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In standing, there are small sways of the body. Our interest is to use an artificial task to illuminate the mechanisms underlying the sways and to account for changes in their size. Using the ankle musculature, subjects balanced a large inverted pendulum. The equilibrium of the pendulum is unstable and quasi-regular sway was observed like that in quiet standing. By giving full attention to minimising sway subjects could systematically reduce pendulum movement. The pendulum position, the torque generated at each ankle and the soleus and tibialis anterior EMGs were recorded. Explanations about how the human inverted pendulum is balanced usually ignore the fact that balance is maintained over a range of angles and not just at one angle. Any resting equilibrium position of the pendulum is unstable and in practice temporary; movement to a different resting equilibrium position can only be accomplished by a biphasic ‘throw and catch’ pattern of torque and not by an elastic mechanism. Results showed that balance was achieved by the constant repetition of a neurally generated ballistic-like biphasic pattern of torque which can control both position and sway size. A decomposition technique revealed that there was a substantial contribution to changes in torque from intrinsic mechanical ankle stiffness; however, by itself this was insufficient to maintain balance or to control position. Minimisation of sway size was caused by improvement in the accuracy of the anticipatory torque impulses. We hypothesise that examination of centre of mass and centre of pressure data for quiet standing will duplicate these results.
We have investigated the use of the ankle strategy in balancing a large inverted pendulum equivalent in mass and inertia to a human body. What are the advantages of this approach? Standing is a complex activity both mechanically and neurologically. The task of balancing a real inverted pendulum in one plane is much simpler to investigate because there are fewer variables. There is only one joint axis through the ankles and the angular position of the pendulum mass can be precisely measured. Investigation of the strategy used to balance a real pendulum should illuminate the mechanisms used in standing and provide a hypothesis against which standing can be tested.
If the inverted pendulum is to be stabilised, then the change of ankle torque per unit change of angle must on average be greater than the toppling torque per unit angle of the pendulum (the so called ‘gravitational spring’ or ‘load stiffness’). If this were not the case the pendulum would fall to the floor. How are these changes in torque produced? At one extreme the ankles might possess sufficient mechanical stiffness to produce stability in the manner of a tree or tall building. This mechanism would stabilise the pendulum at one angle, effectively converting it into a tall stable object with a narrow base. At the other extreme, the ankles might have zero mechanical stiffness as in the case of a person balancing on stilts. In this case the pendulum would have to be balanced by an impulsive mechanism in the form of intermittent ballistic-like adjustments.
For quiet standing, some authors regard the intrinsic elastic properties of the activated ankle musculature alone as sufficient to achieve stabilisation (like a tall building, above) (Horak & MacPherson, 1996; Winter et al. 1998, 2001). The operation of a sufficiently stiff reflex servo is also a theoretical solution, but it has been shown that the reflex loop has a gain close to unity which is insufficient for effective position control based on negative feedback (Fitzpatrick et al. 1996). Others regard predictive neural modulation of ankle torque to be necessary for quiet standing (impulsive mechanism, above) (Morasso & Schieppati, 1999). In the present investigation we show that the ankle torque used for balancing the pendulum can be apportioned into intrinsic mechanical and neurally controlled elements.
It is possible to balance the pendulum (or the body) at any reasonable desired position and to move the pendulum or body from one position to another. Explanations regarding balance of the human inverted pendulum have tended to ignore the problems associated with providing stability and control over a range of angles. Many of the explanations advanced would produce balance only at a single equilibrium point. In suggesting an answer to the question of positional control we show that control of the pendulum is necessarily associated with repeated, ballistic-like patterns of ankle torque change vs. angle. We further show that the neural modulation associated with this positional control scheme increases the operational stiffness and provides intermittent, reactive damping. This activity ‘tops up’ the intrinsic mechanical contribution of the active ankle musculature which, on its own, is not quite sufficient to counteract the ‘gravitational spring’.
Finally, our previous research has shown that the mean sway size of the pendulum could be systematically reduced but this result was not achieved by increasing the change in ankle torque per unit angle (Loram et al. 2001). This result was contrary to theories that sway is altered by controlling operational ankle stiffness or viscosity (Fitzpatrick et al. 1992a, b; Winter et al. 1998; Carpenter et al. 1999; Gatev et al. 1999). In this paper we investigate how the pendulum sway size was minimised not by making the ‘tall building’ stiffer, but by refining the performance of the impulsive mechanism, by improving the accuracy of intermittent, reactively triggered, ballistic patterns of torque.
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The quasi-regular variation of ankle torque with pendulum position showed the same general characteristics with all subjects under all conditions. A representative example is shown in Fig. 3A. The pendulum was not confined to one angle and there was no one position of equilibrium. Rather the pendulum made small sways to and fro of irregular duration, size and speed. Movement of the pendulum was controlled by an ankle torque that always kept close to that required for balance and which attained equilibrium transiently every time the line of equilibrium was crossed. Most of these line crossings were spring-like (torque increases as angle increases) with a positive gradient. Less commonly, negative gradient line crossings could also be seen (X).
Figure 3. Pendulum sway
A, a 12 s record from one subject is plotted as combined ankle torque against pendulum position. Data points are at 40 ms intervals. The starting point (diamond) and finishing point (square) are indicated. The line of equilibrium (gravitational torque on the pendulum) is shown as a continuous straight line. For each trial condition, B shows the mean sway size and C shows the mean sway duration. For both panels, values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % simultaneous confidence intervals for the mean values. As described in Methods, a sway is the angular movement between successive reversal points of the pendulum. Trial conditions were (1) stand still with visual feedback, (2) stand still with no visual feedback, (3) stand easy with visual feedback, and (4) stand easy with no visual feedback.
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When subjects gave their full attention to keeping the pendulum still, as opposed to giving minimal attention to keeping it still, there was a clear reduction in mean sway size (Fig. 3B). A significant reduction in mean sway size occurred both when visual feedback was available (condition 1 vs. 3) and when it was not (condition 2 vs. 4) and generally the effect of intention was significant (two-way ANOVA, effect of intention, n= 40, F= 10.0, P= 0.003). Interestingly, there were no significant differences in mean sway duration between any of the four trial conditions (Fig. 3C) (two-way ANOVA, n= 40, F= 0.61, P= 0.61). Combining both results, the mean sway velocity mirrored the mean sway size for all four conditions.
In balancing the pendulum, an underlying process was repeated over and over again. Figure 4 shows the entry into and departure from equilibrium averaged for all occurrences for all 10 subjects. The pendulum falling, positive-gradient cases were selected under ‘still’ and ‘easy’ conditions with visual feedback available (condition 1 vs. 3). Looking at the ankle torque vs. angle plots (Fig. 4A) we see the same biphasic ‘drop and catch’ pattern for both conditions as the pendulum swayed from one reversal point (a) to the next (b). If torque error is the difference between torque applied and that required for equilibrium, then from a there was initially an increase in torque error (the drop), followed by a decrease in torque error leading to equilibrium and the maximum speed of the pendulum (starred). This was followed by an opposite increase then decrease in torque error which decelerated the pendulum to rest at b (the ‘catch’).
Changes in ankle torque during an individual sway relate to both neural modulation and changes in ankle angle. A notional best-fit line through the changes in torque vs. angle (Fig. 4A) would indicate a generally spring-like characteristic with a gradient approximately twice that of the line of equilibrium. Yet on top of that there are changes in torque which are clearly not spring-like in origin. The decrease in torque immediately after the sway begins at a was not caused mechanically/elastically because the muscle-tendon was being stretched at this point (dorsiflexion). Neither was mechanical viscosity the cause because the speed was increasing. In any case the mechanical and reflex viscosity (∼0.02 and ∼0.07 N m s deg−1, respectively, per leg operating at 15 N m; Mirbagheri et al. 2000) are too small to produce major changes in torque for these averaged sways, which only reached maximum speeds of 0.2 and 0.5 deg s−1, respectively. Thus the decrease in torque must have been caused by neural modulation and this is confirmed in Fig. 4b where it can be seen that the soleus EMG was decreasing in the 140 ms preceding the reversal point at a. The nervous system did not anticipate the initiation of the sway as the soleus EMG only rose as the sway began at a, leading to a delayed increase in ankle torque that will catch the falling pendulum.
Similarly, the decrease in torque after equilibrium (star) and before the reversal point (b) was also not mechanically elastic or mechanically or reflexly viscous in origin for the same reason as above. This decrease in torque corresponded to the decrease in soleus EMG after equilibrium (star) followed by an inevitable electromechanical delay of approximately 140 ms. Note that this decrease in soleus EMG anticipated the end of the sway, indicating predictive behaviour on the part of the nervous system.
The averaged soleus EMG showed the same pattern in each leg for every trial of every subject. Tibialis anterior EMG was usually close to the noise floor with no sign of modulation. Sometimes, such as when the subject was fatigued or when the pendulum swayed close to the vertical, the tibialis anterior EMG would be modulated either antagonistically with soleus or synergistically with soleus and in these cases the tibialis anterior signal could be comparable in magnitude to that of soleus. However, this modulation was not consistent from trial to trial, from subject to subject or even from leg to leg. The lack of averaged modulation in tibialis anterior compared with soleus is shown in Appendix A (Fig. 8A).
Figure 8. Comparison of soleus, tibialis anterior and gastrocnemius EMG
A and B, averaged data from 1 s before to 1 s after all positive-gradient, equilibrium line crossings while the pendulum was falling. Data points are at 40 ms intervals and proceed from reversal points a to b via the arrow. The asterisk marks the instant of equilibrium and maximum velocity. A, soleus (S) and tibialis anterior (TA) EMG from both legs vs. pendulum position. These data were averaged over all 10 subjects and over all 40 trials including ‘still’ and ‘easy’ conditions both with and without visual feedback. B (for one leg only), soleus (S), tibialis anterior (TA), gastrocnemius medialis (GM) and gastrocnemius lateralis (GL) EMG vs. pendulum position as well as ankle torque vs. pendulum position. This data set is different from the one reported in the body of this paper although the methods used were identical. Twelve subjects were asked to balance the pendulum for 200 s.
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One can clearly see that the sway size between a and b was less for the ‘still’ condition than the ‘easy’ condition. There was no change in line crossing gradient, which indicates no change in the operational stiffness or viscosity of the ankle mechanisms (Loram et al. 2001). Since the dots occurred at 40 ms intervals it was clear that through the line crossing there is a smaller rate of growth of torque error per second in the still condition and this would have minimised the absolute acceleration of the pendulum. It can also be seen that at the reversal point (b) there was less torque error in the ‘still’ condition than the ‘easy’ condition and this would have altered the initial acceleration of the subsequent sway. These two observations are relevant to sway minimisation but are complicated by the confounding fact that the ‘still’ and ‘easy’ conditions contain different distributions of sway velocities. In order to contrast like with like, sways of equal velocity should be compared.
For the averaged spring-like line crossing data, changes in ankle torque have been decomposed into changes that result from neural modulation and changes that derive from mechanical stretching and releasing of the elastic structures surrounding the ankle joint (activated muscle fibres, aponeurosis, tendon, foot). For illustration, Fig. 5c shows the actual changes in ankle torque for trial condition 3 averaged over all 10 subjects (cf. Fig 4A, ‘easy’). The changes in torque predicted by the model are also shown (percentage variance accounted for, %VAF = 98.5 %). The predicted changes in torque resulting from stretching of the activated elastic structures and neural modulation are shown in Fig. 5A and B, respectively. It can be seen that neural modulation makes the greatest contribution to changes in torque and adds operational stiffness to the torque changes at the spring-like line crossing. Neural modulation also adds changes in torque orthogonal to changes in position that cause additional acceleration at a (the ‘drop’) and cause additional braking as b (the ‘catch’) is approached.
Figure 5. Decomposition of ankle torque according to our model
The decomposition of ankle torque into intrinsic elastic and neurally modulated components is shown. The model is described in Appendix A. A-C, illustrative data averaged from all subjects for falling, positive-gradient equilibrium line crossings in condition 3. A, the changes in torque arising from stretching and releasing of the elastic components. B, the variation in torque resulting from neural modulation (continuous line) and the preceding variation in soleus EMG (dotted line). C, the actual variation in torque as well as the modelled variation (dotted line). A-C, the same position, EMG and torque data as Fig. 4 (‘easy’). Data points are at 40 ms intervals and the reversal points a and b correspond to those in Fig. 4. The model was applied to averaged, positive-gradient, line crossing data from each trial. Values of parameters for falling and rising line crossings were averaged. For each trial condition, D shows the mean intrinsic mechanical stiffness, E shows the mean neural gain and F shows the mean electromechanical delay between changes in soleus EMG and changes in torque. The neural gain is expressed relative to the isometric neural gain (N m V−1). Parameter values were averaged over 10 subjects for each of the four trial conditions. Error bars show 95 % simultaneous confidence intervals for the mean values.
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The average intrinsic mechanical stiffness of all subjects was ∼8 ± 4 N m deg−1 (±s.d.) and this value is just less than the gravitational toppling torque per unit angle of the pendulum (10.2 N m deg−1, Fig. 5D). The averaged neural gain of all subjects was ∼0.5 ± 0.3 (±s.d.), which was expressed relative to the isometric gain in newton metres per volt measured for each subject (Fig. 5E). This shows that small rapid fluctuations in EMG produce relatively less change in torque than the large slow changes in EMG that were required for the isometric calibration. The average electromechanical delay between changes in EMG and changes in torque was 140 ± 40 ms (±s.d.) (Fig. 5F). For the mechanical stiffness, the neural gain and the electromechanical delay, there were no significant differences resulting from the intention of the subject (two-way ANOVA, n= 80, F= 2.8, P= 0.1; F= 0.8, P= 0.4; F= 3.6, P= 0.06 respectively).
When subjects were minimising pendulum movement they minimised the torque error as the pendulum was brought to rest at the end of a ‘catch’. Figure 6A shows that at all velocities, apart from the lowest, the torque error when the pendulum was first brought to rest following the line crossing was significantly and substantially less in the ‘still’ condition. The mechanical consequence is that the initial acceleration of ‘the sway after the catch’ will be less in the ‘still’ condition than the ‘easy’ condition.
Figure 6. Effect of intention on binned sways
A, the effect of intention on sways sampled and grouped according to their velocity at the first positive-gradient line crossing. A, the acceleration at the end of the sway. B and C, the effect of intention on sways sampled and grouped according to their initial acceleration. B, the sway size to the first positive-gradient line crossing equilibrium and to the reversal point at the end of the sway. C, the duration to the first line crossing and to the end of the sway. For all panels the ‘still’ results (crosses on continuous line) and the ‘easy’ results (dots on dashed line) were averaged over falling and rising sways and over with and without visual feedback for all subjects. The abscissa values are the mean binned values. The error bars represent 95 % confidence intervals in the mean ordinate values for each bin. Two-way ANOVA for ascending bins in A gives n= 80, F= 1.0, 8.7, 16.1, 9.7, 11.7, 9.1, P= 0.3, 0.004, 0.0001, 0.003, 0.001, 0.003.
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To assess the benefit of minimising the initial acceleration of a sway, we needed to sample sways according to their initial acceleration (torque error). Figure 6b shows clearly that on average the size of a complete sway (and the sway to equilibrium) increased with the initial acceleration for both the ‘still’ and ‘easy’ conditions. For the three lowest acceleration bins the difference between the conditions was not significant, though taking all five bins together, the fact that the sway sizes were always less for the ‘still’ condition is significant (n= 5, P= (1/2)5= 0.03). This figure confirms that in minimising pendulum movement there is a benefit from minimising the initial acceleration of a sway. Figure 6c shows that for all initial accelerations the duration of a complete sway was virtually unchanged at 1 s, and the duration till equilibrium was unchanged at 0.4 s; the intention of a subject made no difference to either of these times. Clearly, in the ‘still’ condition there was an improvement in the efficacy but not the rapidity of the movement minimising process.
The size of a sway was clearly associated with the maximum speed of the pendulum at the spring-like line crossing in the middle of the sway and this relationship was unaffected by the intention of the subject (Fig. 7A). This result is unsurprising given the large inertia of the pendulum. After each spring-like line crossing the pendulum was eventually brought to rest and then there was another sway in the opposite direction. By calculating the size of the subsequent sway in the reverse direction one sees a fascinating result (Fig. 7b). For each velocity bin, the subsequent sway size in the opposite direction was significantly and substantially less in the ‘still’ compared to the ‘easy’ condition. The intention of the subject to minimise movement had great effect by minimising the initial acceleration and maximum speed of the ‘rebounding’ sway. Figure 7c shows that for all speeds apart from the very slowest, half the minimisation in sway size resulted purely from minimising the initial acceleration of the sway; and half also resulted from the intention of the subject during the whole course of the subsequent sway.
Figure 7. Effect of intention on the current sway and subsequent return sway
For every sway the pendulum starts from transient rest, passes a positive-gradient line crossing (a speed maximum) and ultimately comes to a reversal point where it changes direction. For each bin, A shows the mean size of a sway vs. the velocity at the first positive-gradient line crossing. After the reversal point, the pendulum executes a return sway in the opposite direction to the current sway. B (continuous and dashed line) shows the mean size of the subsequent return sway vs. the velocity at the first positive-gradient line crossing of the current sway. The dotted line shows the size of the return sway calculated by interpolation from Fig. 6A and B. The ‘still’ results (crosses on continuous line) and the ‘easy’ results (dots on dashed line) were averaged for all subjects over falling and rising sways including with and without visual feedback conditions. The abscissa values are the mean binned values. The error bars represent 95 % confidence intervals in the mean ordinate values for each bin. Two-way ANOVA for ascending bins in B gives n= 80, F= 7.3, 7.3, 11.5, 8.9, 11.2, 11.0, P= 0.008, 0.009, 0.001, 0.004, 0.001, 0.001. C, for a range of current line crossing speeds the continuous line (tot.) shows the total difference in the size of the subsequent return sway caused by the intention of the subject. The dashed (i. a.) and dotted (oth.) lines show respectively the component differences caused by reducing the initial acceleration of the subsequent sway and by other minimisations occurring during the subsequent sway. All three lines were calculated by interpolation from Fig. 6A and B, not Fig. 7B.
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