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Appendix 1
Following Kargo and Giszter (2000) A force-field primitive can be represented by
  
 
(r , r , t )  Aa(t ) (r , r )
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:
  
  

F (r , r , t )   Ai ai (t   i )i (r , r )
i
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 equation 1. 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
  
  
T

i (r , r )   J Tij (r , r )
j
3
JT is the transpose of J, Tij are the torques exerted by the jth muscle contributing to force


field primitive i in configuration r , with velocity r .
The instantaneous torque due to
the jth muscle is given by:


 
T j   j ( r ) f ( E j (t ), r (t ), r (t ))
4
where E j (t ) is the time history of the EMG drive responsible for torque evolution in the


jth muscle, r (t ) is the time history of the limb configuration, r (t ) is the endpoint


velocity,  (r ) is the instantaneous moment arm matrix at configuration r , and f is a
potentially nonlinear function representing the neuromechanical coupling behavior of the
muscle.
T

At a given limb state r , the Jacobian transpose J is fixed and the moment

arm matrix  (r ) is fixed. We assume that a muscle E j (t ) may participate in more than
one primitive, so during activation of several primitives, its amplitudes fractional
contribution to primitive i is  ij (t ) .
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  ij (t ) should be balanced with other EMG drives within the primitive. Although a
particular force balance might be achieved by different  ij (t ) over time, it is simplest to
assume that the balance among component EMG’s fractions  ij (t ) should be constant for
the duration of a primitive’s activation.
The simplest way to achieve a constant ratio
among a set of  ij (t ) is to have some premotor drive C s 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  ij (t ) (i.e. a fixed EMG basis vector) form such a
drive where
 ij (t )  wij Ci (t )
5
where wij represents the strength of the jth muscle in the ith drive, and Ci (t ) 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
E j (t )    ij (t )   wijCi (t )
i
i
6,
i.e., the sum of premotor drives contributes to E j (t ) , where is the index of each
drive/primitive.
Our goal was then to identify the putative synchronous premotor drives ( Ci (t ) ) and
muscle balances ( wij ) 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
(http://www.cat.csiro.au/cmst/staff/pic/robot/) for MATLAB simulink.
toolbox
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.