Where the memory model comes from

What is conventional

The baseline is a driven oscillator with a chosen internal-state law. Chua–Kang supplies the broad memristive-systems lineage; the scalar relaxation belongs to an established family. The contribution here is an inspectable teaching model and comparison workflow. No new dynamical class or empirical discovery is claimed.

An exact embedding with restricted functions

dm/dt = αu²(1 − m) − m/τ
       = [m∞(u) − m] / τeff(u)
τeff(u) = 1 / (αu² + 1/τ)
m∞(u) = αu² / (αu² + 1/τ) = 1 − τeff(u)/τ

This algebra is our derivation. It embeds the scalar equation in a generic first-order relaxation family. It restricts the equilibrium and rate to a linked, even-in-u pair of functions. At a single pulse amplitude, exponential kinetics alone do not identify a distinct mechanism.

What CALM adds

Rivera-Sierra and Bisquert’s equations 3–5 use internal voltage u, terminal voltage V = RₛI + u, current I = [gL + (gH − gL)x]u + Cₘdu/dt, and relaxation toward xeq(u) with time τk(u). Their circuit and chosen voltage functions are more general than the restricted scalar embedding above. Primary paper

Using I ≈ [G₀ + ΔGm]V is a reduced observation model only after justifying u ≈ V and neglecting capacitive current over the analyzed interval. Conductance and voltage require physical units and calibration. Under that reduction, q and p do not enter the current fit; a fit cannot validate the oscillator coupling. Identical parameterizations must count as one model family, not two competitors.

Finite memory is still memory

A volatile physical record can be useful. Its existence does not by itself establish learning or persistent storage. The original scalar state contracts under identical input and decays exponentially at u = 0. Boyd–Chua proves approximation results for operators meeting fading-memory conditions; it does not make all such systems exactly a finite linear filter. A separate stability argument is needed before assigning the full parametrically driven oscillator that property for every input. Fading-memory reference

A discriminating test

A feature near a training amplitude can arise in a nonlinear fading system. It is not a universal signature against all relaxation models. Compare explicit equations with calibrated nuisance parameters and matched histories; reject only the specified alternative. Read the protocol · Inspect state-written candidates