Pipeline or Loop

Two units, A and B, each described by one differential equation. On its own, each unit simply relaxes back to rest after a kick: a single first-order variable can never oscillate. Couple the two equations and the pair can ring at a period neither unit has, or hold activity far longer than either unit could.

The solid traces are the two equations solved together. The dashed traces are the pipeline prediction: A solved on its own, then A's output handed to B, the way a box-and-arrow diagram would have you trace it. When no arrow runs back from B to A, the two agree exactly.

Solved together Pipeline prediction Kick

Circuit

Click an arrow to connect or cut it. Arrowhead excites, bar inhibits.

ModelτA dA/dt = −A + wBA·B
τB dB/dt = −B + wAB·A

Normal speed shows 100 ms of model time per second (10× slow motion).

Activity

Each lane rescales to fit; the vertical bar on the right gives its scale. Hover for values.

Phase plane

Horizontal: A. Vertical: B. Each straight line marks where one unit stops changing. Click anywhere to start the pair from that point.

Each part alone
The pair, solved together
Loop gain

Things to try

  1. Uncoupled, then kick A and kick B. Each unit decays straight back to zero at its own rate and never overshoots. Kicking one has no effect on the other.
  2. Feedforward A→B, then kick B. A does not move, because nothing downstream reaches upstream. The dashed prediction sits exactly on the solid trace: this system really can be solved one stage at a time.
  3. Negative loop, then kick A. Both units ring, with a period that neither unit has. Kick B and A responds. The pipeline prediction misses both traces.
  4. Starting from feedforward, drag the B → A weight slowly from 0 toward −1. The mismatch grows smoothly from zero. While the loop is weak the pipeline is a fair approximation; the readout shows the loop gain at which the pair starts to oscillate.
  5. Positive loop. Units with 10 and 30 ms time constants now hold activity for most of a second, and with the repeated kicks they add them up. Loops like this are the core of models of persistent activity, such as the oculomotor integrator.
  6. With feedback on, switch which unit the pipeline treats as upstream. The prediction depends on where you cut the loop. A loop has no first stage.