Coupling Constants: a Flip-Flop arrangment

A "Flip-Flop" is the central element of a computer. It is discussed here for several reasons: Firstly, it illustrates the idea of a coupling constant; secondly, it shows how the behaviour of a structure can vary radically with a minor change in a coupling constant; thirdly, the generalization of the concept of a flip-flop can be important in explaining other phenomena.

The illustration shows the generic connection of a flip-flop. Two elements (which we will later identify with the perceptual input functions of two control units) each have two inputs, one of which comes from outside, while the other comes as an inhibitory input from the output of the opposite element. The two elements are saturating amplifiers, which cannot go more positive or negative than their saturation values. When input A is high, it tends to depress the output of B, which relases A from inhibition, increasing its output. If the gains of the two cross-connection amplifiers are high, only one of the two outputs can be "high" but if the cross-coupling gains are low, each is only moderately affected by the output of the other.

The relationship between the outputs of the two elements, their external inputs, and the coupling constant can be shown as two related 3-D diagrams. The x-axis (left-to-right in the diagram) represents the difference between the A and B external inputs (assuming their sum stays constant). When the A external input is much higher than the B input, the A output is always high and the B output low, but the degree to which this is true depends on the gain of the cross-coupling amplifiers (the coupling constant), shown on the y-axis (in and out of the viewing plane). The outputs are shown separately in the two halves of the diagram, A on the left and B on the right.

The arrows shown on a path on the near edges of the two diagrams illustrates how changes in data values may affect the values of the two outputs. Suppose A is high and B low with a high coupling constant (red dots), and that the B input is gradually raised. The output traces a path rightward across the upper sheet of the A diagram and the lower sheet of the B diagram until it comes to the reverse curve, at which point it switches abruptly, to the lower sheet of A and the upper sheet of B. If external data are then returned to their former values, the outputs do not. They are at the positions marked by the blue dots, with A staying low and B staying high.

The diagram above shows a characteristic phenomenon of such coupled systems--a change in the dynamics that depends on the coupling constant. If the coupling constant is low, as at the condition shown by the green dots, the output values are determined by the external input values. If, however, the coupling constant is above some transition value, there are input values for which the output values split into two possible pairs (shown by the red and blue dots). For such intermediate data values, The transition is called a "bifurcation point" in the dynamics of the coupled system (the diagram at the left shows a slice through the two 3-D diagrams for a particular set of input data values that slightly favours A, but the bifucation point still exists, as shown by the split curves). A is high and B low if that was the situation earlier, or the contrary might also be the case for the same external data input values. The situation will not change until either the coupling constant is reduced, or the external input data values change.

The arrangement here is a single flip-flop. Next we will look at a more complex arrangement, in which two flip-flops are intercoupled so that each can bias the other.