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Neuronal codes
Spiking models:
• Hodgkin Huxley Model (small regeneration)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
• Spike Response Model
Neuronal codes
Spiking models:
• Hodgkin Huxley Model (small regeneration)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
• Spike Response Model
Neuronal Codes – Action potentials as the elementary units
voltage clamp from a brain cell of a fly
after band pass filtering
Neuronal Codes – Action potentials as the elementary units
voltage clamp from a brain cell of a fly
after band pass filtering
generated electronicallyby a threshold discriminatorcircuit
Neuronal Codes – Probabilistic response and Bayes’ rule
stimulus
)(|}{ tstP i
stimulusspiketrains
conditional probability:
Neuronal Codes – Probabilistic response and Bayes’ rule
conditionalprobability
)(tsPensembles of signals
natural situation:
)(},{ tstP ijoint probability:
experimental situation:
• we choose s(t)
)()(|}{)(},{ tsPtstPtstP ii prior
distributionjoint
probability
Neuronal Codes – Probabilistic response and Bayes’ rule
• But: the brain “sees” only {ti}• and must “say” something about s(t)
• But: there is no unique stimulus in correspondence with a particular spike train• thus, some stimuli are more likely than others given a particular spike train
experimental situation: )()(|}{)(},{ tsPtstPtstP ii
response-conditional ensemble
}{}{|)()(},{ iii tPttsPtstP
Neuronal Codes – Probabilistic response and Bayes’ rule
)()(|}{)(},{ tsPtstPtstP ii
}{}{|)()(},{ iii tPttsPtstP
)()(|}{}{}{|)( tsPtstPtPttsP iii
}{
)()(|}{}{|)(
iii tP
tsPtstPttsP
Bayes’ rule:
what we see:
what ourbrain “sees”:
Neuronal Codes – Probabilistic response and Bayes’ rule
motion sensitive neuron H1 in the fly’s brain:
average angular velocityof motion across the VF
in a 200ms window
spike count
determined by the experimenter
property of theneuron
)()(, vPnPvnP correlation
Neuronal Codes – Probabilistic response and Bayes’ rule
}{|)( ittsP
spikes
determine the probability of astimulus from given spike train
stimuli
Neuronal Codes – Probabilistic response and Bayes’ rule
}{|)( ittsPdetermine the probability of astimulus from given spike train
Neuronal Codes – Probabilistic response and Bayes’ rule
)(|}{ tstP i
determine probability ofa spike trainfrom a given stimulus
Neuronal Codes – Probabilistic response and Bayes’ rule
)(|}{ tstP i
)(tr
determine probability ofa spike trainfrom a given stimulus
Neuronal Codes – Probabilistic response and Bayes’ rule
)(trHow do we measure this time dependent firing rate?
Neuronal Codes – Probabilistic response and Bayes’ rule
SO, WHAT?
We can characterize the neuronal code in two ways:
translating stimuli into spikes translating spikes into stimuli
}{|)( ittsP )(|}{ tstP i
}{
)()(|}{}{|)(
iii tP
tsPtstPttsP Bayes’ rule:
(traditional approach)
-> If we can give a complete listing of either set of rules, than we can solve any translation problem
• thus, we can switch between these two points of view
(how the brain “sees” it)
Neuronal Codes – Probabilistic response and Bayes’ rule
We can switch between these two points of view.
And why is that important?
These two points of view may differ in their complexity!
Neuronal Codes – Probabilistic response and Bayes’ rule
average number of spikes
depending on stimulus amplitude
average stimulus depending on
spike count
Neuronal Codes – Probabilistic response and Bayes’ rule
average number of spikes
depending on stimulus amplitude
average stimulus depending on
spike count
non-linear relation
almost perfectly linearrelation
That’s interesting, isn’t it?
Neuronal Codes – Probabilistic response and Bayes’ rule
For a deeper discussion read, for instance, that nice book:
Rieke, F. et al. (1996). Spikes: Exploring the neural code. MIT Press.
Neuronal codes
Spiking models:
• Hodgkin Huxley Model (small regeneration)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
• Spike Response Model
Hodgkin Huxley Model:
)()( tItIdt
dVC inj
kk
m
)()()( tItItIk
kCinj withu
QC and
dt
dVC
dt
duCIC
)()()( 43LmLKmKNamNa
kk VVgVVngVVhmgI
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
charging current
Ionchannels
Hodgkin Huxley Model:
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
huhuh
nunun
mumum
hh
nn
mm
)()1)((
)()1)((
)()1)((
(for the giant squid axon)
)]([)(
10 uxx
ux
x
1
0
)]()([)(
)]()([)(
uuu
uuux
xxx
xx
x
with
• voltage dependent gating variables
time constant
asymptotic value
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
• If u increases, m increases -> Na+ ions flow into the cell• at high u, Na+ conductance shuts off because of h• h reacts slower than m to the voltage increase• K+ conductance, determined by n, slowly increases with increased u
)]([)(
10 uxx
ux
x
action potential
General reduction of the Hodgkin-Huxley Model
)()()()( 43 tIVugVungVuhmgdt
duC LlKKNaNa
stimulus
NaI KI leakI
1) dynamics of m are fast2) dynamics of h and n are similar
General Reduction of the Hodgkin-Huxley Model: 2 dimensional Neuron Models
)(),( tIwuFdt
du
stimulus
),( wuGdt
dww
Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
)8.07.0(08.0 wudt
dw
u: membran potentialw: recovery variableI: stimulus
0dt
du
0dt
dww
uI(t)=0
Iwu
udt
du
3
3
)( wudt
dw
For I=0: • convergence to a stable fixed point
FitzHugh-Nagumo Model
nullclines
0dt
du
0dt
dww
uI(t)=I0
limit cycle
- unstable fixed point
limit cycle
FitzHugh-Nagumo Model
Iwu
udt
du
3
3
)( wudt
dw
stimulus
FitzHugh-Nagumo Model
nullclines
The FitzHugh-Nagumo model – Absence of all-or-none spikes
• no well-defined firing threshold• weak stimuli result in small trajectories (“subthreshold response”)• strong stimuli result in large trajectories (“suprathreshold response”)• BUT: it is only a quasi-threshold along the unstable middle branch of the V-nullcline
(java applet)
The FitzHugh-Nagumo model – Excitation block and periodic spiking
Increasing I shifts the V-nullcline upward
-> periodic spiking as long as equilibrium is on the unstable middle branch-> Oscillations can be blocked (by excitation) when I increases further
The Fitzhugh-Nagumo model – Anodal break excitation
Post-inhibitory (rebound) spiking:transient spike after hyperpolarization
The Fitzhugh-Nagumo model – Spike accommodation
• no spikes when slowly depolarized• transient spikes at fast depolarization
Neuronal codes
Spiking models:
• Hodgkin Huxley Model (small regeneration)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
• Spike Response Model
iuij
Spike reception
Spike emission
Integrate and Fire model
models two key aspects of neuronal excitability:• passive integrating response for small inputs• stereotype impulse, once the input exceeds a particular amplitude
iui
Spike reception: EPSP
)()( tRItudt
dum
tui Fire+reset threshold
Spike emission
resetI
j
Integrate and Fire model
ui
-spikes are events-threshold-spike/reset/refractoriness
I(t)
I(t)
Time-dependent input
Integrate and Fire model
)()( tRItudt
du
resetuu If firing:
I=0
dt
du
u
I>0
dt
du
u
resting
t
u
repetitive
t
Integrate and Fire model (linear)
u-80 -40
0
)()( tRIuFudt
d
tu Fire+reset
non-linear
threshold
I=0
dt
du
u
I>0
dt
du
u
Quadratic I&F:
02
2)( cucuF
Integrate and Fire model (non-linear)
I=0
dt
du
u
Integrate and Fire model (non-linear)
critical voltagefor spike initiation
(by a short current pulse)
)()( tRIuFudt
d
tu Fire+reset
non-linear
threshold
I=0u
dt
d
u
I>0u
dt
d
u
Quadratic I&F:
02
2)( cucuF
)exp()( 0 ucuuF
exponential I&F:
Integrate and Fire model (non-linear)
Strict voltage threshold - by construction - spike threshold = reset condition
There is no strict firing threshold - firing depends on input - exact reset condition of minor relevance
Linear integrate-and-fire:
Non-linear integrate-and-fire:
Neuronal codes
Spiking models:
• Hodgkin Huxley Model (small regeneration)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
• Spike Response Model
Spike response model (for details see Gerstner and Kistler, 2002)
= generalization of the I&F model
SRM:
• parameters depend on the time since the last output spike
• integral over the past
I&F:
• voltage dependent parameters
• differential equations
allows to model refractoriness as a combination of three components:
1. reduced responsiveness after an output spike
2. increase in threshold after firing
3. hyperpolarizing spike after-potential
iuij
fjtt
Spike reception: EPSP
fjtt
Spike reception: EPSP
^itt
^itt
Spike emission: AP
fjtt ^
itt tui j f
ijw
tui Firing: tti ^
Spike emission
Last spike of i
All spikes, all neurons
Spike response model (for details see Gerstner and Kistler, 2002)
time course of the response to an incoming spike
synaptic efficacy
form of the AP and the after-potential
iuij
fjtt ^
itt tui j f
ijw
Spike response model (for details see Gerstner and Kistler, 2002)
0
^ )(),( dsstIsttk exti
external driving current
CglgKv1gNa
I
gKv3
Comparison: detailed vs SRM
I(t)
detailed model
Spike
threshold model (SRM)
<2ms
80% of spikescorrect (+/-2ms)
References
• Rieke, F. et al. (1996). Spikes: Exploring the neural code. MIT Press.
• Izhikevich E. M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press.
• Fitzhugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1:445-466
• Nagumo J. et al. (1962) An active pulse transmission line simulating nerve axon. Proc IRE. 50:2061–2070
• Gerstner, W. and Kistler, W. M. (2002) Spiking Neuron Models. Cambridge University Press. online at: http://diwww.epfl.ch/~gerstner/SPNM/SPNM.html