Although the concept of transfer function is intrinsically related to an input-output relationship, the traditional and widely used estimation method merges both feedback and feedforward interactions between the two analyzed signals. This limitation may endanger the reliability of transfer function analysis in biological systems characterized by closed loop interactions. In this study, a method for estimating the transfer function between closed loop interacting signals was proposed and validated in the field of cardiovascular and cardiorespiratory variability. The two analyzed signals x and y were described by a bivariate autoregressive model, and the causal transfer function from x to y was estimated after imposing causality by setting to zero the model coefficients representative of the reverse effects from y to x. The method was tested in simulations reproducing linear open and closed loop interactions, showing a better adherence of the causal transfer function to the theoretical curves with respect to the traditional approach in presence of non-negligible reverse effects. It was then applied in ten healthy young subjects to characterize the transfer functions from respiration to heart period (RR interval) and to systolic arterial pressure (SAP), and from SAP to RR interval. In the first two cases, the causal and non-causal transfer function estimates were comparable, indicating that respiration, acting as exogenous signal, sets an open loop relationship upon SAP and RR interval. On the contrary, causal and traditional transfer functions from SAP to RR were significantly different, suggesting the presence of a considerable influence on the opposite causal direction. Thus, the proposed causal approach seems to be appropriate for the estimation of parameters, like the gain and the phase lag from SAP to RR interval, which have a large clinical and physiological relevance.
A method to evaluate the direction and strength of causal interactions in bivariate cardiovascular and cardiorespiratory series is presented. The method is based on quantifying self and mixed predictability of the two series using nearest-neighbour local linear approximation. It returns two causal coupling indexes measuring the relative improvement in predictability along direct and reverse directions, and a directionality index indicating the preferential direction of interaction. The method was implemented through a cross-validation approach that allowed quantification of directionality without constraining the embedding of the series, and fully exploited the available data to maximise the prediction accuracy. Validation on short simulated bivariate time series demonstrated the ability of the method to capture different degrees of unidirectional and bidirectional interaction. Moreover, application to representative examples of heart rate, systolic arterial pressure and respiration series allowed the inference of causal relationships related to known physiological mechanisms and experimental conditions.
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