Nonlinear dynamic system analyzing for heart rate variability mathematical model A.Martynenko & M. Yabluchansky Kharkov National University (Ukraine)

2 HRV – naturally observing phenomenon of nonlinear dynamic behavior of the cardiovascular system HRV – naturally observing phenomenon of nonlinear dynamic behavior of the cardiovascular system Non Linear Mathematical Model (NL MM) of HRV Non Linear Mathematical Model (NL MM) of HRV Why do we need this model? Why do we need this model? What is the advantage of NL MM for investigation of ANS? What is the advantage of NL MM for investigation of ANS?

3 Mathematical model differential equations Regulatory (ANS) group: Nonlinear dynamic quasi-periodical processes in humoral (1), sympathetic (2), parasympathetic (3) branches of nervous system Regulatory (ANS) group: Nonlinear dynamic quasi-periodical processes in humoral (1), sympathetic (2), parasympathetic (3) branches of nervous system d2(ANS i )/dt2 = F i (R i  d(ANS i )/dt, A i  ANS i, S j  ANS j, BioMechanics), i,j = 1,2,3, d2(ANS i )/dt2 = F i (R i  d(ANS i )/dt, A i  ANS i, S j  ANS j, BioMechanics), i,j = 1,2,3, BioMechanics group: equations that describe function of biomechanical parameters forming ANS activity BioMechanics group: equations that describe function of biomechanical parameters forming ANS activity d2(BM i )/dt2 = B i (R i  d(BM i )/dt, A i  BM i, HR ñ, ANS), i = 4,5,6, d2(BM i )/dt2 = B i (R i  d(BM i )/dt, A i  BM i, HR ñ, ANS), i = 4,5,6, HR equation - describe HR changes from cycle to cycle HR equation - describe HR changes from cycle to cycle d2(HR ñ )/dt2 = f i (R  d(HR ñ )/dt, A  HR ñ,S j  ANS j ), j = 1,2,3. d2(HR ñ )/dt2 = f i (R  d(HR ñ )/dt, A  HR ñ,S j  ANS j ), j = 1,2,3.

4 Scheme of cardiovascular regulation (cross-linkage in mathematical model)

5 HRV (corr=0.993) and spectrum (corr=0.999) (registration vs. model)

6 First result of NL MM – new technique of spectral domain separation

7 ‘Nlyzer’ by TU Darmstadt

8 Standard nonlinear analyzes Fractal Dimension (D2) HRV – 4.74 – 5.3 ECG – 2.65 – 3.5 N points for embedding N= D2 =10000 Autocorrelation for time delay Entropy and mutual information

9 Lorentz attractor (D2=2.06)

10 Attractor reconstruction (20 min)

11 Attractor reconstruction (2500 heartbeat)

12 Attractor reconstruction (5000 heartbeat)

13 Attractor reconstruction (10000 heartbeat)

14 Attractor reconstruction (15000 heartbeat)

15 Attractor reconstruction (20000 heartbeat or about 5 hours)

16 Attractor reconstruction (3min+MM)

17 Attractor reconstruction (15000 heartbeat)

18 Poincare map (3 min+ MM)

19 Attractor reconstruction (3 min + MM)

20 Attractor reconstruction (3 min + MM)

21 HRV attractor and its P-map

22 Conclusion Cardiovascular regulation is nonlinear dynamic system, and then we need nonlinear mathematical modeling for their investigation. Cardiovascular regulation is nonlinear dynamic system, and then we need nonlinear mathematical modeling for their investigation. Advantages of NL MM: Advantages of NL MM: New technique of spectra domain separation New technique of spectra domain separation Great time compression: We don’t need 4-5 hours of registration for attractor reconstruction – only 3 min of registration and NL MM Great time compression: We don’t need 4-5 hours of registration for attractor reconstruction – only 3 min of registration and NL MM Attractor visualization in Humoral - Sympathetic - Parasympathetic phase space is very good Attractor visualization in Humoral - Sympathetic - Parasympathetic phase space is very good