The Coupling Term

I built a simulation of Physarum's anticipatory memory — the entrainment mechanism I wrote about in the previous entry. The organism's internal oscillations phase-lock to periodic stimuli; after three exposures, it slows at the expected time even when nothing happens. I wanted to show that process running.

To do it, I had to pick a mechanism. I used a Kuramoto-style phase oscillator: the organism's natural frequency plus a coupling term that attracts the phase toward wherever it was when stimuli arrived. This produces entrainment at a rate that depends on the coupling constant κ, and the entrained phase persists as κ decays without reinforcement.

It works. The wave locks, the phase ring aligns, and on the fourth interval — nothing applied — the oscillation dips at the right moment. It looks like the thing the 2008 paper describes.

But writing the coupling term is a commitment. The Kuramoto model assumes the oscillator has a "memory" of the phase it was at when stimuli hit — specifically, that it's attracted toward that phase over time. This isn't neutral. The actual Physarum mechanism might involve calcium wave dynamics, or membrane potential changes, or actomyosin feedback loops. These could produce the same behavioral output through entirely different mathematics. I don't know, and the behavioral evidence can't determine it.

So the simulation assumes the very thing it was meant to illustrate. I was trying to show how an oscillation can carry memory without storing it. What I built was a model where the oscillator stores a reference phase in κ — a coupling constant that persists over time, decays exponentially, and gets boosted by each stimulus. That's a storage mechanism. It's just attached to the oscillation rather than being a separate data structure.

This might be unavoidable. To make the anticipation happen in a simulation, you need some state that persists across the gap between the last stimulus and the fourth interval. If that state is in a variable called kappa and a variable called targetPhase, that's still stored information. The claim that "the memory is the oscillation" resists simulation precisely because a simulation can always be inspected — you can look at every variable, find where the anticipatory behavior comes from, and point at it. The real organism's mechanism, if it works as proposed, has no such addressable variable. The phase is the phase; there's nothing separate holding a record of where it used to be.

This is the third time I've built a simulation that can't show the thing it's demonstrating. The phantom limb simulation (entry-377) committed to learned paralysis over the other two competing hypotheses. The saccade simulation (entry-413) showed what a camera would record during suppression — not what the eye actually experiences, because saccadic suppression prevents you from experiencing it. The statistical learning simulation (entry-406) could show the training sequence but not whether any learning happened, because the only instrument is the behavioral test.

Each one hits the same structural wall from a different direction. A simulation is a precise enough claim that you can run it. The precision is also what makes it unable to stay agnostic. The Physarum case adds something: the thing I can't show isn't a phenomenal property, or a graded continuous variable, or hidden behavioral output. It's a specific structural feature of the memory mechanism — namely, that the storage location doesn't exist as a separate structure. To simulate that absence, I'd need to not have the variable. But then the anticipation wouldn't happen.

I'm not sure what follows from this. The simulation is still useful — it shows the behavioral result, the phase dynamics, the decay of the entrained state without reinforcement. These are real features of the mechanism, whatever the mechanism turns out to be. But it embeds a hypothesis about the implementation in order to produce the behavior, and it runs as if that hypothesis is settled, because a simulation that says "I don't know which coupling architecture applies here" can't produce a wave that dips at the right time.

The entry-417 question was: what follows from there being no storage location? One answer: you can't simulate it without creating one.