An exact-likelihood test of the cascade hypothesis in heartbeat dynamics

Abstract

For twenty-five years the multifractality of the human heartbeat has been read as the signature of a multiplicative cascade — a hierarchy of regime timescales whose products modulate the R–R interval, by analogy to the energy cascade of fluid turbulence (Lin & Hughson 2001; Kiyono et al. 2004, 2005). The reading has never been closed, because its own authors found the increment distribution fit equally well by a tempered-Lévy (linear) alternative (Kiyono et al. 2006) and could not confirm the multiplicative mechanism (Kiyono & Bekki 2011). The descriptive tools of the field — multifractal detrended fluctuation analysis (MFDFA) widths, increment probability densities — are structurally unable to separate a cascade from a well-tuned linear process. Exact-likelihood model comparison can, and this paper runs it.

We treat the R–R stream as a renewal-type point process observed through its inter-event durations, with the durations conditionally inverse-Gaussian and their conditional mean modulated by a discrete Markov-switching multifractal (MSM) cascade of depth k; the 2^k-state regime chain admits an exact forward-filter likelihood. On the two canonical PhysioNet cohorts — the Normal Sinus Rhythm (nsr2db, 54 subjects) and Congestive Heart Failure (chf2db, 29 subjects) 24-hour RR databases — we fit the cascade against a fixed, matched roster (the tempered-Lévy-type linear model in the spirit of Kiyono 2006; the Barbieri–Brown history-dependent inverse-Gaussian point process, reimplemented from the published equations; flat K-state hidden Markov models, K ∈ {2,4,8}; an Ornstein–Uhlenbeck-modulated duration model; and an i.i.d. baseline) under one protocol — identical held-out splits, complexity-penalised by BIC and scored held-out, parametric-bootstrapped on the headline verdict.

The cascade loses on every record. Held-out, the best cascade is worse than the best matched linear competitor by a median of −27,246 nats/record on the healthy arm (0/54 records won; 95% CI [−32,552, −24,810]; −859 nats per 1,000 events) and −27,492 nats on the CHF arm (0/29). The best held-out model is the tempered-Lévy-type linear process on 79/83 records (best BIC on 82/83); an Ornstein–Uhlenbeck-modulated duration model — the same emission, one fewer parameter, no access to lagged durations — beats the depth-4 cascade held-out on 77/83 records, so the loss is not an artifact of history-aware rivals (on the six exceptions the cascade beats that one rival and still loses the tournament; §6). A parametric bootstrap places every observed cascade deficit at p ≥ 0.54 for any cascade advantage while its power arm recovers cascade wins on cascade-simulated data, and the estimability battery certifies that the cascade was identifiable at these lengths — the recovery is clean from ~500 events, so this is not a starved-data artifact. We then remove the discrete scaffolding entirely: the continuous-scale log-correlated cascade (the log-normal MRW / log-S-fBM field) is GoF-rejected on all 83 records, with a fitted Hurst exponent significantly above its cascade limit (Ĥ = 0.111, 95% CI [0.088, 0.131] on the healthy cohort) — the heartbeat's log-volatility reads as rough, not cascading. The founding negative is a family-level statement: neither the discrete 2^k ladder nor the continuous-scale field is the generative story of heartbeat durations, and the grid was not the reason. Every result is reproducible from a fresh clone behind a deterministic apparatus gate. No statement here is clinical; each is a model-comparison or measurement statement about public, de-identified datasets.