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    CD

    Correlation dimension D2: complexity of heart-rhythm regulation.

    CD — the correlation dimension (D₂) — is a nonlinear heart-rate-variability index borrowed from chaos theory. It estimates how many independent "degrees of freedom" — interacting regulatory mechanisms — are needed to describe the dynamics of the heart rhythm. A healthy heart is controlled by many systems at once and shows high complexity; ageing and disease make regulation simpler and reduce CD.

    Heart rate variabilityNonlinear dynamicsComplexityFractal analysis
    MethodGrassberger–Procaccia
    Unitdimensionless
    Typical short-term values, healthy≈ 3–5
    Reduced instress, ageing, heart disease

    Why measure complexity

    The heart rhythm results from the interplay of breathing, the baroreflex, the vagus and sympathetic nerves, hormones, thermoregulation and the brain. These mechanisms work on different time scales and influence each other, so the sequence of heartbeat intervals is neither random nor periodic: it is complex. Linear indices (SDNN, RMSSD, LF, HF) measure the amount of variability; nonlinear indices such as CD, entropy and DFA describe its structure. Loss of complexity is regarded as a general sign of reduced adaptability of a physiological system.

    How CD is calculated

    Steps of correlation dimension: the RR series is embedded into vectors in phase space, the correlation integral C of r is computed, and CD is the slope of log C against log r 1. RR series2. Phase space3. Scaling slope xᵢ = (RRᵢ, RRᵢ₊τ, …, RRᵢ₊₍ₘ₋₁₎τ) C(r) = share of point pairscloser than r log rlog C(r) slope = CD (D₂) CD = lim (r→0) log C(r) / log r — estimated from the linear part of the log–log curve
    Figure 1. The Grassberger–Procaccia method. The RR series is converted into points in a multidimensional "phase space"; the faster the number of close point pairs grows with the radius r, the higher the dimension.
    C(r) = 2 / (N(N−1)) × Σᵢ<ⱼ Θ(r − ‖xᵢ − xⱼ‖) · CD = d log C(r) / d log r

    Θ is the step function (1 if two points are closer than r, otherwise 0). The result depends on the embedding dimension m, the delay τ, the recording length and the quality of the data; reliable estimation requires sufficiently long, artefact-free recordings.

    Interpretation

    CDMeaningTypical situations
    Low (≈ < 2–3)Simplified, less adaptive regulationAcute stress, exhaustion, high heart rate, ageing; heart failure, after myocardial infarction, autonomic neuropathy
    Typical (≈ 3–5)Normal complexity of heart-rhythm controlHealthy adults at rest (short-term recordings)
    Very highMay reflect noise rather than complexityArtefacts, ectopic beats, atrial fibrillation, too short recording
     Values depend strongly on calculation settings; compare only results obtained with the same method and device.

    What affects CD

    • Physical and mental load, standing up, rapid breathing — complexity falls.
    • Sleep stage: complexity differs between deep sleep and REM sleep.
    • Age and fitness: complexity declines with age and is preserved better in physically active people.
    • Recording quality: artefacts and arrhythmias distort the estimate.
    When to see a doctor

    CD is a research-oriented index and is not used alone for diagnosis. If it is persistently low together with low SDNN or RMSSD, or with symptoms such as palpitations, breathlessness, chest pain or fainting, discuss the results with a cardiologist.

    References

    1. Grassberger P, Procaccia I. Characterization of strange attractors. Phys Rev Lett. 1983;50(5):346–349. doi:10.1103/PhysRevLett.50.346
    2. Voss A, Schulz S, Schroeder R, Baumert M, Caminal P. Methods derived from nonlinear dynamics for analysing heart rate variability. Philos Trans A Math Phys Eng Sci. 2009;367(1887):277–296. doi:10.1098/rsta.2008.0232
    3. Tarvainen MP, Niskanen JP, Lipponen JA, Ranta-aho PO, Karjalainen PA. Kubios HRV — heart rate variability analysis software. Comput Methods Programs Biomed. 2014;113(1):210–220. doi:10.1016/j.cmpb.2013.07.024
    4. Shaffer F, Ginsberg JP. An overview of heart rate variability metrics and norms. Front Public Health. 2017;5:258. doi:10.3389/fpubh.2017.00258

    This article is for general information and does not replace a consultation with a doctor. Nonlinear HRV indices depend on calculation settings and recording quality.

    Published on 7 July 2019