Draft:Time delay stability

Network Physiology seeks to understand how diverse physiological systems interact to generate integrated organism-level functions and states. A major challenge arises from the complex, nonlinear, and nonstationary nature of these interactions. Different systems—such as cardiovascular, respiratory, neural, and endocrine—operate on distinct time scales and through diverse regulatory pathways. Their couplings vary dynamically across physiological states, making it difficult to characterize coordination using conventional statistical or correlation-based approaches [citation needed].

Traditional physiology typically examines organs or subsystems in isolation, an approach that cannot explain how global physiological states, such as sleep or exercise, emerge from cross-system integration. Network Physiology addresses this limitation by treating the human organism as a network of dynamically interacting systems, emphasizing the evolving patterns of connectivity and synchronization among them

According to Rossella Rizzo et al.:

The Time Delay Stability (TDS) method is a novel approach specifically developed to identify and quantify pair-wise coupling and network interactions of diverse dynamical systems (Bashan et al., 2012). This approach is inspired by observations of coordinated bursting activity in the output dynamics of diverse systems (Figure 1).The TDS method is based on the concept of time delay stability. Integrated physiologic systems are coupled by non-linear feedback and/or feed forward loops with a broad range of time delays. Thus, bursting activities in one system are always followed by bursts in signals from other coupled systems. TDS quantifies the stability of the time delay with which bursts in the output dynamics of a given system are consistently followed by corresponding bursts in the signal output of other systems — periods with a constant time delay between bursts in two systems indicate stable interactions. Correspondingly stronger coupling between systems results in longer periods of TDS . Thus, the links strength in the physiologic networks we investigate is determined by the percentage of the time when TDS is observed: higher percentage of TDS (%TDS) corresponds to stronger links.

The TDS method (Bashan et al., 2012) to quantify the interaction between distinct physiologic systems A and B consists of the following steps

1. Consider the output signals {a} of system A and the output signal {b} of system B, each of length N. Divide both signals {a} and {b} into N_L overlapping segments ν of equal length L = 60s. Here we choose an overlap of L/2 = 30s, which corresponds to the time resolution of conventional sleep-stage-scoring epochs, and thus N_L = [2N/L]−1, where [2N/L] is the largest integer k such that k ≤ 2N/L.
2. Normalize the signals separately in each segment ν to zero mean and unit standard deviation in order to remove constant trends in the data and to obtain dimensionless signals. This normalization procedure assures that the estimated coupling between the signals {a} and {b} is not affected by their relative amplitudes.
3. Then, calculate the cross-correlations$$C_{ab}^{\nu }(\tau )={\frac {1}{L}}\sum _{i=1}^{L}a_{i+(\nu -1)L/2}^{\nu }\,b_{i+(\nu -1)L/2+\tau }^{\nu }$$between {a} and {b} in each segment ν using periodic boundary conditions. For each segment ν, we estimate the time delay

$\tau _{0}^{\nu }$ as the maximum in the absolute value of the cross-correlation function $C_{ab}^{\nu }(\tau )$ in the segment These steps result in a new temporal series of time delays $\{\tau _{0}^{\nu }\mid \nu \in \{1,\ldots ,N_{L}\}\}$ describing the temporal evolution of the cross-talk between the signals {a} and {b}. Time periods of stable interrelation between two signals are represented by segments of approximately constant $\tau _{0}$ in the series of time delays. In contrast, the absence of stable coupling between the signals corresponds to large fluctuations in τ₀. To identify periods of stable coupling, the series of time delays is scanned using a 5 points sliding window (corresponding to a window of 5 × 30 s consecutive segments ν) with step size 1. Periods are labeled as stable when at least four out of five points the time delay remains in the interval ${\textstyle [\tau _{0}-1,\,\tau _{0}+1]}$ (Figure 1). The %TDS is finally calculated as the fraction of stable points in the time series $\{\tau _{0}^{\nu }\}$, and is a measure of the coupling strength between the two systems A and B.