The Baroreflex Mechanism Revisited

Abstract

We state that the autonomic part of the brain controls the blood pressure (BP) and the heart rate (HR) via the baroreflex mechanism in all situations of human activity (at sleep, at rest, during exercise, fright etc.), in a way which is not, as was hitherto assumed, a mere homeostatic tool or even a resetting device, designed to bring these variables on the road to preset values. The baroreflex is rather a continuous feedback mechanism commanded by the autonomic part of the brain, leading to values appropriate to the situation at hand. Feasibility of this assertion is demonstrated here by using the Seidel–Herzel feedback system outside of its regular practice. Results show indeed that the brain can, and we claim that it does, control the HR and BP throughout life. New responses are demonstrated, e.g., to a sudden fear or apnea. In this event, large BP and HR overshoots are expected before the variables can relax to a new level. Response to abrupt downward change in the controlling parameter shows an undershoot in HR and just a gradual resetting in the BP. The relaxation from sudden external changes to various expected states are calculated and discussed and properties of the Rheos test are explained. Experimental findings for orthostatic tests and for babies under translations and rotations reveal complete qualitative agreement with our model and show no need to invoke the operation of additional body systems. Our method should be the preferred one by the Occam Razor approach. The outcomes may lead to beneficial clinical implication.

Appendix: The Seidel–Herzel Model and Its Numerical Integration Method

1.1 Equations

We include here the Seidel–Herzel model equations, both to make the paper self-contained and to mark the changes made to its structure and parameter values. Besides, we present the model in such a way that conforms to the simplified feedback system of Fig. 1. For a more detailed explanation, see Dudkowska and Makowiec (2008).

The model consists of 14 coupled nonlinear equations, some of them differential, as follows:

$$\begin{aligned} v_b =k_1 \left( {p-p^{(0)}+90} \right) +k_2 \frac{\text {d}p}{\text {d}t} \end{aligned}$$

(3)

The first equation (3) brings the most important contribution of our approach.

It defines $v_b$ the nervous signal in the brain at point $\Delta$ of Fig. 1. In addition to being a combination of the measured BP, p, and its derivative $\frac{\text {d}p}{\text {d}t}$, this signal includes, in response to p, a constant BP value: $p^{(0)}=140$ mmHg. In our model, this constant is replaced (see Eq. 1 above) by $p^{*(0)}$, a variable controlled by the ANS. The values of the constants $k_{1}$ and $k_{2}$ as well as all other constants in all remaining equations below are given in Table 1. The blood pressure, p, in Eq. (3) is measured by the pressure sensors in the body (baroreceptors, $\nu _s$ M, Fig. 1) and delivered to the brain by afferent neurons carrying the signal. This signal is modified in the brain’s medulla (the lower half of the brainstem, marked BM in Fig. 1) to modulate the brain-to-heart signal and also to include in this signal the breathing influence, which is known to regulate the HR in a rhythmic way. This regulation is usually called the respiratory sinus arrhythmia. The result of the brain action comprises two output signals originating from the sympathetic (Symp., Fig. 1) and the parasympathetic (Para., Fig. 1) parts of the autonomic nervous system in the following manner (Eqs. 4 and 5): The sympathetic $\nu _s$ signal is decreased from a reference value, while the parasympathetic one $\nu _p$ is increased, with an increase in the baroreceptor signal (3) in a negative feedback manner. The zero threshold is inserted to avoid negative HRs.

$$\begin{aligned} v_s (t)= & {} \max \left( {0, v_s ^{\left( 0 \right) }- k_s^b v_b (t)+k_s^r \left| {\sin \left( {\pi f_r t+{\Delta \phi }_s^r } \right) } \right| } \right) \end{aligned}$$

(4)

$$\begin{aligned} v_p (t)= & {} \max \left( {0, v_p ^{\left( 0 \right) }+ k_p^b v_b (t)+k_p^r \left| {\sin \left( {\pi f_r t+{\Delta \phi }_p^r } \right) } \right| } \right) \end{aligned}$$

(5)

A better model for the respiratory influence on HR is possibly that of Koepchen et al. (1981).

Both $\nu _s$ and $\nu _p$ are transferred to the heart (H, Fig. 1). Both govern the HR by controlling its “phase” (7). The $\nu _s$ signal induces a release of noradrenaline (Na, or norepinephrine) according to (6). The release of Na concentration $c_{c,Na}$ occurs by $\nu _s$ after a natural delay and decays exponentially. This concentration is transformed into a phase controlling signal by (8). The $\nu _p$ signal, on the other hand, is considered to directly affect the phase through (9, 10). Its delay is different than that of $\nu _s$ due to the different neurons involved.

$$\begin{aligned} \frac{dc_{cNa} }{\text {d}t}=-\frac{c_{cNa} }{\tau _{cNa} }+k_{C_{cNa}}^s v_s \left( {t-\vartheta _{cNa} } \right) \end{aligned}$$

(6)

The phase ($\varphi$, 7) is a measure of the time-point within the heart cycle. It increases between 0 and 1 during a heartbeat. When reaching 1 at the end of the beat, it is reset to 0, and the cycle repeats. This method of cycle length control is called an “integrate and fire” process. The HR at beat number i equals 60 s divided by the time period, $T_{i}$, between $\varphi =0$ and $\varphi =1$ at this beat.

$$\begin{aligned} \frac{\text {d}\varphi }{\text {d}t}= & {} \frac{1}{T^{\left( 0 \right) }}f_s f_p \end{aligned}$$

(7)

$$\begin{aligned} f_s= & {} 1+k_\varphi ^{cNa} \left( {c_{cNa} +\left( {\hat{{c}}_{cNa} -c_{cNa} } \right) \frac{c_{cNa}^{n_{cNa} } }{\hat{{c}}_{cNa}^{n_{cNa} } +c_{cNa}^{n_{cNa} } }} \right) \end{aligned}$$

(8)

$$\begin{aligned} f_p= & {} 1-k_\varphi ^{p}\left( {v_{p,\vartheta _p } +\left( {\hat{{v}}_p -v_{p,\vartheta _p } } \right) \frac{v_{p,\vartheta _p }^{n_p } }{\hat{{v}}_p^{n_p } +v_{p,\vartheta _p }^{n_p } }} \right) F\left( \varphi \right) \end{aligned}$$

(9)

where $v_{p,\theta _p } =v_p \left( {t-\theta _p } \right)$ is the delayed parametric activity.

Table 1 The parameters

Phase effectiveness curve:

$$\begin{aligned} F\left( \varphi \right) =\varphi ^{1.3}\left( {\varphi -0.45} \right) \frac{\left( {1-\varphi } \right) ^{3}}{\left( {1-0.8} \right) ^{3}+\left( {1-\varphi } \right) ^{3}} \end{aligned}$$

(10)

The translation of HR into BP in the model (see Fig. 1) is carried out separately for the systolic and diastolic parts of the heartbeat (15, 16, respectively). The process uses is called the “Windkessel” (literally “air pipe”) effect, which describes the blood flow through the blood vessels, where the latter are considered as elastic conduits. The effect is manifested by the appearance of a variable $S_i$ in the systolic part of the cycle (15) and the variable $\tau _v \left( t \right)$ in the diastolic part. On its part, $S_i$ depends on the previous heartbeat duration, $T_{i-1}$, and on the noradrenaline concentration (11, 12), while the Windkessel time constant of the exponential decay of pressure during the diastolic period, $\tau _v \left( t \right)$ (13, 14), depends on the concentration, $c_{vNa}$, of vascular sympathetic neurotransmitters.

$$\begin{aligned} {S}'_i =S^{\left( 0 \right) }+k_S^c c_{cNa} +k_S^t T_{i-1} \end{aligned}$$

(11)

The prime symbol does not indicate a derivative.

$$\begin{aligned}&S_i ={S}'_i +\left( {\hat{{S}}-{S}'_i } \right) \frac{{{S}'_i} {\,}^{n_s}}{{{S}'_i}{\,}^{n_s }+\hat{{S}}^{n_s }} \end{aligned}$$

(12)

$$\begin{aligned}&\tau _\nu =\tau _\nu ^{(0)}-\bar{{\tau }}_v \left( {c_{vNa} +\left( {\hat{{c}}_{vNa} -c_{vNa} } \right) \frac{c_{vNa}^{n_v Na} }{\hat{{c}}_{vNa}^{n_v Na} +c_{vNa}^{n_v Na} }} \right) \end{aligned}$$

(13)

$$\begin{aligned}&\frac{\text {d}c_{vNa} }{\text {d}t}=-\frac{c_{vNa} }{\tau _{vNa} }+k_{cvNa}^s v_s \left( {t-\vartheta _{vNa} } \right) \end{aligned}$$

(14)

Thus, the BP increase (15) during the systolic (blood pushing) part continues for a $\tau _\mathrm{sys} /T_i$ fraction of the cycle, while its decay during the diastolic (relaxing) part takes place in the rest of the heartbeat duration (16).

$$\begin{aligned} p= & {} d_{i-1} +S_i \frac{t-t_i }{\tau _\mathrm{sys} }\exp \left\{ {1-\frac{t-t_i }{\tau _\mathrm{sys} }} \right\} \end{aligned}$$

(15)

$$\begin{aligned} \frac{\text {d}p}{\text {d}t}= & {} -\frac{p}{\tau _v \left( t \right) } \end{aligned}$$

(16)

The BP is continually measured (M, Fig. 1) by the baroreceptors (areas sensitive to pressure) in the body, and the value, p, is transmitted to the brain and appears in Eq. (3).

All constant parameters of the model are collected in Table 1.

1.2 Integration Method

The set of 14 equations was integrated using the first-order Euler’s method. The integration should be processed in the following order: firstly Eq. (6), Eq. (8), Eq. (11), Eq. (12); then Eq. (15) or Eq. (16)—depending on whether one is calculating during the systolic or the diastolic periods, respectively; further continue with Eq. (3), Eq. (4), Eq. (5), Eq. (7), Eq. (10), Eq. (9), Eq. (14), and finally Eq. (13).

The error produced by using our finite differences method with time step $\Delta t$ is obviously O $(\Delta t)$. Its value was selected by halving consecutively and checking that the difference between successive values was $<$1 %. Thus, for $p^{*\left( 0 \right) } = 90$, t = 100 s, $\Delta t$ = 0.004, 0.002, 0.001 the following values of the average systolic BP were obtained: $p\left( t \right) =101.3, 98.3,$ and 99.1, respectively. The chosen value was $\Delta t=0.001$ throughout all calculations.

The initial conditions at t = 0 were:

$$\begin{aligned} p(0)=p^{*(0)}, \frac{\text {d}p}{\text {d}t}(0)=0,c_{cNa} (0)=0, \varphi (0)=0, c_{vNa} (0)=1 \end{aligned}$$

Also, $T_{0 }= 0.83$ (Eq. 11); $d_{0}=p^{*\left( 0 \right) }$ (Eq. 15); and

$v_s \left( {t-\theta _{cNa} } \right) =\quad 0$ in Eqs. 6 and 14, during the corresponding time delays: $0 < t < \vartheta _{cNa}$ .