My model had to learn a simple task: the Exclusive Or (XOR) task. This task was to discriminate between these four patterns:

In fact it is a very well known logical relation, and the real-world interpretation would be that, let's say, you run from the lion if it is either alive or if the door of the cage is open. But if both are true you stay (ehm. . . because you are so terrified you parallize). And, also counter-intuitively, if both are false, meaning that the lion is dead in an open cage you also run away (ehm . . . maybe to warn the circus director). Well allright the example is bad but hey it is just an example!

Anyway, in the above figure you see how the network learned this task. On the horizontal axis we see developmental time. Each step on this scale indicates a change in the connections of the network. The network consisted of 6 neurons that converged onto a stable pattern everytime one of the XOR-patterns was presented to 3 of these units (these units were called visible units because they received the input patterns and produced the output patterns that were used to calculate performance. The other units are called hidden units). On the vertical axis we see how bad my network was doing. It is the normalized error which means it is some aggregate measue of performance summed squared and averaged over all the units taken together. Ideally, this error should be zero. When it is zero, the system displays a pattern of behavior that is perfectly suited to the task it was supposed to learn: it learns to discriminate all four XOR-patterns. In this particular run it took about 400 developmental timesteps to learn the task (however, you could speed this baby up to 20 timesteps or less depending on some general parameters! The pattern, however, always remained the same).

Moreover, we the network actually developed this regularity going through stages of other regular behavioral patterns, (depicted by the relatively horizontal parts of the trajectory) with sudden jumps connecting them (depicted by point B, C and D respectively). You can see how the network learned to map each XOR-pattern ("the input") onto a stable behavior (the "output") plotted in a 3-dimensional graph.

The x,y and z-axis in this graph represent the activations of the 3 visible units of the network. Walking through these snap-shots you see the sudden emergence of new stable patterns in the activation space. Ultimately, four of the eight stable patterns end up in the right corners of the cube and error approaches zero.

This specific network, a recurrent backpropagation network, was designed by Fernando Pineda, who already noticed these dicontinuities (Pineda, 1988; 1995). We wanted to test somehow whether these discontinuities were actually the real developmental jumps, and not merely very fast but continuous decreases in error. In order to do that, we had to test whether these jumps were the result of mathematical phenomena called bifurcations. Bifurcations are one of the most important principles by which complex systems self-organize. As it turns out (thank God) the jumps are real. This was because they show several accompanying qualitative phenomena that could be detected after some testing of the data, called catastrophe flags. Among these flags we have the effect that no matter how small you make your time-steps, you would still find that the jumps are discontinuous, that convergence upon each stable pattern takes longer on the behavioral level if you are close to a developmental discontinuity, and that the stable patterns are localized patterns (in this case points) in the activation space instead of distributed zones with no particular regularity.

Now for the bad part

You may have noted that there were 8 stable patterns instead of 4 in the cube. And this turned out to be quite problematic. If the task was really interpreted as an action scheme like the lion-cage example, one could say that the network learned to do all the desired things as in my example above, and at the same time to do all the opposite things as well. So run away when the lion is alive but the door is closed; and stay where you are when the lion is alive but the door is closed. Kind of schizophrenic action planning I hear you saying. And inconsistent with the pattern of staying where you are when the lion is alive and the door is open (if you follow me). Worse still, it was purely the random factors (like the random connectivity before learning) that determined what the result would be in a particular run in a particular situation. This lead me to think that something was badly missing in this developmental model.