Teach-in '98 on Linear Synergy: What people said...

Hi all,

In a recently published paper we discussed the relationship/synergy of hip/knee moments during gait.

Eng, JJ, Winter, DA, Patla, AE (1997). Intralimb dynamics simplify reactive control strategies during locomotion. J. Biomechanics, 30, 581-588.

For normal walking trials, we plotted the mean hip and mean knee moment magnitudes for several subjects during the swing phase and found a linear relationship (see figure 5 in the paper). I actually did similar plots using area and found similar results. In response to a tripping perturbation, subjects also demonstrated the same relationship except the magnitude of the hip and knee moments were greater in response to the perturbation, but the relationship between the hip and knee moments was invariant. The acual equation of the plot was: y = 1.64x - .03 where y = hip extensor moment and x = knee flexor moment

We suggested that this relationship between the hip and knee motor patterns was constrained which reduced the available degrees of freedom and facilitated the selection of a unique set of motor patterns. Such constraints have been reported previously for the lower extremities for a variety of activities including standing tasks (Eng et al., 1992; Yang et al., 1990) and during rapid kicking tasks (Young and Marteniuk, 1995). Iím sure others will mention many more.

We suggested that part of this relationship was due to the biarticulate hamstring muscles which have an estimated hip/knee moment arm magnitude equal to 1.5-2 when the limb is in the late swing position (moment arm estimates from White, SC (1986)).

Hope this adds to the discussion,

Janice Eng

References

Eng, JJ, Winter, DA, MacKinnon, CD, and Patla, AE (1992). Interaction of the reactive moments and centre of mass displacement for postural control during voluntary arm movements. Neuroscience Research Communications, 11, 73-80.

Yang, JF, Winter, DA, and Wells, RP (1990). Postural dynamics in the standing human. Biological Cybernetics, 62, 309-320.

Young, RP, Marteniuk, RG (1995). Changes in inter-joint relationships of muscle moments and powers accompanying the acquisition of a mult-articular kicking task. J. Biomechanics, 28, 701-713.

White, SC (1986). A deterministic model using EMG and muscle kinematics to predict individual muscle forces during normal human gait. PhD thesis, University of Waterloo, Waterloo

Janice Eng, PhD, PT Assistant Professor School of Rehabilitation Sciences Faculty of Medicine University of British Columbia T325 - 2211 Wesbrook Mall Vancouver, British Columbia, Canada V6T 2B5 Tel: (604) 714-4105 Fax: (604) 822-7624 E-Mail: JENG@REHAB.UBC.CA


Dear all,

Many thanks to Janice Eng for that very comprehensive and insightful contribution to the discussion of linear synergy. I confess I missed the paper by her, David Winter and Aftab Patla - it's certainly well worth reading. It's fascinating that the synergy is maintained during a perturbation.

Many years ago Dudley Childress, at Northwestern, and I tried to figure out (as I'm sure many other have previously - I even found a German A/K prosthesis from World War I that used this idea to control the knee using the hip) a kinematic linkage that would simplify control of the lower-limb, using the 2-joint muscles as four-bar linkages, but we mistakenly tried to find it in angle-angle diagrams - in case you're interested, I've put a 3D angle-angle diagram of normal gait at /teach-in/synergy/angles.gif - the linkage seemed frustratingly close at hand but not quite there! It's gratifying to find it in the moments.

I would still be interested in hearing - especially from the mechanical engineers out there - whether the correlation in moments indicates a synergy at the level of the motor control or the biomechanics. Janice seems to be leaning on the side of the mechanics - that ratio of the proximal and distal moment arms certainly does nicely explain the regression of 1.64 that you found.

However, Gerry Gottlieb and Esther Thelen, as I mentioned, have been working on similar synergies in the upper-limb, and they seem to be leaning more towards neurolological explanation. I'm hoping Esther or Gerry might comment on this.

Chris -- Dr. Chris Kirtley (Kwok Kei Chi) MD PhD Assistant Professor Department of Rehabilitation Sciences The Hong Kong Polytechnic University


I think its worth just taking a step back and asking how these moment values are obtained. There are no ground reaction forces during swing so any moments arise from inverse dynamics i.e. are the moments required to control the accelerations and decelerations of the segments distal to the joint in question. From about half way through the swing phase the pelvic tilt and hip flexion are both fairly constant i.e. the thigh is at a fixed orientation (no angular acceleration or decceleration). The "measured" hip moment is therefore a consequence of accelerations of the distal segments (shin and to a probably negligible effect the foot). The accelerations of these segments are what gives rise to the knee moment so we should have been able to guess merely from looking at the kinematic data that there would be a relationship between the hip and knee moments in the second half of swing.

On the question of stumbling and the preservation of this correlation. As long as the hip flexes first in early "swing" and remains fixed while the knee "catches up" in late swing then the relationship between hip and knee moments must occur. The ratio is determined by the length of the segments and the inertial properties specified in the biomechanical model used for the inverse dynamics.

This doesn't advance us a huge amount however as it remains interesting that this relationship also appears to exist for the first half of swing.

Or does it?

I assume that the points on Chris's graph of hip against knee moment (2-d) graph are equally spaced in time. There's 18 of them and we can probably agree that the ten to the right of the vertical axis (approx second half of swing) show a linear relationship. But are we really convinced that there is a linear relationship between the points to the left of the axis? Cover the right half of the graph with a piece of paper and ask if there's a straight line there. You might see a line through points 4-8 (numbered from left) but I'd suggest that this actually has a different gradient than the right half of the graph. I think we're oversimplifying this graph (makes a change to be under-interpretting gait analysis data doesn't it?)

Anyone care to defend the graph?

Richard Baker Musgrave Park Hospital Belfast <baker@unite.co.uk>


Today, I read the following book which reminded me of the last "case of the week: Rt. hemiplegia" at /archives/21-11-97

"Uncertainty in the person's ability to control and balance the body may cause the subject to protectively lower the center of gravity, even though this may require greater energy or promote further loss of balance... People who attempt to walk in a dark and unfamiliar place usually tend to flex at the hips and knees, as do patients who are unsure or frightened." ( Smith LK, Weiss EL, Lehmkuhl LD: Brunnstrom's clinical kinesiology, 5th ed.FA Davis company, 1996, p51 )

As Dr. Kirtley mentioned in his answer, I think that we should think about the question; "Do hemiplegics lower their COG by flexing the good hip and knee joint for better balance ?" And I would like to present the following topics for further discussion.

Reviewing papers about this topics will be worthwhile for our CGA mailing list.

With my best regards,

Sang-hyun Cho MD.PhD.(Rehabilitation medicine specialist) Full time lecturer at Dept. of Rehabilitation Therapy Yonsei University Wonju campus, College of Health Science Wonju-si, Kangwon-do, ZIP 220-710, Rep. of KOREA *E-mail=davinci@interpia.net *My home page=http://www.interpia.net/~davinci *Owner of rehab-kr mailing list at majordomo@mailinglist.net


Dear all,

Dr. Cho's thoughtful reminder of the hemiplegic Case of the Week prompts me to link it up with the current Teach-in.

Bearing in mind what Richard Baker says - like me, he wonders whether the correlation in moments is truly a "synergy" or rather just a by-product of the laws of mechanics of the swinging leg, I propose a test with the current case.

Hypothesis: since this case is hemiplegic, I think we would we expect the hip and knee moments to be correlated on the unaffected (left) side but less correlated on the plegic (right) side.

I have asked Andreas in Vienna if he would mind graphing the swing moments on each side to answer this question. Meanwhile, I'd be keen to hear whether you think it is a satisfactory test for the presence of a neurological synergy.

This discussion seems to be shaping up nicely!

Chris


Dear all,

As promised, Andreas and I have plotted the graphs for the hemiplegic, which you can see at:

/teach-in/synergy/left.gif

and /teach-in/synergy/right.gif

The scales are the same on both. To remind you - the right side is the plegic side, the left unaffected. The data were ensemble averaged from 4 trials.

I've run the correlations and they are 0.93 for the left and 0.94 for the right. The gradients are 0.56 and 0.36 respectively.

I'm not sure what this tells us! By eye, the unaffected side certainly looks more "normal" - the plegic side is all bunched up (which gives it the high correlation).

Interested to hear what you think.

Chris

Dr. Chris Kirtley (Kwok Kei Chi) MD PhD Assistant Professor Department of Rehabilitation Sciences The Hong Kong Polytechnic University


Baker raises two important points: The first asks whether, given the kinematics this isn't 'just physics?' and the second asks whether the data is really all that linear.

Addressing the second point first, in this case, a linear relation is like a null hypothesis. You never prove it to be true, you just dont find compelling reasons for a more complex relationship. All my experiments have been of arm motion which is analogous, I guess to the swing phase of gait. Elbow-shoulder plots lie close to a straight line over a wide varity of movement conditions. Our most recent publication is -

Directional Control of Planar Human Arm Movement G. L. Gottlieb, Q. Song, G. L. Almeida, D.-A. Hong, and D. Corcos Journal of Neurophysiology, December 1997/Volume 78, Number 6

which also provides references to several other works. In the appendix of this paper we provide a quantitative way of measuring linearity (using principal components analysis) but such an analysis does not address the validity of the contention that this reveals some underlying rule. We have speculated that interjoint linearity is 'built in' and that the CNS must learn to violate it, which it can and does, when the task requires it. Linearity is found when it is a sufficient control rule to accomplish the kinematic task. This is one way the CNS might deal with its excess degrees of freedom.

As for the first point, of course the laws of physics are obeyed and in some (but relatively few) cases, we can do the inverse dynamics in our heads and be "able to guess merely from looking at the kinematic data that there would be a relationship between the hip and knee moments in the second half of swing." But I think this is irrelevant. We assume that the limb is subject to two kinds of forces, those produced by muscle contraction (controlled forces) and those produced by kinematics and gravity (dynamic forces). Chris's plot as well as my own data show that under certain circumstances, the dynamic torques of one joint are almost linearly related to those at another. This is certainly 'physics.' But this implies that the muscle forces (sum of the torques of all one and two joint muscles and stuff acting about the joints) are equal and opposite to the dynamic forces and are therefore also linearly related - hip to knee, shoulder to elbow.

So how does that happen? Does the CNS detect the angles, velocities and acceleratations of all joints and solve the inverse dynamic equation and use this to activate the muscles. I doubt it and I think this logic reverses cause and effect. Let us suppose that the CNS starts with a plan that uses the same torque patterns at all joints. Then physics takes care of the kinematics. Finding the right pattern for the task is another problem but at least it is not as hard as finding a different pattern for every joint. Appreciate small favors.

It is hardly intuitive that linearity would work but that is our own cognitive shortcoming. And sometimes linearity does not work. Like during stance phase. But let's remember that babies start off kicking their legs and waving their arms and it takes quite a lot of practice before they can actually stand and walk.

Anyway, that's my opinion and I may be wrong.

Gerald Gottlieb <glg@bu.edu>

NeuroMuscular Research Center Boston University Boston MA 02215 fax 353-5737 http://nmrc.bu.edu/MCL/glg.html


Dear Gerald and all,

I'm afraid our local library can take weeks to get hold of articles so my comments are without the benefit of having read your paper.

My first minor point is the division of forces into controlled forces and dynamic forces. Dynamic forces are a shorthand method for referring to the effects of movement. What we really have are movements (particularly accelerations) and the forces which give rise to them (muscles and gravity). In the absence of externally applied forces, such as a ground reaction, there must always be a relation between the movements (whether viewed as kinematics [joint displacements] or velocities or accelerations).

If we look at a simple two segment model (thigh and shin or upper and lower arm) it is fairly straightforward to obtain equations for the moments occurring at the proximal (shoulder or hip) and distal (elbow or knee) joints in terms of the accelerations of the segments and their inertial properties and work out the ratio of the two. From examination of these equations we find that the ratio will be a constant for at least three important cases:

1 That the proximal joint is held in a fixed position (the case which possibly has a bearing on the behaviour of the lower limb in late swing).

2 That the distal joint is held in a fixed position (i.e. the limb swings as a single rigid body).

3 That the movements of the proximal and distal joints are in phase (or 180 degrees out of phase).

It will obviously be approximately constant for conditions which approximate to any of these.

I have two points from this. Firstly that quite a range of movements are covered by these conditions (particularly condition 3) and secondly that regardless of how control is exerted on the joints to achieve this pattern of movement a linear relationship between the moments will be obtained.

For example take the first case. Position control would appear to be the simplest way to hold the proximal joint in a fixed position. Given this, however control is effected at the distal joint, a fixed ratio will be observed between the moments generated at distal and proximal joints. Similarly the achievement of movements which are in phase would be simple at the level of position, velocity or acceleration (moment). Evidence of the fixed ratio of moments alone is not sufficient to determine at what level control is effected.

My analysis does throw up another interesting point. It assumes that the moments required to counteract gravity can be neglected in comparison to those required to cause movement. This is dependent on the frequency of movement (inertial effects are proportional to the frequency squared). Back of an envelope calculations suggest that at frequencies of around 0.5Hz the effects are roughly equal for the sorts of magnitudes of movement occurring during swing phase of gait. Therefore for frequencies above 2Hz this assumption isn't unreasonable (this probably includes lower limb movement in late swing). I'd be interested to know if the fixed relationship also applies to low frequency movements (in a vertical plane).

A further point is that the fixed ratio will be different for the three case above and for the case of in phase movements of proximal and distal joints will depend on the relative amplitude of movement at proximal and distal joints (but not its frequency I think). Evidence of a fixed ratio for a wide range of movements over different ranges would suggest to me that the control is based on maintaining that fixed ratio.

Richard Baker, Belfast <baker@unite.co.uk>


My response to Ricahrd Baker is as follows:

My first minor point is the division of forces into controlled forces and dynamic forces...

no no no. UNLESS control is exerted over the joint moments in this way, linear relations will not be obtained. It is not enough to WANT to perform movements in this way. You must know how to do it, i.e. how to generate the right torques. I think there is a definite directionality in terms of causality here. Also, linear synergy is not confined to these special kinematic cases.

I'd be interested to know if the fixed relationship also applies to low frequency movements (in a vertical plane)...

It is not necessary to assume that gravitational torques are negligible. It is necessary to assume that POSITIONALLY dependent torques (which control posture) are dealt with on a separate level from dynamic forces. As low frequencies and low velocities the dynamic forces approach zero and it is the static forces that will dominate and linear synergy will probably not appear. Analogously, during the stance phase of gait, the demands of support against gravity are not negligible.

A further point is that the fixed ratio will be different for the three case above...

Well read the paper. The ratio is directionally dependent. Not speed or load dependent.

Gerald Gottlieb <glg@bu.edu>


Dear all,

Fresh from my Chinese New Year holiday (Kung Hei Fat Choy, by the way!), I'd like to continue the debate on linear synergy just a little bit further, if you don't mind.

To sum up so far, the linear relationship between hip and knee moments, found only in swing phase, may be a neurological sysnergy or simply reflect passive biomechanical properties of the lower-limb. Gerry Gottlieb (and, I presume, Janice Eng) are still in favour of the former, while Richard Baker tends to the latter opinion. I must say I'm still not at all certain either way.

I'd like to bring in some more data, though. This time concerning stance phase. Although the linear relation does not hold here, David Winter has pointed out (p 82 of The Biomechanics & Motor Control of Human Gait) that when we look at the VARIABILITY of the stance moments, a pattern emerges, in that there is coupling between hip and knee, and knee and ankle. He demonstrates this by using a covariance measure (COV), which he finds (for one subject) to be 89% and 76%, respectively. He says:

"These high covariances are extremely strong evidence of tight coupling between the motor patterns at adjacent joints... not surprising when we consider the opposite and cancelling functions of the hamstrings and rectus at the hip and knee, and gastroc at the knee and ankle."

He goes on to say that "the single joint muscles constitute 2/3 of the physiological cross-sectional are of all the muscles crossing the hip and knee. Thus the magnitude of the COV is well in excess of that possible anatomically, and therefore must be part of a neural control pattern."

A second subject tested on different days manifested COVs of only 73% and 49% respectively, and David suggests that this is evidence for "a plasticity of the motor control system".

Well, I wonder what you make of all that?! Seems to me to be a surprisingly profound piece of work that has been neglected since it was published. Or is it...?

Over to you - I'm sure you'll have your views! In the meantime, Andreas and I are calculating covariances on some of his data, and it will be interesting to see if this confirms Winter's findings.

Dr. Chris Kirtley (Kwok Kei Chi) MD PhD The Hong Kong Polytechnic University


Dear Gerald,

First of all I'm not for one moment suggesting there is no substance in what you say. I am simply saying that it is vitally important to distinguish exactly what is a consequence of the mathematical derivation of the quantities we are measuring and what is not.

Perhaps the crux is this section of the comments:

secondly that regardless of how control is exerted on the joints to achieve this pattern of movement a linear relationship between the moments will be obtained.

no no no. UNLESS control is exerted over the joint moments in this way, linear relations will not be obtained. It is not enough to WANT to perform movements in this way. You must know how to do it, i.e. how to generate the right torques. I think there is a definite directionality in terms of causality here. Also, linear synergy is not confined to these special kinematic cases.

I agree entirely that you must know what torques to apply to control a joint. The important question is how you know? For example how do you know what torques to exert to hold a joint stationary? The simplest mechanism (if it is available, I'm only an engineer!) is to use position feedback. ie use position feedback to control the force generated in the appropriate muscles. If this is achieved at the proximal joint (as in the late swing phase of Chris's original example), I am telling you that whatever happens at the distal joint, you will observe a linear relation between the moments measured at both joints. Unfortunately the evidence for this only exists on the back of an envelope (quite literally). I'll try and tidy it up over the next couple of days and e-mail you a copy. The notation will require something other than just text can you handle a Word 7.0 file, if not give me a fax number and I'll fax it to you.

Perhaps my most serious limitation in contributing sensibly to this debate is not having a copy of your paper. Not working in an academic environment this can be a slow process for me. Is there any chance you could post me a reprint (and include any other relevant papers you've written as well.)

Richard

Gait Analysis Laboratory Musgrave Park Hospital Stockman's Lane Belfast Northern Ireland BT9 7JB Fax: +44 1232 683816


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