Download The Hodgkin-Huxley model

The Hodgkin-Huxley model Equivalent circuit model of a neuron  The cell membrane • separates inside and outside of cell
• defines voltage difference
• separates impenetrable to the flow of ions
• stores charge
Passive membrane proper8es RC circuits The charge flowing into one element must equal the charge flowing out C
R
At a junc8on of wires, the total current is zero Across a wire, the poten8al is the same The poten8al changes by a variable amount across a resistor symbol: Ohm: V = IR or I = Vg The poten8al changes by a fixed amount across a baEery symbol RC circuit dynamics Ohm’s law:
Capacitor: C = Q/V
C
R
Kirchhoff:
Passive membrane Res8ng poten8al: electrochemical equilibrium RC circuits of neurons inside The movement of ions is driven by the Nernst poten8al, R
C
ENernst
outside How does the baEery change the dynamics? Compact cell model (isopoten8al voltage) t Common ionic concentra8ons EK = −75 mV
ENa = +55 mV
ECl = −70 mV
ECa ≈ +150 mV
Out of equilibrium • For V > VX current flows into cell:
IX > 0
• For V < VX current flows out of cell: IX < 0
• I-V curve (current-voltage):
(
IK = γ K V − VK
)
RC circuits of neurons inside The movement of ions is driven by the Nernst poten8al, R
C
ENernst
outside Charge is carried by several different types of ions Mul8ple kinds of ions Several I-­‐V curves in parallel: New equivalent circuit: How does the cell actually behave? The ac8on poten8al Methods: Squid giant axon Methods: Voltage clamp Currents are voltage-gated, so must control voltage
Methods: Voltage clamp Capacitive current when applying voltage command
Inward and outward currents when depolarizing Separa8on of currents Two sources of voltage dependence ( )
INa V, t
( )
=
g Na V, t =
( ) (
g Na V, t i V − VNa
conductance
( )
INa V, t
V − VNa
)
driving force
( )
g K V, t =
( )
IK V, t
V − VK
Measuring conductances • INa activates, then inactivates
• IK has shallow S-shaped activation, but exponential inactivation
Temporal dynamics • Ansatz:
( )
( )
g K V, t = g K n V, t
decay:
()
(
4
n t  exp − t τ
)
Ac8va8on variable: n(V,t) • For a fixed voltage, V:
τn
()
dn t
dt
=
()
n∞ − n t
Temporal dynamics • Rise: 1.0
Activation Variable
1.0
Activation Variable
• Decay:
0.8
0.6
n
4
n
0.4
0.2
n
4
n
0.8
0.6
0.4
0.2
0.0
0.0
0
1
2
3
Time (s)
4
5
0
1
2
3
Time (s)
4
5
Equa8ons for the ac8ve currents • Conductances:
gK = gK n4
3
g Na = g Na m h
• Currents:
(
IK = g K V − VK
(
)
INa = g Na V − VNa
)
PuTng all the channels together dV
C
= − g L V − VL − g K n 4 V − VK − g Na m3 h V − VNa + I ext
dt
(
)
(
)
(
)
Current (µA/cm^2)
Results: single spike 20
0
40
Voltage
g_Na
g_K
Voltage (mV)
20
30
0
20
-20
-40
10
-60
-80
0
3
4
5
6
7
8
9
Time (ms)
10
11
12
13
Conductance (mmho/cm^2)
40
Current
Results: activation variables
20
0
Voltage (mV)
40
20
0
-20
-40
-60
-80
0
5
10
15
Time (ms)
20
Activation Variable
1.0
25
n
m
h
0.8
0.6
0.4
0.2
0.0
0
5
10
15
Time (ms)
20
25