Modeling the afferent dynamics of the baroreflex control system

Abstract

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.

Figure 1. Schematic representation of the BR feedback system.

Stretch sensitive BR neurons originate in the carotid sinuses and the aortic arch. In these arteries, dynamic changes in blood pressure cause vessel deformation, modulating stretch of mechanoreceptors channels found in the BR nerve endings. Stimulation of these receptors modulates frequency of action potential formation, a signal integrated in the NTS. From the NTS, efferent sympathetic and parasympathetic outputs are generated determining the concentrations of neurotransmitters acetylcholine and noradrenaline, which stimulate or inhibit heart rate, cardiac contractility, vascular resistance and compliance, the latter via activation of smooth muscle cells constricting or dilating the radius of arteriolar vessels.

Figure 2. Various types of BR input pressure.

To test our models we applied a number of pressure stimuli: (A) sinusoidal, (B) step increases, (C) square (step increase followed by a step decrease), (D) ramp and triangular. The above stimuli were used for testing the models' responses both qualitatively and quantitatively.

Figure 3. Block diagram used to describe the BR firing in response to an applied blood pressure stimulus.

Applied changes in blood pressure induce changes in the arterial wall strain, which induce changes sensed by stretch sensitive mechanoreceptors found in BR within the arterial wall. This stimulus modulates frequency of action potential formation, which can be used to determine the BR firing rate.

Figure 4. A schematic illustration of the strain sensed by the mechanoreceptors.

The spring and Voigt bodies (a parallel spring and dashpot) in series shown here describes the strain sensed by the mechanoreceptors relative to the deformation of the arterial wall. The spring represents the elasticity of the BR nerve endings, whereas the Voigt bodies reflect the viscoelastic properties of the surrounding connective tissue. Each element provides a timescale adaptation of BRs firing rate in response to a step increase in pressure observed in experiments. This study compares the cases .

Figure 5. Diagram for leaky integrate-and-fire model.

The circuit diagram (left) represents the schematic layout of the integrate-and-fire components. The graph (right) depicts voltage vs time for a neuron stimulated by a constant current.

Figure 6. The optimized response of linear BR models (left), and the corresponding hysteresis loop (right).

We present the fits for three linear BR models , and (denoted in the legend as V1, V2, and V3, respectively), listed in Table 3. The optimized parameter values, the and the RMSE errors are reported in Table 4.

Figure 7. The optimized response of linear BR models.

We show the ability of three linear models , and (denoted in the legend as V1, V2, and V3, respectively) to reproduce four types of increases in pressure: ((A) 128 mmHg, (B) 134 mmHg, (C) 137 mmHg, and (D) 143 mmHg) published by Brown . The parameters of each model have been optimized for each data set individually and are listed in Table 4 together with the and the RMSE errors.

Figure 8. The optimized response of (A) , and (B) to a PED profile of BR firing rate.

The parameters of each model have been optimized for each data set individually and are given in Table 4 together with the and the RMSE errors.

Figure 9. Simultaneous response with a linear and a nonlinear BR model.

(A) Predictions obtained estimating one parameter set for all four pressure step-increases using the linear model with two Voigt bodies . Note, that the overshoot is diminished for responses to smaller step-increases in pressure, and that the baseline firing rate is not reproduced accurately. (B) Predictions obtained with the nonlinear model accounting for nonlinear stiffening with increased pressure allowed us to accurately fit all four responses using one set of parameter values.

Figure 10. Qualitative responses.

We present a qualitative response of the two Voigt body BR model to various pressure stimuli including sinusoidal (A), ramp up (B), step-increase (C), and trianglular (D) showing the model's ability to reflect rectification (A), saturation (B), two time-scale adaptation (C), and asymmetry (D).