R. Port, November, 2001
What is a mass-spring system?
A Slinky hanging from your hand is a good example. It is a simple system of a mass and a spring (and, of course, gravity). The Slinky will settle at some position, the neural position where gravity acting on the mass balances the spring. The system is controlled by a small number of parameters: the MASS, spring STIFFNESS and DAMPING. If you shake it, it will oscillate at some frequency, f, determined mostly by the stiffness of the spring. If you start the system by pulling the spring above or below the neutral position, it will oscillate with an amplitude that initially matches the initial deviation. The farther you pull it up or down when starting it, the
larger the amplitude will be.
If you increase the MASS,
it will hang at a lower neutral position and oscillate more slowly (other
things being equal). If you increase the STIFFNESS of the spring, it will
oscillate faster. Reduce the STIFFNESS and it will slow down. If
you increase the DAMPING, its oscillations will die out more quickly and
settle at its neutral position. If the DAMPING is reduced to zero,
it will oscillate forever at frequency f. The term CRITICAL DAMPING
means sufficient damping to prevent overshoot as the mass approaches
the neutral position - that is, it will not oscillate but just asymptotically
approach the neutral position. For controlling damping in the Slinky
model, imagine oscillating it under water (which will provide greater damping
than using it in air). For even greater damping, imagine using it in a
bath of oil or honey. In honey, Im sure the Slinky would be critically
damped: pull it away from its neutral position and it will slowly return
to that position without oscillating.
Now, if the monkey's arm in the Pollit and Bizzi experiment behaves like a mass-spring system with a controllable neutral position, how does this help us understand how the monkey can move its arm to point the right way without being able to see the arm or get any sensory feedback from it about where it is located and no matter where you artificially place it.
The point is that if you assume that feedback (from vision or from joints) is logically essential for control of the arm, then you are basically assuming that the brain (or central nervous system) issues commands to muscles of the form `Start pulling and pull until I tell you to stop'. In this case, feedback to the brain (or CNS) is needed to inform the command to stop pulling. But actually the command says `Adopt a position pointing in a certain direction.' So the target angle is specified by the value of the resting angle of the limb - which can be adjusted by changing the stiffness of the antagonistic flexor and extensor muscles. So the command says `Adopt these stiffnesses of the antagonist muscles controlling the limb'. (Presumably these have been learned through earlier experience controlling the limbs.) Then, whatever the angle might be at the moment the command is issued (and no matter where the experimenter may have surreptitiously moved the animal's arm), the command is the same and has exactly the same effect. After sending the command to the peripheral nervous system to set the rest lengths and/or stiffnesses, then the arm just `settles' to its new fixed point - just like a displaced pendulum in honey (that is, a critically damped pendulum) settles gently to the vertical position from any possible initial state. Just like a mass-spring system (or a damped pendulum), the maximum velocity is obtained at the midpoint of the gesture, and is greater when the movement is farther and smaller when only small movement is required.