EJ:Â So when someone scribbles between two points, itâ€™s not known whether the

deceleration and reversal of direction is linear, power law, or some other

function?Â Iâ€™m wondering whether change of direction along a line is a

simplified paradigm for then extrapolating to velocity along a curve.Â It

sounds like itâ€™s been hard to specify the controlled variables for 2D

drawings.

AM:

Maybe that is a good idea. There is a lot of research on point-to-point movements, and Bill never published his analysis done with the LittleMan simulation.Â

Â

EJ:Â I suppose it wouldnâ€™t work to have the Little Man follow an ellipse

target, and see what happens.Â If power-law dynamics are involved in mouse

movements to get the target going in an ellipse, that seems like putting the

dynamics into the reference itself.Â Maybe Bruce Abbott is the one to ask

here.Â Can a point moving in an ellipse be programmed as a target in Little

Man V2, for instance at a constant speed rather than a power-law varying one?

AM:

Yes, you can also take a position control model (even without arm dynamics simulated) and see what happens. I’ve been doing that, and it seems position control is not enough here, I don’t have a complete solution.

Â .

EJ:Â I donâ€™t understand the â€œat any overall speedâ€? portion.Â Your illustration

of the three ellipses with equal time-distant points showed an ellipse on the

left where the points were also equally distant spatially (beta of 1), so it

seems an ellipse can be traveled at a constant speed without it being forced

into a power law relationship.Â But you also emphasized, in your discussion

with Rick, that the 2/3 power law regularity emerged when the speed of the

drawing was fast enough to not deliberately counteract what may be happening

on the curves.

AM:

Movement at overall speed means that there might be variation in local speed, but the average speed of the total movement is relatively constant across cycles. People can traverse ellipses at low overall speed without following the power law, yes. For trajectories generated with orthogonal sinusoids, at any overall speed, speed will be correlated with curvature.

Â

EJ:Â Iâ€™m trying to consider what happens when various repetitive phenomena –

e.g., a line between two points, a sine wave, an ellipse – are produced not

by a formula but by a living control system.Â The formula for an ellipse comes

out of the two radii.Â A drawn version of an ellipse may relate to the

endpoints of greatest inflection on the curve.Â I donâ€™t have equipment to

measure those things myself, so I have to rely on what others may have done.

I am trying (perhaps simplistically) to apply step-changes in Position

control, with two articulation points, to see whether a power-law relationship

â€˜falls outâ€™ of the model as an un-controlled outcome, i.e., a side effect.

Â Â

EJ:Â From what Iâ€™ve heard so far in this discussion, this possibility is not

(yet?) ruled out.

AM:

If you have only point-to-point movement, then there is no curvature, right? So, you’d need something else controlled to see what falls out.

Best,

Adam