Hodgkin and Huxley
Taken from: http://icwww.epfl.ch/~gerstner/SPNM/node14.html
1
Hodgkin Huxley Model:
charging
current
Ion
channels
I inj (t )  I C (t )   I k (t )
k
dVm
C
  I k (t )  I inj (t )
dt
k
P
with
Q
C
u
and
du
dV
IC  C
C
dt
dt
I x  g x (Vm  Vx )
I k = gNa( Vm à VNa) + gK ( Vm à VK ) + gL ( Vm à VL )
k
General Membrane Equation (a very important Equation, used everywhere!)
m
2
=
à
(
à
)
à
(
à
)
à
(
à
)
+
C dV
g
V
V
g
V
V
g
V
V
I i nj
Na
m
Na
K
m
K
L
m
L
dt
Hodgkin Huxley Model:
Introducing time-dependence so as to get an Action Potential modelled
P
I k = gNaf 1( t )( Vm à VNa) + gK f 2( t )( Vm à VK ) + gL f 3( t )( Vm à VL )
k
Following Hodgkin and Huxley (using rising AND falling functions):
I
 g Na m h(Vm  VNa )  g K n (Vm  VK )  g L (Vm  VL )
3
k
4
k
Resulting time-dependent Membrane Equation
dVm
C
  g Na m 3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )3  I inj
dt
V mV
Hodgkin-Huxley Model: Action Potential / Threshold
40
20
0
20
40
60
80
Iinj = 0.42 nA
V mV
0
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.43 nA
0
V mV
5
Short, weak current pulses depolarize the cell only a
little.
5
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.44 nA
An action potential is elicited when crossing the
threshold.
0
5
10
t ms
15
20
Action Potential
5
Action Potential
6
Hodgkin Huxley Model:
dVm
C
  g Na m 3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
• voltage dependent gating variables
m   m (u )(1  m)   m (u )m
n   n (u )(1  n)   n (u )n
h   (u )(1  h)   (u )h
h
h
asymptotic
value
1
x  
[ x  x0 (u )]
 x (u )
time
constant
with
 x (u)
x0 (u ) 
[ x (u )   x (u )]
(for the giant squid axon)
 x (u )  [ x (u )   x (u )]1
7
1
x  
[ x  x0 (u )]
 x (u )
Solution:
x = exp(à üt) + x 0
Derivative
1
t
ç
=
à
exp(à
x
ü
ü)
=
à
1
ü
exp(à üt) + x 0 à x 0
8
dVm
C
  g Na m 3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
1
x  
[ x  x0 (u )]
 x (u )
action potential
• If u increases, m increases -> Na+ ions flow into the cell
• at high u, Na+ conductance shuts off because of h
• h reacts slower than m to the voltage increase
• K+ conductance, determined by n, slowly increases with increased u
9
Hodgkin Huxley Model:
dVm
C
  g Na m 3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
Let’s see it in action!
HHsim (seminar thema!)
10
Your neurons surely don‘t like this guy!
11
Voltage clamp method
• developed 1949 by Kenneth Cole
• used in the 1950s by Alan Hodgkin and Andrew Huxley to measure
ion current while maintaining specific membrane potentials
12
Voltage clamp method
Large depolarization
Small depolarization
Ic: capacity current
Il: leakage current
13
The sodium channel (patch clamp)
14
The sodium channel
15
V mV
Hodgkin-Huxley Model: Firing Latency
40
20
0
20
40
60
80
Iinj = 0.45 nA
V mV
0
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.65 nA
0
V mV
5
5
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.85 nA
0
5
10
t ms
15
20
A higher current reduces the time until an action
potential is elicited.
V mV
Hodgkin-Huxley Model: Firing Latency
40
20
0
20
40
60
80
Iinj = 0.45 nA
V mV
0
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.65 nA
0
V mV
5
5
40
20
0
20
40
60
80
10
t ms
15
20
Iinj = 0.85 nA
0
5
10
t ms
15
20
A higher current reduces the time until an action
potential is elicited.
Function of the sodium channel
18
V mV
Hodgkin-Huxley Model: Refractory Period
40
20
0
20
40
60
80
Iinj = 0.5 nA
V mV
0
10
40
20
0
20
40
60
80
15 20
t ms
25
30
Iinj = 0.5 nA
0
V mV
5
5
10
40
20
0
20
40
60
80
15 20
t ms
25
30
Iinj = 0.5 nA
0
5
10
15 20
t ms
25
30
Longer current pulses will lead to more action
potentials.
However, the next action potential can only occur
after a “waiting period” during which the cell return
to its normal state.
This “waiting period” is called the refractory period.
V mV
Hodgkin-Huxley Model: Firing Rate
40
20
0
20
40
60
80
Iinj = 0.2 nA
V mV
0
40
60
t ms
80
100
Iinj = 0.3 nA
40
20
0
20
40
60
80
When injecting current for longer durations an
increase in current strength will lead to an increase of
the number of action potentials per time.
Thus, the firing rate of the neuron increases.
The maximum firing rate is limited by the absolute
refractory period.
0
V mV
20
20
40
20
0
20
40
60
80
40
60
t ms
80
100
Iinj = 0.6 nA
0
20
40 60
t ms
80
100
Varying firing properties
Rhythmic burst
in the absence of synaptic inputs
???
Influence of
steady hyperpolarization
Influence of the
neurotransmitter Acetylcholin
21
Action Potential / Shapes:
Squid Giant Axon
Rat - Muscle
Cat - Heart
22
Propagation of an Action Potential:
Action potentials propagate without being
diminished (active process).
Open channels per
mm2 membrane area
Local current loops
All sites along a nerve fiber will be
depolarized until the potential passes
threshold. As soon as this happens a new
AP will be elicited at some distance to the
old one.
Main current flow is across the fiber.
Time
Distance
23
Structure of a Neuron:
At the dendrite the incoming
signals arrive (incoming currents)
At the soma current
are finally integrated.
At the axon hillock action potential
are generated if the potential crosses the
membrane threshold
The axon transmits (transports) the
action potential to distant sites
CNS
At the synapses are the
outgoing signals transmitted
onto the dendrites of the
target neurons
Systems
Areas
Local Nets
Neurons
Synapses
24
Molekules
Chemical synapse
Neurotransmitter
Receptors
25
Neurotransmitters
Chemicals (amino acids, peptides, monoamines) that
transmit, amplify and modulate signals between neuron and
another cell.
Cause either excitatory or inhibitory PSPs.
Glutamate – excitatory transmitter
GABA, glycine – inhibitory transmitter
26
Synaptic Transmission:
Synapses are used to transmit signals from the axon of a source to the dendrite of a target
neuron.
There are electrical (rare) and chemical synapses (very common)
At an electrical synapse we have direct electrical coupling (e.g., heart muscle cells).
At a chemical synapse a chemical substance (transmitter) is used to transport the signal.
Electrical synapses operate bi-directional and are extremely fast, chem. syn. operate unidirectional and are slower.
Chemical synapses can be excitatory or inhibitory
they can enhance or reduce the signal
change their synaptic strength (this is what happens during learning).
27
Structure of a Chemical Synapse:
Axon
Synaptic cleft
Motor Endplate
(Frog muscle)
Active
zone
vesicles
Muscle fiber
Presynaptic
membrane
Postsynaptic
membrane
Synaptic cleft
28
What happens at a chemical synapse during signal transmission:
Pre-synaptic
action potential
The pre-synaptic action potential depolarises the
axon terminals and Ca2+-channels open.
Ca2+ enters the pre-synaptic cell by which the
transmitter vesicles are forced to open and release
the transmitter.
Concentration of
transmitter
in the synaptic cleft
Thereby the concentration of transmitter increases
in the synaptic cleft and transmitter diffuses to the
postsynaptic membrane.
Post-synaptic
action potential
Transmitter sensitive channels at the postsyaptic
membrane open. Na+ and Ca2+ enter, K+ leaves the
cell. An excitatory postsynaptic current (EPSC) is
thereby generated which leads to an excitatory
postsynaptic potential (EPSP).
29
Neurotransmitters and their (main) Actions:
Transmitter
Channel-typ
Ion-current
Action
Acetylecholin
nicotin. Receptor
Na+ and K+
excitatory
Glutamate
AMPA / Kainate
Na+ and K+
excitatory
GABA
GABAA-Receptor Cl-
inhibitory
Cl-
inhibitory
Glycine
Acetylecholin
muscarin. Rec.
-
metabotropic, Ca2+ Release
Glutamate
NMDA
Na+, K+, Ca2+
voltage dependent
blocked at resting potential
30
Synaptic Plasticity
31
Structure of a Neuron:
At the dendrite the incoming
signals arrive (incoming currents)
At the soma current
are finally integrated.
At the axon hillock action potential
are generated if the potential crosses the
membrane threshold
The axon transmits (transports) the
action potential to distant sites
CNS
At the synapses are the
outgoing signals transmitted
onto the dendrites of the
target neurons
Systems
Areas
Local Nets
Neurons
Synapses
32
Molekules
Chemical synapse
Neurotransmitter
Receptors
33
Neurotransmitters
Chemicals (amino acids, peptides, monoamines) that
transmit, amplify and modulate signals between neuron and
another cell.
Cause either excitatory or inhibitory PSPs.
Glutamate – excitatory transmitter
GABA, glycine – inhibitory transmitter
34
Synaptic Transmission:
Synapses are used to transmit signals from the axon of a source to the dendrite of a target
neuron.
There are electrical (rare) and chemical synapses (very common)
At an electrical synapse we have direct electrical coupling (e.g., heart muscle cells).
At a chemical synapse a chemical substance (transmitter) is used to transport the signal.
Electrical synapses operate bi-directional and are extremely fast, chem. syn. operate unidirectional and are slower.
Chemical synapses can be excitatory or inhibitory
they can enhance or reduce the signal
change their synaptic strength (this is what happens during learning).
35
Simple Computational Operations that can be Performed with Neurons
The system to be considered first:
One Neuron receiving
2 Synapses.
Input 1
Input 2
Soma
=
CPU
Axon = Output
What are the computations that can be performed with such a simple system ?
First things first: Basic Operations
Arithmetical:
Locigal
+ Summation
- Subtraction
. Multiplication
/ Division
AND
OR
NOT, etc.
More Compex Operations
Calculus:
Integration
dx/dt Differentiation
Linear Algebra:
Vector Operations
36
y=Ax Matrix Operations
Believe it or not: With a single neuron and 2 input you can compute all
alrithmetic, many logic and some of the more complex operations !
Required Requisits:
Kei
ne
Ko
1) Resting Potential (ca. -70 mv, constant)
2) Firing Threshold
3) Equilibrium Potential of different ions
4) Time-constants of the ion-channels.
gni
ti
Summation
on
ohn
Transmitter release at a synapse
e A leads to an excitatory postsynaptic
ddi are opening.
potential (EPSP) because ion channels
tion
mV
EPSP
rest.
pot.
t
37
Necessary conditions for optimal summation:
1) synapses have to be closely adjacent
2) pre-synaptic signals have to arrive simultaneously
3) resting potential and reversal potential(s) have to be very different.
A
B
mV
EPSP
r e s
= EPSP + EPSP
A
B
A
rest.
pot.
t
B
The little “shoulder” shows that the
EPSPs were not truely simultaneous.
Consider 1:
If the synapses are far from each other the amplitude will be
less at the first summing point. It will then further decay
until reaching the soma.
Summation
point
A
mV
simultaneous
inputs !
B
AB
Soma
Dendrite
Spatial Summation
EPSP
r e s
< EPSP + EPSP
A
rest.
pot.
t
38
B
A
B
Direction of signal propagation
The signal propagates essentially
in all directions. The direction
towards the soma is (usually) the
one which is functionally relevant.
Soma
A more complicated situation
1) The signal from B arrives later
at the summation point because
B is farther from it than A.
2) The signal from B is smaller at
the summation point (same
reason).
A
Dendrite
Soma
B
incomplete spatial summation
EPSPres= a EPSPA + b EPSPB ; b<a<1.0
mV
rest.
pot.
A
B
How will the signal
look like at the
summation point ?
39
t
Consider 2: If the signals are not simultaneous then the sum will be smaller
mV
A
A
B
rest.
pot.
B
t
The early signal (A) facilitates the later signal (B). Together the firing threshold
might be reached but not alone.
Temporal Summation
mV
If the difference in arrival times is
too large, temporal summation
does not occur anymore !
A
B
rest.
pot.
t
40
Consider 3: If the equilibrium potential of the involved ions is close to the
resting potential then only incomplete summation is observed.
Even a plateau is possible.
mV
A
A
B
rest.
pot.
B
t
The potential of the involved ions can never exceed their own
equilibrium potential. (“Clipping”).
Conclusion: Summing with neurons is a rather complex process.
Spatial and temporal phenomena and the potential levels will
influence the result of the “summation” substantially.
41
The same conditions apply as for summation. Then one can regard
an IPSP as a sign-inverted EPSP. “Summation” becomes “Subtraction”.
42
Special case: “shunting inhibition”
The equilibrium potential of the ions “B” is very close (”indentical”)
to the resting potential ! (A is excitatory as usual.)
A
B
mV
mV
EPSP
rest.
pot.
rest.
pot.
t
(almost)
change
does no
thepotential
membrane
How
potential change ?
t
When the purple channels are opening (almost) no ion current is obsered
and thus the potential stays (almost) the same.
-
Cl
open channel
-
Cl
This case is commonly observed for the
Chloride ion.
What is the functional significance of this
behavior ?
43
Functional significance of “shunting inhibition”
Consider the case were Cl-channels are already open when the
excitatory channels A are opening and an EPSP is elicited there.
Cl
to the
soma
A
rite
d
n
De
to the peripheral
dendrite
The EPSP travels to the soma. The membrane potential will be depolarized
along the way.
A Cl-current
is the
consequence.
The
pot. fluctuation (viz.
What
happens
at location
Cl positive
with themembrane
relation between
EPSP) willmembrane
be immediately
compensated
for. Thus,
at the open
Cl channels
potential
and Cl-equilibrium
potential
?
no more depolarization is observed. The EPSP is electrically shunted !
44
45
The physiological transmitter is Glutamate (Glu).
out
in
out
in
46
47
48