Romain BretteEcole Normale Supérieure, Paris

romain.brette@ens.fr

What is the most realisticsingle-compartment neuron model?

The simplicity-realism trade-off

Izhikevich (2004). Which model to use for cortical neurons? IEEE Neural networks 15(5)

What is a « realistic »model?

- lots of features named after the real object- it flies

sodium channelspotassium channels

spike threshold

- abstract

The Hodgkin-Huxley model

Hodgkin & Huxley(Nobel prize 1963)

nVndt

dnV

hVhdt

dhV

mVmdt

dmV

VEngVEhmgVEgdt

dVC

n

h

m

KKNall

)()(

)()(

)()(

)()()( 43

The Hodgkin-Huxley model

Hodgkin & Huxley(Nobel prize 1963)

nVndt

dnV

hVhdt

dhV

mVmdt

dmV

VEngVEhmgVEgdt

dVC

n

h

m

KKNall

)()(

)()(

)()(

)()()( 43

Model of the space clamped giant squid axon

In what sense is it « realistic »?

nVndt

dnV

hVhdt

dhV

mVmdt

dmV

VEngVEhmgVEgdt

dVC

n

h

m

KKNall

)()(

)()(

)()(

)()()( 43

several equations and variables

although phenomenological!

but you can make predictions about Vm, and about Na and K currents

In what sense is it « realistic »?

nVndt

dnV

hVhdt

dhV

mVmdt

dmV

VEngVEhmgVEgdt

dVC

n

h

m

KKNall

)()(

)()(

)()(

)()()( 43

several equations and variables

although phenomenological!

you can predict voltage-clamp measurements

The empirical content of a modelKarl 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 Quadratic model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes spikes

Vm

INa

IK

Empirical content of neuron models

multicompartmental Hodgkin-Huxley model

single-compartment Hodgkin-Huxley model

integrate-and-fire model exponential integrate-and-fire model

quadratic integrate-and-fire model

empirical content

Correctness of a modelIntegrate-and-fire model Quadratic model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes spikes

Vm

INa

IK

Can make more predictions,but are predictions correct?

Membrane potential (Vm)

Naundorf, Wolf, Volgushev (Nature 2006)

1) Spikes have sharp onsets in recordings,unlike in single compartment Hodgkin-Huxley models

Badel et al. (J Neurophysiol 2008)

V

2) I-V relationship at soma is very sharp,unlike in single compartment Hodgkin-Huxley models

Sodium current (INa)

Milescu, Bean, Smith (J Neurosci 2010)

-45 mV

-25 mV

I Na (

nA)

Somatic voltage-clamp

Cortical pyramidal neuron Inferior olivary neuron Motoneuron

Voltage-clamp recordings are not as predicted by the single-compartment HH model

mV

soma

total

Correctness of the HH modelIntegrate-and-fire model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes

Vm

INa

IK

The additional empirical content is not correct!

Correctness of the HH modelIntegrate-and-fire model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes

Vm

INa

IK

The additional empirical content is not correct!

How about spikes?

Predicting spikesThe 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

Fourcaud-Trocme et al. (J Neurosci 2003)Brette & Gerstner (J Neurophysiol 2005)

is the « slope factor » = Na activation slope ka

Patch-clamp measurements: ka = 6 mV

Best prediction of spikes: = 0 mV

the best fit is an IF model!

Transfer function of cortical neurons

Ilin et al. (J Neurosci 2013)

3-6 Hz

Prediction of single-compartment HH models:cut-off frequency should be an order of magnitude lower

Transfer function of cortical neurons

Ilin et al. (J Neurosci 2013)

3-6 Hz

Fast response to current changes,as in IF models!

Brunel et al. (PRL 2001)

Correctness of the HH modelIntegrate-and-fire model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes

Vm

INa

IK

Correctness of the HH modelIntegrate-and-fire model Hodgkin-Huxley model

Vm (subthreshold)

spikes spikes

Vm

INa

IK

Spikes are also not well predicted by the single-compartment HH model!

Correctness of neuron models

single-compartment Hodgkin-Huxley model

integrate-and-fire model exponential integrate-and-fire model

quadratic integrate-and-fire model

correctness

But how is this possible ???

Spikes are initiated in the axon

Stuart, Schiller, Sakmann (J Physiol 1997)

Spikes are initiated in the axon

Stuart, Schiller, Sakmann (J Physiol 1997)Debanne et al. (Phys Rev 2011)

Compartmentalization

Most of the time, Vm is the same in soma and AIS

At spike initiation,the two compartments are decoupled

Kole, Letzkus, Stuart (Neuron 2007)

AIS

Compartmentalization

Va – Vs = axonal current * Ra

Va

Vs

Milescu, Bean, Smith (J Neurosci 2010)

axonal current in voltage-clamp

Model of axonal initiation

soma

I

Vs

Va

Ra.I

The soma is a current sink

(Ohm’s law)

Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comp Biol (in press)

Model of axonal initiation

soma

I

Vs

Va

Ra.I

The soma is a current sink

(Ohm’s law)

Model of axonal initiation

soma

I=f(Va)

Vs

Va

Ra.I

Na activation

Model of axonal initiation

soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

I (nA)

I=f(Va)

I=(V a-V s)/

Ra

Vs

Lateral and Na currents must match

Discrete opening of Na channels

Na channels near the soma

soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

I (nA)

I=f(Va)I=

(Va-V

s)/Ra

Vs

Continuous opening of Na channels

A view from the soma

m

Lateral current flows abruptly when a voltage threshold is exceeded

Na channels open in an all-or-none fashion

A view from the soma

m

single compartment HH modelwith axonal initiation

A fairly good phenomenological description:- below Vt, no sodium current- when Vm reaches Vt: all channels open (spike)

Vt

a.k.a. the integrate-and-fire model !

Value of spike threshold

soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

Lateral and Na currents must match

f(Va) = (Va-Vs)/RaFixed point equation

Na activation

Spike threshold = bifurcation point(= Vs when solution jumps)

I (nA)

aNaaNaaas kVERgkkVV /)(log 2/12/1

The threshold equation

V1/2ka

Conclusion

looks good but doesn’t fly! it flies!

Single-compartment Hodgkin-Huxley model Integrate-and-fire model

Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Comp Biol (in press)