Physiological data, such as EEG, provides an objective perspective into conditions like major depressive disorder (MDD) and suicidality, and is an important biomarker for psychiatrists to identify mental health disorders (Badrakalimuthu et al., 2011; Lebiecka et al., 2018; Liu et al., 2022). MDD affects more than 180 million people worldwide and is a leading cause of disability, with suicide representing one of its most devastating outcomes (Marx et al., 2023). Discriminating between MDD with and without suicidal tendencies, as well as distinguishing both from healthy states, is therefore critical for timely intervention and prevention strategies. Recent methods in biosignal analysis have significantly improved the identification and understanding of psychiatric diseases by including advanced techniques such as multimodal signal decomposition, directional coupling analysis, and fractal and entropy measures (Salankar et al., 2021; Marzbanrad et al., 2020; Jiang et al., 2024; Bartsch et al., 2012; Platiša et al., 2020). These methods provide a detailed perspective of physiological interactions, which are important for understanding the autonomic dysregulation associated with psychiatric conditions.

Network physiology emphasizes the complex, directional, and multiscale interactions between physiological subsystems, offering a system-level lens to understand how distributed organ networks coordinate to maintain homeostasis (Salankar et al., 2021; Chen B. et al., 2022; Borovkova et al., 2022; Khalaf et al., 2015; Bartsch and Ivanov, 2014; Krohova et al., 2019). A disruption in these dynamic interactions reflects a loss of physiological complexity and adaptability, which can serve as an early marker of systemic dysfunction and disease (Ivanov, 2021; Ivanov and Bartsch, 2014; Bashan et al., 2012). Approaches grounded in complexity science (Morandotti et al., 2025), particularly measures of information flow such as transfer entropy, provide a powerful framework for quantifying these network interactions and detecting early deviations from healthy dynamics. In psychiatry, this perspective has gained importance through the rise of digital psychiatry, which leverages physiological signals to complement symptom-based assessments and provide objective insights into mental states (Torous et al., 2021; Vignapiano et al., 2023; Zhao et al., 2019). Within this framework, cardio-respiratory coupling (CRC) has emerged as a paradigmatic example of network physiology, as it reflects the continuous exchange of information between cardiac and respiratory systems mediated by vagal pathways to the sinoatrial node and its modulation of heart rate dynamics (Zhao et al., 2019; Kontaxis et al., 2021). Transfer entropy and related complexity metrics allow the directional information flow within CRC to be quantified, thereby capturing the degree of coordination and adaptability in autonomic regulation. Reduced vagal modulation, commonly indexed by diminished heart rate variability (HRV) and impaired respiratory sinus arrhythmia (RSA), has been repeatedly associated with MDD (Koch et al., 2019; Kemp et al., 2010) and, in some studies, suicidality (Sheridan et al., 2021; Adolph et al., 2018). These findings support the view that impaired autonomic regulation represents a core feature of psychiatric morbidity, while also contributing to the elevated cardiovascular risk observed in MDD populations.

Synchronization between physical subsystems has been widely investigated in physics (Sobiech et al., 2017), and, while early applications to biology, neuroscience, and psychopathology lagged behind (Rosenblum et al., 2002), this has changed considerably over the past 2 decades. Numerous studies have since demonstrated synchronization of physiological rhythms, including heart rate, respiration (particularly during slow breathing), blood pressure, cerebral vascular flow, sympathetic muscle activity, brain oscillations, and pupil dilation, often using phase-locking approaches and synchrograms to characterize coupling (Folschweiller and Sauer, 2021; Andrews et al., 2025; Melnychuk et al., 2021). Coupling of multimodal systems refers to the interaction of two or more oscillatory processes that influence one another through the exchange of information or energy, thereby shaping each other’s dynamics. Coupling can take different forms, strong or weak, unidirectional or bidirectional, linear or nonlinear (Dick et al., 2014). Synchronization, by contrast, describes the temporal alignment of oscillations across systems, such as phase locking, frequency locking, antiphase relations, or synchronization with a delay. Both coupling and synchronization are central features of complex network physiology, including heart-to-brain interaction (HBI), which has been investigated using several correlation and causality algorithms such as Granger causality (GC), transfer entropy (TE), and controlled time delay stability (Marzbanrad et al., 2020; Valenza et al., 2018a; Faes et al., 2013; Porta et al., 2013; Ivanov et al., 2016).

Coupling of biosignals first became popular with the use of Granger causality to characterize functional circuits associated with cognition and behavior in health and disease (Pichot et al., 2024). GC identifies directed functional interactions by implementing a statistical, predictive notion of causality (Seth et al., 2015). Porta and colleagues discussed GC as part of the autonomic network linking heart rate with respiration (Porta et al., 2013). Temporal causality features have also been extended to frequency-based measures, including cross-spectral and information-theoretic approaches, showing cardiorespiratory information coupling (Faes et al., 2021). Other nonlinear approaches have also been applied to cardiorespiratory coupling. For example, cross-sample entropy and multiscale entropy analyses have revealed statistically significant coupling between respiration and interbeat interval variability (Ahmed and Mandic, 2011). By contrast, one recent study employing higher-order detrended moving-average cross-correlation analysis (DMCA) in a small sample (n = 8) failed to detect significant long-range correlations between breathing patterns and interbeat interval variability (Nakata et al., 2021), underscoring the need for larger studies and methodological triangulation.

Transfer entropy (TE) then extended the time and frequency-based unidirectional coupling and improved on the limitations of Granger causality. TE quantifies the direction and strength of coupling between two signals (Bossomaier et al., 2016). It can be applied to non-stationary and nonlinear systems such as the cardiorespiratory system and quantifies the information transferred from a past signal process (e.g., respiration) to a current target signal (e.g., heart rate), independent of the past information obtained from the target signal (heart rate) (Valenza et al., 2018a; Stokes and Purdon, 2017). In contrast, Granger causality (GC) is inherently limited by its reliance on linear autoregressive models and the assumption of Gaussian-distributed noise, making it less effective in detecting nonlinear interactions and dynamics commonly observed in physiological systems (Chen et al., 2023).

Although TE was developed to address several limitations of GC, including sensitivity to bias or high variance, difficulties in interpretation, and inability to capture nonlinear interactions, such issues in GC arise particularly when the underlying data exhibits strong nonlinear correlations, a scenario in which GC tends to perform poorly (Chen Y. et al., 2022). Moreover, TE is computed using an information-theoretic framework based on Shannon entropy and conditional entropy. To address potential bias due to self-matching, surrogate TE was introduced by shuffling the driver time series and subtracting this value from the original TE estimate, thereby ensuring robustness. Alternative bias-reduction strategies, such as corrected conditional entropy (Porta et al., 2001) or permutation-based TE, have been proposed in the literature, and cross-sample entropy remains a valuable complementary method for avoiding self-matching altogether. TE, by contrast, can account for nonlinear and directional dependencies, but it requires substantially larger datasets, is computationally intensive, and its results may be challenging to interpret (Stokes and Purdon, 2017). Thus, while RSA and HRV measures have proven sensitive to depressive phenotypes, traditional coupling metrics such as GC and TE remain limited by assumptions of stationarity, the need for long recordings, or difficulties in disambiguating directionality in nonlinear systems (Pichot et al., 2024; Vicente et al., 2011). To overcome these challenges, we introduce an extended phase-based bidirectional coupling algorithm based on the Niizeki–Saitoh model, which estimates both the strength and direction of influence with lower computational demands (Niizeki and Saitoh, 2018; Niizeki and Saitoh, 2016).

Cardiovascular and respiratory systems exhibit multiscale oscillatory behavior driven by autonomic nervous system dynamics, including low-frequency oscillations associated with sympathetic activity, baroreflex control, and central neural pacemaking drive from the pons (Pfurtscheller et al., 2017; Pfurtscheller et al., 2020), high-frequency components reflecting parasympathetic modulation during respiration, and intermediate bands indicating dynamic interactions between cortical and subcortical regulatory centers (Wang et al., 2025). Swarm Decomposition (SwD) is a novel approach in non-stationary signal decomposition that utilizes swarm intelligence algorithms inspired by biological swarm behaviors, such as predator-prey dynamics, to extract intrinsic frequency bands from physiological time series. This method adaptively separates signals into constituent oscillatory components (OC) based on amplitude, frequency content, and local structural features, which provides better spectral specificity, reduced mode mixing, and better preservation of signal integrity across time (Apostolidis and Hadjileontiadis, 2017; Baltatzis et al., 2017; Ganiti-Roumeliotou et al., 2023). In psychiatric populations, particularly those with MDD and MDD with suicidal ideation (MDDSI), alterations in autonomic regulation often manifest as blunted or chaotic oscillatory patterns (Lehrer and Eddie, 2013). By isolating the OCs, SwD enables frequency-specific coupling analysis, which can identify small disruptions in physiological regulation that standard time-domain HRV metrics may miss and may provide clinicians with information on the link between specific frequency disruptions to functional domains and enable personalized physiological profiling and targeted interventions such as vagal nerve stimulation or paced breathing therapies.

Fractal time series analysis provides a framework for quantifying the complexity and self-similarity of physiological signals across multiple time scales (Yamamoto and Hughson, 1993; Porcaro et al., 2024). In the context of autonomic regulation, it enables the detection of long-range correlations and dynamic fluctuations that are often obscured in conventional linear analyses (Peng et al., 1992; Stanley et al., 1999; Ivanov et al., 1999; Valenza et al., 2018b). Techniques such as entropy analysis, Higuchi and Katz fractal dimension analysis capture the geometric intricacy and temporal irregularity of signals like electrocardiography (ECG) and photoplethysmogram (PPG), that describe the adaptability or rigidity of the underlying network physiology (Cysarz et al., 2000; Raghavendra and Narayana Dutt, 2009; Higuchi, 1988). In MDD, reduced fractal complexity has been associated with diminished autonomic flexibility and impaired emotional regulation (Zitouni et al., 2022; Mandarano et al., 2022). By applying fractal metrics to oscillatory components derived from the Swarm Decomposition, nonlinear dynamical analysis is integrated with network physiology to investigate changes in cardio-respiratory coupling that distinguish MDD from suicidal ideation (Khandoker et al., 2017a).

Organ networks display bi-directional coherency including between heart rate and respiration (Figure 1). The proposed study extends the phase coherency algorithm discussed by Niizeki and Saito to a bi-directional coupling algorithm that provides phase and directionality to gain a better understanding of the cardio-respiratory coherency in MDD with and without suicidal ideation. To further delineate the biosignal characteristics associated with disease progression, the time series were decomposed into oscillatory components and analyzed using fractal geometry-based methodology.

Subjects in the control, MDD, and MDDSI groups had average ages of 28, 34, and 32 years (Table 2). No significant difference in age was observed between the three groups. Most participants were female (66%), with male representation at 47.2%, 27.3%, and 15.8% for the control, MDD, and MDDSI groups, respectively. A significant difference was observed for gender (p = 0.014).

Significant differences were observed for waist circumference (WC) (p-value = 0.001), suicidal score (p-value <0.001), and PHQ-9 (p-value = 0.041). For WC, the significant difference was between control and MDD and between control and MDDSI, whereas the suicidal score and PHQ- 9 showed a significant difference between MDD and MDDSI.

The present study provides novel insights into physiological network changes associated with major depressive disorder (MDD) and suicidal ideation (MDDSI), with a focus on frequency-specific bidirectional cardio-respiratory coupling and providing a framework for psychiatric network physiology. Our findings were consistent with ongoing evidence in literature that there is robust cardiorespiratory coupling in adults, underscoring the relevance of physiological synchronization, even in non-pathological populations (Sobiech et al., 2017). The integration of Swarm Decomposition (SwD) with phase-based coupling metrics, fractal geometry, and entropy analysis enabled a detailed, scale-sensitive characterization of autonomic dysfunction of depressive phenotypes. Moreover, although transfer entropy is an established information-theoretic approach for quantifying directed interactions, it was not directly employed as a feature in this study. Instead, we extracted measures characterizing fractal properties, entropy, and frequency-domain parameters, in addition to basic statistical descriptors. This choice reflects our focus on fractal and oscillatory properties of phase coupling, while acknowledging that transfer entropy represents a complementary approach for future analyses. In addition to these nonlinear descriptors, we also calculated linear parameters (mean and standard deviation) for each oscillatory component (OC1–OC4). This allowed us to evaluate both linear and nonlinear properties of the coupling dynamics when comparing groups.

Previous research has reported reduced heart rate variability (HRV) and impaired respiratory sinus arrhythmia (RSA) as hallmarks of MDD (Brunoni et al., 2013; Wainsztein et al., 2020). However, unlike traditional HRV or RSA indices, which are limited to global measures and unidirectional influences, this study demonstrates that mid- and high-frequency oscillatory components (OC3 and OC4) in cardio-respiratory signals—particularly pulse rate (PR) and pulse transit time (PTT) that carry discriminative value in differentiating MDD from MDDSI. Specifically, the bidirectional coupling strength (λ_bi) of PR in OC3 yielded significant group differences, while the reduced Higuchi fractal dimension associated with PTT in MDDSI suggests a collapse in physiological complexity. These results extend previous findings of diminished fractal scaling in subthreshold or overt depressive states (Mandarano et al., 2022; Valenza et al., 2015; Bartsch et al., 2014). The directional entropy differences observed between MDD and MDDSI agree with prior entropy-based studies on affective states, where reduced entropy reflected diminished variability between both groups and increased physiological rigidity (Sheridan et al., 2021; Čukić et al., 2023). This is particularly relevant when considering the role of entropy in predicting suicidal ideation linked to altered photoplethysmographic entropy dynamics (Khandoker et al., 2017a; Khandoker et al., 2017b). The introduction of both oscillatory components and spectral indices provided a multifaceted view of coupling dynamics. Although this finer stratification makes direct physiological interpretation more complex, it also offers exploratory value by uncovering patterns that may support state discrimination. Such an approach complements established methodologies and may open avenues for refining feature selection in future studies. Moreover, λ(tk) and Biλ(tk) are nonlinear indices of cardiorespiratory coupling75. The additional features extracted, e.g., fractal dimensions, entropy, and spectral indices, were therefore considered as descriptors of the variability and structural properties of the λ(tk) time series. This approach was adopted to maximize discriminatory potential between groups and provide a summary of these time series in the form of extracted features.

While earlier approaches like Granger causality and transfer entropy have been instrumental in identifying directionality in physiological networks, they often suffer from high computational demands and limited applicability to nonstationary data (Abdul Razak and Jensen, 2014). The current study addresses these issues by extending the Niizeki-Saitoh phase coherency algorithm into a bidirectional coupling framework that is both computationally efficient and physiologically interpretable (Niizeki and Saitoh, 2012). The suggested λ_bi, further improves inconclusive or weak cardiorespiratory interactions using traditional long-range correlation techniques (Valenza et al., 2018a; Faes et al., 2013). Furthermore, the use of SwD provided an advantage over conventional decomposition methods such as empirical mode decomposition or variational mode decomposition, which may introduce mode mixing or fail to adapt to signal-specific frequency characteristics (Saleem et al., 2022). SwD preserved signal integrity and allowed for more accurate identification of functionally relevant oscillatory components.

The adaptive decomposition used by SwD was essential in isolating frequency bands that correspond to autonomic nervous system components, and integrative cortical-subcortical dynamics, which reflect hierarchical models of autonomic regulation in psychiatric illness (Lahey et al., 2021). The current results suggest that MDDSI is characterized not only by a further degradation of vagal regulation and signal complexity compared to MDD, but also by a specific pattern of disrupted bidirectional interaction across oscillatory bands. This reinforces the hypothesis that suicidality represents a physiologically distinct phenotype within depressive disorders, with implications for diagnostic stratification and targeted intervention.

This study provides novel insights into frequency-specific bidirectional cardio-respiratory coupling in depression and suicidal ideation. However, several limitations need to be mentioned. However, there are limitations. The sample size was relatively small, particularly in the MDDSI group, potentially limiting statistical power and generalizability. Additionally, the use of only resting-state data and the available two-minute signal segments influence the detection of longer-range dynamics. Despite these constraints, the findings indicate the clinical potential of Swarm Decomposition and bidirectional coupling metrics, specifically λ_bi in OC3 and OC4, as physiological biomarkers for stratifying depressive phenotypes for clinical decision making and treatment options. Use of multimodal and bidirectional coupling has shown their importance in differentiating MDD from MDDSI through entropy and fractal features and incorporating multiscale network physiology into digital psychiatry. Moreover, we did not examine spectral measures of GC, which extend GC into the frequency domain and may provide additional insights into oscillatory interactions between physiological systems (Faes et al., 2022; Chicharro, 2011; Bilgin et al., 2009). Future work is aimed at replicating these results in larger, longitudinal cohorts, integrating multimodal data (e.g., EEG, behavioral indices), and exploring how these metrics respond to therapeutic interventions.

In this study, we examined coupling dynamics across multiple oscillatory components to differentiate between MDD subtypes and healthy controls. This study shows that frequency-specific disruptions in bidirectional cardiorespiratory coupling, together with reductions in signal complexity and entropy, distinguish MDDSI from other groups. These findings suggest impaired autonomic adaptability and emotional regulation in this high-risk depressive state. By combining phase-based coupling metrics with SwD analysis, our work highlights promising physiological markers that may support early identification and stratification of depressive subtypes in digital psychiatry. Future research should aim to validate these markers in larger cohorts and refine their clinical applicability.