Biophysical Foundations
BENG/BGGN 260 Neurodynamics
University of California, San Diego
Week 1
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
1 / 15
Reading Material
B. Hille, Ion Channels of Excitable Membranes, Sinauer, 2001, Ch. 1
and 10, pp. 1-21 and 309-326.
C. Koch, Biophysics of Computation, Oxford Univ. Press, 1999, Ch.
1, pp. 5-13.
P. Dayan and L. Abbott, Theoretical Neuroscience, MIT Press, 2001,
Ch. 5, pp. 153-162.
E.M. Izhikevich, Dynamical Systems in Neuroscience, MIT Press,
2007, Ch. 2, pp. 25-28.
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
2 / 15
Neurodynamics: Overview
Dendrite
Axon Terminal
Node of
Ranvier
Cell body
Axon
Myelin sheath
Nucleus
Membrane
Dynamics
Temporal Coding
Signal Propagation
Adaptation and
Learning
Action Potential
Generation
Axon Dynamics
STDP, reinforcement, ...
∼ 1 ms
∼ 10 − 100 ms
∼ 1 s − years
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
3 / 15
Nernst Potential
intracellular
extracellular
[K+ ]i
Vi
Eo −Ei
[K+ ]o
no
=
= e − kT
+
ni
[K ]i
[K+ ]o
energies
per mol
energies
per molecule
density
ratio
= e−
Boltzman
absolute
constant temperature
Vo
=e
Vm = Vi − Vo
−
=
kT
ln 10
q
| {z }
gas constant
charge per molecule
= 1.6 10−19 C per valence index
E
energy
25mV @ room T
⇒ Erest = Vm |equilibrium =
Vo −Vi
kT /q
membrane potential
z}|{
kT
q
Uo −Ui
RT
+
[K ]o
[K+ ]i
[K+ ]o
log10 +
[K ]i
ln
Boltzman
“tail”
T
Fermi
level
n
density
62mV @ room T
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
4 / 15
Nernst Potential
TABLE 1.3
Ion
Na+
K+
Ca2+
Cl−
a
Free Ion Concentrations and Equilibrium
Potentials for Mammalian Skeletal Muscle
Extracellular
Equilibrium
Intracellular
concentration concentration
potentiala
Iono
Ioni
(mM)
(mV)
(mM)
145
12
12
+67
4
155
0.026
−98
1.5
100 nM
15, 000
+129
123
4.2b
29b
−90b
Calculated from Equation 1.11 at 37◦ C (Erest =
kT
q
o
ln [Ion]
[Ion]i ).
b
Calculated assuming a −90-mV resting potential for the muscle
membrane and that Cl− ions are a equilibrium at rest.
Hille 2001, p. 17
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
5 / 15
Nernst-Planck
i
Equilibrium between diffusion and drift:
o
inside
outside
0
−
→
JD
+
[K ]i
(chemical)
x
L
(electrical)
Diffusion:
→
−
→
−
JD = qD · − ∇[K+ ]
Drift:
→
−
→
→
−
− JE = q[K+ ]µ · E = qµ[K+ ] − ∇V
[K+ ]o
x
Vo
Vm
Vi
−
→
JE
0
L
x
→
−→ −
→ −
⇒ JK+ = JD + JE = 0
+
]
+ dV
⇒ −qD d[K
dx − qµ[K ] dx = 0 ⇒
⇒
D
µ
+
]o
ln [K
= − (Vo − Vi ) = Vm
[K+ ]
i
indeed, D =
BENG/BGGN 260 Neurodynamics (UCSD)
+
D d[K ]
µ [K+ ]
= −dV
d ln [K+ ]
= Erest with
kT
q
·µ
Biophysical Foundations
equilibrium
D
µ
=
kT
q
Dissipation-fluctuation
(Einstein)
Week 1
6 / 15
Membrane Capacitance
−
→
E
+Q+
−Q
−
+
−
+
−
Vi
+
−
0
Charge
Density
ρ
−
Electric →
E
Field
Vo
capacitance = differential charge
accumulation with voltage across
insulator (membrane)
x
L
Q
A
x
−Q
A
1Q
εA
x
Vi
Potential V
Vm =
Vo
−E
LQ
εA
−
→
−→
dE
ρ
ρ
=
∇ E = , or
ε
dx
ε
→
−
→
−
dV
∇V = − E , or
= −E
dx
Q = Cm · Vm with Cm = ε ·
A
L
ε
L
≈ 0.01 F/m2
Vm
Vi
Dynamics:
BENG/BGGN 260 Neurodynamics (UCSD)
Vo
I
Cm
I or
voltage
Biophysicaldependent
Foundations
m
: Cm dV
dt = I (Vm , ...)
Week 1
7 / 15
Membrane Dynamics
→
−
→
−
→
− ∇
ln[K]
+
∇V
K+ : current density JK = −qµ[K] kT
q
⇓ approximate
arguments ... more rigorous
later
[K]o
kT
A
current IK ≈ −qµ[K] L
q ln [K]i + Vo − Vi = gK (Vm − EK )
(
Vm
average
membrane
effective cross-section
gK = qµ[K] AL
concentration thickness
gK
[K]o
Cm
EK = kT
q ln [K]i
EK
Ohmic approximation
K+ , Na+ :
Vm
gK
gNa
EK
ENa
Equilibrium: Erest = Vm =
Cm
BENG/BGGN 260 Neurodynamics (UCSD)
gK EK + gNa ENa
gK + gNa
Not quite...
Biophysical Foundations
Week 1
8 / 15
Goldman-Hodgkin-Katz
i
o
inside
outside
0
[K]i
x
L
[K](x)
[K]o
x
→
−
→
→
− −
[K]
∇V
∇[K]
JK = −qµ kT
+
q
CURRENT
DENSITY, µA/cm2
DIFFUSION
DRIFT
1-D, along x perpendicular to membrane:
kT d[K]
dV
JK = −qµ
+ [K]
q dx
dx
Vo
Vm
Vi
0
JK = constant (flow conservation)
L
x
JK
BENG/BGGN 260 Neurodynamics (UCSD)
dV
dx
= constant (assumption!)
= − VLm
kT d[K]
JK
Vm
⇒
=−
+
· [K]
q dx
qµ
L
Biophysical Foundations
Week 1
9 / 15
Goldman-Hodgkin-Katz Continued
kT d[K]
dx
JK
Vm
d[K]
= kT ⇒
=−
+
· [K] ⇒
J
V
K
m
q dx
qµ
L
− qµ + L · [K]
q
m
−JK + qµV
JK
Vm
dx
L
Vm
L [K]o
d ln −
+
· [K]
= kT ⇒ kT = ln
⇒
Vm
qµ
L
−JK + qµVm [K]i
q
q
L
JK 1 − e
−Vm /
kT
q
qµVm −V / kT
=
[K]i − e m q [K]o ⇒
L
−V / kT
qµVm [K]i − e m q [K]o
JK =
·
−V / kT
L
1−e m q
“GHK current equation”
Ohm’s limit: [K]i = [K]o = [K]
qµVm
⇒ JK =
· [K]
L
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
10 / 15
Goldman-Hodgkin-Katz Current Law
Hille 2001, Fig. 14.2, p. 447
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
11 / 15
Goldman-Hodgkin-Katz Limits
JK =
qµK Vm [K]i − e −Vm /Vt [K]o
·
L
1 − e −Vm /Vt
Vt =
kT
q
≈ 25.9mV
o
1 JK = 0 for [K]i = e −Vm /Vt [K]o , or Vm = Vt ln [K]
[K]i = EK (OK!)
Vm 2 JK
Vt
Vm −
Vt
qµK
L [K]i Vm
qµK
L [K]o Vm
= PK [K]i Vm
= PK [K]o Vm
PK = qµLK
“permeability”
⇒ Current asymptotes to a linear conductance limited by
the sourcing concentration at active terminal.
3 JK ≈ gK (Vm − EK ) for JK ≈ 0
BENG/BGGN 260 Neurodynamics (UCSD)
(Vm ≈ EK )
Biophysical Foundations
?
Week 1
12 / 15
GHK Ohmic Approximation
V
E
M
K
− kT
− kT
−
kT [K]o
Vm ≈ EK =
ln
⇒e q =e q e
q
[K]i
qµEK
⇒ JK ≈
·
L
[K]i
≈
[K]o
1−
Vm − EK
!
kT
q
[K]i
−EK
Z
[K]
(1A − VmkT
) ·
[[K]
o
[K]
Zi − o
q
1−
[K]i
[K]o (1
−
or: JK ≈ gK (Vm − EK )
with
VM −EK
kT
q
gK
|{z}
channel
conductance
BENG/BGGN 260 Neurodynamics (UCSD)
=
qµ
L
|{z}
permeability
Vm −E
K
kT
q
)
[K]o
[K]i [K]o
· ln
·
[K]i [K]o − [K]i
|
{z
}
average concentration
Biophysical Foundations
Week 1
13 / 15
Goldman-Hodgkin-Katz Equilibrium
qµK Vm
JK =
L
JNa
qµNa Vm
=
L
[K]i − e
Vm
− kT
/q
1−e
[K]o
Vm
− kT
/q
[Na]i − e
Vm
− kT
/q
1−e
[Na]o
Vm
− kT
/q
Equilibrium : JK + JNa ≡ 0, neglecting common terms :
− Vm
− Vm
µK [K]i − e kT /q [K]o + µNa [Na]i − e kT /q [Na]o ≡ 0
Erest = Vm =
kT µK [K]o + µNa [Na]o
ln
q
µK [K]i + µNa [Na]i
usually expressed in terms of permeabilities:
PK =
BENG/BGGN 260 Neurodynamics (UCSD)
qµK
qµNa
and PNa =
L
L
Biophysical Foundations
Week 1
14 / 15
GHK Neuromorphism
(for the silicon enthusiast)
LIPID BILAYER :
(K+ )
[K]i
G
SILICON MOSFET :
(pMOS)*
S
source
neuromorphisms :
(imperfect)
(
PK [K]i for Vm Vt
[K]o JK ∼
=
PK [K]o for Vm −Vt

J e κ VVgt · e VVst for V V
SiO2
t
0
ds
Jp ∼
=
Vg
Vd
κ
J e Vt · e Vt for V −V
D
gate
channel
(holes)
Si
0
drain
ds
t
V
PK → J 0 e
[K]i → e
κ Vg
t
s
−V
V
t
controlled by Vg , gate voltage
controlled by Vs , source voltage
V
[K]o
Vm
Vi
Vo
→
→
→
→
− Vd
e t
controlled by Vd , drain voltage
Vds = Vd − Vs , i.e. :
−Vs
−Vd
* C.A. Mead, Analog VLSI and Neural Systems, Addison-Wesley, 1989
BENG/BGGN 260 Neurodynamics (UCSD)
Biophysical Foundations
Week 1
15 / 15