Imagine the
Possibilities
Mathematical Biology
University of Utah
Dynamical Systems for Biology Excitability
J. P. Keener
Mathematics Department
University of Utah
Dynamical Systems for Biology – p.1/25
Imagine the
Possibilities
Examples of Excitable Media
Mathematical Biology
University of Utah
• B-Z reagent
• CICR (Calcium Induced Calcium Release)
• Nerve cells
• cardiac cells, muscle cells
• Slime mold (dictystelium discoideum)
• Forest Fires
Features of Excitability
• Threshold Behavior
• Refractoriness
• Recovery
Resting
Excited
Refractory
Recovering
What about flush toilets? – p.2/25
Imagine the
Possibilities
Mathematical Biology
University of Utah
Intracellular Calcium
What about flush toilets? – p.3/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
Dynamical Systems for Biology – p.4/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
with
JRY R Ryanodine Receptor - calcium regulated calcium channel,
Dynamical Systems for Biology – p.4/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
with
JRY R Ryanodine Receptor - calcium regulated calcium channel,
JSERCA Sarco- and Endoplasmic Reticulum Calcium ATPase,
Dynamical Systems for Biology – p.4/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
with
JRY R Ryanodine Receptor - calcium regulated calcium channel,
JSERCA Sarco- and Endoplasmic Reticulum Calcium ATPase,
JL L-type voltage regulated calcium channel,
Dynamical Systems for Biology – p.4/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
with
JRY R Ryanodine Receptor - calcium regulated calcium channel,
JSERCA Sarco- and Endoplasmic Reticulum Calcium ATPase,
JL L-type voltage regulated calcium channel,
JN CX sodium(Na++ )- Calcium eXchanger .
Dynamical Systems for Biology – p.4/25
Imagine the
Possibilities
Calcium Handling
Mathematical Biology
University of Utah
J
v dc
dt
L
JNCX
Extracellular Space
Cytosol
= JRY R − JSERCA + JL − JN CX
(Intracellular Space)
J RYR
JSERCA
Sarcoplasmic Reticulum
(Calcium stores)
with
JRY R Ryanodine Receptor - calcium regulated calcium channel,
JSERCA Sarco- and Endoplasmic Reticulum Calcium ATPase,
JL L-type voltage regulated calcium channel,
JN CX sodium(Na++ )- Calcium eXchanger .
Challenge: Determine the flux terms.
Dynamical Systems for Biology – p.4/25
++
Ca
University of Utah
Mathematical Biology
Imagine the
Possibilities
Ryanodine Receptors
Ca
++
Ca++
Ca
++
Dynamical Systems for Biology – p.5/25
Imagine the
Possibilities
Ryanodine Receptors
Mathematical Biology
University of Utah
Flux through ryanodine receptor is diffusive,
JRY R = gmax Po (c − csr )
3 is the open probability. To determine P , we
where Po = S10
o
must find S10 :
= k1 cS00 + k−2 S11 − k−1 S10 − k2 cS10
and so on.
++
Ca
S
00
"fast"
"slow"
++
Ca
S10
"slow"
dS10
dt
++
Ca
++
S01
Ca
S11
"fast"
Dynamical Systems for Biology – p.6/25
Imagine the
Possibilities
Observation
Mathematical Biology
University of Utah
If the binding sites are independent (no cooperativity), then
S10 = mh, S00 = (1 − m)h, S11 = m(1 − h), S01 = (1 − m)(1 − h),
where
dm
dh
=
φ
(c)(1
−
m)
−
ψ
(c)m,
m
m
dt
dt = φh (c)(1 − h) − ψ(c)h.
Furthermore, m is a fast variable, so can be taken to be in qss,
m = m∞ (c).
Consequently,....
Dynamical Systems for Biology – p.7/25
Imagine the
Possibilities
Calcium Dynamics
Mathematical Biology
University of Utah
dc
= (gmax Po + Jer )(ce − c) − JSERCA ,
dt
dh
= φh (c)(1 − h) − ψh (c)h,
dt
1
0.8
0.6
0.4
where
0.2
Po =
h3 f (c)
0
5
10
15
20
25
time (s)
30
35
40
45
50
1
0.8
0.6
h
JSERCA =
0
c2
Vmax K 2 +c2 ,
s
0.4
0.2
0
−2
10
−1
10
0
c (µ M)
10
Dynamical Systems for Biology – p.8/25
Imagine the
Possibilities
Mathematical Biology
University of Utah
Membrane Electrical Activity
Dynamical Systems for Biology – p.9/25
Imagine the
Possibilities
Mathematical Biology
University of Utah
Membrane Electrical Activity
Transmembrane potential φ is regulated by transmembrane ionic
currents and capacitive currents:
dw
n
Cm dφ
+
I
(φ,
w)
=
I
where
=
g(φ,
w),
w
∈
R
ion
in
dt
dt
(Reminder: This is a conservation equation.)
Dynamical Systems for Biology – p.9/25
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, etc.)
Dynamical Systems for Biology – p.10/25
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
Dynamical Systems for Biology – p.11/25
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 ,
Dynamical Systems for Biology – p.11/25
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 ,
Dynamical Systems for Biology – p.11/25
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 .
Dynamical Systems for Biology – p.11/25
Imagine the
Possibilities
Ionic Currents
Mathematical Biology
University of Utah
Ionic currents are typically of the form
I = g(φ, t) Φ(φ)
Dynamical Systems for Biology – p.12/25
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,
Dynamical Systems for Biology – p.12/25
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.
Dynamical Systems for Biology – p.12/25
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)
Dynamical Systems for Biology – p.13/25
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
Dynamical Systems for Biology – p.14/25
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
Dynamical Systems for Biology – p.15/25
Imagine the
Possibilities
Two Variable Models
Mathematical Biology
University of Utah
Following is a brief 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
Dynamical Systems for Biology – p.16/25
Imagine the
Possibilities
Mathematical Biology
University of Utah
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 < 12 , and “small”.
(Remark: This model is used these days only by
mathematicians.)
Dynamical Systems for Biology – p.17/25
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
Dynamical Systems for Biology – p.18/25
Imagine the
Possibilities
Mitchell-Schaeffer-Karma
Mathematical Biology
University of Utah
dv
Cm
dt
dh
τh
dt
= gN a hm2 (VN a − v) + gK (VK − v),
= h∞ (v) − h
where



0,
v<0


m(v) =
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 )),
Dynamical Systems for Biology – p.19/25
Imagine the
Possibilities
MSK Phase Portrait
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
Dynamical Systems for Biology – p.20/25
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
Dynamical Systems for Biology – p.21/25
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.
(-8)
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
Dynamical Systems for Biology – p.22/25
Imagine the
Possibilities
Mathematical Biology
University of Utah
Features of Excitable Systems
Threshold Behavior, Refractoriness
Alternans
Wenckebach Patterns
Dynamical Systems for Biology – p.23/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
From last time:
• Changing quantities are tracked by following the
production/destruction rates and their influx/efflux rates;
In case you forgot. – p.24/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
From last time:
• Changing quantities are tracked by following the
production/destruction rates and their influx/efflux rates;
• Because reactions can occur on many different time scales,
quasi-steady state approximations are often quite useful;
In case you forgot. – p.24/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
From last time:
• Changing quantities are tracked by following the
production/destruction rates and their influx/efflux rates;
• Because reactions can occur on many different time scales,
quasi-steady state approximations are often quite useful;
• For two variable systems, much can be learned from the
"phase portrait".
In case you forgot. – p.24/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
For excitable systems:
• Changing quantities are tracked by following their
influx/efflux rates;
Pretty much the same as before! – p.25/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
For excitable systems:
• Changing quantities are tracked by following their
influx/efflux rates;
• Because channel dynamics can occur on many different
time scales, quasi-steady state approximations are often
quite useful;
Pretty much the same as before! – p.25/25
Imagine the
Possibilities
Summary
Mathematical Biology
University of Utah
For excitable systems:
• Changing quantities are tracked by following their
influx/efflux rates;
• Because channel dynamics can occur on many different
time scales, quasi-steady state approximations are often
quite useful;
• For two variable systems, much can be learned from the
"phase portrait".
Pretty much the same as before! – p.25/25