Appendix 1
Following Kargo and Giszter (2000) A force-field primitive can be represented by
1
where A is some scaling factor, a(t) is the activation time course, t is time, r is a position vector representing the limb’s configuration and is the base vector force field. Superposition of primitives to effect more complicated motions can then be described by:
2,
where i represents the phase of the ith primitives.
This force behavior can be created by continuously balanced muscle activation. Force-field primitives then arise from spinal circuits generating the balanced action of muscle groups and reflexes. Synchronous muscle synergies and common drive forming “independent controllers” of limb force are thus embodied in the notion of the force-field primitive. This means that relative balances among muscles forces (or the driving EMG amplitudes) participating in a single synergy must remain constant over time. In contrast, the relative balance of muscle forces (or EMG amplitudes) from muscles participating in different synergies and associated force-field primitives can be variable.
Mathematically, the forces exerted on a limb by force field primitives can be described by the equation1. At each instant of time, as a single primitive is activated, the forces generated in each force field i can be related to the torques exerted by a set of component muscle activations through the jacobian matrix of the limb (J) by
3
JT is the transpose of J, Tij are the torques exerted by the jth muscle contributing to force field primitive i in configuration , with velocity . The instantaneous torque due to the jth muscle is given by:
4
where is the time history of the EMG drive responsible for torque evolution in the jth muscle, is the time history of the limb configuration, is the endpoint velocity, is the instantaneous moment arm matrix at configuration , and f is a potentially nonlinear function representing the neuromechanical coupling behavior of the muscle. At a given limb state , the Jacobian transpose is fixed and the moment arm matrix is fixed. We assume that a muscle may participate in more than one primitive, so during activation of several primitives, its amplitudes fractional contribution to primitive i is . To obtain the behavior of a force field primitive defined in equations 1 and 2, the relative contribution to the ith primitive of the jth EMG drive should be balanced with other EMG drives within the primitive. Although a particular force balance might be achieved by different over time, it is simplest to assume that the balance among component EMG’s fractions should be constant for the duration of a primitive’s activation. The simplest way to achieve a constant ratio among a set ofis to have some premotor drive contributing to several motor pools (and therefore to recorded EMGs). This will cause balanced torques and forces of the form observed in a force field primitive. We thus expect a fixed ratiometric balances among the driven EMGs in a primitive (i.e. a fixed EMG basis vector) form such a drive where
5
where represents the strength of the jth muscle in the ith drive, and the time history of the ith drive. Thus any two muscles remain in fixed activity ratios in their contributions to a primitive. The net EMG for the jth muscle is then
6,
i.e., the sum of premotor drives contributes to , where is the index of each drive/primitive.
Our goal was then to identify the putative synchronous premotor drives () and muscle balances () from the set of EMGs recorded.
Appendix 2
ICA has generally been applied to linear, feedforward systems. It is unclear the extent to which ICA unmixing can be used to extract drives in systems with feedback and/or intrinsic nonlinearities. To test the robustness of ICA in these conditions, we constructed a simple model of limb control in MATLAB ™ Simulink™. We devised a model of a two-link limb controller using the ROBOT toolbox ( for MATLAB simulink. Two distinct fixed postures and associated viscoelastic corrections were governed by feedback from separate controllers. A 2 link limb formed a common physical plant. A schematic of the model can be seen in Fig. 10 A. Control of the actuation at each joint of the model limb was assigned to 2 proportional derivative (PD) controllers, one controller for each joint posture. Thus a total of 4 PD controllers (2 joints X 2 postures) operated in parallel in the entire system. Actuation dependent noise was added at the level of each PD controller output. Strength of outputs from each PD controller were varied in common for the controllers governing the same multijoint posture using multiplicative drives. These inputs, Drive 1 and Drive 2 in Fig. 10A, multiplied the output of associated PD controllers, effectively gating them to form ‘premotor drives’. Thus for posture 1: PD1,1 controlled joint 1 and PD1,2 controlled joint 2. Drive 1 multiplied the outputs of both PD1,1 and PD1,2. The linkage posture and velocity feedback signals provided to all the PD controllers were perturbed by common, additive Gaussian noise sources. These sources simulated common execution and sensor noise sources. This noise variance was varied between 1% and 20% of signal variance for both sources. Multiplicative gating by drives was formed of a series of random phase half-cosine wave pulses. Pulse activation modulating each controller occurred at random intervals in order to simulate the action of independent control modules (a la motor primitives). Multiplicative (‘motor’) noise was added to the gated controller output torque signals. The torques resulting from the summed PD controller output torques which drove the linkage (simulating the ‘EMGs’) were saved in a single matrix for analysis by ICA.
ICA analysis proved able to isolate the waveforms of the gating drives to the PD controllers attached to the physical plant for an extensive range of input noise magnitudes. The two largest components identified by ICA explained >91% of variance. This was true both for actuator-dependent noise between 1% and 20% of signal variance and for endpoint position noise between 1% and 20% of variance. That two components explain significant variance in the 4 channels is unsurprising: limb position is controlled by 2 PD controllers. However, robustness of this result in the face of additive feedback and multiplicative output noise, though, is significant here. ICA may extract premotor drives from EMG signals despite nonlinear motor noise from different premotor drives and effects of feedback.