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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
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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
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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
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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
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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
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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
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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
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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
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Single SA Cell Dynamics
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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
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SA-MC Cell Pair Dynamics
two neighbouring cells have equal and opposite effects on each other due to gap junction coupling
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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
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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
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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
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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
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A String of Cells Recall: dSA,MC = dMC,SA dSA,SA dSA,MC dSA,SA dMC,MC
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String Dynamics: Equal Coupling 2 SA - 8 MC cell string
All couplings d = 0.1
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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
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SA Node Oscillations Fail to Drive MC String (weak SA-SA coupling)
dSA,SA = dSA,MC = dMC,MC = 0.25
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SA Node Drives Stable Oscillations of MC String (moderate SA-SA coupling)
dSA,SA = 0.1 dSA,MC = 0.2 dMC,MC = 0.25
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Biological Death of the String System (strong SA-SA coupling)
dSA,SA = dSA,MC = 0.2 dMC,MC = 0.25
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A Ring of Cells dSA,SA dSA,MC SA SA dSA,MC MC MC dMC,MC dMC,MC MC MC
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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
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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
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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
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A Sheet of Cells
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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
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Cell Sheet Dynamics: A “Dysfunctional” SA Node 4 SA - 32 MC Cell Sheet
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Cell Sheet Dynamics: A “Functional” SA Node 5 SA - 31 MC Cell Sheet
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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
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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
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Thank you Jim Keener, & Alex
Acknowledgements Thank you Jim Keener, & Alex
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