Fig 1.
Ankle stiffness and spinal feedback.
A. Spinal feedback is probed by electrically stimulating the sensory fibres of a muscle and measuring the resulting change in muscle contraction, called the H-reflex. B. Decrease in soleus H-reflex in challenging balance conditions: when standing facing a cliff (left) [4], when standing on a narrow support (middle)[5] and standing with the eyes closed (right)[6]. Co-contraction of antagonist ankle muscles is observed in each of these three cases. C. Decrease in soleus H-reflex in older adults in normal stance [7–9]. Co-contraction of antagonist ankle muscles during quiet standing is observed in older but not in young subjects [7,12].
Fig 2.
Stability limits of the proportional and derivative gains.
A. for stiffness increasing from 0 (black) to critical stiffness (turquoise) with a delay of 0.14 s; B. for a delay increasing from 0.14 s (black) to 0.28 s (yellow) without stiffness; C. for combined increases in delay and stiffness (maintaining a constant S), from a delay of 0.14 s with a stiffness of 0 (black) to a delay of 0.28 s with a stiffness of 0.75 Kcrit (yellow). The critical feedback gains for each condition are indicated as dots. In the three panels, the black curves are identical and correspond to zero stiffness and a delay of 0.14 s.
Fig 3.
The time to recover balance (A, B, C), the corresponding critical proportional (D, E, F) and derivative (G, H, I) gains and the peak excursions (J,K) of the CoM (full line) and CoP (dashed line) are shown as a function of stiffness and neural delay. The strategy of increasing ankle stiffness with constant τdelay = 0.14 s is illustrated in blue. The effect of an increase in neural delay with fixed ankle stiffness k = 50% Kcrit is illustrated in yellow. The strategy of increasing ankle stiffness to compensate for an increased delay (maintaining a constant S) is illustrated in green. The reference system used in section V. is indicated as a black dot.
Fig 4.
Co-adaptation of ankle stiffness and sensorimotor gains to improve immobility.
Simulated response to a perturbation in ankle angle of arbitrary amplitude 1, showing the torque of weight (black, corresponding to the position of the centre of mass), and the ground reaction torque (grey, corresponding to the position of the centre of pressure), with its two components respectively due to stiffness (dark blue) and feedback contraction (light blue). All torques are normalised to weight. A. Reference system without stiffness, and with the corresponding critical feedback. B. Co-adaptation strategy: system with critical stiffness and critical feedback. C. Gain decrease strategy: system without stiffness, and with the same feedback controller as in B. D. Stiffness increase strategy: system with critical stiffness, and with the same feedback controller as in A. The response delay (between the perturbation and the first change in contraction) is shaded in grey. In panels A and B, the CoP is not shown as it overlaps with the torque due to contraction (light blue).
Fig 5.
Co-adaptation of ankle stiffness and sensorimotor gains to compensate for increased delays.
Simulated response to a perturbation in ankle angle of arbitrary amplitude 1, showing the torque of weight (black), and the ground reaction torque (grey), with its two components due to stiffness (dark blue) and feedback contraction (light blue). All torques are normalised to weight. A. Reference system with a delay of 0.14 s (shaded in grey in all panels), intermediate stiffness k = 50% Kcrit, and the corresponding critical gains. B.-D. Increase in delay by 0.14 s (shaded in red). B. No adaption: same stiffness and gains as A. C. Gain adaptation strategy: same stiffness as A, and critical gains corresponding to the increased delay. D. Co-adaptation strategy: increase in stiffness to maintain a constant relative speed S, and corresponding critical gains.
Fig 6.
Generalisation to more complex neuromechanical models.
A-F Mechanical damping dM. The time to recover balance (A, D), critical proportional gain (B, E) and critical derivative gain (C, F) are shown for τdelay = 0.14 s and increasing ankle stiffness (A–C), and for k = 50% Kcrit ankle stiffness and increasing neural delay (D–F). G-N Acceleration (green) and integral (yellow) feedback. The time to recover balance (G, L), critical proportional gain (H, M), critical derivative gain (I, N), critical acceleration gain (J, O) and critical integral gain (K, P) are shown for τdelay = 0.14 s and increasing ankle stiffness (G–K), and for k = 50% Kcrit ankle stiffness and increasing neural delay (L–P).
Table 1.
Standing in challenging balance conditions.
For each of the Reference, Gain decrease and Co-adaptation strategies, the Table shows the values of stiffness, neural delay and sensorimotor gains used for the simulation, as well as the peak CoM and CoP excursion after a perturbation of arbitrary amplitude 1. All changes are shown as a percent of the reference value, except stiffness changes, which are reported in % K_crit.
Table 2.
Compensating for increased delay.
The table reports the sensorimotor control parameters and performance of the Reference, and the three strategies for increased delay (No adaptation, Gain adaptation and Co-adaptation). All changes are shown as a percent of the reference value, except stiffness changes, which are reported in % K_crit.
Fig 7.
Single inverted pendulum model of stance.
The external forces acting on the body are the weight (green arrow), whose torque around the ankle depends on ankle angle θ, and the ground reaction force (red arrow) whose torque depends on both ankle angle and muscle contraction.
Fig 8.
A. Stability limits. The range of stable normalised proportional and derivative gains (P and D) is plotted for different values of relative speed S ranging from S = 0 to the maximal . B-D Critical feedback. B. The critical proportional gain Pcrit is plotted in blue as a function of S. The range of stable proportional gains is shaded in grey. C. The critical derivative gain Dcrit is plotted in blue as a function of S. The range of stable derivative gains is shaded in grey. D. The time required to recover balance with critical feedback is plotted as a function of S.
Fig 9.
Illustration of critical feedback.
Simulated response to a perturbation with critical derivative gain and various values of the proportional gain (A-C); and with critical proportional gain and various values of the derivative gain (D-E); for a system with τdelay = 0.14 s (A, B, D, E) and either critical stiffness (A, D) or no stiffness (B, E); and for a system with a 20% increase in delay and no stiffness (C, F).
Fig 10.
Mechanical damping and acceleration and integral feedback.
Time to recover balance (A, D), critical proportional gain (B, E), critical derivative gain (C, F) critical acceleration gain (G) and critical integral gain (H) for increasing values of damping (A-C) and with acceleration (D-G, green) and integral (D, E, F, H, yellow) feedback.