Analyzing entrainment of heartbeat and respiration with surrogates

Entrainment between two rhythms is a very well-known phenomenon in the theory of nonlinear dynamics. Although heart and respiration influence each other by several mechanisms, and though modulation of heart rate by respiration is a very well-known and investigated phenomenon, there are surprisingly few indications of true entrainment between the two rhythms. This absence might be due to an insufficient coupling strength or to disturbances by other physiological rhythms. Nonetheless, we sometimes observe intermittent phases where cardiac and respiratory rhythms run in parallel; i.e., where both rhythms seem to be entrained. However, it is not obvious how to decide whether this effect is true entrainment or whether it is just quasi-entrainment that occurs when two rhythms have an approximate frequency ratio of for example, 4:1. Therefore, we use a surrogate-data technique to determine the probability of quasi-entrainment as a function of its duration. This probability can be used to obtain a significance level for true entrainment.