Transcript Document 7431424

Pacemaker Mathematics:
Have a Heart
Jen Lindquist, University of Victoria
Rolina van Gaalen, University of Western Ontario
PIMS Mathematical Biology
Summer Workshop 2007
University of Alberta
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
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
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
FitzHugh Nagumo Equations
dv 1
 (v(v   )(1  v)  w  I )
dt 
dw
 v  w
dt
• 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
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
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
Single SA Cell Dynamics
• if we look at :
dv
1

(v(v  0.1)(1  v)  w  I )
dt 0.01
dw
 v  0.5w
dt
• 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
x ' = (1/epsilon) (x (x - alpha) (1 - x) - y + ln)
y ' = x - gam y
epsilon = 0.01 alpha = 0.1
gam = 0.5
ln = 0.1
0.4
0.3
0.1
x ' = (1/epsilon) (x (x - alpha) (1 - x) - y + ln)
y ' = x - gam y
0
epsilon = 0.01
gam = 0.5
alpha = 0.1
ln = 0.11
0.4
-0.1
0.3
-0.2
0.2
-0.2
0
0.2
0.4
0.6
0.8
1
x
0.1
x ' = (1/epsilon) (x (x - alpha) (1 - x) - y + ln)
y ' = x - gam y
0
epsilon = 0.01
gam = 0.5
alpha = 0.1
ln = 0.12
0.4
-0.1
0.3
-0.2
-0.4
-0.2 0.2
0
0.2
0.4
0.6
0.8
1
x
y
-0.4
y
y
0.2
0.1
0
-0.1
-0.2
-0.4
-0.2
0
0.2
0.4
x
0.6
0.8
1
Single SA Cell Dynamics
I = 0.11
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
SA-MC Cell Pair Dynamics
• two neighbouring cells have equal and opposite
effects on each other due to gap junction
coupling
dv1 1
 (v1 (v1   )(1  v1 )  w1  I1  d1, 2 (v2  v1 ))
dt 
dw1
 v1  w1
dt
dv2 1
 (v2 (v2   )(1  v2 )  w2  I 2  d 2,1 (v1  v2 ))
dt 
dw2
 v2  w2
dt
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
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
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
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
A String of Cells
SA
dSA,SA
Recall: dSA,MC = dMC,SA
dSA,SA
SA
dSA,MC
MC
dMC,MC
dMC,SA
dMC,MC
MC
dMC,MC
dMC,MC
String Dynamics: Equal Coupling
2 SA - 8 MC cell string
All couplings d = 0.1
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
SA Node Oscillations Fail to Drive MC String
(weak SA-SA coupling)
dSA,SA = 0.03 dSA,MC = 0.14 dMC,MC = 0.25
SA Node Drives Stable Oscillations of MC String
(moderate SA-SA coupling)
dSA,SA = 0.1 dSA,MC = 0.2 dMC,MC = 0.25
Biological Death of the String System
(strong SA-SA coupling)
dSA,SA = 0.13 dSA,MC = 0.2 dMC,MC = 0.25
A Ring of Cells
dSA,SA
dSA,MC
MC
dMC,MC
SA
SA
dSA,MC
MC
MC
MC
dMC,MC
dMC,MC
dMC,MC
MC
MC
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
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
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
A Sheet of Cells
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
Cell Sheet Dynamics:
A “Dysfunctional” SA Node
4 SA - 32 MC Cell Sheet
Cell Sheet Dynamics:
A “Functional” SA Node
5 SA - 31 MC Cell Sheet
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
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
Acknowledgements
Thank you Jim Keener, & Alex