For that purpose, I consider the task of maintaining a pendulum in the reverse position by controlling the position of its axis along a horizontal axis.

The controlling mechanism is a Kohonen self organizing map with 64 cells, which inputs are the angle and angular speed of the pendulum. The angle is 0 at bottom, pi/2 at right, and pi at top. When the pendulum is at left, we consider its symetric, that is we negate the angle and the angular speed of the pendulum and the acceleration of its axis. The weights of the inputs are termed "receptive field".

The motion of the pendulum is controlled by giving an acceleration to its axis in function of the activated network cell (we know that only one is activated at a time in a self organizing map). The value of the acceleration in function of the activated cell is termed "projective field".

My theory is that only the receptive field is involved in motor task
learning. The behavior of the system is modified by tuning the input weights
(through the regular, unsupervised learning mechanism of a self organizing
map), while the set of possible actions remains the same during the whole
process : the result of the process depends on what the system *can* do
(by its constitution), not on what it *has to* do as in supervised
learning processes.

The projective field being not involved in the learning process, one has to determine it by hand before start. For the present experiment, it has been found by groping. The landscape below represents the accelation in function of the coordinates of the activated cell in the self organizing map.

Another way to compute the projective field would be by genetic algorithm, with for instance the time spent by the pendulum above its axis as the fitness function. If we include the size and topology of the network as additional parameters to be optimized, we get a fairly plausible model of what is at work in real living organisms, knowing that the self organizing map itself already has a good biological plausibility.

Why should the projective field coefficients arise genetically ? Well, in our model, a coefficient in the projective field (as strength of the action triggered by a neuron's activation) is actually undistinguishable from a physical feature as such : it may be the strength of a muscle, for instance. This is why it mussn't be included in the learning session's weight tuning, but rather in some evolutionnary process that will decide what the system will be able to do (walk ? one needs legs, ...), and how (how strongly, how quickly, ...).

Here is a demonstration applet of the discussed system. Sometimes we can see an interesting behavior : the ability of the system to raise the pendulum and keep it in equilibrium for a while.

Here is some quantitative material about the system's behavior without and with learning. The time spent above the axis is calculated over 1000 individual sessions of 1000 iterations (100 seconds) each. The sessions with learning consist in a preliminary 1000 iterations proper learning session, followed by an additional 1000 iterations session for the evaluation.

Here is the source code of the demonstration applets.

created on Sat Jan 6 2001