Strategy of arm movement control is determined by minimization of neural effort for joint coordination

Abstract

Optimality criteria underlying organization of arm movements are often validated by testing their ability to adequately predict hand trajectories. However, kinematic redundancy of the arm allows production of the same hand trajectory through different joint coordination patterns. We therefore consider movement optimality at the level of joint coordination patterns. A review of studies of multi-joint movement control suggests that a ‘trailing’ pattern of joint control is consistently observed during which a single (‘leading’) joint is rotated actively and interaction torque produced by this joint is the primary contributor to the motion of the other (‘trailing’) joints. A tendency to use the trailing pattern whenever the kinematic redundancy is sufficient and increased utilization of this pattern during skillful movements suggests optimality of the trailing pattern. The goal of this study is to determine the cost function minimization of which predicts the trailing pattern. We show that extensive experimental testing of many known cost functions cannot successfully explain optimality of the trailing pattern. We therefore propose a novel cost function that represents neural effort for joint coordination. That effort is quantified as the cost of neural information processing required for joint coordination. We show that a tendency to reduce this ‘neurocomputational’ cost predicts the trailing pattern and that the theoretically developed predictions fully agree with the experimental findings on control of multi-joint movements. Implications for future research of the suggested interpretation of the trailing joint control pattern and the theory of joint coordination underlying it are discussed.

Appendix: Sequential approximation

The sequential approximation control mechanism applies to a motor system that includes at least two actuators (e.g., joints of a limb). This mechanism involves subordinated control of the actuators. Here we show that, under reasonable assumptions, this mechanism can significantly reduce the cost of information processing for control.

Assume that a control command U is a sum of control commands generated by two actuators:

$$U = F_{1} (S) + F_{2} (S).$$

(7)

In general, there may be a significant degree of redundancy between U ₁ = F ₁(S) and U ₂ = F ₂(S), while neither actuator alone can generate U precisely. The sequential approximation control mechanism consists in producing a crude approximation of U by using one actuator and fine-tuning that approximation by complementing it with the other actuator. Mathematically, this approach can be expressed as follows:

$$U_{1} = F_{1} (S),$$

(8)

$$U_{2} = F_{2} (S, U_{1} ),$$

(9)

$$U = U_{1} + U_{2} .$$

(10)

Note that U ₁ is included in the input for the second actuator as an additional state component. The cost of information processing can be saved (according to Eq. 4) by significantly lowering the precision of U ₁ and the amplitude (magnitude) of U ₂, while the precision of U ₂ is sufficiently high. Indeed, if the two actuators equally contribute to U with the same precision ∆U ₁ = ∆U ₂ = ∆U and amplitude h ₁ = h ₂ = |U|/2, the total amount of information required for encoding U in Eq. 7 is

$$Q = 2\log_{2} (1 + |U|/(2\Delta U)).$$

(11)

When the sequential approximation mechanism (Eqs. 8–10) is employed, ∆U ₁ ≠ ∆U ₂, and the total amount of information is equal to

$$Q_{\text{SA}} = \log_{2} (1 + |U_{1} |/\Delta U_{1} ) + \log_{2} (1 + |U - U_{1} |/\Delta U_{2} ),$$

(12)

where the first actuator’s precision is significantly lowered, i.e.,

$$\Delta U_{1} >\Delta U_{2} ,$$

(13)

and U ₁ is a fair approximation of U, i.e.,

$$|U - U_{1} | \ll |U|,$$

(14)

meaning that the magnitude of the control command sent to the second actuator (U ₂ = U − U ₁) is relatively low.

The sequential approximation mechanism saves the cost of information processing if

$$Q_{\text{SA}} < Q.$$

(15)

This condition can be expressed in terms of control vectors by substituting the right-hand parts of Eqs. 11 and 12 into Eq. 15. Namely, since log₂ is a monotonic function, for Eq. 15 to hold it is necessary and sufficient that

$$\left( {|U - U_{1} |/|U|} \right) \cdot \left( {|U_{1} |/|U|} \right) \cdot (\Delta U_{2} /\Delta U_{1} ) < 0.25,$$

(16)

which is feasible provided that the following conditions of sequential approximation applicability are satisfied. First, as already mentioned above, it is assumed that U ₁ is a fair approximation of U, and therefore, the condition described by Eq. 14 is satisfied, meaning that the first component of the product in the left-hand part of (16) is rather small (|U − U ₁|/|U| ≪ 1). Second, for the same reason, |U ₁|/|U| is close to 1. Third, assuming the condition represented by Eq. 13 is satisfied, ∆U ₂/∆U ₁ < 1. Thus, the entire product in the left-hand part of Eq. 16 is likely to be quite small, meaning that the sequential approximation control mechanism is likely to allow a significant reduction of information processing cost.