A recent study extends a classic property of linear systems—complete monotonicity—to nonlinear dynamics. For any two past inputs where one stays above the other, the difference in future outputs follows a pattern that is itself completely monotone. This property, called *completely monotone memory*, is a strong form of monotonicity and also imposes stability constraints similar to fading memory, contraction, and incremental stability. In linear time‑invariant (LTI) systems the new definition is equivalent to the traditional notion of a completely monotone impulse response, a result that recovers classical relaxation theory. However, the equivalence does not hold in general for higher‑dimensional or nonlinear systems: stability and monotonicity alone are inadequate to guarantee complete monotone memory in those cases.

The authors formulate the property using the Hankel operator, which maps past inputs to future outputs. By requiring that positive past increments produce completely monotone future increments, they translate nonlinear relaxation into an input‑output framework that collapses to the classical definition when the system is linear.

To illustrate relevance, the authors examine gating dynamics in conductance‑based neuronal models, including the Hodgkin‑Huxley model. Their analysis shows that the monotonicity and stability assumptions behind the framework naturally capture key features of neuronal excitability.

The work clarifies the relationship between monotonic behavior, stability, and memory decay in nonlinear systems. It also highlights limitations: in multi‑dimensional, strongly nonlinear, or non‑output‑affine systems, additional conditions beyond monotonicity and stability are required to guarantee completely monotone memory.