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Sliding Mode Control of Cardiac Rhythms in the Sinoatrial Node using Gaussian Process Regression
Gabriel da Silva Lima, Marcelo Amorim Savi, Wallace Moreira Bessa
TL;DR
Pathological activity at the sinoatrial node can produce abnormal cardiac rhythms, motivating control strategies that stabilize the natural pacemaker. The paper combines sliding mode control with Gaussian process regression for uncertainty compensation, models the heart with three coupled nonlinear oscillators, and reports normalized rhythms with critical cardiac behaviors avoided.
Problem
Cardiac rhythm control must stabilize sinoatrial-node activity despite nonlinearities, modeling inaccuracies, external disturbances, and uncertainty about the cardiac dynamics.
Method
The paper combines sliding mode control with an online Gaussian process regressor and uses a three-oscillator cardiac model with delayed coupling among the SA, AV, and HP nodes.
Results
The controller regularizes pathological heart rhythms, while Lyapunov analysis proves boundedness and convergence of the SA-node closed-loop signals.
Takeaways & Limitations
Gaussian process regression learns, predicts, and compensates modeling inaccuracies and disturbances, enabling successful heart-rhythm normalization even with high uncertainty about system dynamics.
Takeaways & Limitations
The implemented model uses unidirectional SA-to-AV-to-HP transmission, although the general representation allows bidirectional coupling, and the external excitation is a reduced-order representation of spatiotemporal effects.
Abstract
from arXiv · showhide
The Sinoatrial node (SA), also called natural pacemaker, is responsible to initiate the heart electrical activity, usually represented by electrocardiograms (ECGs). Abnormalities at the SA node can produce disordered heart rhythms or, in other words, cardiac arrhythmia that are visualized in the ECGs. The development of control strategies to stabilize the cardiac rhythm at the natural pacemaker can provide efficient ways to deal with and avoid some heart pathology. This paper investigates the use of a robust controller based on sliding modes for cardiac rhythms at the SA node in order to induce normal rhythms from pathological responses. Embedded into this controller, a Gaussian process regressor is utilized to predict and compensate modeling uncertainties and disturbances. A mathematical model that presents close agreement with experimental measurements is employed to represent the heart functioning. The adopted model comprises a network of oscillators formed by sinoatrial node, atrioventricular node (AV) and His-Purkinje complex (HP). Three nonlinear oscillators are employed to represent each one of the nodes that are connected by delayed couplings. The boudedness and convergence properties are investigated with a Lyapunov-like stability analysis. In order to evaluate the ability of the control law to deal with interpatient variability, the heart model is assumed to be not available to the controller designer, being used only in the simulator to assess the control performance. The results show that, by applying the proposed control scheme, abnormal rhythms can be avoided, turning the ECG closer to the expected normal behavior and preventing critical cardiac responses.
1 Introduction
Cardiac rhythm control must address nonlinear dynamics, modeling inaccuracies, disturbances, and the chattering–precision trade-off of conventional sliding mode control. The paper proposes combining sliding mode control with Gaussian process regression to regularize pathological ECG signals while adapting online to unknown cardiac dynamics.
- 1 Introduction: Cardiac rhythm control has been studied for applications including artificial pacemakers and implantable defibrillators, using methods such as OGY control, time-delayed feedback, high-order control, and feedback linearization.
- 1 Introduction: A suitable controller must handle cardiac nonlinearities together with modeling inaccuracies and external disturbances.
- 1 Introduction: Conventional sliding mode control offers robustness to modeling uncertainty but can cause chattering, while boundary-layer smoothing sacrifices perfect tracking for bounded precision.
- 1 Introduction: Gaussian process regression predicts unknown system dynamics and quantifies prediction uncertainty, supporting compensation of structured and unstructured modeling errors.
- 1 Introduction: The paper proposes an online sliding mode controller with Gaussian process regression to regularize ECG signals produced by pathological natural-pacemaker dynamics.The Gaussian process regressor compensates unknown cardiac dynamics and estimates uncertainty dispersion for controller robustness.
- 1 Introduction: The proposed method uses supervised online training with overlapping rolling windows, proves error boundedness and convergence through Lyapunov analysis, and evaluates performance in simulation.
2 Mathematical Modeling
The cardiac model represents SA, AV, and HP activity with coupled nonlinear oscillators and delayed transmission, combining their states into an ECG. Simulations reproduce normal ECG features and generate pathological rhythms through external excitation of the SA node.
- Cardiac system model: Three nonlinear oscillators represent the SA node, AV node, and HP complex, with asymmetrical bidirectional delayed couplings modeling cardiac electrical transmission.Each oscillator is a modified van der Pol oscillator, while delayed terms represent transmission time between nodes.
- ECG representation: The ECG is formed as a linear combination of the three oscillators’ state variables, supporting qualitative assessment through ECG phase space.The combination uses coefficients β0, β1, β2, and β3 for the SA, AV, and HP contributions.
- Cardiac rhythms: The implemented physiological model uses unidirectional transmission from SA to AV to HP, while external excitation in the SA node represents pathological behavior.Although the general model allows bidirectional coupling, the simulations adopt the clinically motivated propagation direction.
- Cardiac rhythms: The simulated normal ECG closely agrees with real data and reproduces the principal P, QRS, and T waves, while also exposing the individual node components.The model is numerically implemented with fourth-order Runge–Kutta integration at a 1 kHz sampling rate and scaled to approximately 90 bpm.
- Cardiac rhythms: Changing the SA excitation amplitude and frequency produces pathological ECG patterns, including double R peaks, alternating T waves, and highly dispersed high-frequency activity.For ρSA = 9.6 and ωSA = 2.1, the SA frequency increases and coupling dramatically alters the AV and HP signals, producing the pathological ECG.
3 Sliding Mode Control with Gaussian Process Regression
The controller combines sliding-mode control with Gaussian-process regression to regulate the SA node despite unknown dynamics and modeling inaccuracies. Lyapunov analysis establishes boundedness, attraction to the boundary layer, and exponential convergence of tracking errors to a specified region.
- 3.1 Sliding Mode Control for the Natural Pacemaker: The SA-node model separates known dynamics, control input, and unknown dynamics that include parametric uncertainties and perturbations.The three-oscillator cardiac model is rewritten so the controller uses an estimated vector field while treating unmodeled effects as a disturbance term.
- 3.1 Sliding Mode Control for the Natural Pacemaker: The sliding-mode design uses a tracking-error surface and saturation-based boundary layer to reduce conventional chattering.The control law estimates the disturbance, applies a control gain, and replaces the discontinuous sign function with a saturation function.
- 3.2 Gaussian Process Regression: Gaussian-process regression estimates total uncertainty from noisy observations of the sliding surface and supports probabilistic disturbance compensation.The predictive mean supplies the disturbance estimate, while posterior variance bounds the disturbance for control-gain design; a rolling window updates the training set.
- 3.3 Boundedness and Convergence Analysis: The Lyapunov analysis shows that all initial states are attracted to and remain within the sliding boundary layer.The Lyapunov function is nonincreasing and has a negative derivative outside the layer, while its derivative is zero inside.
- 3.3 Boundedness and Convergence Analysis: The controller ensures exponential convergence of tracking errors to the region |ũ| ≤ ϕ/λ and |u̇̃| ≤ 2ϕ.This region follows from the boundary-layer condition |s| ≤ ϕ and the definition of the sliding surface.
4 Rhythm Control
The proposed controller is tested numerically on a severe SA-node pathology without prior model knowledge. It stabilizes the expected normal rhythm, improves ECG morphology and tracking error, and uses GPR uncertainty estimates to moderate control effort.
- 4 Rhythm Control: The simulation uses a squared-exponential kernel, hyperparameters σf = 1, ℓ = 10^-6, σε = 0.2, and a rolling training window of N = 50.The controller parameters are λ = 3, ϕ = 20, η = 3, and ϑ = 2.
- 4 Rhythm Control: The proposed controller stabilizes the expected normal SA-node rhythm, whereas the conventional controller fails under the tested pathology.The simulation uses desired states from an expected normal cycle and assumes no prior knowledge of the cardiac model, with f̂ = 0.
- 4 Rhythm Control: The regulated ECG shows identifiable P, QRS, and T waves and a heart rate near 90 bpm, while the conventional scheme produces abnormal ECG features.The conventional result includes double R peaks, T-wave alternans, and ventricular tachycardia.
- 4 Rhythm Control: The proposed scheme drastically reduces control error compared with the conventional scheme.The comparison is reported in Figure 7(e).
- 4 Rhythm Control: GPR predictive means compensate unknown dynamics, while predictive standard deviations set disturbance bounds that reduce control gain and control effort.The uncertainty bounds remain sufficient for trajectory tracking even when the regressor cannot reproduce the disturbance exactly.
5 Concluding remarks
The paper concludes that an SMC controller embedded with GPR can normalize pathological cardiac rhythms under uncertain dynamics, while analytically establishing boundedness and convergence and addressing patient variability.
- The three-oscillator cardiac model represents both normal and pathological rhythms, supporting evaluation from the natural pacemaker’s perspective.
- The controller accurately tracks the normal cardiac rhythm at the SA node despite the nonlinear heart dynamics.
- GPR learns, predicts, and compensates modeling inaccuracies and disturbances, improving controller performance.
- Online adaptation allows the regressor to follow dynamical changes and address intra- and interpatient variability.
- Using one input signal and limited rolling training windows reduces computational complexity for potential low-power pacemaker circuits.
- The proposed controller normalizes heart rhythms under high system uncertainty and avoids critical cardiac behaviors.The authors report successful rhythm normalization using sliding mode control with a Gaussian process regressor.