, Boston, MA, USA) were recorded (left soleus, gastrocnemius medialis, tibialis anterior and gastrocnemius lateralis), amplified (10 000×) and band-pass filtered (20–450 Hz). All signals were sampled at 1000 Hz and recorded to 16-bit resolution on computer (Measurement computing PCI-DAS6036, MATLAB).

Procedure

A common sequence was followed for each subject.

1. We wanted to establish the angle of the CoM at the ankle joint as accurately as possible. The angle of the footplate relative to the vertical was known precisely. In order to establish the angle of the body accurately, the horizontal position of the vertical supporting board was adjusted at hip and shoulder level to support the subject as closely to the true vertical as possible. A plumb line was used to ensure a vertical through the subject's shoulders, hips and ankles.

2. The subject was instructed to relax the leg muscles completely and maintain even weight on both legs. The experimenter monitored the left and right ankle torques while visual and audio feedback of tibialis anterior and soleus EMG helped the subject minimize their muscle activity. The secure support at the hip made it possible to relax the leg muscles while maintaining the upright configuration.

3. The footplates were rotated at constant speed from −1.5 deg (toes down) to 5.5 deg (toes up) and back to −1.5 deg, and were repeated giving four unidirectional rotations each of 20 s in the 80 s trial. This trial was repeated once.

   Procedures (1)–(3) enabled us to alter the angle between the CoM and the footplate and determine the resultant passive torque.

4. For subsequent experiments we wanted the subject to be maintained passively at their normal standing configuration since it is known that ankle stiffness depends on ankle angle (Mirbagheri et al. 2000). The subject was released and the vertical supporting board was withdrawn so that the subject stood freely. The tonic ankle torque in both legs was recorded. The board was then moved forwards to the subject and adjusted at hip and shoulder level so that the subject was strapped to the board at their normal standing configuration including their normal ankle angle. The footplates were adjusted to be horizontal.

5. The subject was again instructed to relax their legs using EMG feedback and maintain even weight on both legs.

6. A series of small, slow ankle rotations was applied. Each 40 s trial contained continuous, symmetrical, up and down, ramp rotations of one of three sizes (0.06, 0.15 and 0.4 deg), and one of three unidirectional durations (1, 2 and 5 s). Since the response was immeasurable the smallest rotation was not applied at the durations of 2 and 5 s. These trials were not repeated (as reported in the companion paper, we also recorded ultrasound from the calf muscles, the analysis of which imposed severe data processing limitations on the total number of trials). The size and duration of these stretches were chosen to correspond to the typical size and duration of unidirectional sways about the ankle during quiet standing which are 0.1–0.15 deg over 1–1.5 s (Loram et al. 2005b).

Measurement of gravitational load stiffness ‘mgh’

In a separate experiment on a different occasion we measured the load stiffness (gravitational toppling torque per unit angle) of each subject. Subjects stood on a board, freely pivoted coaxially with the ankle joints, with their back against a body-length, lightweight splint rigidly attached to the footplate, and were supported at various angles backwards from the vertical by a steel cable attached to the wall. Their angle from the vertical was measured using a laser rangefinder, reflected off the splint. Their torque was measured using a very stiff load cell (K25 Inscale Technology Ltd, UK) in series with the steel cord and at a known moment arm from the ankle joints. The load stiffness was calculated from regression of torque against angle.

Data analysis

The EMG signals were digitally rectified and subsequently low-pass filtered using the second order transfer function

where s is the Laplace variable and the time constant τ is 0.16 s. This time constant is longer than is sometimes used, but is justified as giving the best fit between the observed changes in filtered EMG and changes in ankle torque (separate paper under preparation). After filtering of EMG, all signals were down-sampled to 100 Hz.

For the long (7 deg) ankle rotations we calculated the angle of passive balance from the intersection of the left torque versus ankle angle curve from the line of gravitational torque versus angle divided by 2 for the left leg. The normal angle of standing was computed by dividing the tonic standing torque by the gravitational load stiffness.

Also, for the long ankle rotations we divided each trial into four unidirectional stretches demarcated by the zero crossings in footplate velocity. From the start of each unidirectional stretch we calculated the magnitude change in torque, the magnitude change in footplate angle and we calculated the chord stiffness as

where T is left ankle torque, T₀ is left ankle torque at the preceding reversal, A is footplate angle, A₀ is footplate angle at the preceding reversal and K_chord(A – A₀) is chord stiffness as a function of angular rotation from the preceding reversal.

Parallel model of whole joint ankle stiffness

For each trial, including long and short ankle rotations, the stiffness, viscosity and friction of the entire ankle joint was calculated. We used a simple model (Fig. 1B), reconstructing the left ankle torque from the applied footplate rotation. The model is defined by the equation:

where the variables are T, left ankle torque and θ, footplate angle, and the coefficients are K, whole joint ankle stiffness, B, whole joint viscosity and F, whole joint friction. The sign function has values 1, 0 and −1 for arguments which are positive, zero and negative, respectively. The viscosity term was not used for the long ankle stretches.

Visual inspection of the data showed the existence of occasional irregular fluctuations in EMG that were clearly related to corresponding changes in ankle torque. To test whether neural modulation accounted for the changes in torque that we were attributing to footplate rotation, we constructed a second, slightly more complex, parallel model incorporating the measured EMG signals. The model is given by the equation:

where E_i are the filtered EMG for each of the four muscles described above and k_i are the coefficients for each filtered EMG. The mechanical coefficients were compared with the values calculated using the non-EMG model.

Multiple linear regression was used to calculate best fit values of the coefficients K, B, F and k_i. These models assume that all variables are independent, i.e. orthogonal without co-linear variation. Co-linearity between the EMG signals is of no consequence for this analysis since we are only interested in the combined EMG contribution to changes in ankle torque and so no attempt was made to decompose the four EMG signals into orthogonal components.

For the smallest rotations (0.06 deg), the regression model yielded considerably lower values of R² (46 ± 25% for the ‘without EMG’ model) than other rotation sizes and so this rotation category was excluded from further analysis.

In a preliminary experiment to test the veracity of the apparatus and measurement procedure, springs of known stiffness varying from 0.4 to 1.3 Nm deg⁻¹ were attached to the footplates from the apparatus frame. The stiffness of the springs was measured with a mean error of 7% and a coefficient of variation of 7%. The friction and viscosity of the left footplate was measured as 0.002 Nm and 0.12 Nm s deg⁻¹, respectively, at the prevailing 3 Nm torque.

Statistical analysis

Using multiple linear regression, all parameters calculated for the short rotation trials (< 0.5 deg) were regressed against rotation size and rotation speed. Where the combined regression model was significant, significant variation individually with size and speed was reported at the 95% confidence level. Throughout the paper the values of all measurements and coefficients are quoted as the mean ± standard deviation (s.d.) unless otherwise stated.

Results

Using surface EMG, it is impossible to know whether or not the muscles were completely inactive. However, subjects achieved levels of activity that were indistinguishable visually and aurally from the background noise level and which were clearly distinguishable from any active movement or slight generation of force. This passive or near-passive state was also associated with a greatly reduced neural responsiveness to footplate motion. The passive state was also distinguishable from the active state using ultrasound. When the muscle is inactive and the foot unsupported, the ankle is naturally plantar flexed and the white streaks of collagen in the ultrasound image appear fuzzy. When the subject is passively maintained in the standing configuration, the ankle joint is slightly dorsi-flexed which induces stretch of the muscles and the white streaks of collagen appear relatively well defined and very still (cf. movie in Supplemental material for companion paper). This was distinguishable from the active state in which there is considerable movement in the live images. One subject had a consistent, low level activity in tibialis anterior which they could not reduce.

Long passive stretches

Typical features

We observed consistent short and long range torque responses from all subjects to 7 deg ankle rotations (Figs 2 and 3). The constant velocity, 20 s duration, ankle rotations (Fig. 2A) were generally unaccompanied by modulation of EMG with either time (Fig. 2B) or angle. The irregular low-frequency ripples in EMG were distinguishable from the noisier signals of active muscle. With the subject vertical, starting at a footplate angle of −1.5 deg (toes down) relative to the horizontal, there was invariably an initial steep rise in left torque of about 1 Nm over 2 s duration and through a rotation of 1 deg (Figs 2C and 3A). This was followed by a brief shoulder for less than a second, followed by a more uniform and less rapid increase in torque with both time and angle. When the footplates reverse at 20, 40 and 60 s, there was an initial increased change in torque with time and angle graduating continuously to a more constant, lesser rate of change in torque with time and angle (Figs 2C and 3A). Inspection of the initial rise and first reversal in torque for all subjects (Fig. 2D) confirms that the initial increased change in torque is observed generally. As shown by the dotted lines in Figs 2C and 3A the linear regression model, without EMG, provides a good, coarse representation of the long range gradient and the torque hysteresis.

Figure 2. Representative long stretches through time.

Slow, 7 deg ramp, ankle rotations were applied. A, footplate rotation ankle. The subject was at the true vertical. Zero degrees represents a horizontal footplate and positive angles are toes up. B, EMG from left leg, soleus (upper dotted), gastrocnemius medialis (lower continuous), gastrocnemius lateralis (lower dotted), tibialis anterior (upper continuous). C, left ankle torque (continuous) and model predicted torque (dotted). D, change in left ankle torque with time for the first two unidirectional ankle rotations is shown for the remaining 9 subjects. The changes from the previous footplate reversal, starting at times 0 and 20 s are shown as starting at the same time from zero torque. For display purposes each subject is offset along the time axis.

Figure 3. Representative long stretches with angle.

Slow, 7 deg ramp, ankle rotations were applied. Panels show same data as Fig. 1 but as changes with footplate angle: A, left ankle torque (continuous), gravitational torque scaled for one leg (inclined dashed), normal standing angle (vertical dotted). B, in order to compare each unidirectional rotation, all four unidirectional rotations are overlaid. The 2nd and 4th rotation have been reversed so that all changes in torque and angle proceed from bottom left to top right. Thus the log–log axes show magnitude change in ankle torque since the preceding footplate reversal (0, 20, 40 60 s) versus magnitude change in footplate angle. A straight line on a log–log plot represents a non-linear power law relationship. The fourth rotation is truncated at the point where an irregular fluctuation in EMG occurs. C, chord stiffness versus magnitude change in angle. Chord stiffness means the average stiffness over a certain rotation, thus chord stiffness = magnitude change in torque/magnitude change footplate angle. All four unidirectional rotations are superimposed and the fourth is truncated. D, magnitude change in ankle torque versus magnitude change in footplate angle. For all 9 remaining subjects all four unidirectional rotations are shown. The origins are offset for display purposes.

It is interesting to compare the first with subsequent unidirectional rotations. For the representative subject, and for most subjects, ankle torque was lower at the end of the second unidirectional rotation than it was at the start of the first (Figs 2C and 3A) However, during subsequent unidirectional rotations the change in torque was equal to the second rotation provided there was no irregular burst of EMG (Figs 2C and 3A). When each of the four unidirectional rotations are overlaid to proceed from left to right by plotting the magnitude change in torque against the magnitude rotation, it can be seen that the torque–angle relationship is consistent for all rotations, particularly for the first 0.5 deg (Fig. 3B). Compared with subsequent unidirectional rotations, the first rotation reveals a slight reduction in torque beyond 0.5 deg. These observations were consistently observed in all subjects as shown by Fig. 3D, which plots all four unidirectional rotations overlaid for each remaining subject. Considering each unidirectional rotation and calculating the chord stiffness, i.e. the mean stiffness acting over an angular change, Fig. 3C shows that for each unidirectional rotation, chord stiffness decreases non-linearly as angular rotation increases.

For this representative subject, the angle at which passive balance is possible is shown by the intersections between the inclined load stiffness line and passive torque–angle curves. The angle of passive balance ranged from 0.5 to 1 deg corresponding to a 2 Nm disparity in the ascending versus descending passive torque–angle relationship (Fig. 3A). The self-chosen standing angle of this subject (vertical dotted line) was 2.3 deg, requiring a continuous 7–9 Nm of active torque.

Summary of all subjects for long stretches

When all unidirectional rotations are combined and shown on a log–log plot, subjects show a consistent relationship between magnitude change in torque and ankle rotation progressing smoothly from 0.03 to 7 deg rotations (Fig. 4A). The similarity in gradient of different subjects is striking. Likewise, chord stiffness of the left leg shows a consistent non-linear relationship progressing smoothly from 5.6 ± 1.4 to 1.0 ± 0.4 Nm deg⁻¹ for 0.03–7 deg rotations, respectively (Fig. 4B). When normalized to the load stiffness (mgh), chord stiffness progresses smoothly from 101 ± 9% to 19 ± 5% for rotations of 0.03–7 deg, respectively (Fig. 4C). The near-straight line on a log–log plot indicates a power relationship between chord stiffness and angle. Each subject was well described by such a relationship (Table 1) and the mean coefficients for all subjects give a numerical approximation

Figure 4. Summary of long stretches.

A, magnitude change in torque versus magnitude footplate rotation. For A and B, each dotted line represents the mean of four unidirectional rotations for one subject. Continuous line represents mean for all subjects. C, normalized chord stiffness versus magnitude footplate rotation. Continuous line represents mean for all subjects. Dashed lines represents 95% confidence limits for mean value. D–F, for all subjects from all trials, panels show the median, interquartile range (box edges), range (whisker ends) and outliers (crosses), for D long range stiffness (K) and friction (F), for E long range stiffness relative to load stiffness mgh, and for F the upper (Pas+) and lower (Pas−) intersection of the passive torque angle curve and load stiffness, and the normal standing angle (stand). Notches represent a robust estimate of the uncertainty about the medians for box-to-box comparison.

Table 1.

Power law approximation for chord stiffness versus ankle rotation

Subject	Load stiffness (Nm deg−1)	a	b	R2	P
1	7.9	−0.533 ± 0.001	0.386 ± 0.002	0.99	< 0.001
2	10.6	−0.538 ± 0.001	−0.389 ± 0.002	0.99	< 0.001
3	12.8	−0.374 ± 0.002	−0.273 ± 0.002	0.97	< 0.001
4	10.3	−0.543 ± 0.001	−0.325 ± 0.001	0.99	< 0.001
5	8.9	−0.487 ± 0.001	−0.354 ± 0.001	1.00	< 0.001
6	15.5	−0.424 ± 0.001	−0.332 ± 0.002	0.99	< 0.001
7	11.0	−0.307 ± 0.001	−0.243 ± 0.001	0.99	< 0.001
8	9.0	−0.496 ± 0.001	−0.359 ± 0.002	0.99	< 0.001
9	13.8	−0.454 ± 0.001	−0.320 ± 0.001	1.00	< 0.001
10	10.0	−0.480 ± 0.001	−0.353 ± 0.002	0.99	< 0.001
Mean	11.0	−0.464	−0.334	0.99	< 0.001
s.d.	2.3	0.073	0.044	0.01

For each subject, the equation log(K) = log(a) –b log(θ) was fitted to the chord stiffness (K) versus rotation angle (θ) shown in Fig. 4B. This logarithmic equation is equivalent to the power law relationship given by . For each subject is shown the load stiffness, the coefficients a and b with ± 95% confidence intervals, the R² and the P values.

Although this formula gives values lower than the empirical ones by 5%mgh in the region 0.15–0.4 deg, nonetheless it is clear that short range and long range stiffness can be approximated as a single, consistent relationship.

Multiple regression was used to represent the torque–angle relationship as a long range stiffness and a zero range friction. From this analysis, without EMG included, the passive long range stiffness (K) in the left leg for all subjects was 0.81 ± 0.3 Nm deg⁻¹ corresponding to a normalized stiffness (K/mgh) of 15 ± 3. Although the short range stiffness is not really a friction but is a stiffness acting for a limited angular extent, the passive friction coefficient (F), 2.0 ± 1.2 Nm (mean ±s.d.), provides a measure of the change in torque associated with short range stiffness. The mean coefficient of determination (R²) was 93 ± 6%. Including EMG in the regression model increased R² to 97 ± 2% and gave values of 0.73 ± 0.2 Nm deg⁻¹, 13 ± 2% and 0.7 ± 0.3 Nm, respectively, for K, K/mgh and F (Fig. 4D and E). The similar values of long range stiffness, K, for the ‘with EMG’ and ‘without EMG’ models confirms that mechanical rotation, not neural modulation is responsible for the estimated value.

Passive extensor torque alone is capable of sustaining unstable balance at upper and lower angles of 1.5–1.9 ± 0.4 deg whereas subjects chose to stand at 3.1 ± 0.9 deg corresponding to a continuous active torque of 6 to 9 ± 4 Nm (Fig. 4F). One subject, with minimal muscle activity in all four muscles, stood normally at an angle of purely passive balance.

Short passive stretches

The passive torque response to small, slow ankle rotations is shown in Figs 5–7. Figures 5 and 6 show representative changes in torque versus time and angular rotation versus time, respectively. Figure 7 summarizes the results for all subjects. All three figures show that passive stiffness is substantial for small slow stretches and that the stiffness decreases as the size of ankle rotation increases.

Figure 5. Representative short stretches through time varying size.

Short rotations of varying size at constant speed were applied. A, B and C show 0.06, 0.15 and 0.4 deg rotations. The duration of single stretches is 1, 2 and 5 s, respectively. The steady state response is shown since 10 s of ramps have already elapsed. Rows in descending order show: (1), left ankle torque (continuous) and model prediction (dotted); (2) EMG from left leg, soleus (upper dotted), gastrocnemius medialis (upper continuous), gastrocnemius lateralis (lower dotted), tibialis anterior (lower continuous – electrode accidentally detached); (3) footplate rotation ankle. The subject was at their normal standing angle. Zero degrees represents a horizontal footplate and positive angles are toes up.

Figure 7. Comparison of short and large ankle rotations.

A, left ankle stiffness (K); B, normalized ankle stiffness (K/mgh). Values are plotted for all 0.15 and 0.4 deg rotations from all subjects to show the variation in stiffness with the size of ankle rotation. For both quantities shown there was no significant variation with rotation speed. For A the dashed lines show the best fit and 95% confidence regression lines. For B the continuous line shows the progressive chord stiffness from the long ankle rotations to allow comparison with the short rotations. Continuous line is mean for all subjects and dashed lines are 95% confidence intervals. C, normalized ankle stiffness (bars) and R² (crosses) of ‘with EMG’ model. C shows mean stiffness +s.e.m. for each subject at different rotation amplitudes.

Figure 6. Representative short stretches with angle varying size.

Short rotations of varying size at constant speed showing the same data as Fig. 5 but with respect to ankle angle. A, B and C show 0.06, 0.15 and 0.4 deg rotations. The durations are 1, 2 and 5 s, respectively. The steady state response is shown since 10 s of ramps have already elapsed. Rows in descending order show: (1) left ankle torque (continuous) and model prediction (dotted); (2) EMG from left leg, soleus (upper dotted), gastrocnemius medialis (upper continuous), gastrocnemius lateralis (lower dotted), tibialis anterior (lower continuous – electrode accidentally detached).

Typical changes with size of stretch

With the subject at their normal standing configuration, we applied short, continuous ankle rotation ramps of 0.06, 0.15 and 0.4 deg at a mean speed of 0.08 deg s⁻¹. The left ankle torque showed a largely elastic steady-state response (Figs 5A–C and 6A–C) and, apart from the 0.4 deg rotation, with minimal hysteresis. From inspection of the torque versus angle gradient (Fig. 6A–C), the stiffness is substantial and decreases slightly with the size of rotation. Inspection shows no angle-related modulation of EMG. The linear regression model, without EMG, shows very close agreement with measured torque (Fig. 5).

Summary of all subjects for short stretches

Using the ‘without EMG’ model, left ankle stiffness (K) was 3.0 ± 1.0 and 3.6 ± 0.9 Nm deg⁻¹, respectively, for 0.4 and 0.15 deg rotations (Fig. 7A). Using the ‘with EMG’ model raised the R² from 76 ± 26% to 85 ± 14% (Fig. 7C) and gave values of 3.0 ± 0.9 and 3.6 ± 0.8 Nm deg⁻¹, respectively. The stiffness coefficient reflects mechanical not neural modulation of ankle torque because it changes little when EMG is added to the model. Regression showed significant variation of ankle stiffness with size but not speed of stretch (R²= 16%, P < 0.004).

Relative ankle stiffness (K/mgh) was 54 ± 12% and 67 ± 10% for 0.4 and 0.15 deg rotations, respectively (Fig. 7B). Again, regression showed significant variation of relative ankle stiffness with size but not speed of stretch (R²= 29%, P < 0.0001). Typical ankle rotations during upright stance are 0.13 deg (Loram et al. 2005b). For this size of rotation the normalized stiffness would be close to 67 ± 10%, which is considerably greater than the long range value of 13 ± 2% (Fig. 7C).

The values of normalized stiffness obtained using short rotations may be compared with the relative chord stiffness obtained from the 7 deg rotations. It can be seen (Fig. 7B) that the values derived from short, slow rotations are entirely consistent with decrease in ankle stiffness with rotation size derived from the long slow rotations.

Using the ‘with EMG’ model, the friction of the ankle joint is negligible (0.004 ± 0.007 Nm) and indistinguishable from that of the footplates; and the viscosity is small (0.7 ± 0.6 Nm s deg⁻¹).

Discussion

On the basis of these experiments we are in a position to consider the passive, postural contribution of the calf muscles to (i) the tonic ankle extensor torque and (ii) the dynamic regulation of unstable balance.

The tonic, passive contribution of the calf muscles

Passive ankle extensor torque is capable of balancing the gravitational toppling torque at a mean CoM angle of 1.4–1.8 deg forward from the vertical (Fig. 4C). For a CoM at a typical 90 cm above the ankle joints this equates to a position 2–3 cm in front of the ankle joint. This result is consistent with the finding of Kelton & Wright (1949) who showed that continuous calf activity is not necessary for maintaining upright balance.

Our own unreported observations of periods with minimal activity and minimal modulation of activity (from dataset of Loram et al. 2005a), the muscle activity dead zone reported by Kelton & Wright (1949) and the inherent anatomical stability of the ankle joint at this position (MacConaill & Basmajian, 1977) all support the idea of a small range of CoM angles and velocities at which continuous neural regulation of ankle torque is not required.

In addition to the anatomical stability of the postural ankle configuration, it appears there is an additional physiological adaptation of passive muscle to tonic postural function. Following the first complete up–down rotation of footplate, the resting torque (torque that prevailed before the initiation of the rotation) was decreased by approximately 1 Nm (Fig. 3A). This result parallels the observations of a filamentary resting tension in stationary, passive amphibian muscle (Hill, 1968). Muscle stretch causes a reduction in resting tension below the pre-stretch level and this tension restores with a time constant of several seconds when the muscle is still. This temporal effect is distinct from short range stiffness and under postural conditions may provide additional passive torque which is dispelled during movement.

For one subject, it was normal to sustain the CoM within this passive balance region. Most subjects maintain the CoM an equivalent 2–3 cm further forward from the ankles requiring 6–9 Nm continuous active torque for each leg. Maintenance of the forward position would be more subject to muscle fatigue and is possibly more unstable than the passive balance region. Beyond that, this experiment offers no insight into whether there are advantages to maintaining the forward position or whether the forward position simply represents the imperfections of normal human balance.

The passive muscle contribution to dynamic balance

Long stretches

The 7 deg rotations that we applied revealed a progressive non-linear change in ankle torque (Fig. 4A) and chord stiffness (Fig. 4B and C) such that the short range chord stiffness was high, diminishing from 101%mgh at 0.03 deg to 19% for 7 deg rotations. The long range passive stiffness is low (13%mgh) (Fig. 4E), which is in good agreement with previous studies (Peterka, 2002). This non-linear torque–angle relationship was generally un-associated with changes in EMG or with tonic muscle activity and is thus a property of passive muscle. The enhanced, short range stiffness was reliably repeated on subsequent stretches (Figs 3C, D and 4A, B and C) which implies that it is a steady-state property of passive muscle. The best numerical description of the chord stiffness represents it as proportional to the inverse cube root of rotation size (rotation^−1/3). This tantalisingly suggests the possibility of an explanation rooted in the change in dimensions of the structures affected by ankle rotation rather than a physiological explanation. Sadly, we cannot further substantiate this line of thought.

The relatively abrupt changes in torque associated with the first stretch, and not subsequent stretches, was apparent after 1 deg of rotation (Fig. 3A, B and D). This apparent thixotropic change of potential significance for posture and movement requires closer investigation in a further study.

Short range and long range stiffness

The long stretches that we applied (Figs 2 and 3) revealed an enhanced change in ankle torque, or short range stiffness, when the direction of ankle rotation is reversed. This occurred when the ankle joint angle reversed at angles of −1.5 and +5.5 deg. This information alone does not determine whether the same short range stiffness would be found at the typical ankle angle of standing (+3 deg) using short rotations. By applying small ankle rotations of size, speed, duration, frequency and ankle position typical of those encountered in quiet standing we found that the short range stiffness is consistent through time as shown by the consistent torque responses (Figs 5 and 6), diminishes as the size of rotation increases (Figs 4C and 7 and Table 1) and shows no change with the velocity of ankle rotation. By implication, we predict that changing the frequency of a sinusoidal rotation of certain amplitude should make no difference to the short range stiffness. These stiffness values were explained by mechanical footplate rotation and not by neural modulation of ankle torque and are thus a property of passive muscle. The values of ankle stiffness obtained by short stretches of less than 0.5 deg agree well with the progression in chord stiffness derived from the long 7 deg stretches (Fig. 7B).

At a typical standing rotation of 0.13 deg (Loram et al. 2005b), the normalized stiffness (K/mgh) of 67 ± 10% is sufficient to make an important contribution to stabilization in quiet standing. At a mean value of 0.7 Nm s deg⁻¹ the viscosity of the torque response is low and at a typical standing rotation speed of 0.1 deg s⁻¹ cannot have a meaningful stabilizing effect. Moreover, this value is barely beyond the measured viscosity of the footplate.

Comparison with previous authors

How do these values agree with those provided by previous experiments?

The passive stiffness values of 3–4 Nm deg⁻¹ (Fig. 7A) are similar to values of 4 ± 2 Nm deg⁻¹ per leg calculated for averaged spontaneous sways of similar size during human balancing of an inverted pendulum (Loram & Lakie, 2002a).

For perturbations of 1 deg, our results predict a passive, normalized chord stiffness of 30–40% (Fig. 4C) which is lower than the 65 ± 8% obtained by Casadio et al. (2005) using discrete 1 deg, 7 deg s⁻¹ ramp rotations. The difference might be explained by the 20 times higher rotation speed and higher tonic muscle activity studied by Casadio et al.

For footplate rotations of 0.055 deg, the passive, normalized chord stiffness is 83–93% (Fig. 4C) which is close to the value of 91 ± 23% obtained by Loram & Lakie (2002b) using discrete, raised cosine, 0.055 deg, 0.7 deg s⁻¹, 7 Hz rotations. This agreement between the current 0.025 Hz rotations and the previous 7 Hz rotations implies that frequency is not a significant factor determining short range stiffness. The previous experiments (Loram & Lakie, 2002b) tested active muscle and found that increasing muscle activity or tonic torque level produced only a small increment in what we would now call short range stiffness. The similarity between our current and previous finding adds weight to the idea that muscle activity makes only a little difference to short range stiffness.

Our results show that chord stiffness decreases from 101% to 19% as rotation size increases from 0.03 to 7 deg (Fig. 4B and C). By comparison, using continuous footplate perturbations of peak-to-peak amplitude 0.5 to 8 deg, Peterka (2002) estimated a low value of 0.8 Nm deg⁻¹ per leg which we estimate as approximately 10–15%mgh and moreover estimates that passive stiffness tends to increase as perturbation amplitude increases. We think these two studies can be harmonized according to the following points. (i) Given the size of the continuous perturbations (0.5–8 deg peak to peak) and consequent sway, and given our value for long range stiffness of 13 ± 2%mgh, it is likely that Peterka has identified the long range stiffness rather than the short range stiffness. (ii) The identification method of fitting a transfer function to the non-parametric frequency response function has probably estimated stiffness from a single torque origin which is used for the entire trial. This long range passive stiffness would be low. Our results show enhanced changes in torque when ever ankle rotation reverses; thus the torque origin is always the previous reversal which we find relocates approximately once per second (Loram & Lakie, 2002a, 2005b). In effect there is a region of enhanced stiffness about any point of temporary rest. (iii) The identification method did not use EMG information to discriminate active from passive torque, but discriminated between active and passive torque on the basis that active torque acts with a time delay and passive torque acts without a time delay. This will give a different answer to discrimination on the basis of neural modulation because the nervous system can anticipate the partially predictable position stimulus to produce a component of the neurally modulated response that acts without delay.

Our results are substantially higher than the estimate of 0%mgh provided by Maurer and Peterka (Maurer & Peterka, 2005b) in their analysis of spontaneous sway. These authors optimized a model containing a passive spring and damper as well as an active torque from a proportional-integral-derivative (PID) controller with a time delay. Reasons (ii) and (iii) above may account for the difference in results.

Our values of short range stiffness are qualitatively similar but substantially higher than those of Hufschmidt & Schwaller (1987) (10–15%mgh). The lower values measured by these authors may result from the fact that their subjects had the knee flexed at 90 deg such that the gastrocnemius muscles would have been slack whereas our subjects were in the normal standing configuration.

Our conclusion of decreasing stiffness with increasing size of ankle rotation, and invariance of stiffness with frequency is in excellent agreement with the work of Kearney & Hunter (1982). Using a similar ankle torque of 4–6 Nm, these authors found that ankle stiffness increased from 1 to 4 Nm deg⁻¹ as rotation size decreased from 13 to 1 deg and that this stiffness was constant for frequencies of 1–10 Hz. The slightly higher values of these authors might be attributed to the fact that they contain an undetermined active reflex contribution.

Our conclusion that short range stiffness depends only weakly on activation and torque level stands in contrast to most studies using larger ankle rotations which show that ankle stiffness increases convincingly as ankle torque increases (Hunter & Kearney, 1982; Mirbagheri et al. 2000; Casadio et al. 2005). Ankle stiffness is largely determined by the weakest link in the series chain comprising tendon and contractile tissue. This contrasting result predicts that at short range the weakest series link is the Achilles tendon which varies more weakly with ankle torque whereas at long range the weakest link is the contractile tissue which varies considerably in stiffness with activation. This prediction is considered in the following companion paper.

Implications for sway referencing, perturbations and altered support surfaces

For the typical, quiet standing sways of 0.13 deg (Loram et al. 2005b) the passive stiffness (67%± 10%mgh) is not sufficient for static stability, but it does reduce the effect of gravity on the human inverted pendulum. It increases the unstable time constant of the CoM stabilized by a low stiffness spring, giving the nervous system more time to respond to transitory losses of balance. From the equation of motion of the human inverted pendulum

where θ is pendulum angle, c is the normalized stiffness, m is the mass, g is the acceleration due to gravity, h is the height of the centre of mass, and I is the moment of inertia, the time constant τ can be written:

If the moment of inertia is written as I = kmh² where k is a shape factor of value approximately 1.3 (Morasso & Sanguineti, 2002), the time constant becomes

The results presented here imply that when angular sway about the ankles increases from 0.1 deg to 1 deg or more, then using mean values, the passive stiffness decreases from 67% to 13%. With a mean centre of mass height of 0.92 m (Loram & Lakie, 2002b) the time constant decreases from 0.6 to 0.37 s. This change may not seem much but it makes a large difference to how easily a subject can balance their inverted pendulum. We have previously found that manually balancing a human proportioned inverted pendulum via a compliant spring using only visual feedback and feeling the force in the spring becomes increasingly difficult as the stiffness of the spring decreases. Normalized stiffness springs of less than 50%mgh were impossible for most subjects and sway doubled as spring stiffness decreased from 95% to 58% (Lakie et al. 2003). More recently we balanced a real inverted pendulum manually using a sensitive joystick with only visual information available. We found the task became impossibly difficult as the time constant decreased below 0.5 s (Loram et al. 2006). Using pedal balance of a virtual pendulum with a constant angle at the ankle joint, Fukuoka et al. (1999, 2001) deduced that the visual feedback system alone does not produce sufficient phase advance to allow a subject to maintain upright balance. The interesting point is that their subjects were balancing an inverted pendulum equivalent to an adult, but without the assistance of passive stiffness.

Applying sensory perturbations, sway referencing the support platform or changing the support surface are means commonly used to study the effect of altered sensory information on human balance. However, standing on a foam support, or sway referencing the support surface is effectively degrading or removing the benefit of passive stiffness: by doing this the unstable time constant is potentially reduced to a minimum of 0.35 s corresponding to 0% passive stiffness. Thus it is possible that the observed increase in sway is a consequence of changing the biomechanical system time constant in relation to the human bandwidth of control rather than a sensory phenomenon (Jeka et al. 2004). Equally, applying any sensory perturbations that increase sway will alter the short range stiffness regime that the subject is operating under. Thus, the time constant of the load in relation to the human bandwidth will be decreased with a significant increase in instability. An assumption of perturbation experiments is that noise is added to a system which is invariant and this principle allows the system to be identified by the response to the noise. If the alteration in system characteristics is not accounted for, the system identification procedure will lead to an erroneous interpretation of the system. For example, if the time constant of the load is reduced in relation to the human control bandwidth, it is likely that the human will adapt their control, possibly adopting shorter latency control processes which are cruder and less accurate. As a result, the control adopted in relation to perturbations may be necessarily different to that used in spontaneous sway.

An appropriate model for human standing would either have local parameters for passive stiffness appropriate for the mean sway size in a trial or would have a stiffness that decreases with angular rotation from the previous reversal (Fig. 4C).

Conclusion

We have found that the passive calf muscles provided: (i) an extensor torque capable of sustaining unstable balance without tonic activity at a mean CoM–ankle angle of 1.6 deg; (ii) a stiffness that diminishes non-linearly from 101% to 19% for 0.03–7 deg rotations; and (iii) a short range stiffness appropriate to quiet standing of 67 ± 10%mgh. Thus, passive muscle properties have a substantial postural role eliminating the need for continuous muscle activity and decreasing the unstable time-constant of the human inverted pendulum. Ignoring the dependence of passive stiffness on sway size could lead to misinterpretation of experiments using perturbations, sway referencing and altered support surfaces.