1. Pacemaker Mathematics: Have a Heart
PIMS Mathematical Biology
Summer Workshop 2007
University of Alberta
Jen Lindquist, University of Victoria
Rolina van Gaalen, University of Western Ontario

2. Overview
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA/MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

3. The Sinoatrial (SA) Node
the SA node is the “pacemaker” of the heart
receives constant input from the ANS
biochemical messages interpreted as input current, I
SA cells are excitable cells that can display stable oscillatory behaviour (versus pancreatic β cells, which “burst”)
each cell may be modelled with FitzHugh-Nagumo equations – derived from the Hodgekin-Huxley model of voltage dynamics

4. Overview FitzHugh-Nagumo equations
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA-MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

5. FitzHugh Nagumo Equations
v represents the membrane potential, a measure cell excitation
w is a recovery variable
α and γ are excitation parameters, and I is an applied current

6. FitzHugh Nagumo Equations
SA cells: I > 0 (represents ANS input)
MC cells: I = 0
this models the primary difference between SA and MC cells: SA cells can show stable oscillations on their own

7. Overview cell system dynamics: single cell model
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA-MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

8. Single SA Cell Dynamics
if we look at :
i.e. γ = 0.5, ε = 0.01, α = 0.1
then slowly increasing I moves the cell from damped firing, to stable oscillatory behaviour, and finally back to damped behaviour

10. Single SA Cell Dynamics

11. Overview cell system dynamics: SA-MC pair dynamics
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA-MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

12. SA-MC Cell Pair Dynamics
two neighbouring cells have equal and opposite effects on each other due to gap junction coupling

13. SA-MC Cell Pair Dynamics
Assumptions throughout this project:
α,γ,ε constant between cells
all SA cells receive equal ANS input, I, at the same time

14. SA-MC Cell Pair Dynamics
in coupling an SA to a MC cell, the SA cell is effectively “drained” by the MC cell
a greater input, I, is thus required to produce the stable oscillations seen in a single cell system

15. SA-MC Cell Pair Dynamics
Recall: A single SA cell
requires an applied
current of only I = 0.11
in order to maintain
stable oscillations
I = 0.15
I = 0.16
I = 0.17

16. Overview cell system dynamics:
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA-MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

17. A String of Cells Recall: dSA,MC = dMC,SA dSA,SA dSA,MC dSA,SA dMC,MC

18. String Dynamics: Equal Coupling 2 SA - 8 MC cell string
All couplings d = 0.1

19. String Dynamics: Variable Coupling
Given a system with different coupling coefficients between cells, as SA-SA coupling increases the string system will:
1) maintain SA oscillations but not drive atrial
oscillations: SA beating, but no atrial pulse
2) show stable oscillations: drive the atrium
3) die off: biological death

20. SA Node Oscillations Fail to Drive MC String (weak SA-SA coupling)
dSA,SA = dSA,MC = dMC,MC = 0.25

21. SA Node Drives Stable Oscillations of MC String (moderate SA-SA coupling)
dSA,SA = 0.1 dSA,MC = 0.2 dMC,MC = 0.25

22. Biological Death of the String System (strong SA-SA coupling)
dSA,SA = dSA,MC = 0.2 dMC,MC = 0.25

23. A Ring of Cells dSA,SA dSA,MC SA SA dSA,MC MC MC dMC,MC dMC,MC MC MC

24. Ring Dynamics: Symmetry Effects
The dynamics of the 10 cell ring are the same as the dynamics of a 5 cell string with one SA cell: the system dies off
This may be explained by noticing that a ring formation effectively doubles the initial SA effects

25. Ring Dynamics 2 neighbouring SA cells, in a ring with 8 MC cells
dSA,SA = 0.1 dSA,MC = 0.2 dMC,MC = 0.25

26. Complex Networks of Cells: Sheets
the SA node can be thought of as an area of cells within a larger sheet of cells, the atrial surface
Assumption: cells are arranged in a matrix such that each cell has 4 neighbours with which it interacts

27. A Sheet of Cells

28. Cell Sheet Dynamics: System Requirements
variable coupling is essential for maintaining oscillations in a sheet of cells (vs. string system)
there is a minimum number of SA cells required to maintain oscillations
4SA/32MC does not show stable oscillations
5SA/31MC does show stable oscillations
this minimum SA number may not be linearly related to the size of the sheet
e.g. a system of 40SA/320MC may have emergent properties which allow for stable oscillations

29. Cell Sheet Dynamics: A “Dysfunctional” SA Node 4 SA - 32 MC Cell Sheet

30. Cell Sheet Dynamics: A “Functional” SA Node 5 SA - 31 MC Cell Sheet

31. Overview future research and nagging, yet interesting, questions
primer on the sinoatrial (SA) node and surrounding myocardial (MC) tissue
FitzHugh-Nagumo equations
cell system dynamics:
single cell model
SA-MC pair dynamics
systems of cells: strings, rings, & sheets
future research and nagging, yet interesting, questions

32. Future Research (& nagging, yet interesting questions)
Optimal size/shape of SA node
Connectivity: hexagonal vs. rectangular
What are the effects of modifying the SA-SA coupling in the sheet model?
Sinoatrial cells in the adult organism are not replaced: how do dead/dying or uncoupled cells affect the system?
Biologically, the system must be able to control frequency of oscillation, and adjust as necessary – can the model account for this?
3-dimensional questions regarding heart physiology and emergent properties of tissues

33. Thank you Jim Keener, & Alex
Acknowledgements
Thank you Jim Keener, & Alex