Download What is the most realistic single

What is the most realistic
single-compartment neuron model?
Romain Brette
Ecole Normale Supérieure, Paris
[email protected]
The simplicity-realism trade-off
Izhikevich (2004). Which model to use for cortical neurons? IEEE Neural networks 15(5)
What is a « realistic »model?
sodium channels
potassium
channels
spike threshold
- lots of features named after the real object
- abstract
- it flies
The Hodgkin-Huxley model
dV
 g l ( El  V )  gm3 h( E Na  V )  g K n 4 ( EK  V )
dt
dm
 m (V )
 m (V )  m
dt
dh
 h (V )  h (V )  h
dt
dn
 n (V )  n (V )  n
dt
C
Hodgkin & Huxley
(Nobel prize 1963)
The Hodgkin-Huxley model
dV
 g l ( El  V )  gm3 h( E Na  V )  g K n 4 ( EK  V )
dt
dm
 m (V )
 m (V )  m
dt
dh
 h (V )  h (V )  h
dt
Model of the space clamped
dn
 n (V )  n (V )  n
giant squid axon
dt
C
Hodgkin & Huxley
(Nobel prize 1963)
In what sense is it « realistic »?
dV
 g l ( El  V )  gm3 h( E Na  V )  g K n 4 ( EK  V )
dt
dm
 m (V )
 m (V )  m
dt
several equations and variables
dh
 h (V )  h (V )  h
dt
although phenomenological!
dn
 n (V )  n (V )  n
dt
C
but you can make predictions about
Vm, and about Na and K currents
In what sense is it « realistic »?
dV
 g l ( El  V )  gm3 h( E Na  V )  g K n 4 ( EK  V )
dt
dm
 m (V )
 m (V )  m
dt
several equations and variables
dh
 h (V )  h (V )  h
dt
although phenomenological!
dn
 n (V )  n (V )  n
dt
C
you can predict voltage-clamp
measurements
The empirical content of a model
Karl Popper:“I define the empirical content of a statement p as the class of its
potential falsifiers” (The logic of scientific discovery, 1935).
In other words: what a model can potentially predict
Integrate-and-fire model
spikes
Vm (subthreshold)
Quadratic model
spikes
Hodgkin-Huxley model
spikes
Vm
INa
IK
Empirical content of neuron models
empirical content
multicompartmental Hodgkin-Huxley model
single-compartment Hodgkin-Huxley model
integrate-and-fire model
exponential integrate-and-fire model
quadratic integrate-and-fire model
Correctness of a model
Integrate-and-fire model
spikes
Vm (subthreshold)
Quadratic model
spikes
Hodgkin-Huxley model
spikes
Vm
INa
IK
Can make more predictions,
but are predictions correct?
Membrane potential (Vm)
1) Spikes have sharp onsets in recordings,
unlike in single compartment Hodgkin-Huxley models
2) I-V relationship at soma is very sharp,
unlike in single compartment Hodgkin-Huxley models
V
Naundorf, Wolf, Volgushev (Nature 2006)
Badel et al. (J Neurophysiol 2008)
Sodium current (INa)
Somatic voltage-clamp
INa (nA)
-45 mV
Voltage-clamp recordings are not as
predicted by the single-compartment
HH model
-25 mV
Milescu, Bean, Smith (J Neurosci 2010)
soma
total
mV
Cortical pyramidal neuron
Inferior olivary neuron
Motoneuron
Correctness of the HH model
Integrate-and-fire model
spikes
Vm (subthreshold)
Hodgkin-Huxley model
spikes
Vm
INa
IK
The additional empirical content is
not correct!
Correctness of the HH model
Hodgkin-Huxley model
Integrate-and-fire model
spikes
spikes
Vm (subthreshold)
Vm
INa
IK
The additional empirical content is
not correct!
How about spikes?
Predicting spikes
The INCF competition
Gerstner & Naud (Science 2009)
Results for IF models
cortical neuron
MAT2 (86%)
AIF (84%)
Izhikevich (62%)
Rossant et al. (Frontiers Neurosci 2011)
And the Hodgkin-Huxley model?
• We did not manage to fit it
• However, we did fit an approximation of it, the adaptive
exponential IF model
C
dV
V -VT
= -gL (V - EL ) + gL DT exp(
)+ I
dt
DT
D T is the « slope factor » = Na activation slope ka
Patch-clamp measurements: ka = 6 mV
Best prediction of spikes: D T = 0 mV
the best fit is an IF model!
Fourcaud-Trocme et al. (J Neurosci 2003)
Brette & Gerstner (J Neurophysiol 2005)
Transfer function of cortical neurons
3-6 Hz
Ilin et al. (J Neurosci 2013)
Prediction of single-compartment HH models:
cut-off frequency should be an order of magnitude lower
Transfer function of cortical neurons
3-6 Hz
Ilin et al. (J Neurosci 2013)
Fast response to current changes,
as in IF models!
Brunel et al. (PRL 2001)
Correctness of the HH model
Integrate-and-fire model
spikes
Vm (subthreshold)
Hodgkin-Huxley model
spikes
Vm
INa
IK
Correctness of the HH model
Integrate-and-fire model
spikes
Vm (subthreshold)
Hodgkin-Huxley model
spikes
Vm
INa
IK
Spikes are also not well predicted by the single-compartment HH model!
Correctness of neuron models
correctness
integrate-and-fire model
exponential integrate-and-fire model
single-compartment Hodgkin-Huxley model
quadratic integrate-and-fire model
But how is this possible ???
Spikes are initiated in the axon
Stuart, Schiller, Sakmann (J Physiol 1997)
Spikes are initiated in the axon
Debanne et al. (Phys Rev 2011)
Stuart, Schiller, Sakmann (J Physiol 1997)
Compartmentalization
AIS
Most of the time,
Vm is the same in
soma and AIS
Kole, Letzkus, Stuart (Neuron 2007)
At spike initiation,
the two compartments are decoupled
Compartmentalization
axonal current in voltage-clamp
Va – Vs = axonal current * Ra
Va
Vs
Milescu, Bean, Smith (J Neurosci 2010)
Model of axonal initiation
I
The soma is a current sink
soma
Va
Vs
(Ohm’s law)
Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comp Biol (in press)
Model of axonal initiation
I
The soma is a current sink
soma
Va
(Ohm’s law)
Vs
Model of axonal initiation
Na activation
I=f(Va)
soma
Va
Vs
Model of axonal initiation
I=f(Va)
I=(Va-Vs)/Ra
Lateral and Na currents must match
soma
Vs
Vs
I=f(Va)
I (nA)
Discrete opening of Na channels
Na channels near the soma
I=f(Va)
I=(Va-Vs)/Ra
soma
Vs
Vs
I=f(Va)
I (nA)
Continuous opening of Na channels
A view from the soma
Lateral current flows abruptly when a
voltage threshold is exceeded
m
Na channels open in an all-or-none fashion
A view from the soma
A fairly good phenomenological description:
- below Vt, no sodium current
- when Vm reaches Vt: all channels open (spike)
a.k.a. the integrate-and-fire model !
with axonal
initiation
Vt
m
single compartment HH model
Value of spike threshold
I=f(Va)
I=(Va-Vs)/Ra
Lateral and Na currents must match
soma
Vs
Fixed point equation
f(Va) = (Va-Vs)/Ra
I (nA)
Spike threshold = bifurcation point
(= Vs when solution jumps)
Na activation
The threshold equation
Vs  V1/ 2  ka  ka log g Na Ra ( ENa  V1/ 2 ) / ka 
ka V1/2
Conclusion
Single-compartment Hodgkin-Huxley model
looks good but doesn’t fly!
Integrate-and-fire model
it flies!
Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization.
PLoS Comp Biol (in press)