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Introduction to Mathematical Physiology III:
The Dynamics of Excitability
J. P. Keener
Mathematics Department
University of Utah
Introduction to Mathematical Physiology III: The Dynamics of Excitability – p.1/33
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Examples of Excitable Media
• 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 RefractoryResting Excited Recovering
• Recovery What about flush toilets? – p.2/33
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Modeling Membrane Electrical
Activity
Excitable Cells – p.3/33
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Modeling Membrane Electrical
Activity
Transmembrane potential φ is regulated by transmembrane ionic
currents and capacitive currents:
Cm
dφ
dt
+ Iion(φ, w) = Iin where
dw
dt
= g(φ, w), w ∈ Rn
Excitable Cells – p.3/33
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Examples
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.4/33
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The Squid Giant Axon...
What does a squid look like? – p.5/33
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is not the Giant Squid Axon
Excitable Cells – p.6/33
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The Hodgkin-Huxley Equations
K
Cm I
ion
φ
φ
φ = φ − φ
e
i
i e
Extracellular Space
Intracellular Space
I
Na
I
l
I
Cm
dV
dt
+ INa + IK + Il = 0,
Excitable Cells – p.7/33
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The Hodgkin-Huxley Equations
K
Cm I
ion
φ
φ
φ = φ − φ
e
i
i e
Extracellular Space
Intracellular Space
I
Na
I
l
I
Cm
dV
dt
+ INa + IK + Il = 0,
with sodium current INa,
Excitable Cells – p.7/33
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The Hodgkin-Huxley Equations
K
Cm I
ion
φ
φ
φ = φ − φ
e
i
i e
Extracellular Space
Intracellular Space
I
Na
I
l
I
Cm
dV
dt
+ INa + IK + Il = 0,
with sodium current INa, potassium current IK,
Excitable Cells – p.7/33
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The Hodgkin-Huxley Equations
K
Cm I
ion
φ
φ
φ = φ − φ
e
i
i e
Extracellular Space
Intracellular Space
I
Na
I
l
I
Cm
dV
dt
+ INa + IK + Il = 0,
with sodium current INa, potassium current IK, and leak
current Il.
Excitable Cells – p.7/33
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Ionic Currents
Ionic currents are typically of the form
I = g(φ, t) Φ(φ)
Excitable Cells – p.8/33
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Ionic Currents
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.8/33
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Ionic Currents
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.8/33
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Voltage Dependent Conductance
Example: Sodium and Potassium channels - Voltage clamp
experiments
Excitable Cells – p.9/33
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K+
Channel Gating
Four independent subunits:
C
α(V )
−→
←−
β(V )
O.
so that
S0
4α(V )
−→
←−
β(V )
S1
3α(V )
−→
←−
2β(V )
S2
2α(V )
−→
←−
3β(V )
S3
α(V )
−→
←−
4β(V )
S4
One can show that x4 = n4 where
dn
dt
= α(V )(1 − n) − β(V )n
Excitable Cells – p.10/33
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Na+
Channel Gating
Two types of subunits
β
β
α
α
γ δ γ δ γ δ
S00
S21S11S01
S20S10
2α
2α
2β
2β
Conducting state is S12. Then X12 = m2h, where
dm
dt
= α(1 − m) − βm
dh
dt
= γ(1 − h) − δh
Excitable Cells – p.11/33
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Currents
Hodgkin and Huxley found that
Ik = gkn4
(φ − φK), INa = gNam3
h(φ − φNa),
where
τu(φ)
du
dt
= u∞(φ) − u, u = m, n, h
1.0
0.8
0.6
0.4
0.2
0.0
100806040200
Potential (mV)
h (v)
m (v)
n (v)
10
8
6
4
2
0
ms
80400
Potential (mV)
τh(v)
τn(v)
τm(v)
Excitable Cells – p.12/33
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Hodgkin-Huxley Equations
Cm
dV
dt
= −¯gNam3
h(V − VNa) − ¯gK n4
(V − VK) − gL(V − VL) + Iapp,
where
du
dt
= αu(1 − u) − βuu, u = m, n, h.
The specific functions α and β proposed by Hodgkin and Huxley were (in units
of ms−1
)
αm = 0.1
25 − v
exp
`25−v
10
´
− 1
, βm = 4 exp
“−v
18
”
,
αh = 0.07 exp
“−v
20
”
, βh =
1
exp(30−v
10
) + 1
,
αn = 0.01
10 − v
exp(10−v
10
) − 1
, βn = 0.125 exp
“−v
80
”
.
Excitable Cells – p.13/33
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Action Potential Dynamics
120
100
80
60
40
20
0
-20
Potential(mV)
20151050
time (ms)
1.0
0.8
0.6
0.4
0.2
0.0
Gatingvariables
20151050
time (ms)
m(t)
n(t)
h(t)
35
30
25
20
15
10
5
0
Conductance(mmho/cm
2
)
20151050
time (ms)
gNa
gK
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
h
n
Excitable Cells – p.14/33
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Fast-Slow Subsystem Dynamics
Observe that τm << τn, τh
1.0
0.8
0.6
0.4
0.2
0.0
m
100806040200
v
ve
vs
vr
solution trajectories
stable manifold of vs
dm/dt = 0
dv/dt = 0
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
m
80400
v
1
4
3
2
(Go Back) Excitable Cells – p.15/33
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Two Variable Reduction of HH Eqns
Set m = m∞(φ), and set h + n ≈ N = 0.85.
This reduces to a two variable system
C
dφ
dt
= ¯gK n4
(φ − φK ) + ¯gNam3
∞(φ)(N − n)(φ − φNa) + ¯gl(φ − φL),
τn(φ)
dn
dt
= n∞(φ) − n.
1.0
0.8
0.6
0.4
0.2
0.0
n
12080400
v
dn/dt = 0
dv/dt = 0
(Go Back) Excitable Cells – p.16/33
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Two Variable Models
Following is a summary of two variable models of excitable
media. The models described here are all of the form
dv
dt
= f(v, w) + I
dw
dt
= g(v, w)
Typically, v is a “fast” variable, while w is a “slow” variable.
v
w
V-(w) V0(w) V+(w)
f(u,w) = 0
g(v,w) = 0
W*
W*
Excitable Cells – p.17/33
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Cubic FitzHugh-Nagumo
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 < 1
2 , and “small”.
Excitable Cells – p.18/33
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FitzHugh-Nagumo Equations
0.20
0.15
0.10
0.05
0.00
-0.05
w
1.20.80.40.0-0.4
v
dw/dt = 0
dv/dt = 0
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
1.00.80.60.40.20.0
time
v(t)
w(t)
0.20
0.15
0.10
0.05
0.00
-0.05
w
0.80.40.0-0.4
v
dw/dt = 0
dv/dt = 0
0.8
0.4
0.0
-0.4
3.02.52.01.51.00.50.0
time
v(t)
w(t)
(Go Back)
Excitable Cells – p.19/33
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Mitchell-Schaeffer
Mitchell-Schaeffer two-variable model (also in a slightly different
but equivalent form by Karma)
F(v, w) =
1
τin
wv2
(1 − v) −
v
τout
,
G(v, w) =



1
τopen
(1 − w) v < vgate
− 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.20/33
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Mitchell-Schaeffer Revised
To make the Mitchell-Schaeffer look like an ionic model, take
Cm
dv
dt
= gNahm2
(VNa − v) + gK(VK − v),
τh
dh
dt
= 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.21/33
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Mitchell-Schaeffer Revised-II
0 100 200 300 400 500 600 700 800 900 1000
0
0.2
0.4
0.6
0.8
1
time
φ,h
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
φ
h
(Go Back)
Excitable Cells – p.22/33
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Morris-Lecar
This model was devised for barnacle muscle fiber.
F(v, w) = −gcam∞(v)(v − vca) − gkw(v − vk) − gl(v − vl) + Iapp
G(v, w) = φ cosh(
1
2
v − v3
v4
)(w∞(v) − w),
m∞(v) =
1
2
+
1
2
tanh(
v − v1
v2
), w∞(v) = (1 + tanh(
v − v3)
2v4
)).
0 50 100 150 200 250 300 350 400 450 500
−60
−40
−20
0
20
40
time
V
−60 −40 −20 0 20 40 60
−0.2
0
0.2
0.4
0.6
0.8
1
V
W
(Go Back)
Excitable Cells – p.23/33
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McKean
McKean suggested two piecewise linear models with
F(v, w) = f(v) − w and G(v, w) = (v − γw). For the first,
f(v) =
8
>><
>>:
−v v < α
2
v − α α
2
< v < 1+α
2
1 − v v > 1+α
2
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
v
f(v)
where 0 < α < 1
2 .
The second model suggested by McKean had
f(v) =
8
<
:
−v v < α
1 − v v > α
(-12)
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
v
f(v)
and γ = 0. Excitable Cells – p.24/33
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Barkley
A model devised to give very fast 2D computations (the code is
known as EZspiral)
F(v, w) = v(1 − v)(v −
w + b
a
),
G(v, w) = (v − w).
Excitable Cells – p.25/33
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Puschino
A piecewise linear model devised to match cardiac restitution
properties
F(v, w) = f(v) − w
G(v, w)
1
τ(v)
(v − w)
where
f(v) =



−30v, v < v1
γv − 0.12, v1 < v < v2,
−30(v − 1), v > v2
, τ(v) =



2 v < v1
16.6 v > v1
with v1 = 0.12
30+γ , v2 = 30.12
30+γ . (Go Back)
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Aliev
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)
where (v, w) = 1 + µ1
w
v+µ2
.
Reasonable parameter values are β = 0.0001 , α = 0.05, ga = 8.0,
gs = 8.0, µ1 = 0.05, µ2 = 0.3, 1 = 0.03, 2 = 0.0001.
Excitable Cells – p.27/33
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Tyson-Fife
These dynamics describe the oxidation-reduction of malonic
acid. For this system,
F(v, w) = v − v2
− (fw + φ0)
v − q
v + q
(-13)
G(v, w) = (v − w) (-13)
with typical parameter values = 0.05, q = 0.002, f = 3.5, φ0 =
0.01.
Excitable Cells – p.28/33
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APD Alternans
Action Potential Duration Restitution Curve
APDn + DIn = BCL.
where APDn = A(DIn−1) is the restitution curve. It follows that
DIn = BCL − A(DIn−1),
APD Map Animated
Excitable Cells – p.29/33
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Features of Excitable Systems
Threshold Behavior, Refractoriness
Alternans
Wenckebach Patterns
Excitable Cells – p.30/33
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Cardiac Models
All cardiac models are of the form
Cm
dφ
dt
+ Iion(φ, w, [Ion]) = Iin
with currents, gating and concentrations for sodium, potassium,
calcium, and chloride ions.
Excitable Cells – p.31/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
Excitable Cells – p.32/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
with sodium current INa,
Excitable Cells – p.32/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
with sodium current INa, time independent potassium current
IK,
Excitable Cells – p.32/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
with sodium current INa, time independent potassium current
IK, gated potassium current Ix,
Excitable Cells – p.32/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
with sodium current INa, time independent potassium current
IK, gated potassium current Ix, and (slow) calcium current Is.
Excitable Cells – p.32/33
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The Beeler-Reuter Model
Cm
dφ
dt
+ INa + IK + Ix + Is = 0,
Intracellular Space
I
Na
II
K s
I
x
Extracellular Space
with sodium current INa, time independent potassium current
IK, gated potassium current Ix, and (slow) calcium current Is.
-80
-60
-40
-20
0
20
Potential(mV)
4003002001000
Time (ms)
(Go Back) Excitable Cells – p.32/33
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Detailed Ionic Models- Luo-Rudy
Excitable Cells – p.33/33