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View Full Version : Re: Summer science quiz #2... The answer!!



Pietro-luciano Buono
05-24-2001, 06:22 AM
Dear all,

On Fri, 18 May 2001, Jon Dingwell wrote:

[cut text]

>
> A number of people have worked on the idea that central pattern generators
> (CPGs) in the spinal cord can be modeled as sets of coupled nonlinear
> oscillators. As it turns out, when CPGs are modeled as *SYMMETRIC* rings
> of coupled oscillators, they exhibit all of the gaits (and the appropriate
> gait transitions) exhibited by terrestrial animals in nature, regardless of
> the number of pairs of legs. It is the symmetry conditions on these rings
> of coupled oscillators that produces pairs (even numbers) of legs.

It seems to me that pairs of legs arise not because of the
symmetry of some network of coupled oscillators, but because
of the bilateral symmetry of the animal. The networks of coupled
oscillators found in our work (see below) are models built using
the natural symmetries found in the animals and their gaits.

In an animal like the starfish appendages do not necessarily come in pair.
For instance, there are starfish with five appendages. My guess is that at
a certain stage of development the starfish is circularly symmetric, it is
a well known fact that in systems with circular symmetry, the
symmetry-breaking can lead to a symmetry group with any regular polyhedral
symmetry. Thus, any number of appendages could appear. Of course, other
factors enter the development which would prescribe a given number. But as
far as I know, this number can be odd or even.

Best
Luciano

> There are a number of published articles on this, which may not be familiar
> to many in the biomechanics community, the most prominent of which are (in
> my opinion) the following:
>
> Collins, J.J. and Stewart, I.N. (1993). "Coupled Nonlinear Oscillators and
> The Symmetries Of Animal Gaits." Journal of Nonlinear Science, 3: 349-392.
>
> Golubitsky, M., et al. (1999). "Symmetry in Locomotor Central Pattern
> Generators and Animal Gaits." Nature, 401 (6754; Oct. 14): 693-695.
>
>
> For those of you who wish to dig around into the details a bit further:
>
> Buono, P.-L. (2001). "Models of Central Pattern Generators for Quadruped
> Locomotion I. Secondary Gaits." Journal of Mathematical Biology, 42 (4):
> 327-346.
>
> Buono, P.-L. and Golubitsky, M. (2001). "Models of Central Pattern
> Generators for Quadruped Locomotion I. Primary Gaits." Journal of
> Mathematical Biology, 42 (4): 291-326.
>
> Collins, J.J. and Richmond, S.A. (1994). "Hard-Wired Central Pattern
> Generators For Quadrupedal Locomotion." Biological Cybernetics, 71: 375-385.
>
> Collins, J.J. and Stewart, I.N. (1993). "Hexapodal Gaits And Coupled
> Nonlinear Oscillator Models." Biological Cybernetics, 68: 287-298.
>
> Collins, J.J. and Stewart, I.N. (1994). "A Group-Theoretic Approach to
> Rings of Coupled Biological Oscillators." Biological Cybernetics, 71: 95-103.
>
> Golubitsky, M., et al. (1998). "A Modular Network for Legged Locomotion."
> Physica D, 115: 56-72.
>
>
> I am forwarding this message also to Pietro-Luciano Buono and Marty
> Golubitsky because I think they have done the most recent work on this
> topic and because I am not sure if they receive Biomech-L. I would be very
> interested to hear their comments on this issue.
>
>
> Regards,
> Jon Dingwell
>
> ----------------------------------------------------------------------------
> Jonathan Dingwell, Ph.D.
> Postdoctoral Research Associate
>
> Sensory Motor Performance Program
> Rehabilitation Institute of Chicago & Northwestern University
> 345 East Superior, Room 1406
> Chicago, Illinois, 60611
>
> Phone: (312) 238-1233 [Office] / (312) 238-1232 [Lab]
> FAX: (312) 238-2208
> E-Mail: j-dingwell@northwestern.edu
> Web: http://manip.smpp.northwestern.edu/dingwell/
> ----------------------------------------------------------------------------
>
>
> > -----Original Message-----
> > From: Dr. Chris Kirtley [SMTP:kirtley@CUA.EDU]
> > Sent: Tuesday, May 15, 2001 2:12 PM
> > To: BIOMCH-L@NIC.SURFNET.NL
> > Subject: Summer Science Quiz #2
> >
> > Dear all,
> >
> > My little quiz seems to have proven very popular. So, since it's Summer
> > (at least in the Northern hemisphere), when academics languish like
> > chimnies, here's another...
> >
> > This one is from none other than Aristotle himself, who in 350 BC asked
> > the following question in his "On the Gait of Animals" <
> > http://classics.mit.edu/Aristotle/gait_anim.html >
> >
> > "Why do all animals have an even number of feet?", or in other words,
> > why are there no three-legged animals, for example?
> >
> > Unlike the previous question, this one has no solution (apart from
> > Aristotle's, which I find personally rather dissatisfying).
> >
> > Chris
> > --
> > Dr. Chris Kirtley MD PhD
>
>

------------------------------------------------
Pietro-Luciano Buono
Centre de Recherches Mathematiques
Universite de Montreal
www.crm.umontreal.ca/~buono

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