Memory in a complex system may be visible not only in correlations between waiting times, but also in what follows a particular event. A preprint combines a mathematical theory of history-dependent jump processes with an analysis of BK ion-channel recordings and reports an exact way to reconstruct fluctuations from the full observed history. In the channel data, two BK classes ranked differently depending on whether memory was measured by dwell correlations or by a response residual.
The central question is whether a fluctuation-response theory can remain exact when a jump rate depends on the realized past rather than only on the present state. The paper asks whether the record of past events can be used directly to describe the fluctuations produced after the starting point, without leaving an unexplained remainder.
A response can carry history
To build that description, the authors use conditional intensities, the history-dependent tendency for a transition to occur, together with the Doob-Meyer decomposition and martingale representation. In practical terms, each transition count is split into an event tendency based on the observed history and a martingale increment, the part left after that tendency is removed. Those increments are orthogonal across transition channels and times, so they can serve as a basis for trajectory fluctuations.
The theory then assigns each event an event-consequence kernel: a record of the effect that event has on a chosen trajectory observable under the relevant history. That same object serves as the coefficient of the event's fluctuation increment; multiplied by the event's intensity, it becomes the response associated with that transition. The result gives one coordinate that links an event, its consequence and the measured response.
With the full history retained, these response coordinates reconstruct the fluctuations generated after the initial condition exactly. The representation has no unexplained residual, according to the theory.
Once histories are averaged, the reconstruction becomes a lower bound rather than an equality. The missing, nonnegative part is an intensity-weighted response-heterogeneity gap. It is not a single fixed property of a system: it depends on the observable being measured, the family of rate perturbations being considered and the time horizon.
That gap also gives the authors a test for coarse-graining, the practice of replacing a rich history with a shorter description. A candidate retained history is response-sufficient for a chosen observable and edge-perturbation family when the residual gap is zero; keeping more information cannot increase that residual. In a solvable coarse-grained network, a state-only surrogate matched standard kinetic statistics but failed to preserve a directional response. The directional observable required the current state plus the residence age and the previous state.
The channel data reverse the ranking
The empirical test used thresholded open-closed trajectories from 30 cell-BK patches and 26 mitoBK patches. The recordings were made at membrane voltages of -60, -40, -20, 20, 40 and 60 mV, with a sampling frequency of 10 kHz. Across the records, the analysis contained 433,523 cell-BK dwells and 100,395 mitoBK dwells, with a dwell representing a period spent in one of the channel's open or closed states.
Quality control reduced the fixed-horizon comparison from 56 patches to 52: 27 cell-BK and 25 mitoBK. The state-wise shuffled controls kept the marginal dwell distributions, open fraction and total recording time, while removing the serial order of events.
Against those order-shuffled controls, cell-BK had the larger excess same-state dwell-correlation median, 0.2077 versus 0.1290 for mitoBK. The ranking flipped for the state-and-age excess response residual: 0.0373 for cell-BK versus 0.1954 for mitoBK. The dwell-correlation comparison was reported with p = 0.00671 in a one-sided Mann-Whitney test.
The finding is not that one channel class simply has more memory. It is that the answer changes with the question: cell-BK showed stronger conventional dwell memory, while mitoBK showed the larger residual after the present state and its age were accounted for.
The useful memory depends on the question
The researchers next asked how much of the mitoBK residual could be explained by retaining preceding dwell durations. Its median excess residual fell from 0.1954 when no preceding dwell was retained, to 0.1529 with one, and 0.1038 with two. The earlier dwell history therefore explained part of the response-relevant memory, but did not exhaust the reference residual. The between-class residual difference was reported with p = 0.00000279 in a one-sided Mann-Whitney test.
A frequency-resolved analysis pointed to slower dynamics as the strongest part of the missing response history. For mitoBK, the median excess residual was 0.2214 at angular frequency 0.5, 0.1436 at 2 and 0.0989 at 16. It remained positive throughout the resolved band, although the analysis was limited to that band and to the finite-history reference used.
Taken together, the results make a case for treating memory as a response property as well as a correlation property. For a selected observable and perturbation family, the proposed gap can show whether a retained history is enough for that task. In the BK recordings, the two memory rankings went in opposite directions, and the response-relevant residual in mitoBK was most pronounced at low frequency.
Paper data and sources
Original title: The Memory Hidden in Response Fluctuations: Trajectory-Level Fluctuation-Response Theory and Inequalities for Non-Markovian Jump Dynamics
Authors: Jiming Zheng, Zhiyue Lu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text