Imagine the
Possibilities
Mathematical Biology
University of Utah
Coupling and Propagation in Excitable Media
J. P. Keener
Mathematics Department
University of Utah
Coupling and Propagation in Excitable Media – p.1/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Spatially Extended Excitable Media
Neurons and axons
Coupling and Propagation in Excitable Media – p.2/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Spatially Extended Excitable Media
Mechanically stimulated Calcium waves
Coupling and Propagation in Excitable Media – p.3/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Conduction system of the heart
Coupling and Propagation in Excitable Media – p.4/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Conduction system of the heart
• Electrical signal originates in the SA node.
Coupling and Propagation in Excitable Media – p.4/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Conduction system of the heart
• Electrical signal originates in the SA node.
• The signal propagates across the atria (2D sheet), through
the AV node, along Purkinje fibers (1D cables), and
throughout the ventricles (3D tissue).
Coupling and Propagation in Excitable Media – p.4/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Spatially Extended Excitable Media
The forest fire analogy
Coupling and Propagation in Excitable Media – p.5/22
Imagine the
Possibilities
Spatial Coupling
Mathematical Biology
University of Utah
Conservation Law:
d
(stuff in Ω) = rate of transport + rate of production
dt
Z
Z
Z
d
udV =
J · nds + f dv
dt Ω
∂Ω
Ω
becomes
Flux
∂u
= ∇ · (D∇u) + f (u)
∂t
J
production
f(u)
Ω
Coupling and Propagation in Excitable Media – p.6/22
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Re
φ 1e
Cardiac Cells
φ 2e
ie
Cm
I
ion
Cm
I
ig
gap junctions
φ1
i
Rg
ion
φ2
i
1
dφ1
1
+ Iion (φ , w) = −ii =
Cm
(φ2 − φ1 )
dt
Re + R g
1
dφ2
2
+ Iion (φ , w) = ii =
Cm
(φ1 − φ2 )
dt
Re + R g
Question: Can anything interesting happen with coupled cells that
does not happen with a single cell?
Take a wild guess. – p.7/22
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with slightly elevated potassium - uncoupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1
Who could have guessed? – p.8/22
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with slightly elevated potassium - coupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1
Who could have guessed? – p.8/22
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with moderately elevated potassium uncoupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1 could have guessed? – p.8/22
Who
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with moderately elevated potassium coupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1 could have guessed? – p.8/22
Who
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with greatly elevated potassium - uncoupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1
Who could have guessed? – p.8/22
Imagine the
Possibilities
Coupled Cells
Mathematical Biology
University of Utah
Normal cell and cell with greatly elevated potassium - coupled
normal cell
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
500
1000
0
1500
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6
"ischemic" cell
1
0.8
1
0
0
0
500
0.2
1000
0.4
1500
0.6
0.8
1
Who could have guessed? – p.8/22
Imagine the
Possibilities
Axons and Fibers
Mathematical Biology
University of Utah
Ve (x)
It dx
Cm dx
re dx
Ie(x)
Cm dx
Iion dx
Ve (x+dx)
Extracellular
space
It dx
Iion dx
Cell
membrane
Ii(x)
Vi (x)
ri dx
Vi (x+dx)
Intracellular
space
From Ohm’s law
Vi (x+dx)−Vi (x) = −Ii (x)ri dx, Ve (x+dx)−Ve (x) = −Ie (x)re dx,
In the limit as dx → 0,
1 dVi
,
Ii = −
ri dx
1 dVe
Ie = −
.
re dx
Coupling and Propagation – p.9/22
Imagine the
Possibilities
The Cable Equation
Mathematical Biology
University of Utah
Ve (x)
It dx
Cm dx
re dx
Ve (x+dx)
It dx
Ie(x)
Cm dx
Iion dx
Extracellular
space
Iion dx
Cell
membrane
Ii(x)
Vi (x)
ri dx
Vi (x+dx)
Intracellular
space
From Kirchhoff’s laws
Ii (x) − Ii (x + dx) = It dx = Ie (x + dx) − Ie (x)
In the limit as dx → 0, this becomes
∂Ie
∂Ii
=
.
It = −
∂x
∂x
Coupling and Propagation – p.10/22
Imagine the
Possibilities
The Cable Equation
Mathematical Biology
University of Utah
Combining these
∂
It =
∂x
1 ∂V
ri + re ∂x
,
and, thus,
∂
∂V
Cm
+ Iion = It =
∂t
∂x
1 ∂V
ri + re ∂x
.
This equation is referred to as the cable equation.
Coupling and Propagation – p.11/22
Imagine the
Possibilities
Modelling Cardiac Tissue
Mathematical Biology
University of Utah
Cardiac Tissue The Bidomain Model:
• At each point of the cardiac domain there are two comingled
regions, the extracellular and the intracellular domains with
potentials φe and φi , and transmembrane potential
φ = φi − φe .
Coupling and Propagation – p.12/22
Imagine the
Possibilities
Modelling Cardiac Tissue
Mathematical Biology
University of Utah
Cardiac Tissue The Bidomain Model:
• At each point of the cardiac domain there are two comingled
regions, the extracellular and the intracellular domains with
potentials φe and φi , and transmembrane potential
φ = φi − φe .
• These potentials drive currents, ie = −σe ∇φe , ii = −σi ∇φi ,
where σe and σi are conductivity tensors.
Coupling and Propagation – p.12/22
Imagine the
Possibilities
Modelling Cardiac Tissue
Mathematical Biology
University of Utah
Cardiac Tissue The Bidomain Model:
• At each point of the cardiac domain there are two comingled
regions, the extracellular and the intracellular domains with
potentials φe and φi , and transmembrane potential
φ = φi − φe .
• These potentials drive currents, ie = −σe ∇φe , ii = −σi ∇φi ,
where σe and σi are conductivity tensors.
• Total current is
iT = ie + ii = −σe ∇φe − σi ∇φi .
Coupling and Propagation – p.12/22
Imagine the
Possibilities
Mathematical Biology
University of Utah
Kirchhoff’s laws:
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
Extracellular Space
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
Extracellular Space
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
surface to volume ratio,
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
Extracellular Space
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
surface to volume ratio, capacitive current,
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
Extracellular Space
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
surface to volume ratio, capacitive current, ionic current,
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Extracellular Space
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
surface to volume ratio, capacitive current, ionic current,
and current from intracellular space.
Coupling and Propagation – p.13/22
Imagine the
Possibilities
Kirchhoff’s laws:
Mathematical Biology
University of Utah
• Total current is conserved: ∇ · (σi ∇φi + σe ∇φe ) = 0
• Transmembrane current is balanced:
φe
χ ( Cm ∂φ
∂τ + Iion ) = ∇ · (σi ∇φi )
Extracellular Space
Cm
I
ion
Intracellular Space
φ = φ i− φ e
φi
surface to volume ratio, capacitive current, ionic current,
and current from intracellular space.
• Boundary conditions:
n · σi ∇φi = 0,
n · σe ∇φe = I(t, x)
R
and ∂Ω I(t, x)dx = 0 on ∂Ω.
+
−
Coupling and Propagation – p.13/22
Imagine the
Possibilities
The Bistable Equation
Mathematical Biology
University of Utah
∂2u
∂u
= D 2 + f (u)
∂t
∂x
with f (0) = f (a) = f (1) = 0, 0 < a < 1.
• There is a unique traveling wave solution u = U (x − ct),
• The solution is stable up to phase shifts,
√
• The speed scales as c = c0 D,
• U is a homoclinic trajectory of DU 00 + cU 0 + f (U ) = 0
1
3
0.9
2.5
0.8
2
0.7
1.5
dU/dx
U(x)
0.6
0.5
1
0.5
0.4
0
0.3
−0.5
0.2
−1
0.1
0
−3
−2
−1
0
x
1
2
3
−1.5
−0.2
0
0.2
0.4
U
0.6
0.8
1
1.2
Coupling and Propagation – p.14/22
Imagine the
Possibilities
Discreteness
Mathematical Biology
University of Utah
Cardiac Cells
gap junctions
Gap junctional coupling
Cardiac Cell
Calcium release sites
Calcium Release through CICR Receptors
Coupling and Propagation – p.15/22
Imagine the
Possibilities
Discrete Effects
Mathematical Biology
University of Utah
Discrete Cells
dvn
= f (vn ) + d(vn−1 − 2vn + vn−1 )
dt
Discrete Calcium Release
Discrete Release Sites
∂u
∂2u
= D 2 + g(x)f (u)
∂t
∂x
Coupling and Propagation – p.16/22
Imagine the
Possibilities
Fire-Diffuse-Fire Model
Mathematical Biology
University of Utah
h
Suppose a diffusible chemical u is released from
• a long line of evenly spaced release sites;
• Release of full contents C occurs when concentration u
reaches threshold θ.
∂u
∂2u X
Source(x − nh)δ(t − tn )
=D 2 +
∂t
∂x
n
Coupling and Propagation – p.17/22
Imagine the
Possibilities
Fire-Diffuse-Fire-II
Mathematical Biology
University of Utah
Recall that the solution of the heat equation with δ-function initial
data at x = x0 and at t = t0 is
(x − x0 )2
exp(−
u(x, t) = p
)
4D(t − t0 )
4π(t − t0 )
1
3
0.26
0.24
2.5
0.22
0.2
2
u
u
0.18
1.5
0.16
0.14
1
0.12
0.1
0.5
0.08
0
−5
−4
−3
−2
−1
0
x
1
2
3
4
5
0.06
0
1
2
3
4
5
6
t with x fixed
7
8
9
10
Coupling and Propagation – p.18/22
Imagine the
Possibilities
Fire-Diffuse-Fire-III
Mathematical Biology
University of Utah
Suppose known firing times are tj at position xj = jh,
j = −∞, · · · , n − 1. Find tn . At x = xn = nh,
=
(nh − jh)2
p
exp(−
)
4D(t − tj )
4π(t − tj )
C
j=−∞
0.26
0.24
0.22
0.2
2
≈
h2
C D∆t
exp(−
) = f( 2 )
p
4D(t − tn−1 )
h
h
4π(t − tn−1 )
C
F(D∆ t/h )
u(nh, t)
n−1
X
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0
1
2
3
4
5
D∆ t/h2
6
7
8
9
10
Coupling and Propagation – p.19/22
Imagine the
Possibilities
Fire-Diffuse-Fire-IV
Mathematical Biology
University of Utah
Solve the equation
θh
D∆t
= f( 2 )
C
h
0.26
10
0.24
9
0.22
8
0.2
7
0.18
6
vh/D
2
F(D∆ t/h )
This is easy to do graphically:
0.16
5
0.14
4
0.12
3
0.1
2
0.08
1
0.06
0
1
2
3
4
5
D∆ t/h2
6
7
8
9
10
0
0.06
0.08
0.1
0.12
0.14
0.16
θ h/C
0.18
0.2
0.22
0.24
0.26
∗ ≈ 0.25 (i.e. if h is too
Conclusion: Propagation fails for θh
>
θ
C
large, θ is too large, or C is too small.)
Coupling and Propagation – p.20/22
Imagine the
Possibilities
With Recovery
Mathematical Biology
University of Utah
Including recovery variables
∂v
∂2v
= D 2 + f (v, w),
∂t
∂x
∂w
= g(v, w)
∂t
Solitary Pulse
Periodic Waves
Skipped Beats
Coupling and Propagation – p.21/22
Imagine the
Possibilities
Periodic Ring
Mathematical Biology
University of Utah
To be continued ...
Coupling and Propagation – p.22/22