EXPERIMENTS 03 & 04 · COMPUTING WITH FEEDBACK
Give persistence
a job to do.
Hold a choice. Remember a sequence.
Two uses of the same sine feedback rule: a bit that keeps its state, and a driven system that carries traces of recent inputs.
03 / A WRITABLE BIT
A choice that survives the pulse.
A brief phase pulse writes 0 or 1. Release it and let feedback hold the state. Try a small disturbance, then one large enough to switch sides.
Try this with the default settings: click Write 1, Hold +50, Apply disturbance, then Hold +50 again. Does the state recover? Set the kick to −1.00 and repeat the disturbance and hold. Then choose “let it fade,” click Write 1 and Hold +50, and compare.
Not yet written
State · Update
Write a bit to begin.
State through time
Last 300 events · shaded read regionsRead 1 at w ≥ 0.3 and 0 at w ≤ −0.3. The middle region has no reliable decoded bit. Vertical jumps show a direct state kick.
Write 0 or 1, then hold the state.
Why this bit holds
At β = 1.2 and zero noise, the two attracting states are approximately ±0.855615. A positive write produces cos(1.2w), which is positive for every w in [−1, 1]; a negative write does the reverse. With phase noise bounded by 0.02, the intervals [0.3, 1] and [−1, −0.3] remain invariant during hold. Direct state kicks can leave these regions. At β = 0.8, an unforced noiseless state instead approaches zero.
04 / MEMORY OF A SEQUENCE
How much of the past is still readable?
Send in a new −1 or +1 at each step. This system keeps receiving inputs, while the bit above holds a choice after its input ends. Its 32 values can carry traces of recent inputs. We learn how to combine those values to guess an earlier input, then test the guesses on a new sequence.
Preparing the first experiment…
Recall versus delay
All scores use unseen inputsRead the earlier input
Sine readout · final 60 test samplesRecall uses a fit score, R², rather than a percentage correct: 1 is perfect; 0 matches always guessing the test average; negative is worse. XOR asks whether the bits one and two steps earlier differ. Its score is the percentage of correct answers.
| Model | Recall R² | Recall MSE | XOR accuracy |
|---|
The delay buffer stores the required past inputs explicitly and computes the answers directly. It shows what ordinary storage can do. The linear bank is a reference for linear recurrent processing; it also uses different decay rates, so more than the sine function changes. The no-feedback bank keeps the sine units and removes their recurrence.
The equation and evaluation method
Each state follows w′ⱼ = sin(cⱼ + βwⱼ + γmⱼuₜ), with fixed phase biases cⱼ and input weights mⱼ. States do not interact. Only the final weighted readout is trained. The linear baseline uses a range of stable signed decay coefficients. All trained models see the same training and test inputs; both streams start from zero states and process 200 warm-up inputs before scoring.
With identical inputs and |β| < 1, differences between starting states shrink toward zero. A finite warm-up does not guarantee that startup effects are negligible: after 200 updates, the remaining fraction is at most |β|²⁰⁰, about 7.1 × 10⁻¹⁰ at β = 0.9 but 0.134 at β = 0.99. The benchmark uses the specified zero start in every run.
Recall at delay d targets uₜ₋d after processing uₜ. Delayed XOR targets the two bits at t−1 and t−2. Feature scaling and readout fitting use training data only. The exported result records the seeds, weights, settings, target definitions and test predictions. A single seeded run is an experiment, not a claim of general superiority.
TWO KINDS OF PERSISTENCE
The bit keeps a choice after its input ends. The sequence processor gradually loses dependence on its starting state while retaining traces of recent inputs. Each turns feedback into a different, measurable task.
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