romain brette computational neuroscience of sensory systems [email protected] dynamics of...
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Romain BretteComputational neuroscience of sensory systems
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)
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.
Model of axonal initiation
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
Lateral and Na currents must match
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
Model of axonal initiation
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
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
KaaNaaNaaas IRkVERgkkVV /)(log 2/12/1
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)
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:
Testing the model
In vivo intracellular recordings in barn owl IC (JL Peña)
VmӨ
We fit a threshold model to predict spikes
Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol
Testing the model
Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol
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
Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo. PLoS Comp Biol
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
KaaNaaNaaas IRkVERgkkVV /)(log 2/12/1
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.