Romain Brette Computational neuroscience of sensory systems [email protected] Dynamics of neural excitability

Romain BretteComputational neuroscience of sensory systems

[email protected]

Dynamics of neural excitability

The spike threshold

firing threshold

action potential

postsynaptic potential (PSP)

temporal integration

Questions

1. Is there a voltage threshold? If yes, is it equal to the onset voltage?

If yes, which one?

Kole & Stuart (2008)

Questions

2. What determines the value of the threshold?

3. How does the spike threshold vary?

4. What difference does it make?

IS THERE A VOLTAGE THRESHOLD FOR SPIKE INITIATION?

Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Computational Biology

The « sharpness » of spike initiation

Naundorf et al (2006)

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

Badel et al. (J Neurophysiol 2008)

V

2) I-V relationship at soma is very sharp

The « sharpness » of spike initiation

Naundorf et al (2006)

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

2) I-V relationship at soma is very sharp

3) Integrate-and-fire models can predict precise spike trains of neurons

4) Cortical neurons transmit high frequency inputs (>200 Hz)

Spikes are initiated in the axon

Stuart, Schiller, Sakmann (J Physiol 1997)

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.


soma

I

Vs

Va

Ra.I

The soma is a current sink

(Ohm’s law)


soma

I=f(Va)

Vs

Va

Ra.I

Na activation


soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

I (nA)

I=f(Va)I=

(Va-V

s)/Ra

Vs

Lateral and Na currents must match

Distal axonal initiation

soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

I (nA)

I=f(Va)

I=(V a-V s)/

Ra

Vs


Discrete 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 !

Answers

1. Is there a voltage threshold?If yes, is it equal to the onset voltage?

If yes, which one?

Kole & Stuart (2008)

yes

yes

X

WHAT DETERMINES THE VALUE OF THE THRESHOLD?


soma

I=f(Va)

I=(Va-Vs)/Ra

Vs


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

How about other channels?

soma

I=f(Va)

I=(Va-Vs)/Ra

Vs

IKThere are also K+ channels!

Vs

VaRa.IK

(Ohm’s law)

If the axonal threshold is unchanged,then the somatic threshold increases by –Ra.IK

KaaNaaNaaas IRkVERgkkVV /)(log 2/12/1

The threshold equation

HOW DOES THE SPIKE THRESHOLD VARY?

Platkiewicz, J. & Brette, R. A Threshold Equation for Action Potential Initiation. PLoS Comp Biol 6(7): e1000850. doi:10.1371

Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol

The spike threshold is not fixed

The spike threshold is variable in vivo

Voltage (mV)

Membrane potential Threshold

(Azo

uz &

Gra

y, 2

000)

Large threshold variability (>10 mV)

The threshold depends on depolarization speed

(Wile

nt &

Con

trer

as, 2

005)

In the visual cortex(Azouz & Gray 2003)

2 ms

Spike threshold is inversely correlated with depolarization speed

Mean membrane potential (mV)

The threshold adapts to the membrane potential

Two possible mechanisms


Inactivation of Na channels decreases threshold

Activation of K+ channels increases threshold

Na inactivation: gNa proportional to h (= non-inactivated channels)

hkV aT logThe « threshold equation »

Huguenard

et

al. (

19

88)

Validation in a multicompartmental model

Prediction: AxonaT hkV log

Validation in a multicompartmental model

AxonaT hkV log

Threshold dynamics

hkV aT log

dt

dh

hk

dt

da

1

dt

d

)()( Vdt

dVh

)(log)( VhkVV aT

« Steady-state threshold »

Example with linear membrane equation:

The steady-state threshold

where ka/ki 1

h

Inactivation curve

hkV aT log

Testing the model

In vivo intracellular recordings in barn owl IC (JL Peña)

VmӨ

We fit a threshold model to predict spikes


Testing the model


WHAT DIFFERENCE DOES IT MAKE?

Platkiewicz, J. & Brette, R. Impact of sodium channel inactivation on spike threshold dynamics and synaptic integration. PLoS Comp Biol 7(5): e1001129. doi:10.1371Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol

Synaptic integration with adaptive threshold

Above Vi, threshold is a low-pass filtered version of the membrane potential

VL*0

Threshold

PSP

VT

« threshold PSP »

The effective PSPThreshold PSP

PSP V

q

V - q

Fixed threshold

« Effective PSP »

Time (ms)

Distance to threshold:

))(*(0 ittPSPLPSPVU

shorter integration time constant

Sharpening and noise reductionPSPs Autocorrelation

« Noise » reduction

Effective time constant


Summary

« Effective PSP »

1) There is a sharp voltage threshold because of compartmentalization.2) Spike threshold depends on AIS geometry, Na channel properties, K+ currents

3) Spike threshold adapts on a short timescale

4) Threshold adaptation shortens integrationtime constant and reduces effective variability


Thank you!

Jonathan Platkiewicz

Platkiewicz, J. & Brette, R. A Threshold Equation for Action Potential Initiation. PLoS Comp Biol 6(7): e1000850. doi:10.1371

Platkiewicz, J. & Brette, R. Impact of sodium channel inactivation on spike threshold dynamics and synaptic integration. PLoS Comp Biol 7(5): e1001129. doi:10.1371

Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol, 10(4): e1003560.

Bertrand Fontaine

Experimental collaborators: Jose Peña (New York; in vivo electrophysiology in barn owls)Philip Joris (Leuven; in vivo electrophysiology in cats)

Brette, R. (2013). Sharpness of spike initiation in neurons explained by compartmentalization. PLoS Computational Biology

Fontaine B, Benichoux V, Joris PX and Brette R (2013). Predicting spike timing in highly synchronous auditory neurons at different sound levels. J Neurophysiol 110(7):1672-88.

Alternative mechanisms of threshold variability. 1 – Remote initiation site

Spikes are initiated in the axon, but usually recorded at the soma.

Vm

Time (ms) Depolarization slope (mV/ms)

Thre

shol

d

V1

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Romain Brette Computational neuroscience of sensory systems [email protected] Dynamics of neural excitability