Lecture 4: Diffusion and the Fokker-Planck equation

1
Lecture 4 Diffusion and the Fokker-Planck
equation

• Outline
• intuitive treatment
• Diffusion as flow down a concentration gradient
• Drift current and Fokker-Planck equation

2
Lecture 4 Diffusion and the Fokker-Planck
equation

• Outline
• intuitive treatment
• Diffusion as flow down a concentration gradient
• Drift current and Fokker-Planck equation
• examples
• No current equilibrium, Einstein relation
• Constant current, out of equilibrium

3
Lecture 4 Diffusion and the Fokker-Planck
equation

• Outline
• intuitive treatment
• Diffusion as flow down a concentration gradient
• Drift current and Fokker-Planck equation
• examples
• No current equilibrium, Einstein relation
• Constant current, out of equilibrium
• Goldman-Hodgkin-Katz equation
• Kramers escape over an energy barrier

4
Lecture 4 Diffusion and the Fokker-Planck
equation

• Outline
• intuitive treatment
• Diffusion as flow down a concentration gradient
• Drift current and Fokker-Planck equation
• examples
• No current equilibrium, Einstein relation
• Constant current, out of equilibrium
• Goldman-Hodgkin-Katz equation
• Kramers escape over an energy barrier
• derivation from master equation

5
Diffusion
Ficks law
6
Diffusion
Ficks law
cf Ohms law
7
Diffusion
Ficks law
cf Ohms law
conservation
8
Diffusion
Ficks law
cf Ohms law
conservation
gt
9
Diffusion
Ficks law
cf Ohms law
conservation
gt
diffusion equation
10
Diffusion
Ficks law
cf Ohms law
conservation
gt
diffusion equation
initial condition
11
Diffusion
Ficks law
cf Ohms law
conservation
gt
diffusion equation
initial condition
solution
12
Diffusion
Ficks law
cf Ohms law
conservation
gt
diffusion equation
initial condition
solution
http//www.nbi.dk/hertz/noisecourse/gaussspread.m
13
Drift current and Fokker-Planck equation
Drift (convective) current
14
Drift current and Fokker-Planck equation
Drift (convective) current
Combining drift and diffusion Fokker-Planck
equation
15
Drift current and Fokker-Planck equation
Drift (convective) current
Combining drift and diffusion Fokker-Planck
equation
16
Drift current and Fokker-Planck equation
Drift (convective) current
Combining drift and diffusion Fokker-Planck
equation Slightly more generally, D can
depend on x
17
Drift current and Fokker-Planck equation
Drift (convective) current
Combining drift and diffusion Fokker-Planck
equation Slightly more generally, D can
depend on x gt
18
Drift current and Fokker-Planck equation
Drift (convective) current
Combining drift and diffusion Fokker-Planck
equation Slightly more generally, D can
depend on x gt
First term alone describes probability cloud
moving with velocity u(x) Second term alone
describes diffusively spreading probability cloud
19
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m

20
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)

21
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)
Stationary case
22
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)
Stationary case Gas of Brownian particles in
gravitational field u0 µF -µmg
23
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)
Stationary case Gas of Brownian particles in
gravitational field u0 µF -µmg µ
mobility
24
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)
Stationary case Gas of Brownian particles in
gravitational field u0 µF -µmg µ
mobility Boundary conditions (bottom of
container, stationarity)
25
Examples constant drift velocity
http//www.nbi.dk/hertz/noisecourse/gaussspreadmo
ve.m
Solution (with no boundaries)
Stationary case Gas of Brownian particles in
gravitational field u0 µF -µmg µ
mobility Boundary conditions (bottom of
container, stationarity)
drift and diffusion currents cancel
26
Einstein relation
FP equation
27
Einstein relation
FP equation Solution
28
Einstein relation
FP equation Solution But from equilibrium
stat mech we know
29
Einstein relation
FP equation Solution But from equilibrium
stat mech we know So D µT
30
Einstein relation
FP equation Solution But from equilibrium
stat mech we know So D µT
Einstein relation
31
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel
Pumps maintain different inside and outside
concentrations of ions
32
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell

33
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field

34
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field
• Question At a given Vm, what current flows
  through the channel?

35
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field
• Question At a given Vm, what current flows
  through the channel?

x0
xd
x
inside
outside
36
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field
• Question At a given Vm, what current flows
  through the channel?

x0
xd
Vout 0
x
inside
outside
V(x)
Vm
37
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field
• Question At a given Vm, what current flows
  through the channel?

x0
xd
?out
?in
Vout 0
x
inside
outside
V(x)
Vm
38
Constant current Goldman-Hodgkin-Katz model of
an (open) ion channel

• Pumps maintain different inside and outside
  concentrations of ions
• Voltage diff (membrane potential) between
  inside and outside of cell
• Can vary membrane potential experimentally by
  adding external field
• Question At a given Vm, what current flows
  through the channel?

x0
xd
?out
?
?in
Vout 0
x
inside
outside
V(x)
Vm
39
Reversal potential
If there is no current, equilibrium gt
?in/?outexp(-ßV)
40
Reversal potential
If there is no current, equilibrium gt
?in/?outexp(-ßV) This defines the reversal
potential at which J 0.
41
Reversal potential
If there is no current, equilibrium gt
?in/?outexp(-ßV) This defines the reversal
potential at which J 0. For Ca, ?outgtgt
?in gt Vr gtgt 0
42
GHK model (2)
Vmlt 0 both diffusive current and drift current
flow in
x0
xd
?out
?
?in
Vout 0
x
inside
outside
V(x)
Vm
43
GHK model (2)
Vmlt 0 both diffusive current and drift current
flow in Vm 0 diffusive current flows in, no
drift current
x0
xd
?out
?
?in
V(x)
Vout 0
x
inside
outside
44
GHK model (2)
Vmlt 0 both diffusive current and drift current
flow in Vm 0 diffusive current flows in, no
drift current Vmgt 0 diffusive current flows in,
drift current flows out
x0
xd
?out
?
Vm
V(x)
?in
Vout 0
x
inside
outside
45
GHK model (2)
Vmlt 0 both diffusive current and drift current
flow in Vm 0 diffusive current flows in, no
drift current Vmgt 0 diffusive current flows in,
drift current flows out At Vm Vr they cancel
x0
xd
?out
?
Vm
V(x)
?in
Vout 0
x
inside
outside
46
GHK model (2)
Vmlt 0 both diffusive current and drift current
flow in Vm 0 diffusive current flows in, no
drift current Vmgt 0 diffusive current flows in,
drift current flows out At Vm Vr they cancel
x0
xd
?out
?
Vm
V(x)
?in
Vout 0
x
inside
outside
47
Steady-state FP equation
48
Steady-state FP equation
49
Steady-state FP equation
Use Einstein relation
50
Steady-state FP equation
Use Einstein relation
51
Steady-state FP equation
Use Einstein relation
Solution
52
Steady-state FP equation
Use Einstein relation
Solution
We are given ?(0) and ?(d). Use this to solve for
J
53
Steady-state FP equation
Use Einstein relation
Solution
We are given ?(0) and ?(d). Use this to solve for
J
54
Steady-state FP equation
Use Einstein relation
Solution
We are given ?(0) and ?(d). Use this to solve for
J
55
Steady-state FP equation
Use Einstein relation
Solution
We are given ?(0) and ?(d). Use this to solve for
J
56
Steady-state FP equation
Use Einstein relation
Solution
We are given ?(0) and ?(d). Use this to solve for
J
57
GHK current, another way
Start from
58
GHK current, another way
Start from
59
GHK current, another way
Start from
60
GHK current, another way
Start from Note
61
GHK current, another way
Start from Note Integrate from 0 to
d
62
GHK current, another way
Start from Note Integrate from 0 to
d
63
GHK current, another way
Start from Note Integrate from 0 to
d
64
GHK current, another way
Start from Note Integrate from 0 to
d
65
GHK current, another way
Start from Note Integrate from 0 to
d (as before)
66
GHK current, another way
Start from Note Integrate from 0 to
d (as before) Note J 0 at
Vm Vr
67
GHK current is nonlinear
(using z, Vr for Ca)
J
V
68
GHK current is nonlinear
(using z, Vr for Ca)
J
V
69
GHK current is nonlinear
(using z, Vr for Ca)
J
V
70
GHK current is nonlinear
(using z, Vr for Ca)
J
V
71
GHK current is nonlinear
(using z, Vr for Ca)
J
V
72
GHK current is nonlinear
(using z, Vr for Ca)
J
V
73
GHK current is nonlinear
(using z, Vr for Ca)
J
V
74
Kramers escape
Rate of escape from a potential well due to
thermal fluctuations
P2(x)
P1(x)
V1(x)
V2(x)
www.nbi.dk/hertz/noisecourse/demos/Pseq.mat www.nb
i.dk/hertz/noisecourse/demos/runseq.m
75
Kramers escape (2)
V(x)
a b c
76
Kramers escape (2)
V(x)
J ?
a b c
77
Kramers escape (2)
V(x)
J ?
a b c
Basic assumption (V(b) V(a))/T gtgt 1
78
Fokker-Planck equation
Conservation (continuity)
79
Fokker-Planck equation
Conservation (continuity)
80
Fokker-Planck equation
Conservation (continuity) Use Einstein
relation
81
Fokker-Planck equation
Conservation (continuity) Use Einstein
relation Current
82
Fokker-Planck equation
Conservation (continuity) Use Einstein
relation Current
If equilibrium, J 0,
83
Fokker-Planck equation
Conservation (continuity) Use Einstein
relation Current
If equilibrium, J 0,
84
Fokker-Planck equation
Conservation (continuity) Use Einstein
relation Current
If equilibrium, J 0, Here almost
equilibrium, so use this P(x)
85
Calculating the current
(J is constant)
86
Calculating the current
(J is constant)
integrate
87
Calculating the current
(J is constant) (P(c) very small)
integrate
88
Calculating the current
(J is constant) (P(c) very small)
integrate
89
Calculating the current
(J is constant) (P(c) very small)
integrate
If p is probability to be in the well, J pr,
where r escape rate
90
Calculating the current
(J is constant) (P(c) very small)
integrate
If p is probability to be in the well, J pr,
where r escape rate
91
Calculating the current
(J is constant) (P(c) very small)
integrate
If p is probability to be in the well, J pr,
where r escape rate
92
Calculating the current
(J is constant) (P(c) very small)
integrate
If p is probability to be in the well, J pr,
where r escape rate
93
calculating escape rate
In integral
integrand is peaked near x b
94
calculating escape rate
In integral
integrand is peaked near x b
95
calculating escape rate
In integral
integrand is peaked near x b
96
calculating escape rate
In integral
integrand is peaked near x b
97
calculating escape rate
In integral
integrand is peaked near x b
98
calculating escape rate
In integral
integrand is peaked near x b
99
calculating escape rate
In integral
integrand is peaked near x b
100
calculating escape rate
In integral
integrand is peaked near x b
________
101
More about drift current
Notice If u(x) is not constant, the probability
cloud can shrink or spread even if there is no
diffusion
102
More about drift current
Notice If u(x) is not constant, the probability
cloud can shrink or spread even if there is no
diffusion (like density of cars on a road where
the speed limit varies)
103
More about drift current
Notice If u(x) is not constant, the probability
cloud can shrink or spread even if there is no
diffusion (like density of cars on a road where
the speed limit varies)
Demo initial P Gaussian centered at x 2 u(x)
.00015x
http//www.nbi.dk/hertz/noisecourse/driftmovie.m
104
Derivation from master equation
105
Derivation from master equation
(1st argument of r starting point 2nd argument
step size)
106
Derivation from master equation
(1st argument of r starting point 2nd argument
step size)
107
Derivation from master equation
(1st argument of r starting point 2nd argument
step size)
108
Derivation from master equation
(1st argument of r starting point 2nd argument
step size)
Small steps assumption r(xs) falls rapidly to
zero with increasing s on the scale on which
it varies with x or the scale on which P varies
with x.
109
Derivation from master equation
(1st argument of r starting point 2nd argument
step size)
Small steps assumption r(xs) falls rapidly to
zero with increasing s on the scale on which
it varies with x or the scale on which P varies
with x.
x
s
110
Derivation from master equation (2)
expand
111
Derivation from master equation (2)
expand
112
Derivation from master equation (2)
expand
113
Derivation from master equation (2)
expand
114
Derivation from master equation (2)
expand
Kramers-Moyal expansion
115
Derivation from master equation (2)
expand
Kramers-Moyal expansion Fokker-Planck eqn if drop
terms of order gt2
116
Derivation from master equation (2)
expand
Kramers-Moyal expansion Fokker-Planck eqn if drop
terms of order gt2
117
Derivation from master equation (2)
expand
Kramers-Moyal expansion Fokker-Planck eqn if drop
terms of order gt2
rn(x)?t nth moment of distribution of step size
in time ?t
118
Derivation from master equation (2)
expand
Kramers-Moyal expansion Fokker-Planck eqn if drop
terms of order gt2
rn(x)?t nth moment of distribution of step size
in time ?t