The study of dynamical systems has the goal of describing the evolution of interconnected variables over time. Non-linear systems are the subject of extensive study because they accurately model many natural phenomena and because they are mathematically intractable. Except in rare cases, the future states of non-linear systems are hard to predict, but knowing the future states is the goal of studying the system. Chaotic systems are non-linear systems that demonstrate aperiodic behavior sensitive to initial conditions. They are often particularly well-suited to describing many natural phenomena, including heart-rate variability, and mathematicians have developed powerful tools for studying them.
Chaotic systems often involve the interplay of so many variables that the evolution of the system appears stochastic (random) but is in fact deterministic. Classifying a system as one or the other determines the type of analysis appropriate for understanding the system. Heartrate is an example of a system that demonstrates both types of behavior. Doctors study heartrate because the heartbeat pattern allows them to indirectly and non-invasively predict and diagnose pathophysiological conditions.1 For example, the peaks and troughs on a heartrate tachogram can indicate mortality risk in cardiac patients. Traditionally, heartrate tachogram analysis was done with time-series models, which assume that the system is stochastic instead of chaotic. These traditional methods are insufficient in many cases like congestive heart failure, which has a distinct chaotic signature separate from the random signatures related to health and aging.2 The pattern of the heartbeat appears random, but in reality is deterministic, which allows for a whole new class of analysis.
Non-linear methods outperform linear methods in explaining changes in the heartbeat interval because they can detect patterns that linear methods cannot. Young and Benton showed that only after measuring heart-rate complexity with non-linear methods was a model of heart-rate behavior statistically significantly related to depression, focused attention reaction times, and perceived stress and anxiety. They also showed that the added heart-rate complexity was necessary to show a difference between the behavior of females and males.3 Another study showed that qualitative measures of state anxiety were positively correlated with a loss in the complexity of heart-rate variability.4
The complexity of the heartbeat refers to the dimension of the dynamical system required to best model the observed signal. An entirely stochastic heartbeat would have infinite dimension. A periodic heartbeat would have dimension one. The signal captured by the non-linear dynamics of the heartbeat more accurately reflects the deep complexity of the process the body undertakes when regulating the heartbeat. Linear measures such as variability can differentiate between old and young, but cannot capture the richness of the signal needed for doctors to infer sinister underlying conditions.5
(1) Shiogai Y., Stefanovska A., McClintock P. V. E. (2010). Nonlinear dynamics of cardiovascular ageing. Phys. Rep. 488, 51–110. 10.1016/j.physrep.2009.12.003, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2853263/.
(2) Wu, Guo-Qiang, et al. “Chaotic Signatures of Heart Rate Variability and Its Power Spectrum in Health, Aging and Heart Failure.” PloS One, Public Library of Science, 2009, www.ncbi.nlm.nih.gov/pmc/articles/PMC2629562/.
(3) Hayley Young, and David Benton. “We Should Be Using Nonlinear Indices When Relating Heart-Rate Dynamics to Cognition and Mood.” Nature News, Nature Publishing Group, 13 Nov. 2015, www.nature.com/articles/srep16619.
(4) Dimitriev, Dimitriy A., et al. “State Anxiety and Nonlinear Dynamics of Heart Rate Variability in Students.” PLOS ONE, Public Library of Science, 26 Jan. 2016, journals.plos.org/plosone/article?id=10.1371\%2Fjournal.pone.0146131.
(5) Kaplan DT, Furman MI, Pincus SM, Ryan SM, Lipsitz LA, Goldberger AL. Aging and the complexity of cardiovascular dynamics. Biophys J. 1991;59(4):945‐949. doi:10.1016/S0006-3495(91)82309-8, https://core.ac.uk/download/pdf/82722142.pdf.