1. 1 Heart rate variability: challenge for both experiment and modelling I. Khovanov, N. Khovanova, P. McClintock, A. Stefanovska Physics Department, Lancaster University

2. 2 Heart rate variability (HRV) Outline ● Motivations ● Experiment ● Modelling ● Summary

3. 3 The object of investigation is heart rate Heart rate variability (HRV) Heart Rate Variability ElectroCardioGram SinoAtrial Node

4. 4 24h RR-intervals Heart rate variability (HRV) RR-intervals, sec Number of interval Medical people: Average over one hour rhythm Physicists: Entropies, Dimensions, Long-range correlation, Scaling, Multifractal etc

5. 5 HRV is the product of the integrative control system Heart rate variability (HRV) Parasympathetic branch Vagus nerve fibres Fast and - Sympathetic branch Postganglionic fibres Slow and + From receptor afferents: baro-receptors, chemo-receptors, stretch receptors etc Vagal Symp Input Nucleous Medulla Hypothalamus Cortex (higher centres) - - + SAN

6. 6 Respiration masks other rhythms Circles corresponds to RR-intervals, Dashed line corresponds to respiration respiration

7. 7 Use of apnoea (breath holding)? 1) The use of breath holding as longer as possible, BUT physiologists discussed a long breath holding as one of unsolved problems with a specific dynamics ( Parkers, Exp. Phys. 2006 ) 2) We can notice: In spontaneous respiration there are apnoea intervals (not very long) An idea is to prolong by keeping normocapnia Physiology literature said 30 sec is fine(!)

8. 8 Specific form of respiration Intermittent type (intervals of fixed duration, 30 sec)

9. 9 The idea is to consider HRV without dominate external perturbation, but keeping all internal perturbation and without modification of a net of regulatory networks The task is to study intrinsic dynamics on short-time scales ● Special design of experiments: relaxed, supine position,records of 45-60 minutes ● Time-series analysis of sets of short time- series (appr. 40Apn.int X 35RR-int) Intrinsic dynamics of regulatory system

10. 10 Decomposition (nonlinear transformation) of heart rate by specific forms of respiration Circles corresponds to RR-intervals, Dashed line corresponds to respiration respiration Object of analysis: a set of RR-intervals {RR i } j corresponded to apnoea intervals

11. 11 RR-intervals Non-stationary dynamics of RR-intervals. Number of interval, i RR i -RR 1 [sec]  RR i -  RR 1 [sec] Increments  RR  RR i =RR i -RR i-1

12. 12 RR-intervals during apnoea intervals is non-stationary. The use of random walk framework. DFA (detrended fluctuational analysis): scaling exponent  (Peng’95) Aggregation analysis: scaling exponent b (West’05) Both methods for the considered time-series estimate a diffusion velocity Non-stationary dynamics of RR-intervals.

13. 13 Scaling exponent  by DFA DFA method (C.-K. Peng, Chaos,1995) (1) Integration of RR-intervals: n (2) Calculation of linear trend y n (k) for time window of length n (3) Calculation of scaling function for set of n (4) Determination of   =1,5 corresponds to Brown noise (free Brownian motion)

14. 14 Scaling exponent b  by aggregation analysis Aggregation method (B. J. West, Complexity, 2006) Invention by L. R. Taylor, Nature, 1961 (1) Creating a set of aggregated time-series: (2) Calculation of the variance and mean for each m=1,2…: (3) Determination of b b=2 corresponds to Brown noise (free Brownian motion) The aggregation method is close to the stability test for the increments  RR

15. 15 Scaling exponents  and b  on the base of 24h RR-intervals DFA and the aggregation method in the presence respiration (the previous published results) Brown noise, Brownian motion White noise

16. 16 Scaling exponents  and b  RR-intervals during apnoea DFA and the aggregation method without respiration Brown noise, Brownian motion White noise

17. 17 RR-intervals during apnoea intervals is non-stationary. So let us use stationary increments  RR i =RR i -RR i-1 then use the modified definition of ACF  (  ) to use non- overlapped windows corresponding apnoea intervals Dynamics of increments  RR. ACF  - time delay k j – number of RR-intervals in each apnoea N –total number of apnoea intervals Finally use fitting by the function

18. 18 Crosses corresponds to calculations using  RR The solid line corresponds to approximation by ACF of  RR-intervals Fast decay of ACF with weak oscillations near 0.1 Hz Oscillations are on-off nature and observed for parts of apnoea intervals and, not in all measurements.

19. 19  =2  =1.5  =1  =0.5 Distribution of increments of RR-intervals P(  RR) Calculate histogram and fit by  -stable distribution. A random variable X is stable, if for X 1 and X 2 independent copies of X, the following equality holds: Means equality in distribution  is a stability index defines the weight of tails  =2 Normal (Gaussian) distribution  =1 Cauchy (Lorentz) distribution

20. 20 Yellow areas and cycles correspond to histograms The solid lines is fit by the normal distribution (  =2) The dashed lines corresponds to the  -stable distributions Distribution of increments of RR-intervals The previous published results for 24h RR-intervals Apnoea intervals

21. 21 HRV intrinsic dynamics Summary of experimental results: RR-intervals show stochastic diffusive dynamics. HRV during apnoea can be described as a stochastic process with stationary increments Conclusion: Intrinsic dynamics is a result of integrative action of many weakly interacting components Increments  RR describes by  -stable process with a weak correlation In zero approximation  RR corresponds to uncorrelated normal random process and RR-intervals show classical free Brownian motion.

22. 22ModellingModelling Heart beat is initiated in SAN Sinoatrial node

23. 23 Isolated heart (e.g. in case of brain dead)ModellingModelling No signal from nervous system Nearly periodic oscillations, but heart rate is 200 beats/min whereas in normal state 60-80 beats/min

24. 24 Vagal activationModellingModelling Parasympathetic branch Vagus nerve fibres Fast and - Threshold potential Potential of hyperpolarization Slope of depolarization ●Decreasing depolarization slope ● Increasing hyperpolarization potential

25. 25 Sympathetic activationModellingModelling Sympathetic branch Postganglionic fibres Slow and + ● Increasing depolarization slope ● Decreasing hyperpolarization potential

26. 26ModellingModelling Integrate & Fire model titi t i+1 t i+2 Threshold potential U t U r Hyperpolarization potential Integration slope 1/  Random numbers having the stable distribution

27. 27ModellingModelling FitzHugh-Nagumo model ● Additive versus multiplicative noise ● Noise properties What kind of noise will produce non-Gaussianity of increments  RR