Post on 11-Jan-2016
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Hodgkin and Huxley
Taken from: http://icwww.epfl.ch/~gerstner/SPNM/node14.html
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Hodgkin Huxley Model:
)()( tItIdt
dVC inj
kk
m
)()()( tItItIk
kCinj withu
QC and
dt
dVC
dt
duCIC
charging current
Ionchannels
)( xmxx VVgI
P
kI k=gNa(Vmà VNa) +gK (Vmà VK ) +gL(Vmà VL)
C dtdVm= à gNa(Vmà VNa) à gK (Vmà VK ) à gL(Vmà VL) + I inj
General Membrane Equation (a very important Equation, used everywhere!)
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Hodgkin Huxley Model:
)()()( 43LmLKmKNamNa
kk VVgVVngVVhmgI
injLmLKmKNamNam IVVgVVngVVhmgdt
dVC )()()( 43
P
kI k=gNaf 1(t)(Vmà VNa) +gK f 2(t)(Vmà VK )+gLf 3(t)(Vmà VL)
Introducing time-dependence so as to get an Action Potential modelled
Following Hodgkin and Huxley (using rising AND falling functions):
Resulting time-dependent Membrane Equation
Hodgkin-Huxley Model: Action Potential / Threshold
Short, weak current pulses depolarize the cell only a little.
An action potential is elicited when crossing the threshold.
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.42 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.43 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.44 nA
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Action Potential
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Action Potential
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Hodgkin Huxley Model:
injLmLKmKNamNam IVVgVVngVVhmgdt
dVC )()()( 43
huhuh
nunun
mumum
hh
nn
mm
)()1)((
)()1)((
)()1)((
(for the giant squid axon)
)]([)(
10 uxx
ux
x
1
0
)]()([)(
)]()([)(
uuu
uuux
xxx
xx
x
with
• voltage dependent gating variables
time constant
asymptotic value
(u)
8
)]([)(
10 uxx
ux
x
Solution:
x = exp(à üt) +x0
xç= à ü1exp(à ü
t)
Derivative
= à ü1 exp(à ü
t) +x0à x0
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injLmLKmKNamNam IVVgVVngVVhmgdt
dVC )()()( 43
• 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
)]([)(
10 uxx
ux
x
action potential
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Hodgkin Huxley Model:
injLmLKmKNamNam IVVgVVngVVhmgdt
dVC )()()( 43
Let’s see it in action!
HHsim (seminar thema!)
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Your neurons surely don‘t like this guy!
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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
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Voltage clamp method
Small depolarization
Ic: capacity currentIl: leakage current
Large depolarization
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The sodium channel (patch clamp)
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The sodium channel
Hodgkin-Huxley Model: Firing Latency
A higher current reduces the time until an action potential is elicited.
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.45 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.65 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.85 nA
Hodgkin-Huxley Model: Firing Latency
A higher current reduces the time until an action potential is elicited.
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.45 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.65 nA
0 5 10 15 20t ms
806040200
2040
VV
m
Iinj = 0.85 nA
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Function of the sodium channel
Hodgkin-Huxley Model: Refractory Period
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.
0 5 10 15 20 25 30t ms
806040200
2040
VV
m
Iinj = 0.5 nA
0 5 10 15 20 25 30t ms
806040200
2040
VV
m
Iinj = 0.5 nA
0 5 10 15 20 25 30t ms
806040200
2040
VV
m
Iinj = 0.5 nA
Hodgkin-Huxley Model: Firing Rate
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 20 40 60 80 100t ms
806040200
2040
VV
m
Iinj = 0.2 nA
0 20 40 60 80 100t ms
806040200
2040
VV
m
Iinj = 0.3 nA
0 20 40 60 80 100t ms
806040200
2040
VV
m
Iinj = 0.6 nA
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Varying firing properties
???
Influence of steady hyperpolarization
Rhythmic burst in the absence of synaptic inputs
Influence of the neurotransmitter Acetylcholin
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Action Potential / Shapes:
Squid Giant Axon Rat - Muscle Cat - Heart
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Propagation of an Action Potential:
Action potentials propagate without being
diminished (active process).
Distance
Time
Local current loops
Open channels per
mm
2 mem
brane area
Action potentials propagate without being
diminished (active process).
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.
Action potentials propagate without being
diminished (active process).
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.
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At the dendrite the incomingsignals arrive (incoming currents)
Molekules
Synapses
Neurons
Local Nets
Areas
Systems
CNS
At the soma currentare finally integrated.
At the axon hillock action potentialare generated if the potential crosses the membrane threshold
The axon transmits (transports) theaction potential to distant sites
At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons
Structure of a Neuron:
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Chemical synapse
NeurotransmitterReceptors
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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
27
Synaptic Transmission:
Synapses are used to transmit signals from the axon of a source to the dendrite of a target
neuron.
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)
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).
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.
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 uni-
directional and are slower.
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 uni-
directional 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).
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Structure of a Chemical Synapse:
Motor Endplate
(Frog muscle)
Axon
Synaptic cleft
Activezone
vesicles
Muscle fiber
Presynapticmembrane
Postsynapticmembrane
Synaptic cleft
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What happens at a chemical synapse during signal transmission:
The pre-synaptic action potential depolarises the
axon terminals and Ca2+-channels open.Pre-synapticaction potential
Concentration oftransmitterin the synaptic cleft
Post-synapticaction 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.
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.
Thereby the concentration of transmitter increases
in the synaptic cleft and transmitter diffuses to the
postsynaptic membrane.
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.
Thereby the concentration of transmitter increases
in the synaptic cleft and transmitter diffuses to the
postsynaptic membrane.
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).
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Neurotransmitters and their (main) Actions:
Transmitter Channel-typ Ion-current ActionTransmitter Channel-typ Ion-current Action
Acetylecholin nicotin. Receptor Na+ and K+ excitatory
Transmitter Channel-typ Ion-current Action
Acetylecholin nicotin. Receptor Na+ and K+ excitatory
Glutamate AMPA / Kainate Na+ and K+ excitatory
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
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
Glycine Cl- inhibitory
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
Glycine Cl- inhibitory
Acetylecholin muscarin. Rec. - metabotropic, Ca2+ Release
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
Glycine Cl- inhibitory
Acetylecholin muscarin. Rec. - metabotropic, Ca2+ Release
Glutamate NMDA Na+, K+, Ca2+ voltage dependent
blocked at resting potential
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Synaptic Plasticity
32
At the dendrite the incomingsignals arrive (incoming currents)
Molekules
Synapses
Neurons
Local Nets
Areas
Systems
CNS
At the soma currentare finally integrated.
At the axon hillock action potentialare generated if the potential crosses the membrane threshold
The axon transmits (transports) theaction potential to distant sites
At the synapses are the outgoing signals transmitted onto the dendrites of the target neurons
Structure of a Neuron:
33
Chemical synapse
NeurotransmitterReceptors
34
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
35
Synaptic Transmission:
Synapses are used to transmit signals from the axon of a source to the dendrite of a target
neuron.
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)
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).
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.
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 uni-
directional and are slower.
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 uni-
directional 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).
36
Simple Computational Operations that can be Performed with Neurons
A xon = O utpu t
Input 1
Input 2S om a =C P U
The system to be considered first:One Neuron receiving2 Synapses.
What are the computations that can be performed with such a simple system ?
Firs t th ings firs t: Basic Operations
A rithm etica l: + S um m ation- S ubtraction. M ultip lication/ D iv is ion
Lociga l AN D O R N O T, e tc.
More Compex Operations
C alcu lus: In tegra tion
dx/d t D iffe rentiation
L inear A lgebra :Vector O perationsy=A x M atrix O perations
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B e lieve it o r not: With a single neuron and 2 input you can compute all alrithmetic, many logic and some of the more complex operations !
R equ ired R equ is its : 1) R esting P oten tia l (ca. -70 m v, constan t)2 ) F iring T hreshold3 ) E quilib rium P oten tia l o f d ifferen t ions4 ) Tim e-constants o f the ion-channels .
Summation
Keine Kognition ohne Addition
Transm itte r re lease a t a synapse leads to an excita tory postsynapticpotentia l (E P S P) because ion channe ls are open ing .
E PS P
m V
t
rest.pot.
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Necessary conditions for optim al sum m ation:1) synapses have to be c lose ly ad jacent2) p re-synaptic s igna ls have to arrive s im ultaneously3) resting potentia l and reversa l potentia l(s) have to be very d iffe rent.
E P S P = E P SP + E P S Pr e s A B
m V
t
rest.po t.
ABA
BThe little “shoulder” show s tha t theE PS Ps were not true ly s im ultaneous.
Spatial Sum mation
E P S P < E P S P + E P S Pr e s A B
m V
t
rest.pot.
A B
A
B
S om a
D e ndrite
If the synapses are far from each o ther the am plitude w ill beless a t the firs t sum m ing poin t. It w ill then further decayuntil reach ing the som a.
Consider 1:
sim ultaneousinputs !
Sum m ationpoint
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H ow w ill the s igna llook like at the sum m ation point ?
ABm V
t
rest.po t.
BS om a
D endrite
A more complicated situationA1) The signal from B arrives la ter
a t the sum m ation po int because B is fa rther from it than A .2) The signal from B is sm alle r a t the sum m ation point (sam e reason).
A
BS om a
Direction of signal propagation
The signal propagates essentia llyin all d irections. The directiontow ards the som a is (usua lly) theone which is functiona lly re levant.
incomplete spatial summ ationEPSP = a EPSP + b EPSP ; b<a<1.0A Bres
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A
B
Consider 2: If the signals a re not s im ultaneous then the sum w ill be sm aller
m V
t
rest.po t.
A B
The ea rly s ignal (A ) fac ilita tes the la ter signal (B ). Toge ther the firing th resholdm ight be reached but not a lone.
Temporal Summ ation
If the d ifference in arriva l tim es is too large, tem pora l sum m ation does no t occur anym ore !
m V
t
rest.po t.
A B
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A
B
Consider 3: If the equilib rium potentia l o f the invo lved ions is close to the resting potentia l then on ly incom plete sum m ation is observed. Even a p la teau is poss ib le .
m V
t
rest.po t.
AB
The po tentia l o f the invo lved ions can never exceed the ir ow n equ ilib rium potentia l. (“C lipp ing”).
Conclusion: Sum m ing with neurons is a rather com plex process. Spatial and tem poral phenom ena and the potential levels will influence the result of the “sum m ation” substantially.
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The sam e conditions apply as for sum m ation. Then one can regardan IPSP as a s ign-inverted E PS P. “Sum m ation” becom es “Subtraction”.
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Special case: “shunting inhibition”The equilib rium potentia l o f the ions “B ” is very c lose (”indentica l”)to the resting potentia l ! (A is exc itatory as usua l.)
E P S Pm V m V
t t
res t.po t.
res t.po t.
A B
H ow does the m em branepotentia l change ?
C l-
C l-open channel
This case is com m only observed for theC hloride ion.
W hat is the functional s ign ificance of th isbehavior ?
(almost)no potential change
W hen the are opening (a lm ost) no ion current is obsered and thus the po tentia l s tays (a lm ost) the sam e.
purp le channels
44
The E PSP trave ls to the som a. The m em brane potentia l w ill be depolarized along the w ay.
W hat happens at location C l w ith the re la tion betweenm em brane po tentia l and C l-equilibrium potentia l ?
A C l-current is the consequence. The positive m em brane pot. fluctua tion (viz . E PS P ) w ill be im m ediate ly com pensated for. Thus, a t the open C l channe ls no m ore depo larization is observed . The E PS P is e lectrica lly shunted !
Functional significance of “shunting inhibition”C onsider the case w ere C l-channels are a lready open w hen theexcita tory channels A are opening and an E PSP is e lic ited there.
A
C lto thesom a
to the periphera ldendrite
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T he physio log ica l transm itte r is G lu tam ate (G lu ).
i n
o u t
i n
o u t
47
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