This circuit is an evolution of the uTone circuit. It uses shift registers to make a ring counter, the oscillation happens when outputs are fed back to the input. In the uTone circuit this happens pretty clearly, the touch pads are linked directly with one input of an XOR gate. In that circuit noise modes are achieved by using a switch to connect another bit to the other input of the XOR gate. Much like Fig a below. See the uTone page for more info on that circuit.
With the walking ring, I wanted to try and get rid of the mode switch. To see what it sounds like when the mixture of bits on the fingertips are decoded using some op-amps in a window comparator configuration. A very even mix of two square waves is shown in Fig b next to the schematic. I began to think about making two thresholds so that a mixture of bits made with fingers can then be un-mixed with the window comparator. Maybe obviously fingers are not perfect resistors so the system oscillates, but not in the same way it would if there were two separate and direct connections to the input of the xor gate. Or even a single connection to an inverter. Fig c shows the scheme that is outlined in the paper circuit link above. The string of resistors along the right side set the thresholds and can be tuned for sensitivity by the potentiometer at the bottom. It is an XNOR gate that works on one input with two thresholds. Initially I had tried to use the exact truth table of an XOR gate as in Fig d, but the results of this were not so great. The circuit in Fig c seems to give a fairly wide variety of waves and feels more intuitive.