Enhancing cardiovascular monitoring: a non-linear model for characterizing RR interval fluctuations in exercise and recovery

Abstract

This work aimed to develop and validate a novel non-linear model to characterize RR interval (RRi) time-dependent fluctuations throughout a rest-exercise-recovery protocol, offering a more precise and physiologically relevant representation of cardiac autonomic responses than traditional HRV metrics or linear approaches. Using data from a cohort of 272 elderly participants, the model employs logistic functions to capture the non-stationary and transient nature of RRi time-dependent fluctuations, with parameter estimation achieved via Hamiltonian Monte Carlo. Sobol sensitivity analysis identified baseline RRi (α) and recovery proportion (c) as the primary drivers of variability, underscoring their critical roles in autonomic regulation and resilience. Validation against real-world RRi data demonstrated robust model performance (R² = 0.868, CI95%[0.834, 0.895] and Root Mean Square Error [RMSE] = 32.6 ms, CI95%[30.01, 35.77]), accurately reflecting autonomic recovery and exercise-induced fluctuations. By advancing real-time cardiovascular assessments, this framework holds significant potential for clinical applications in rehabilitation and cardiovascular monitoring in athletic contexts to optimize performance and recovery. These findings highlight the model’s ability to provide precise, physiologically relevant assessments of autonomic function, paving the way for its use in personalized health monitoring and performance optimization across diverse populations.

Full text

Enhancing cardiovascular
monitoring: a non-linear model
for characterizing RR interval
uctuations in exercise and
recovery
Matías Castillo-Aguilar1, Diego Mabe-Castro1,2, David Medina4,5 &
Cristian Núñez-Espinosa1,3
This work aimed to develop and validate a novel non-linear model to characterize RR interval (RRi)
time-dependent uctuations throughout a rest-exercise-recovery protocol, oering a more precise and
physiologically relevant representation of cardiac autonomic responses than traditional HRV metrics
or linear approaches. Using data from a cohort of 272 elderly participants, the model employs logistic
functions to capture the non-stationary and transient nature of RRi time-dependent uctuations,
with parameter estimation achieved via Hamiltonian Monte Carlo. Sobol sensitivity analysis identied
baseline RRi (α) and recovery proportion (c) as the primary drivers of variability, underscoring
their critical roles in autonomic regulation and resilience. Validation against real-world RRi data
demonstrated robust model performance (R2 = 0.868, CI95%[0.834, 0.895] and Root Mean Square Error
[RMSE] = 32.6 ms, CI95%[30.01, 35.77]), accurately reecting autonomic recovery and exercise-induced
uctuations. By advancing real-time cardiovascular assessments, this framework holds signicant
potential for clinical applications in rehabilitation and cardiovascular monitoring in athletic contexts
to optimize performance and recovery. These ndings highlight the model’s ability to provide precise,
physiologically relevant assessments of autonomic function, paving the way for its use in personalized
health monitoring and performance optimization across diverse populations.
Keywords Heart rate variability, Exercise physiology, Autonomic nervous system, Cardiovascular system,
Models, eoretical, Logistic models
e human cardiovascular system exhibits intricate dynamic responses to physical exertion, reecting the
complex interplay between the autonomic nervous system (ANS) and cardiac function. Understanding these
time-dependent uctuations is crucial for assessing physiological adaptation to exercise, optimizing athletic
performance, and evaluating cardiovascular health1–3. R-R intervals (RRi), representing the beat-to-beat
time intervals between successive heartbeats, provide a direct, high-resolution reection of cardiac electrical
activity. Unlike aggregated measures of heart rate variability (HRV), which summarize autonomic activity over
longer periods and can mask transient uctuations, RRi analysis oers a granular, beat-to-beat perspective on
autonomic modulation during exercise and recovery4–7.
is granular perspective is particularly relevant in dynamic exercise scenarios, where rapid shis in
autonomic balance occur, and in specic populations such as older adults, where age-related changes in
autonomic function may inuence cardiac responses2,3,8. Analyzing RRi allows for examining immediate
cardiac responses to exercise-induced stress, providing valuable insights into the eciency and adaptability of
the cardiovascular system.
1Centro Asistencial Docente e Investigación (CADI-UMAG), Universidad de Magallanes, Punta Arenas, Chile.
2Departamento de Kinesiología, Universidad de Magallanes, Punta Arenas, Chile. 3Escuela de Medicina,
Universidad de Magallanes, Avenida Bulnes 01855, Box 113-D, Punta Arenas, Chile. 4Departamento de Ingeniería
en Computación, Universidad de Magallanes, Punta Arenas, Chile. 5Centre for Biotechnology and Bioengineering,
CeBiB, Universidad de Chile, Santiago, Chile. email: cristian.nunez@umag.cl
OPEN
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While many studies have investigated cardiovascular responses to exercise using quasi-stationary protocols,
simplifying analysis by minimizing non-stationarities9,10, these approaches may not fully capture physiological
responses’ dynamic and continuous nature during real-world activities. Although traditional linear methods like
time-series analysis and linear regression have been employed to model RRi behavior11, they oen fall short in
capturing the complex, non-linear time-dependent uctuations of RRi transitions, particularly during periods of
intense exertion and the subsequent recovery phase12. is limitation is signicant because the ANS undergoes
rapid and non-linear shis between parasympathetic withdrawal and sympathetic activation during exercise,
resulting in intricate RRi uctuations that linear models cannot adequately represent by their nature13. ese
rapid autonomic adjustments, including vagal tone and sympathetic outow changes, contribute to the non-
linear patterns observed in RRi data4,5. Consequently, these simplied models may miss critical physiological
information related to cardiovascular adaptation, such as the speed and extent of recovery14.
Model-based approaches, particularly those employing exponential functions, have been widely used to
estimate heart rate and RRi recovery time constants aer exercise15–20. While these models provide valuable
insights into recovery kinetics, they oen focus on specic phases of the exercise-recovery cycle. ey may not
fully capture the continuous transitions in RRi from rest to exercise and back to baseline. Furthermore, these
models oen rely on simplifying assumptions about the underlying physiological mechanisms, which may limit
their ability to represent individual variability across dierent exercise intensities and populations accurately.
Other models, like advanced non-linear approaches, have been developed to address the limitations of linear
methods like decision tree-based ensemble algorithms and convolutional neural networks21–23. More advanced
techniques, such as non-linear mode decomposition24,25, dynamical modeling26,27, and the explicit consideration
of non-autonomous dynamics28,29, have also been applied to analyze physiological time series.
However, many of these existing non-linear models’ lack of a direct link to underlying physiological
processes is a signicant limitation. While they may provide a better t to the observed data, they oen lack
clear physiological interpretability, limiting their clinical utility and hindering a deeper understanding of the
mechanisms driving RRi changes30–32. Furthermore, few models are designed to capture the continuous, beat-
to-beat transitions in RRi throughout the entire rest-exercise-recovery cycle while simultaneously providing
physiologically meaningful parameters that can explain individual variability across diverse exercise intensities,
durations, and populations33. is gap hinders a comprehensive understanding of how individuals adapt to
exercise and how these adaptations might dier based on age, tness level, or underlying health status. For
example, understanding how RRi time-dependent uctuations dier between trained athletes and sedentary
individuals during and aer exercise could provide valuable insights for personalized training programs and
rehabilitation strategies.
erefore, this paper introduces a novel non-linear model designed to characterize the continuous RRi
transitions from rest to exercise and recovery. is model aims to address the limitations of existing approaches
by (1) accurately capturing the non-linear time-dependent uctuations of RRi uctuations throughout the
entire rest-exercise-recovery cycle, providing a more complete picture of cardiovascular responses to exercise,
and (2) providing physiologically interpretable parameters that reect the underlying autonomic mechanisms,
allowing for a more mechanistic understanding of individual adaptations. By focusing on these key aspects,
this model oers a more detailed and physiologically relevant understanding of cardiovascular adaptation to
exercise compared to traditional HRV metrics, with potential applications in personalized exercise prescription,
performance monitoring, and clinical assessment of cardiovascular health.
Methods
Data collection and preprocessing
To furt her assess the prop osed model’s performance, real-world RRi data were analyzed in addition to the synthetic
data generated through simulation. e dataset consisted of 272 participants who underwent a validated exercise
protocol encompassing rest, exercise, and recovery phases within a single, continuous measurement session2.
Subjects
Participants were recruited from a local community. Subjects were included if (i) they were aged 60 years or
older; (ii) were permanently residing in the Magallanes and Chilean Antarctic region; (iii) had a percentage
greater than 60% on the Karnofsky Performance Status Scale, which allowed us to work with older people who
had a state of autonomy necessary to carry out the study tests; (iv) absence of the following diagnosis: diabetic
neuropathy; use of pacemakers; clinical depression; cognitive or motor disability; and dementia. e exclusion
criteria were: (i) consumption of beta-blockers during the study, (ii) taking drugs or stimulant substances
within 12h before the cardiac assessment, and (iii) having some degree of motor disability that prevented
participants from moving around. No participants met the exclusion criteria. is dataset was derived from a
cohort participating in the FONDECYT Project No. 11,220,116, funded by the Chilean National Association of
Research and Development (ANID). Ethical approval was granted by the Ethics Committee of the University
of Chile (ACTA No. 029 − 18/05/2022) and the Ethics Committee of the University of Magallanes (No. 008/
SH/2022).
Exercise protocol
e exercise protocol consisted of the continuous measurement of RRi before, during, and immediately aer the
2-minute step test, which is a part of the Senior Fitness Test protocol34. is functional cardiorespiratory test
required each subject to march in place as many times as possible for 2min. e participants were monitored
throughout the assessment using cardiovascular measures (i.e., heart rate and blood pressure) to prevent adverse
events during the exercise protocol. e evaluation protocol was estimated to last approximately 20min for
each subject. None of the participants expressed discomfort during the evaluation. Continuous heart rate data,
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including RRi, were collected using the Polar Team2 system (Polar®) application, capable of capturing dynamic
uctuations associated with varying exercise intensities and recovery.
Preprocessing of RRi data
Preprocessing steps were conducted to remove artifacts and ectopic heartbeats, with less than 3% of data
excluded following established guidelines35. e preprocessed RRi data were then aggregated into time intervals
to facilitate analysis, allowing the examination of acute exercise responses and post-exercise recovery patterns.
is real-world dataset provided a critical context for validating the model’s predictive capability against
observed physiological responses, oering a robust foundation for understanding RRi time-dependent
uctuations under physical activity conditions.
Parameter estimation
Parameter estimation was performed using Hamiltonian Monte Carlo (HMC) with the No-U-Turn Sampler
(NUTS) to explore the parameter space36. e parameters
α
,
β
,
c
,
λ
,
φ
,
τ
, and
δ
were estimated by sampling
from the posterior distribution, which was constructed from observed RRi data and model predictions.
e gradient of the log-likelihood function for each parameter was computed during estimation using the
brms R package (v2.21.0), which employs the Stan probabilistic programming language. Convergence of the
HMC chains was assessed using standard diagnostics, including R-hat values, kept below 1.01 for all parameters37,
and eective sample sizes targeted at a minimum of 1,000 for each parameter38. Trace plots were inspected to
conrm stable mixing. ese diagnostics collectively conrmed reliable posterior estimates for each parameter.
e tting process utilized ve Markov Chain Monte Carlo (MCMC) chains, each consisting of 10,000
iterations with a burn-in period of 5,000 iterations, resulting in 25,000 post-warmup samples.
To enhance the exploration of parameter space, we performed a two-stage analysis: We assessed parameter
values at the individual level, which we then used to estimate population-level parameters. is hierarchical
structure enables us to capture individual variability through subject-level random eects while estimating
group-level eects across all parameters, thus providing estimates of subject- and population-level model
parameters.
Individual-level analysis
Firstly, each subject’s RRi data
RRii,t
was standardized against his mean
¯
RRii
and standard deviation
SRRii
to
improve convergence and exploration of the posterior distribution. e standardized RRi data
yi,t
for each time
point
t
was computed as:
y
i,t =
RRi
i,t −
¯
RRi
i
SRRii
(1)
is standardization allowed the model to focus on relative changes in RRi time-dependent uctuations
independent of individual baseline dierences.
e model for each subject
i
was then specied in terms of standardized RRi data
yi,t
:
y
i,t =αi+
β
i
1+e
λi
·
(t
−
τi)+
−c
i
·β
i
1+e
φi
·
(t
−
τi
−
δi)+ϵi,t (2)
where
αi
,
βi
,
ci
,
λi
,
φi
,
τi
,
δi
are the individual-specic model parameters and
ϵ
i,t
∼N(0
,σ
2)
is the residual
error term at each time point
t
.
Aerwards, we transformed the estimated
α
and
β
parameters back to the original RRi scale, ensuring a
physiologically meaningful interpretation. e transformation for each subject
i
is given by:
α
RRi
i
=
αi·SRRii
+¯
RRi
i
β
RRi
i=βi·SRRii
(3)
Group-level analysis
Aer obtaining the posterior distribution for each subject’s parameters, each parameter’s mean (
θobs
) and
standard error (
ϵ
) were calculated. ese estimates were then used as input data to create a univariate hierarchical
model, capturing variability at both the subject and group levels. e modeling process is described as follows:
For each subject
i
, we estimated an interdependent stochastic process in which the true parameter
θk,i
, with
k∈{α, β, c, λ, φ, τ , δ}
with their corresponding standard error
ϵk,i
was used to model the observed parameter
θobs
k,i
as:
θobs
k,i ∼N(θk,i ,ϵ
k,i)
(4)
en, the true parameter
θk,i
was further modeled as:
θ
k,i
∼N(
µk
+
bk,i,σ
2
k
)
(5)
where
µk
is the group-level mean for parameter
k
,
bk,i
represents the subject-level random eect for the subject
i
on parameter
k
and
σ2
k
is the residual variance for the parameter
k
. e subject-level eects
bk,i
were assumed
to be distributed as
b
k,i
∼N(0
,σ
2)
, with
σ
being the standard error of the subject-level eect.
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Model performance
e primary statistical performance metrics, estimated for each subject, included R2, root mean square error
(RMSE), and mean absolute percentage error (MAPE), estimated for each subject. Bootstrap resampling across
each metric was performed to estimate the mean performance of the model and corresponding quantile-based
95% CI.
Also, residual analysis were conducted to evaluate the model’s accuracy in capturing RRi time-dependent
uctuations. Residuals were dened as the dierence between obser ved and predicted RRi values. ese residuals
were analyzed for temporal structure and partial autocorrelation to ensure that no systematic patterns remained
in the errors. is indicates that the model has suciently captured the underlying time-dependent uctuations
of the RRi response to exercise.
Model parameters sensitivity
Once a model that described RRi behavior in response to exercise was obtained, an assessment of the proportion
of the variance explained by each model parameter was then computed.
We implemented a Sobol sensitivity analysis using Monte Carlo simulations to assess the sensitivity of
model parameters inuencing RRi over time. Sobol index (
Sind
) provide a measure of the proportion of the
contribution of each parameter to the variance in RRi at each time point, and it was selected for its robustness in
handling non-linear and non-monotonic relationships, which are intrinsic to RRi time-dependent uctuations
in response to exercise39.
To compute
Sind
, 1000 Monte Carlo simulations were conducted, each involving 1000 randomly sampled
parameter sets (1,000,000 model runs). For each set of parameters, RRi were calculated at each time point
t
across a range from 0 to 20min at intervals of 0.1min. e 95% CI parameter values estimated from HMC-
NUTS were then used as input ranges for
Sind
computation. Finally, the mean values of
Sind
over the 20-minute
time span for each model parameter were estimated and reported, with their corresponding 95% CI using a
normal approximation based on estimated standard errors (SE).
Results
Problem characterization
RRi time-dependent uctuations in response to exercise tend to follow a U-shaped form. e initial decrease in
RRi is associated with exercise onset and an increased heart rate. Aer exercise cessation, an opposite increase in
RRi is observed, associated with the cardiovascular recovery phase. In both cases, the drop and recovery phases
occur at dierent rates; some individuals experience a quick recovery in RRi aer exercise; however, in some
others, this slope is less steep. Additionally, the new baseline reached following exercise cessation is oen below
the RRi baseline before exercise.
ese hallmarks of RRi time-dependent uctuations in response to exercise highlight the complex and non-
linear behavior of the cardiovascular response in the context of rest and exercise conditions. Figure1 shows an
example of RRi record data.
Model construction
e process of deriving the nal equation for modeling RRi uctuations was guided by an iterative exploration
of mathematical functions capable of capturing the observed time-dependent uctuations. Initially, exponential
and logarithmic functions were considered due to their simplicity and broad applicability in describing temporal
changes. Exponential functions were hypothesized to capture the rapid initial adaptations of RRi post-exercise
onset. In contrast, logarithmic functions were explored for their capacity to describe asymptotic behaviors
observed in some physiological variables.
However, neither approach successfully reproduced the non-linear and bidirectional nature of the RRi
uctuations. While eective at modeling monotonic decay or growth, exponential functions could not account
for the observed sigmoidal transitions. Similarly, logarithmic functions, with their inherent monotonicity, failed
to represent the plateauing behavior seen in real-world data.
We shied to logistic functions to address these limitations, which inherently model sigmoidal transitions.
Logistic functions introduce parameters for growth rate and inection point, allowing for precise control over
the shape and timing of the transition between dynamic states. By using two coupled logistic functions, one
representing the initial decrease in RRi and a second inverted logistic function describing the recovery phase, we
achieved a model structure that could exibly reproduce the observed non-linear variations.
is approach provided a biologically plausible representation, with parameters that directly correspond
to identiable physiological features, such as the rate of adaptation and recovery, the time to peak response,
and the extent of deviation from baseline. e logistic function framework emerged as the optimal solution
aer systematic testing and evaluation against empirical data, ensuring that the model accurately captured the
qualitative and quantitative aspects of RRi time-dependent uctuations.
e mathematical model proposed to characterize the RRi response to exercise and recovery is dened by
Eq.6.
RRi (
t
)=
α
+β
1+e
λ(t
−
τ)
+−c·β
1+e
φ(t
−
τ
−
δ) (6)
is model includes two logistic functions representing the RRi time-dependent uctuations across exercise
and recovery phases. e rst logistic term models the decrease in RRi during exercise, where the parameter
β
denotes the magnitude of this decline. e rate of decrease is governed by
λ
, while
τ
represents the onset of the
RRi decrease or the time the physiological shi begins.
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e second logistic term accounts for RRi recovery post-exercise. Here,
c
scales the magnitude of recovery
relative to the initial decline represented by
β
, capturing the proportion of the decline regained during recovery.
e rate at which RRi returns to baseline is controlled by
φ
, and
δ
indicates the lag following the cessation of
exercise, marking the beginning of recovery.
Additionally, the time-dependent uctuations of RRi in response to physical exertion can be represented as
a linear combination of a baseline RRi
α
and two logistic functions denoted as
f1(t)
and
f2(t)
. e function
f1(t)
models the initial decay in RRi following the initiation of exercise while
f2(t)
characterizes the recovery
phase aer exercise cessation.
Essentially, the fundamental structure of both logistic functions can be expressed as:
f(t)= a1
1+e
a
2
(t
−
a
3
) (7)
In this equation,
a1
represents the asymptotic value the logistic function approaches, which can be either positive
(indicating an increase) or negative (indicating a decrease). For
f1(t)
, this parameter is specied as
β
, indicating
the absolute change in RRi at the onset of exercise. In contrast, for
f2(t)
,
a1
is parametrized as
−c·β
, where
c
denotes the proportion of change relative to the initial drop indicated by
β
. is parametrization ensures that,
aer the initial decline, the second logistic function facilitates the return of RRi toward the baseline value
α
.
e parameter
a2
denes the rate at which the specied increase or decrease occurs. is rate parameter
is expressed on a logarithmic scale; to convert it to a percentage change per unit of time, it can be scaled as
1−exp (a2)
.
e parameter
a3
serves as an activation threshold, causing the value within the exponential function, and
consequently, the value in the denominator, to increase signicantly until reaching
a3
. Beyond this point, the
denominator approaches 1, allowing the logistic function to attain the asymptotic level determined by the
numerator. Figure2 illustrates the behavior of the model constituents.
Fig. 1. Example data of RRi recordings of 6 subjects over a 20-minute rest-exercise-recovery protocol in
a sample of elderly individuals. e subject-level data shows the inter-individual variability of RRi time-
dependent uctuations in response to exercised-induced cardiovascular stress, with similar behavior and
recovery trajectories over time.
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Sample characteristics
e sample used to assess RRi time-dependent uctuations consists of 272 subjects selected from a local
community of elderly individuals. e sample characteristics can be seen in Table1.
An initial graphical exploration of RRi time-dependent uctuations (see Fig.3) indicates a clear drop in RRi
around the 5–7min, associated with exercise-induced cardiovascular stress. However, greater variability across
individuals in post-exercise recovery can be observed.
Parameter Estimation
Priors
Given the parameters that reproduced the observed RRi patterns in exercise and rest conditions, priors were
chosen based on physiological constraints and the graphical visualization of standardized RRi data. Hence,
ensuring the identiability of model parameters by constraining the parameter space to plausible values to
improve model convergence and parameter exploration. e prior distributions were dened as follows:
Characteristic Overall Female Male
Sex — 217 (79.8%) 55 (20.2%)
Age 71.14 ± 6.03 70.73 ± 6.27 72.73 ± 4.7
SBP (mm hg) 130.23 ± 17.07 129.58 ± 17.37 132.8 ± 15.69
DBP (mm hg) 77.1 ± 9.58 76.68 ± 9.83 78.75 ± 8.4
MAP (mm hg) 94.81 ± 10.69 94.31 ± 10.95 96.76 ± 9.45
PP (mm hg) 53.14 ± 14.07 52.9 ± 14.26 54.05 ± 13.38
BMI 30.66 ± 5.43 30.7 ± 5.64 30.53 ± 4.53
Weight (kg) 75.06 ± 14.23 73.88 ± 14.09 79.69 ± 13.95
Height (cm) 156.56 ± 9.18 155.29 ± 8.46 161.55 ± 10.24
Tab le 1. Sample characteristics from which continuous RRi monitoring data was collected during the rest-
exercise-rest protocol. Data is presented as Mean ± standard deviation (SD). SBP, systolic blood pressure; DBP,
diastolic blood pressure; MAP, mean arterial pressure; PP, pulse pressure; BMI, body mass index.
Fig. 2. e RRi time-dependent uctuations in response to exercise are expressed as a linear combination
of model constituents based on the baseline RRi
α
and two logistic functions, denoted
f1(t)
and
f2(t),
respectively. e vertical dashed lines represent the time at which the exercise and recovery onset given by
τ=5
and
δ=2
.
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α∼N(1,0.5)
β∼N(−2.5,0.5) withβ≤0
c∼N(0.8,0.2) withc≥0
λ∼N(−2,0.5) withλ≤0
φ∼N(−2,0.5) withφ≤0
τ∼N(5,0.5) withτ≥0
δ∼N(5,0.5) withδ≥0
(8)
Simulated standardized RRi time-dependent uctuations based on prior parameter distributions are shown in
Fig.4.
Parameter estimates
Once subject-level RRi data was tted using the proposed model in Eq.2, a population-parameter value was
estimated based on the proposed group-level methodology. e estimated parameter values can be seen in
Table2.
In Fig.5, the model parameter’s posterior distribution can be observed.
Model evaluation
Model performance
Estimated through bootstrapped resampling, relative statistical performance metrics suggest that the model
tends to deviate by 3.4% (CI95%[3.06, 3.81]) from the observed RRi data. is is equivalent to a 32.6 ms in
the RRi scale (CI95%[30.01, 35.77]). Additionally, the bootstrapped R2 indicates that the model explains 0.868
(CI95%[0.834, 0.895]) of the total variance observed in RRi.
Residuals analysis showed that the estimated partial correlation function (ACF) from the model residuals
indicates a correlation among non-explained errors greater than 0.1 up to the 5th lag. However, the partial ACF
is signicant (CI-wise) and strictly positive or negative until the second lag. Correlations among model residuals
against other time indices remained insignicant (see Fig.6).
Model parameters sensitivity
Sobol sensitivity analysis reveals that the parameter
α
exerts the most substantial inuence on the model’s
output, followed by parameters
c
and
δ
. In contrast, parameters
β
,
λ
, and
φ
demonstrate relatively minor eects,
with some values crossing zero, indicating negligible inuence within the tested parameter ranges.
Individual perturbation of each parameter highlighted that RRi time-dependent uctuations are sensitive to
the baseline RRi parameter,
α
. Conversely, the rate parameters for the initial decay during exercise,
λ
, and the
Fig. 3. (A) Mean and SD from each subject’s RRi recordings were used for the standardization process. (B)
2D kernel density of standardized RRi dynamics over time from a sample of individuals subjected to the rest-
exercise-rest protocol. Darker colors indicate greater probability density. e contrary can be said about lighter
colors.
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recovery post-exercise,
φ
, show lower sensitivity, suggesting that they are not primary sources of variation in
predicted RRi trajectories when assessed in isolation. e results of the sensitivity analysis are in Table3.
Discussion
To our knowledge, this study represents the rst attempt to develop a non-linear model specically designed
to capture RRi time-dependent uctuations continuously across a complete rest-exercise-rest protocol.
Previous studies have either focused on aggregate HRV indices or utilized simplied linear or exponential
models, which are insucient to describe the complex, non-stationary transitions observed during and aer
exercise40. By employing a combination of logistic functions, our model uniquely accounts for the gradual shis
in autonomic regulation denoted by RRi time-dependent uctuations, oering a detailed and physiologically
relevant representation of cardiac dynamics. is continuous modeling framework integrates exercise-induced
RRi decline and post-exercise recovery within a single unied structure, bridging a critical gap in the current
literature. Such an approach advances our understanding of cardiovascular responses and opens new avenues for
real-time monitoring and intervention in clinical and athletic settings.
Parameter EstimateaSEaLowerbUpperb
α
861.78 5.73 850.57 872.85
β
-345.49 7.41 -359.81 -330.97
c
0.84 0.01 0.82 0.86
λ
-3.05 0.06 -3.16 -2.94
φ
-2.60 0.06 -2.71 -2.48
τ
6.71 0.05 6.61 6.81
δ
3.24 0.10 3.05 3.44
σ
27.57 0.57 26.45 28.70
Tab le 2. Population-parameter values estimated from group-level analysis. aEstimates and SE are computed
as the posterior distribution’s median and median absolute deviation, respectively; bLower and Upper bounds
from the quantile-based CI95% of the posterior distribution.
Fig. 4. (A) Simulated standardized RRi time-dependent uctuations based on prior parameter distributions,
illustrating predicted RRi responses to exercise. Shaded areas represent 95%, 80%, and 60% quantile CI,
oering insight into expected physiological variability across parameters. (B) Prior distributions and 95%
CI were used to generate predictions based on physiological constraints and graphical visualization of
standardized RRi data.
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e proposed model demonstrates a precise capacity to reproduce RRi dynamics. Its combination of logistic
functions captures the key transitions of cardiac response, the initial decline during exercise, and the subsequent
recovery. is design accommodates the inherent non-linearity and non-stationarity of RRi time-dependent
uctuations, overcoming the limitations of linear models and exponential functions commonly used in prior
studies8,41.
Compared to previous research, our ndings align with eorts to capture nonlinear dynamics in HRV to
understand cardiac responses during exercise12. Similarly, previous studies have shown that dynamic uctuations
in RRi can serve as critical indicators of cardiorespiratory tness7,8. is supports the need for models to address
the complexity of cardiovascular responses during physical stress8. However, while many existing models focus
primarily on linear metrics or aggregate HRV measures, our study provides a high-resolution analysis of RRi time-
dependent uctuations that enhances interpretability and application across diverse tness levels and exercise
intensities. Critically, many model-based approaches, particularly those employing exponential functions, have
been used to estimate time constants of heart rate and RRi recovery aer exercise15–20. ese models oen
focus on characterizing the recovery phase and may not capture the continuous transitions from rest to peak
exercise and subsequent recovery. Our model, by contrast, provides a unied framework for modeling the entire
rest-exercise-recovery cycle, allowing for the estimation of parameters that reect both the exercise-induced
changes in RRi and the subsequent recovery dynamics. is continuous modeling approach provides a more
comprehensive picture of cardiovascular response to exercise than models focusing solely on recovery kinetics.
e exibility of the logistic components allows for physiologically interpretable parameters, such as baseline
RRi (
α
) and recovery proportion (
c
), which directly correlate with intrinsic cardiac function and autonomic
recovery capacity, respectively. ese features position the model as a robust framework for investigating the
cardiovascular system’s dynamic adaptation to physical stressors. For example, prior studies have highlighted the
inadequacy of linear HRV metrics in capturing transient autonomic shis42; our results align with this critique,
demonstrating the advantages of modeling RRi directly.
Prior studies have examined cardiorespiratory interactions using both deterministic and stochastic
approaches. Deterministic models have demonstrated that respiration-driven heart rate uctuations exhibit
structured, predictable behavior, suggesting an underlying regulatory mechanism of autonomic control43,44.
Conversely, stochastic models emphasize the role of random variability in these interactions, accounting
Fig. 5. (A) Posterior probability distributions of the expectation for each population-parameter estimate
(E[θ])
with quantile-based 95% CI. (B) Transformed rate parameters into a percentage scale using the
1−exp (θ)
transformation.
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Fig. 6. Individual-level performance metrics. (A) Bootstrapped MAPE and RMSE are statistical metrics
of relative and absolute model deviance from observed RRi. (B) Individual-level estimates of model
performance and the relationship between them. (C) Partial autocorrelation function (ACF) of model residuals
with corresponding quantile-based CI. (D) Example data with model estimates of RRi uctuations and
corresponding quantile-based CI initially displayed.
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for inherent physiological uctuations45. Our non-linear model aligns with the deterministic perspective by
employing logistic functions to characterize time-dependent RRi uctuations while also incorporating inter-
individual variability. Although this model does not explicitly integrate stochastic noise, it captures structured
autonomic responses. Future work could explore the incorporation of stochastic elements to further enhance its
applicability in more variable physiological conditions.
Unlike prior research that aggregates HRV measures or applies simple decay models, our approach directly
models RRi changes, oering richer physiological insight. For instance, commonly utilized exponential decay
models for post-exercise recovery are used but fail to incorporate the transition dynamics observed during
exercise itself46. By integrating exercise and recovery phases, our model provides a more comprehensive view
of autonomic regulation. Furthermore, it’s important to note that the traditional “sympathovagal balance”
hypothesis, which posits a reciprocal relationship between sympathetic and parasympathetic activity, may be
oversimplied, especially during exercise9. Recent evidence suggests that parasympathetic control can remain
active even during periods of high sympathetic activation. By capturing the continuous time-dependent
uctuations of RRi, our model may provide insights into these complex interactions, potentially revealing
nuances in autonomic control that are not captured by simpler models that assume a strict sympathovagal
balance.
Moreover, the sensitivity of parameters such as
λ
(decay rate) and
φ
(recovery rate) was found to be relatively
low, suggesting that the model is robust to variability in these rates while remaining sensitive to key physiological
parameters (
α
and
c
). is robustness makes it suitable for individualized monitoring and population-level
analyses, oering versatility in its application across dierent use cases.
e Sobol sensitivity analysis revealed that baseline RRi (
α
) and recovery proportion (
c
) are the primary
drivers of model output variance, emphasizing their physiological importance. ese ndings are consistent
with prior research, which identied baseline cardiac function as a determinant of cardiovascular health and
recovery proportion as a marker of autonomic resilience14.
However, the Sobol method assumes parameter independence, which may overlook interactions common
in biological systems47–49. For example, the interplay between
λ
and
c
, which dictates the rate and magnitude
of recovery, is likely critical but remains unexplored in the current framework. Future studies could explore
Bayesian sensitivity analysis or variance decomposition methods that account for parameter interdep endence50,51.
Furthermore, more advanced techniques, such as non-linear mode decomposition24,25, dynamical modeling26,27,
and the explicit consideration of non-autonomous dynamics28,29, oer powerful tools for analyzing physiological
time series. While these methods can capture complex dynamics, our model provides a more direct link to
physiological interpretation through its parameters related to specic aspects of autonomic control. Future work
could investigate how these approaches could be combined or compared to enhance our understanding of RRi
time-dependent uctuations.
is model demonstrates signicant potential for practical applications in clinical and athletic settings.
In clinical contexts, it could aid in tailoring cardiovascular rehabilitation protocols by monitoring autonomic
recovery in real-time, ensuring safe and eective exercise regimens for at-risk populations52. is aligns
with previous research, highlighting the importance of individualizing rehabilitation programs to optimize
recovery52–54.
e model could guide training strategies in athletic settings, particularly for interval training, where
determining optimal recovery periods is crucial. Similar ndings suggest that precise monitoring of RRi time-
dependent uctuations can prevent overtraining and enhance performance55,56. e model’s ability to integrate
real-time data from wearable devices further enhances its applicability in dynamic, uncontrolled environments,
enabling eld-based monitoring and feedback57.
While the model presents substantial advances, it has limitations that warrant consideration. First, the
assumption of uniform parameter sampling in sensitivity analysis, while practical, may not fully capture the
variability observed in populations with extreme autonomic proles4. Empirical distributions, or Bayesian priors,
could improve parameter estimation and enhance the model’s applicability to diverse populations51. Bayesian
inference could be a valuable extension of this work, particularly dynamic Bayesian inference58–60, specically
designed to model time-evolving dynamics. is approach could allow for the incorporation of prior knowledge
about individual physiological characteristics and provide more robust estimates of the model parameters.
Parameter EstimateaSEaLowerbUpperb
α
0.61329 0.01756 0.57887 0.64771
β
0.06651 0.00286 0.06090 0.07212
c
0.18939 0.00815 0.17342 0.20536
λ
0.00147 0.00007 0.00133 0.00161
φ
0.00160 0.00008 0.00144 0.00176
τ
0.04982 0.00172 0.04645 0.05319
δ
0.07896 0.00239 0.07428 0.08364
Tab le 3. Estimated
Sind
of model parameters. aEstimates and SE are computed as mean and standard
deviation of Monte Carlo samples, respectively. Each parameter’s
Sind
reects a uniform variation from the
95% CIs of the estimated parameter values.
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Another limitation lies in the demographic composition of the sample, which consisted exclusively of elderly
individuals. While this population provides valuable insights into age-specic cardiovascular time-dependent
uctuations, the ndings may not fully generalize to younger populations, whose autonomic responses to
exercise and recovery dier signicantly due to higher baseline vagal tone, greater cardiac plasticity, and distinct
metabolic proles61,62. Previous studies have demonstrated that younger individuals exhibit faster autonomic
recovery and greater adaptability during physical exertion compared to older populations62,63. is suggests that
the parameter estimates derived from this model may vary across age groups62,63. Future research should validate
the model in more diverse cohorts, including younger adults and athletes, to ensure broader applicability and
to explore potential age-dependent modications of the model’s parameters. is would enhance its utility in
clinical and athletic contexts, where age diversity is a critical factor62,63.
Furthermore, the uneven sex ratio in our sample (79.8% female, 20.2% male) is another limitation that
should be addressed in future studies. Sex dierences in autonomic control have been reported61, and this
imbalance could have inuenced our results. Future research should strive for a more balanced sex ratio to
minimize potential bias and explore sex-specic dierences in RRi time-dependent uctuations during exercise
and recovery. is study did not explicitly consider environmental and psychological factors like temperature,
stress, or sleep quality. Future work could integrate these variables into the model, enhancing its robustness and
applicability across varied real-world scenarios. is aligns with calls for more integrative modeling approaches
in cardiovascular research53,55,56.
Conclusion
In summary, this study presents a novel non-linear model for RRi time-dependent uctuations, capturing the
complex and transient autonomic responses during rest-exercise-recovery protocols, overcoming the limitations
of traditional autonomic metrics. e model emphasizes their critical roles in reecting autonomic regulation
and resilience by identifying baseline RRi and recovery proportion as the dominant contributors to variability.
Validated across a cohort of elderly participants, the model demonstrates robust performance in real-time
cardiovascular assessments, oering signicant potential for clinical applications such as rehabilitation and
monitoring in at-risk populations and athletic contexts like fatigue management and performance optimization.
While the model’s applicability is currently constrained by its focus on elderly individuals, future validation in
younger cohorts and under diverse environmental conditions will enhance its generalizability and utility. is
work establishes a foundational framework for advancing personalized cardiovascular health monitoring and
intervention.
Data availability
e authors will make the data supporting this article’s conclusions available without reservation. If any data is
required, please request it from the corresponding author of this work.
Received: 9 December 2024; Accepted: 7 March 2025
References
1. Eser, P. et al. Acute and chronic eects of high-intensity interval and moderate-intensity continuous exercise on heart rate and its
variability aer recent myocardial infarction: A randomized controlled trial. Ann. Phys. Rehabil. Med. 65, 101444 (2022).
2. Castillo-Aguilar, M. et al. Validity and reliability of short-term heart rate variability parameters in older people in response to
physical exercise. Int. J. Environ. Res. Public Health. 20, 4456 (2023).
3. Mabe-Castro, D. et al. Associations between physical tness, body composition, and heart rate variability during exercise in older
people: Exploring mediating factors. PeerJ 12, e18061 (2024).
4. Kristal-Boneh, E., Raifel, M., Froom, P. & Ribak, J. Heart rate variability in health and disease. Scand. J. Work. Environ. Health 21.
85–95 (1995).
5. ayer, J. F., Yamamoto, S. S. & Brosschot, J. F. e relationship of autonomic imbalance, heart rate variability and cardiovascular
disease risk factors. Int. J. Cardiol. 141, 122–131 (2010).
6. Dong, J. G. e role of heart rate variability in sports physiology. Exp. erapeutic Med. 11, 1531–1536 (2016).
7. Lundstrom, C. J., Foreman, N. A. & Biltz, G. Practices and applications of heart rate variability monitoring in endurance athletes.
Int. J. Sports Med. 44, 9–19 (2023).
8. Mongin, D. et al. Decrease of heart rate variability during exercise: An index of cardiorespiratory tness. PLoS ONE. 17, e0273981
(2022).
9. Storniolo, J. L., Cairo, B., Porta, A. & Cavallari, P. Symbolic analysis of the heart rate variability during the plateau phase following
maximal sprint exercise. Front. Physiol. 12, 632883 (2021).
10. Porta, A. et al. On the relevance of computing a local version of sample entropy in cardiovascular control analysis. IEEE Trans.
Biomed. Eng. 66, 623–631 (2018).
11. Lian, J., Wang, L. & Muessig, D. A simple method to detect atrial brillation using RR intervals. Am. J. Cardiol. 107, 1494–1497
(2011).
12. Gronwald, T., Hoos, O. & Hottenrott, K. Eec ts of a short-term c ycling inter val session and active recovery on non-linear dynamics
of cardiac autonomic activity in endurance trained cyclists. J. Clin. Med. 8, 194 (2019).
13. Boettger, S. et al. Heart rate variability, QT variability, and electrodermal activity during exercise. Med. Sci. Sports Exerc. 42,
443–448 (2010).
14. Hautala, A. J., Mäkikallio, T. H., Seppänen, T., Huikuri, H. V. & Tulppo, M. P. Short-term correlation properties of r–r interval
dynamics at dierent exercise intensity levels. Clin. Physiol. Funct. Imaging. 23, 215–223 (2003).
15. Imai, K. et al. Vagally mediated heart rate recovery aer exercise is accelerated in athletes but blunted in patients with chronic heart
failure. J. Am. Coll. Cardiol. 24, 1529–1535 (1994).
16. Borresen, J. & Lambert, M. I. Autonomic control of heart rate during and aer exercise: Measurements and implications for
monitoring training status. Sports Med. 38, 633–646 (2008).
17. Pierpont, G. L., Stolpman, D. R. & Gornick, C. C. Heart rate recovery post-exercise as an index of parasympathetic activity. J.
Auton. Nerv. Syst,. 80, 169–174 (2000).
Scientic Reports | (2025) 15:8628 12
| https://doi.org/10.1038/s41598-025-93654-6
www.nature.com/scientificreports/
Content courtesy of Springer Nature, terms of use apply. Rights reserved

18. Pierpont, G. L. & Voth, E. J. Assessing autonomic function by analysis of heart rate recovery from exercise in healthy subjects. Am.
J. Cardiol. 94, 64–68 (2004).
19. Buchheit, M., Laursen, P. B. & Ahmaidi, S. Parasympathetic reactivation aer repeated sprint exercise. Am. J. Physiol. Heart Circ.
Physiol. 293, H133–H141 (2007).
20. Peçanha, T., Silva-Júnior, N. D. & de Forjaz, C. L. Heart rate recovery: Autonomic determinants, methods of assessment and
association with mortality and cardiovascular diseases. Clin. Physiol. Funct. Imaging. 34, 327–339 (2014).
21. Wang, J. Automated detection of atrial brillation and atrial utter in ECG signals based on convolutional and improved Elman
neural network. Knowl. Based Syst. 193, 105446 (2020).
22. Lee, H. et al. Real-time machine learning model to predict in-hospital cardiac arrest using heart rate variability in ICU. NPJ Digit.
Med. 6, 215 (2023).
23. Berrahou, N., El Alami, A., Mesbah, A., El Alami, R. & Berrahou, A. Arrhythmia detection in inter-patient ECG signals using
entropy rate features and RR intervals with CNN architecture. Comput. Methods Biomech. BioMed. Eng 17. 1–20 (2024).
24. Iatsenko, D., McClintock, P. V. & Stefanovska, A. Nonlinear mode decomposition: A noise-robust, adaptive decomposition
method. Phys. Rev. E. 92, 032916 (2015).
25. Iatsenko, D., McClintock, P. V. & Stefanovska, A. Extraction of instantaneous frequencies from ridges in time–frequency
representations of signals. Sig. Process. 125, 290–303 (2016).
26. Kralemann, B. et al. In vivo cardiac phase response cur ve elucidates human respiratory heart rate variability. Nat. C ommun. 4, 2418
(2013).
27. Stankovski, T., Duggento, A., McClintock, P. V. & Stefanovska, A. Inference of time-evolving coupled dynamical systems in the
presence of noise. Phys. Rev. Lett. 109, 024101 (2012).
28. Clemson, P. T. & Stefanovska, A. Discerning non-autonomous dynamics. Phys. Rep. 542, 297–368 (2014).
29. Lehnertz, K. Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems. Chaos: Interdiscip.
J. Nonlinear Sci. 34, 072102 (2024).
30. Gronwald, T., Hoos, O., Ludyga, S. & Hottenrott, K. Non-linear dynamics of heart rate variability during incremental cycling
exercise. Res. Sports Med. 27, 88–98 (2019).
31. Bacopoulou, F., Chryssanthopoulos, S., Koutelekos, J., Lambrou, G. I. & Cokkinos, D. Entropy in cardiac autonomic nervous
system of adolescents with general learning disabilities or dyslexia. in GeNeDis 2020: Genetics and Neurodegenerative Diseases
121–129 (Springer, 2021).
32. Fonseca, R. X. et al. Post-exercise heart rate recovery and its speed are associated with resting-reactivity cardiovagal modulation in
healthy women. Sci. Rep. 14, 5526 (2024).
33. Kanniainen, M. et al. Estimation of physiological exercise thresholds based on dynamical correlation properties of heart rate
variability. Front. Physiol. 14, 1299104 (2023).
34. Rikli, R. E. & Jones, C. J. Senior Fitness Test Manual (Human kinetics, 2013).
35. Malik, M. Heart rate variability: Standards of measurement, physiological interpretation, and clinical use: Task force of the
European society of cardiology and the North American society for pacing and electrophysiology. Ann. Noninvas. Electrocardiol.
1, 151–181 (1996).
36. Homan, M. D. et al. e no-u-turn sampler: Adaptively setting path lengths in hamiltonian Monte Carlo. J. Mach. Learn. Res. 15,
1593–1623 (2014).
37. Vehtari, A., Gelman, A., Simpson, D., Carpenter, B. & Bürkner, P. C. Rank-normalization, folding, and localization: An improved
r for assessing convergence of MCMC. arXiv. arXiv preprint arXiv:1993.08008 (2019).
38. Bürkner, P. C. brms: An R package for bayesian multilevel models using Stan. J. Stat. Sow. 80, 1–28 (2017).
39. Zhang, X. Y., Trame, M. N., Lesko, L. J. & Schmidt, S. Sobol sensitivity analysis: A tool to guide the development and evaluation of
systems Pharmacology models. CPT: Pharmacometr. Syst. Pharmacol. 4, 69–79 (2015).
40. Molkkari, M., Solanpää, J. & Räsänen, E. Online tool for dynamical heart rate variability analysis. in 2020 Computing in Cardiology
1–4 (IEEE, 2020).
41. Silva, L. R. B. et al. Exponential model for analysis of heart rate responses and autonomic cardiac modulation during dierent
intensities of physical exercise. R. Soc. Open. Sci. 6, 190639 (2019).
42. Grégoire, J. M., Gilon, C., Carlier, S. & Bersini, H. Autonomic ner vous system assessment using heart rate variability. Acta Cardiol.
78, 648–662 (2023).
43. Lotrič, M. B. & Stefanovska, A. Synchronization and modulation in the human cardiorespiratory system. Phys. A Stat. Mech. Its
Appl. 283, 451–461 (2000).
44. Iatsenko, D. et al. Evolution of cardiorespiratory interactions with age. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371, 20110622
(2013).
45. Kenwright, D., Bahraminasab, A., Stefanovska, A. & McClintock, P. e eect of low-frequency oscillations on cardio-respiratory
synchronization: Observations during rest and exercise. Eur. Phys. J. B. 65, 425–433 (2008).
46. Sahoo, K. P. et al. Unanticipated evolution of cardio-respiratory interactions with cognitive load during a go-NoGo shooting task
in virtual reality. Comput. Biol. Med. 182, 109109 (2024).
47. Harenberg, D., Marelli, S., Sudret, B. & Winschel, V. Uncertainty quantication and global sensitivity analysis for economic
models. Quant. Econ. 10, 1–41 (2019).
48. Cheng, K., Lu, Z., Wei, Y., Shi, Y. & Zhou, Y. Mixed kernel function support vector regression for global sensitivity analysis. Mech.
Syst. Signal Process. 96, 201–214 (2017).
49. Herman, J. D., Kollat, J. B., Reed, P. M. & Wagener, T. Method of Morris eectively reduces the computational demands of global
sensitivity analysis for distributed watershed models. Hydrol. Earth Syst. Sci. 17, 2893–2903 (2013).
50. Bornn, L., Doucet, A. & Gottardo, R. An ecient computational approach for prior sensitivity analysis and cross-validation. Can.
J. Stat. 38, 47–64 (2010).
51. Xue, W. & Zaidi, A. Bayesian sensitivity analysis for missing data using the e-value. arXiv preprint arXiv:2108.13286 (2021).
52. Carrasco-Poyatos, M., López-Osca, R., Martınez-González-Moro, I. & Granero-Gallegos, A. HRV-guided training vs traditional
HIIT training in cardiac rehabilitation: A randomized controlled trial. GeroScience 46, 2093–2106 (2024).
53. Takahashi, C. et al. Are signs and symptoms in cardiovascular rehabilitation correlated with heart rate variability? An observational
longitudinal study. Geriatr. Gerontol. Int. 20, 853–859 (2020).
54. Rizvi, M. R., Sharma, A., Malki, A. & Sami, W. Enhancing cardiovascular health and functional recovery in stroke survivors: A
randomized controlled trial of stroke-specic and cardiac rehabilitation protocols for optimized rehabilitation. J. Clin. Med. 12,
6589 (2023).
55. Hebisz, R. G., Hebisz, P. & Zatoń, M. W. Heart rate variability aer sprint interval training in cyclists and implications for assessing
physical fatigue. J. Strength. Conditioning Res. 36, 558–564 (2022).
56. Nuuttila, O. P., Uusitalo, A., Kokkonen, V. P., Weerarathna, N. & Kyröläinen, H. Monitoring fatigue state with heart rate-based and
subjective methods during intensied training in recreational runners. Eur. J. Sport Sci 24, 857–869 (2024).
57. Zimatore, G. et al. Recurrence quantication analysis of heart rate variability during continuous incremental exercise test in obese
subjects. Chaos: Interdiscip. J. Nonlinear Sci. 30, 033135 (2020).
58. Duggento, A., Stankovski, T., McClintock, P. V. & Stefanovska, A. Dynamical bayesian inference of time-evolving interactions:
From a pair of coupled oscillators to networks of oscillators. Phys. Rev. E—Stat. Nonlinear So Matter Phys. 86, 061126 (2012).
Scientic Reports | (2025) 15:8628 13
| https://doi.org/10.1038/s41598-025-93654-6
www.nature.com/scientificreports/
Content courtesy of Springer Nature, terms of use apply. Rights reserved

59. Stankovski, T., Duggento, A., McClintock, P. V. & Stefanovska, A. A tutorial on time-evolving dynamical bayesian inference. Eur.
Phys. J. Special Top. 223, 2685–2703 (2014).
60. Lukarski, D., Stavrov, D. & Stankovski, T. Variability of cardiorespiratory interactions under dierent breathing patterns. Biomed.
Signal Process. Control. 71, 103152 (2022).
61. Castillo-Aguilar, M. et al. Cardiac autonomic modulation in response to muscle fatigue and sex dierences during consecutive
competition periods in young swimmers: A longitudinal study. Front. Physiol. 12, 769085 (2021).
62. Garavaglia, L., Gulich, D., Defeo, M. M., omas Mailland, J. & Irurzun, I. M. e eect of age on the heart rate variability of
healthy subjects. PLoS ONE. 16, e0255894 (2021).
63. Choi, J., Cha, W. & Park, M. G. Declining trends of heart rate variability according to aging in healthy Asian adults. Front. Aging
Neurosci. 12, 610626 (2020).
Author contributions
Conceptualization, MC-A; Data curation, MC-A; Investigation, MC-A, DM-C; Methodology, MC-A, NMD;
Supervision, CN-E; Formal analysis, MC-A; Visualization, MC-A; Writing–original dra, MC-A, CN-E, DM-C;
Writing–review & editing, MC-A, CN-E, DM. All authors have read and agreed to the published version of the
manuscript.
Funding
is work was funded by ANID Proyecto Fondecyt Iniciación Nº11220116.
Declarations
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at h t t p s : / / d o i . o r g / 1
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