Imagine the
Possibilities
Mathematical Biology
University of Utah
The Dynamics of Excitability
J. P. Keener
Mathematics Department
University of Utah
The Dynamics of Excitability – p.1/??
Imagine the
Possibilities
Examples of Excitable Media
Mathematical Biology
University of Utah
• B-Z reagent
• Nerve cells
• cardiac cells, muscle cells
• Slime mold (dictystelium discoideum)
• CICR (Calcium Induced Calcium Release)
• Forest Fires
Features of Excitability
• Threshold Behavior
• Refractoriness
• Recovery
Resting
Excited
Refractory
Recovering
What about flush toilets? – p.2/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
Modeling Membrane Electrical
Activity
Excitable Cells – p.3/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
Modeling Membrane Electrical
Activity
Transmembrane potential φ is regulated by transmembrane ionic
currents and capacitive currents:
dφ
+ Iion (φ, w) = Iin
Cm
dt
dw
= g(φ, w),
where
dt
w ∈ Rn
Excitable Cells – p.3/??
Imagine the
Possibilities
Examples
Mathematical Biology
University of Utah
Examples include:
• Neuron - Hodgkin-Huxley model
• Purkinje fiber - Noble
• Cardiac cells - Beeler-Reuter, Luo-Rudy, Winslow-Jafri, Bers
• Two Variable Models - reduced HH, FitzHugh-Nagumo,
Mitchell-Schaeffer, Morris-Lecar, McKean, Puschino, etc.)
Excitable Cells – p.4/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
The Squid Giant Axon...
What does a squid look like? – p.5/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
is not the Giant Squid Axon
Excitable Cells – p.6/??
Imagine the
Possibilities
The Hodgkin-Huxley Equations
Mathematical Biology
University of Utah
I
K
I
l
I
Na
Extracellular Space
φe
Cm
Intracellular Space
I
φ
ion
φ=φ−φ
i
e
i
dV
Cm
+ INa + IK + Il = 0,
dt
Excitable Cells – p.7/??
Imagine the
Possibilities
The Hodgkin-Huxley Equations
Mathematical Biology
University of Utah
I
K
I
l
I
Na
Extracellular Space
φe
Cm
Intracellular Space
I
φ
ion
φ=φ−φ
i
e
i
dV
Cm
+ INa + IK + Il = 0,
dt
with sodium current INa ,
Excitable Cells – p.7/??
Imagine the
Possibilities
The Hodgkin-Huxley Equations
Mathematical Biology
University of Utah
I
K
I
l
I
Na
Extracellular Space
φe
Cm
Intracellular Space
I
φ
ion
φ=φ−φ
i
e
i
dV
Cm
+ INa + IK + Il = 0,
dt
with sodium current INa , potassium current IK ,
Excitable Cells – p.7/??
Imagine the
Possibilities
The Hodgkin-Huxley Equations
Mathematical Biology
University of Utah
I
K
I
l
I
Na
Extracellular Space
φe
Cm
Intracellular Space
I
φ
ion
φ=φ−φ
i
e
i
dV
Cm
+ INa + IK + Il = 0,
dt
with sodium current INa , potassium current IK , and leak
current Il .
Excitable Cells – p.7/??
Imagine the
Possibilities
Ionic Currents
Mathematical Biology
University of Utah
Ionic currents are typically of the form
I = g(φ, t) Φ(φ)
Excitable Cells – p.8/??
Imagine the
Possibilities
Ionic Currents
Mathematical Biology
University of Utah
Ionic currents are typically of the form
I = g(φ, t) Φ(φ)
where g(φ, t) is the total number of open channels,
Excitable Cells – p.8/??
Imagine the
Possibilities
Ionic Currents
Mathematical Biology
University of Utah
Ionic currents are typically of the form
I = g(φ, t) Φ(φ)
where g(φ, t) is the total number of open channels, and Φ(φ) is
the I-φ relationship for a single channel.
Excitable Cells – p.8/??
Imagine the
Possibilities
Currents
Mathematical Biology
University of Utah
Hodgkin and Huxley found that
Ik = gk n4 (φ − φK ),
where
IN a = gN a m3 h(φ − φN a ),
du
τu (φ)
= u∞ (φ) − u,
dt
10
1.0
0.8
u = m, n, h
n (v)
h (v)
τh(v)
8
0.6
ms
6
0.4
m (v)
4
0.2
2
0.0
0
0
20
40
60
Potential (mV)
80
100
τn(v)
τm(v)
0
40
80
Potential (mV)
Excitable Cells – p.9/??
Imagine the
Possibilities
Action Potential Dynamics
Mathematical Biology
University of Utah
1.0
120
Gating variables
80
60
40
20
0
-20
0
0.8
h(t)
0.6
0.4
n(t)
0.2
5
10
time (ms)
15
20
0
35
0.8
30
0.7
25
20
0.5
15
0.4
gK
10
10
time (ms)
15
20
0.3
5
0.2
0
0.1
0
5
0.6
gNa
h
2
m(t)
0.0
Conductance (mmho/cm )
Potential (mV)
100
5
10
time (ms)
15
20
0
0
0.1
0.2
0.3
0.4
n
0.5
0.6
0.7
0.8
Excitable Cells – p.10/??
Imagine the
Possibilities
Fast-Slow Subsystem Dynamics
Mathematical Biology
University of Utah
Observe that τm << τn , τh
1.0
solution trajectories
stable manifold of vs
ve
0.8
0.6
m
dm/dt = 0
0.4
dv/dt = 0
0.2
vr
0.0
vs
0
20
40
v
60
80
100
1.4
1.2
m
1.0
4
0.8
3
0.6
2
0.4
1
0.2
0.0
0
40
80
v
(Go Back)
Excitable Cells – p.11/??
Imagine the
Possibilities
Two Variable Reduction of HH Eqns
Mathematical Biology
University of Utah
Set m = m∞ (φ), and set h + n ≈ N = 0.85.
This reduces to a two variable system
=
ḡK n4 (φ − φK ) + ḡN a m3∞ (φ)(N − n)(φ − φN a ) + ḡl (φ − φL ),
=
n∞ (φ) − n.
1.0
dn/dt = 0
0.8
0.6
n
dφ
dt
dn
τn (φ)
dt
C
0.4
dv/dt = 0
0.2
0.0
0
40
80
v
120
(Go Back)
Excitable Cells – p.12/??
Imagine the
Possibilities
Two Variable Models
Mathematical Biology
University of Utah
Following is a summary of two variable models of excitable
media. The models described here are all of the form
dv
= f (v, w) + I
dt
dw
= g(v, w)
dt
Typically, v is a “fast” variable, while w is a “slow” variable.
w
f(u,w) = 0
g(v,w) = 0
W*
V-(w)
W*
V0(w)
V+(w)
v
Excitable Cells – p.13/??
Imagine the
Possibilities
Cubic FitzHugh-Nagumo
Mathematical Biology
University of Utah
The model that started the whole business uses a cubic
polynomial (a variant of the van der Pol equation).
F (v, w) = Av(v − α)(1 − v) − w,
G(v, w) = (v − γw).
with 0 < α < 12 , and “small”.
Excitable Cells – p.14/??
Imagine the
Possibilities
FitzHugh-Nagumo Equations
Mathematical Biology
University of Utah
0.20
1.0
dw/dt = 0
0.8
0.15
v(t)
0.6
0.10
w
0.4
dv/dt = 0
0.05
0.2
w(t)
0.0
0.00
-0.2
-0.05
-0.4
0.0
0.4
v
0.8
0.0
1.2
0.2
0.4
0.6
0.8
1.0
time
0.20
0.15
0.8
dw/dt = 0
0.10
0.4
dv/dt = 0
w
v(t)
0.05
w(t)
0.0
0.00
-0.4
-0.05
-0.4
0.0
0.4
v
0.8
0.0
0.5
1.0
1.5
time
2.0
2.5
3.0
(Go Back)
Excitable Cells – p.15/??
Imagine the
Possibilities
Mitchell-Schaeffer
Mathematical Biology
University of Utah
Mitchell-Schaeffer two-variable model (also in a slightly different
but equivalent form by Karma)
1
v
2
F (v, w) =
wv (1 − v) −
,
τin
τout

 1 (1 − w) v < v
gate
τopen
G(v, w) =
w

− τclose
v > vgate
Notice that F (v, w) is cubic in v, and w is an inactivation variable
(like h in HH).
Excitable Cells – p.16/??
Imagine the
Possibilities
Mitchell-Schaeffer Revised
Mathematical Biology
University of Utah
To make the Mitchell-Schaeffer look like an ionic model, take
dv
Cm
dt
dh
τh
dt
= gN a hm2 (VN a − v) + gK (VK − v),
= h∞ (v) − h
where
m(v) =



0,


v<0
v, 0 < v < 1 ,



 1,
v>1
h∞ = 1 − f (v),
τh = τopen + (τclose − τopen )f (v)
f (v) =
1
2 (1
+ tanh(κ(v − vgate )),
Excitable Cells – p.17/??
Imagine the
Possibilities
Mitchell-Schaeffer Revised-II
Mathematical Biology
University of Utah
1
φ, h
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
time
600
700
800
900
1000
1
h
0.8
0.6
0.4
0.2
0
0
0.2
0.4
φ
0.6
0.8
1
(Go Back)
Excitable Cells – p.18/??
Imagine the
Possibilities
Morris-Lecar
Mathematical Biology
University of Utah
This model was devised for barnacle muscle fiber.
G(v, w)
=
m∞ (v) =
−gca m∞ (v)(v − vca ) − gk w(v − vk ) − gl (v − vl ) + Iapp
1 v − v3
φ cosh(
)(w∞ (v) − w),
2 v4
1
1
v − v1
),
+ tanh(
2
2
v2
w∞ (v) = (1 + tanh(
v − v3 )
)).
2v4
40
20
0
V
=
−20
−40
−60
0
50
100
150
200
250
time
300
350
400
450
500
1
0.8
0.6
W
F (v, w)
0.4
0.2
0
−0.2
−60
−40
−20
V
0
20
40
60
(Go Back)
Excitable Cells – p.19/??
Imagine the
Possibilities
McKean
Mathematical Biology
University of Utah
McKean suggested two piecewise linear models with
F (v, w) = f (v) − w and G(v, w) = (v − γw). For the first,
0.6
v−α
>
>
:
α
2
< 1+α
2
1+α
2
0.5
v<
−v
α
2
1−v
<v
v>
0.4
0.3
f(v)
f (v) =
8
>
>
<
0.2
0.1
0
−0.1
−0.2
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
v
0.8
1
1.2
1.4
where 0 < α < 12 .
The second model suggested by McKean had
1
0.8
−v
: 1−v
v<α
0.6
v>α
0.2
0
−0.2
−0.4
−0.6
and γ = 0.
(-9)
0.4
f(v)
f (v) =
8
<
−0.4
−0.2
0
0.2
0.4
v
0.6
0.8
1
1.2
1.4
Excitable Cells – p.20/??
Imagine the
Possibilities
Barkley
Mathematical Biology
University of Utah
A model devised to give very fast 2D computations (the code is
known as EZspiral)
w+b
),
F (v, w) = v(1 − v)(v −
a
G(v, w) = (v − w).
Excitable Cells – p.21/??
Imagine the
Possibilities
Puschino
Mathematical Biology
University of Utah
A piecewise linear model devised to match cardiac restitution
properties
F (v, w) = f (v) − w
1
(v − w)
G(v, w)
τ (v)
where
f (v) =







with v1 =
v < v1
−30v,
γv − 0.12,
v 1 < v < v2 , ,
−30(v − 1), v > v2
0.12
30+γ ,
v2 =
30.12
30+γ . (Go Back)
τ (v) =


2
v < v1
 16.6 v > v1
Excitable Cells – p.22/??
Imagine the
Possibilities
Aliev
Mathematical Biology
University of Utah
For the Aliev model,
F (v, w) = ga (v − β)(v − α)(1 − v) − vw
G(v, w) = −(v, w)(w + gs (v − β)(v − α − 1)
w
where (v, w) = 1 + µ1 v+µ
.
2
Reasonable parameter values are β = 0.0001 , α = 0.05, g a = 8.0,
gs = 8.0, µ1 = 0.05, µ2 = 0.3, 1 = 0.03, 2 = 0.0001.
Excitable Cells – p.23/??
Imagine the
Possibilities
Tyson-Fife
Mathematical Biology
University of Utah
These dynamics describe the oxidation-reduction of malonic
acid. For this system,
v−q
F (v, w) = v − v − (f w + φ0 )
v+q
G(v, w) = (v − w)
2
(-10)
(-10)
with typical parameter values = 0.05, q = 0.002, f = 3.5, φ 0 =
0.01.
Excitable Cells – p.24/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
Intracellular Calcium
Excitable Cells – p.25/??
Imagine the
Possibilities
Mathematical Biology
University of Utah
Features of Excitable Systems
Threshold Behavior, Refractoriness
Alternans
Wenckebach Patterns
Excitable Cells – p.26/??
Imagine the
Possibilities
Cardiac Models
Mathematical Biology
University of Utah
All cardiac models are of the form
dφ
Cm
+ Iion (φ, w, [Ion]) = Iin
dt
with currents, gating and concentrations for sodium, potassium,
calcium, and chloride ions.
Excitable Cells – p.27/??