Biosignal Processing Fundamentals and Recent Applications with MATLAB (Stefan Bernhard, Andreas Brensing etc.)

3 Fundamentals of the Formation of Biosignals

time units

Voltage U / mV

Disturbing Pulse

Van-der-Pol

Fig. 3.22: Action potential x(t) belonging to the perturbed modified Van der Pol oscillator in Fig-

ure 3.20 leading to the complete cessation of oscillation; an externally applied square pulse at time

unit 35 with an amplitude of 8 and a width of 2 time units acts as the perturbation.

lower natural frequency. Due to the coupling, the sinus node works as a pacemaker

node of the AV node and causes the heart cells of the AV node to also generate action

potentials with the frequency of the sinus node. Both oscillations –that of the sinus

node and the delayed one of the AV node –generate an altered signal by weighted

superimposition, and by the shape of which pathological situations can also be re-

cognised in a real ECG signal.

For the mathematical description, the equations (analogous to equation 3.25) for

two mutually coupled modified van der Pol oscillators are used. Initially, only one

coupling is considered, namely that from the sinus node to the AV node [89]:

xSA = −αSA(xSA −vSA1)(xSA −vSA2)^_xSA

−xSAfSA(xSA + dSA)(xSA −eSA)

xAV = −αAV(xAV −vAV1)(xAV −vAV2)^_xAV

−xAVfAV(xAV + dAV)(xAV −eSA)

+ kSA−AVx^τ^SA⁻^AV

−kAV−AVxAV .

(3.26)

Here, the lower index SA denotes the signals and parameters of the sinus node and the

lower index AV denotes the signals and parameters of the atrioventricular node. As for

the coupling, kSA−AV is the attenuation factor from the sinus node to the AV node and

kAV−AV is the feedback factor from the output to the input of the AV node. Furthermore,

x^τ^SA⁻^AV

is the output signal of the sine node at the input of the AV node delayed by the

transit time τSA−AV, since the output signal of the sine node is not directly applied to

the AV node, but must reach it via other nerve cells. In general, let x^τ(t) = x(t −τ)

apply. The corresponding model is shown in Figure 3.23 and Figure 3.24.

If the coupling factor kSA−AV from the sine node to the AV node is not too large

compared to the feedback factor kAV−AV of the AV oscillator, the two oscillators can