To the editor

In a recent paper in Nature Neuroscience, Todorov1 referred to our finding that a motor cortical representation of hand trajectory during spiral drawing precedes the hand's movement by an interval that varies with path curvature2,3. Although there are several possible explanations for this finding, Todorov, using a simplistic model, argued that because cortical cells share common properties with muscles, this relationship could be due to a combination of inertia, viscosity and stiffness acting on the acceleration, speed and position of the arm, respectively. Although simple, his model is flawed and cannot support this conclusion.

The author models a multijoint arm as a simple cantilever that is converted to single point-mass equation using a Jacobian transformation (web supplement A, http://www.nature.com/neuro/web_specials/ ). The arm's properties were derived from a simplified version of muscle whose activity is a linear combination of motor cortical activity. This model was used to reinterpret our results2,3. In our study, monkeys drew spirals on a vertically oriented computer touchscreen. The center of the spiral was located in front of the monkey, between its shoulders. According to Todorov's model, this location corresponded to the equilibrium point of the arm—the location where the parameters in his model would force the arm to rest. Todorov assumed that cortical activity reflects the inertia, viscosity and stiffness of the arm and showed that his model produces the same variable lags as our cortical population vectors. However, any acceleration representation in the cortical activity would actually decrease lags as a function of curvature, which is exactly opposite to our finding (web supplement B, http://www.nature.com/neuro/web_specials/ ).

The increased lag with increasing curvature shown in Todorov's article is due to his positional term. The idea that extrinsic position may be a factor in motor cortical activity is not new4,5,6. However, Todorov's method of equating extrinsic position representation to muscle stiffness is incorrect. This model assumes that muscle viscoelastic properties are independent of muscle activation. Thus, even an inactivated muscle will act as a large spring pulling the arm back to some equilibrium position. In real muscle, the force–length and force–velocity relationships are modulated by muscle activation such that at zero activation, the muscle is essentially a non-force producer. In the real world, the combination of gravity and inactive muscles will force the arm to fall to the side. In Todorov's model, the combined effect of gravity and muscle stiffness on inactive muscles would make the hand float at mid-chest level; muscle activity would be required to force the arm down below chest level. This, of course, is unrealistic. Viscoelastic models like the ones used by Todorov are only valid for perturbation studies where both posture and neural activity are assumed to be constant. Using such equations to solve for time-varying muscle activations violates the basic assumptions of perturbation models. Simple dynamic models can be useful to explain arm mechanics. However, when the models are not consistent with basic physiology, exclude important phenomena, and violate inherent assumptions, they cannot be compared to empirical data.

See“Reply to 'One motor cortex, two different views'” by Tordorov.