Biological Modeling of Neural Networks
Week 2 – Biophysical modeling:
The Hodgkin-Huxley model
Wulfram GerstnerEPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 1: Biophysics of neurons
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 1: Biophysics of neurons
Neuronal Dynamics – 2.1. Introduction
visual cortex
motor cortex
frontal cortex
to motoroutput
Neuronal Dynamics – 2.1. Introduction
motor cortex
frontal cortex
to motoroutput
10 000 neurons3 km wires
1mm
Neuronal Dynamics – 2.1 Introduction
10 000 neurons3 km wires1mm
Signal:action potential (spike)
action potential
Ramon y Cajal
How is a spike generated?
Review of week 1: Integrate-and-Fire models
u
Spike emission
-spikes are events-triggered at threshold-spike/reset/refractoriness
Postsynaptic potential
synapse t
Neuronal Dynamics – week 2: Biophysics of neurons
Signal:action potential (spike)
action potential
Ca2+
Na+
K+
-70mV
Ions/proteins
Cell surrounded by membraneMembrane contains - ion channels - ion pumps
Neuronal Dynamics – week 2: Biophysics of neurons
Ca2+
Na+
K+
-70mV
Ions/proteins
Cell surrounded by membraneMembrane contains - ion channels - ion pumps
Resting potential -70mV how does it arise?
Ions flow through channel in which direction?
Neuron emits action potentials why?
Neuronal Dynamics – 2. 1. Biophysics of neurons
Resting potential -70mV how does it arise?
Ions flow through channel in which direction?
Neuron emits action potentials why?
Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963
Neuronal Dynamics – 2. 1. Biophysics of neurons
Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963
u
...du
dt u
( )I t
Neuronal Dynamics – Quiz In a natural situation, the electrical potential inside a neuron is [ ] the same as outside [ ] is different by 50-100 microvolt [ ] is different by 50-100 millivolt
If a channel is open, ions can [ ] flow from the surround into the cell [ ] flow from inside the cell into the surrounding liquid
Neurons and cells [ ] Neurons are special cells because they are surrounded by a membrane [ ] Neurons are just like other cells surrounded by a membrane [ ] Neurons are not cells
Ion channels are [ ] located in the cell membrane [ ] special proteins [ ] can switch from open to closed
Multiple answers possible!
Week 2 – Biophysical modeling:
The Hodgkin-Huxley model
Wulfram GerstnerEPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 2: Reversal potential and Nernst equation
Biological Modeling of Neural Networks
Neuronal Dynamics – 2.2. Resting potential
Ca2+
Na+
K+
-70mV
Ions/proteins
Cell surrounded by membraneMembrane contains - ion channels - ion pumps
Resting potential -70mV how does it arise?
Ions flow through channel in which direction?
Neuron emits action potentials why?
Neuronal Dynamics – 2. 2. Resting potential
Resting potential -70mV how does it arise?
Ions flow through channel in which direction?
Neuron emits action potentials why?
Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963
100
mV
0
inside
outside
Ka
Na
Ion channels Ion pumpkTE
en
density
Mathetical derivation
Ion pump Concentration difference
Neuronal Dynamics – 2. 2. Reversal potential
E
K
100
mV
0
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2. 2. Nernst equation
kTE
en K
100
mV
0
inside
outside
Ka
Na
Ion channels Ion pump
Reversal potential
1
2
( )1 2 ( )ln n u
n u
kTu u u
q
Neuronal Dynamics – 2. 2. Nernst equation
Concentration difference voltage difference
K
inside
outside
Ka
Na
Ion channels Ion pump
Reversal potential
Neuronal Dynamics – 2. 2. Reversal potential
Ion pump Concentration difference
Concentration difference voltage difference
Nernst equation
K
Reversal potential
Exercise 1.1– Reversal potential of ion channels
1
2
( )1 2 ( )ln n u
n u
kTu u u
q
Calculate the reversal potentialfor Sodium Postassium Calciumgiven the concentrations
What happens if you changethe temperature T from 37to 18.5 degree?
Next Lecture 9:45
Neuronal Dynamics:Computational Neuroscienceof Single NeuronsWeek 2 – Biophysical modeling:
The Hodgkin-Huxley model
Wulfram GerstnerEPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgkin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 3 : Hodgkin-Huxley Model
Neuronal Dynamics – 2. 3. Hodgkin-Huxley Model
Hodgkin-Huxley model Hodgkin&Huxley (1952) Nobel Prize 1963
...du
dt u
( )I t
giant axonof squid
inside
outside
Ka
Na
Ion channels Ion pump
C RlRK RNa
I
Mathematicalderivation
Hodgkin and Huxley, 1952Neuronal Dynamics – 2.3. Hodgkin-Huxley Model
C RlRK RNa
I
Neuronal Dynamics – 2.3. Hodgkin-Huxley Model
100
mV
0
)()()()( 43 tIEugEungEuhmgdt
duC llKKNaNa
)(
)(0u
umm
dt
dm
m
)(
)(0u
unn
dt
dn
n
)(
)(0u
uhh
dt
dh
h
stimulusleakI
inside
outside
Ka
Na
Ion channels Ion pump
C glgK gNa
I
u u
n0(u))(un
Hodgkin and Huxley, 1952Neuronal Dynamics – 2.3. Hodgkin-Huxley Model
NaI KI
1 2n nion ionI g r s
)(
)(0u
urr
dt
dr
r
u u
r0(u))(ur
Neuronal Dynamics – 2.3. Ion channel
, ( )ion kk
duC I I tdt
0 ( )
( )r
s s uds
dt u
)()(21 tIEusrgdt
duC Na
nnion
)(
)(0u
urr
dt
dr
r
u u
r0(u))(ur
Exercise 2 and 1.2 NOW!! - Ion channel
Next lecture at: 10H40
Week 2 – Biophysical modeling:
The Hodgkin-Huxley model
Wulfram GerstnerEPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 4: Threshold in the Hodgkin-Huxley Model
Biological Modeling of Neural Networks
Neuronal Dynamics – 2.4. Threshold in HH model inside
outside
Ka
Na
Ion channels Ion pump
)()()()( 43 tIEugEungEuhmgdt
duC llKKNaNa
leakI
)(
)(0u
umm
dt
dm
m
)(
)(0u
uhh
dt
dh
h
inside
outside
Ka
Na
Ion channels Ion pump
u u
h0(u)
m0(u) )(uh
)(um
C glgK gNa
I
Where is the threshold for firing?
Neuronal Dynamics – 2.4. Threshold in HH model
NaI KI
Constant current inputI0
C glgK gNa
I
Threshold? for repetitive firing
(current threshold)
Neuronal Dynamics – 2.4. Threshold in HH model
inside
outside
Ka
Na
Ion channels Ion pump
pulse inputI(t)
Threshold?
- AP if amplitude 7.0 units- No AP if amplitude 6.9 units
(pulse with 1ms duration)
(and pulse with 0.5 ms duration?)
Neuronal Dynamics – 2.4. Threshold in HH model
)()()()( 43 tIEugEungEuhmgdt
duC llKKNaNa
)(
)(0u
umm
dt
dm
m
)(
)(0u
uhh
dt
dh
h
Stim. NaI KI leakI
u u
h0(u)
m0(u) )(uh
)(um
pulse input
I(t)
Mathematical explanation
Neuronal Dynamics – 2.4. Threshold in HH model
3 ( ) ( )Na Na K leak
duC g m h u E I I I tdt
)(
)(0u
umm
dt
dm
m
)(
)(0u
uhh
dt
dh
h
Stim. NaI
u u
h0(u)
m0(u) )(uh
)(um
pulse input
I(t)
Neuronal Dynamics – 2.4. Threshold in HH model
Where is the threshold?Blackboard!
)()()()( 43 tIEugEungEuhmgdt
duC llKKNaNa
)(
)(0u
umm
dt
dm
m
)(
)(0u
uhh
dt
dh
h
Stim. NaI KI leakI
u u
h0(u)
m0(u) )(uh
)(um
pulse input
I(t)
Neuronal Dynamics – 2.4. Threshold in HH model
0 ( )
( )n
n n udn
dt u
Why start the explanation with m and not h?
What about n?
Where is the threshold?
Neuronal Dynamics – 2.4. Threshold in HH model
3
4
( )
( )
( )
( )
Na Na
K K
l l
duC g m h u Edt
g n u E
g u E
I t
Neuronal Dynamics – 2.4. Threshold in HH model
There is no strict threshold:
Coupled differential equations
‘Effective’ threshold in simulations?
First conclusion:
0 20ms
Strongstimulus
Action potential100
mV
0
strong stimuli
refractoriness
Strongstimulus
100
mV
0
Where is the firing threshold?
Refractoriness! Harder to elicit a second spike
Neuronal Dynamics – 2.4. Refractoriness in HH model
I(t)
100
mV
0
Stimulation withtime-dependentinput current
Neuronal Dynamics – 2.4. Simulations of the HH model
I(t)
0
5
-5
mV
mV
0
100100
mV
0
Subthresholdresponse
Spike
Neuronal Dynamics – 2.4. Simulations of the HH model
Step current inputI2
I
Neuronal Dynamics – 2.4. Threshold in HH model
step input I2I
Where is the firing threshold?
pulse inputI(t)
ramp input
There is no threshold - no current threshold - no voltage threshold
‘effective’ threshold - depends on typical input
...)(3 NaNa Euhmgdt
duC
Neuronal Dynamics – 2.4. Threshold in HH model
Response at firing threshold?
ramp input/constant input
I0
Type I type II
I0 I0
fff-I curve f-I curve
Neuronal Dynamics – 2.4. Type I and Type II
Hodgkin-Huxley modelwith standard parameters(giant axon of squid)
Hodgkin-Huxley modelwith other parameters(e.g. for cortical pyramidalNeuron )
Neuronal Dynamics – 2.4. Hodgkin-Huxley model -4 differential equations
-no explicit threshold-effective threshold depends on stimulus-BUT: voltage threshold good approximation
Giant axon of the squid cortical neurons-Change of parameters-More ion channels-Same framework
Exercise 3.1-3.3 – Hodgkin-Huxley – ion channel dynamics
)()(4 tIEungdt
duC KK
stimulus
inside
outside
Ka
Na
Ion channels Ion pump
C gK
I
u u
n0(u)
)(un
Determine ion channel dynamics
adapted fromHodgkin&Huxley 1952
apply voltage step
voltage step u2u
Next Lecture at:11.38
Week 2 – Biophysical modeling:
The Hodgkin-Huxley model
Wulfram GerstnerEPFL, Lausanne, Switzerland
2.1 Biophysics of neurons - Overview2.2 Reversal potential
- Nernst equation2.3 Hodgkin-Huxley Model2.4 Threshold in the Hodgkin-Huxley Model - where is the firing threshold?2.5. Detailed biophysical models - the zoo of ion channels
Week 2 – part 5: Detailed Biophysical Models
Biological Modeling of Neural Networks
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
b
a
Na+ channel from rat heart (Patlak and Ortiz 1985)A traces from a patch containing several channels. Bottom: average gives current time course.B. Opening times of single channel events
Steps:Different numberof channels
Ca2+
Na+
K+
Ions/proteins
Neuronal Dynamics – 2.5 Ion channels
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
b
a
There are about 200identified ion channels
http://channelpedia.epfl.ch/
How can we know which ones are present in a given neuron?
b
a
a
b
Kv1 Kv2 Kv3
Kv
Voltage Activated K+ Channels
+ +
Kv1.1 Kv1.2 Kv1.3 Kv1.4 Kv1.5 Kv1.6 Kv2.2Kv2.1 Kv3.2Kv3.1 Kv3.3 Kv3.4 KChIP1 KChIP2 KChIP3
KChIP
Kv1 Kv3Kv2 Kv4 Kv5 Kv6 Kv8 Kv9
Kv
Kv4.3Kv4.1 Kv4.2
Ion Channels investigated in the study of
CGRP
CB
PV CR NPY
VIP
SOM
CCK
pENK
Dyn SP CRH
schematic mRNAExpression profile
C glgKv1gNa
I
gKv3
Toledo-Rodriguez, …, Markram (2004)
C glgKv1gNa
I
gKv3
Model of a hypothetical neuron
)()()()()( 233
411
3 tIEugEungEungEuhmgdt
duC llKKvKvKKvKvNaNa
)(
)(0u
umm
dt
dm
m
1 0,11
,1
( )
( )n
n n udn
dt u
)(
)(0u
uhh
dt
dh
h
stimulusNaI KI leakI
inside
outside
Ka
Na
Ion channels Ion pump
Erisir et al, 1999Hodgkin and Huxley, 1952
1 0,33
,3
( )
( )n
n n udn
dt u
How many parameters per channel?
C glgKv1gNa
I
gKv3
Model of a hypothetical neuron
I(t)
Spike
Subthreshold
Detailed model, based on ion channels
C glgKv1gNa
I
gKv3
Model of a hypothetical neuron (type I)
Biophysical model, based on ion channels
Current pulse
constantcurrent
- Delayed AP initiation- Smooth f-I curve type I neuron
Neuronal Dynamics – 2.5 Adaptation
Functional roles of channels?- Example: adaptation
I
u
Neuronal Dynamics – 2.5 Adaptation: IM -current
M current: - Potassium current - Kv7 subunits - slow time constant
IM current is one of many potential sources of adaptationYamada et al., 1989
( )M M KI g m u E
Neuronal Dynamics – 2.5 Adaptation – INaP current current: - persistent sodium current - fast activation time constant - slow inactivation ( ~ 1s)
INaP current - increases firing threshold- source of adaptationAracri et al., 2006
( )NaP NaP NaI g m h u E
inside
outside
Ka
Na
Ion channels Ion pump
Neuronal Dynamics – 2.5 Biophysical models
b
Hodgkin-Huxley modelprovides flexible framework
Hodgkin&Huxley (1952) Nobel Prize 1963
Exercise 4 – Hodgkin-Huxley model – gating dynamics
( ) (1 ) ( )m m
dmu m u m
dt
0 ( ) 0.5{1 tanh[ ( )]m u u
1( ) ( )
1 exp[ ( ) / ]m mu uu a b
A) Often the gating dynamics is formulated as
B) Assume a form
How are a and b related to and in the equations 0 ( )
( )m
m m udm
dt u
C ) What is the time constant ? ( )m u
0 ( )
( )m
m m udm
dt u
0 ( ) ( )mm u and uCalculate
Now Computer Exercises:
Play with Hodgkin-Huxley model
Neuronal Dynamics – References and Suggested Reading
Reading: W. Gerstner, W.M. Kistler, R. Naud and L. Paninski,Neuronal Dynamics: from single neurons to networks and models of cognition. Chapter 2: The Hodgkin-Huxley Model, Cambridge Univ. Press, 2014OR W. Gerstner and W. M. Kistler, Spiking Neuron Models, Chapter 2, Cambridge, 2002
- Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 117(4):500-544. -Ranjan, R.,et al. (2011). Channelpedia: an integrative and interactive database for ion channels. Front Neuroinform, 5:36.
-Toledo-Rodriguez, M., Blumenfeld, B., Wu, C., Luo, J., Attali, B., Goodman, P., and Markram, H. (2004). Correlation maps allow neuronal electrical properties to be predicted from single-cell gene expression profiles in rat neocortex. Cerebral Cortex, 14:1310-1327.
-Yamada, W. M., Koch, C., and Adams, P. R. (1989). Multiple channels and calcium dynamics. In Koch, C. and Segev, I., editors, Methods in neuronal modeling, MIT Press.
- Aracri, P., et al. (2006). Layer-specic properties of the persistent sodium current in sensorimotor cortex. Journal of Neurophysiol., 95(6):3460-3468.
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