Neural codes and spiking models. Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions.

Neural codes and spiking models

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:





Neuronal Codes – Action potentials as the elementary units

voltage clamp from a brain cell of a fly



after band pass filtering



after band pass filtering

generated electronicallyby a threshold discriminatorcircuit

Neuronal Codes – Probabilistic response and Bayes’ rule

stimulus

)(|}{ tstP i

stimulusspiketrains

conditional probability:


conditionalprobability

)(tsPensembles of signals

natural situation:

)(},{ tstP ijoint probability:

experimental situation:

• we choose s(t)

)()(|}{)(},{ tsPtstPtstP ii prior

distributionjoint

probability


• 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


)()(|}{)(},{ tsPtstPtstP ii

}{}{|)()(},{ iii tPttsPtstP

)()(|}{}{}{|)( tsPtstPtPttsP iii

}{

)()(|}{}{|)(

iii tP

tsPtstPttsP

Bayes’ rule:

what we see:

what ourbrain “sees”:


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


}{|)( ittsP

spikes

determine the probability of astimulus from given spike train

stimuli


}{|)( ittsPdetermine the probability of astimulus from given spike train


)(|}{ tstP i

determine probability ofa spike trainfrom a given stimulus


)(|}{ tstP i

)(tr

determine probability ofa spike trainfrom a given stimulus


)(trHow do we measure this time dependent firing rate?


Nice probabilistic stuff, but

SO, WHAT?


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)


We can switch between these two points of view.

And why is that important?

These two points of view may differ in their complexity!



average number of spikes

depending on stimulus amplitude

average stimulus depending on

spike count


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?


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:

)()( 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:


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


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

dw

Iwu

udt

du

3

3

)( wudt

dw

nullclines

0dt

du

0dt

dww

uI(t)=I0

Iwu

udt

du

3

3

)( wudt

dw


nullclines

stimulus

0dt

du

0dt

dww

uI(t)=0

Iwu

udt

du

3

3

)( wudt

dw

For I=0: • convergence to a stable fixed point


nullclines

0dt

du

0dt

dww

uI(t)=I0

limit cycle

- unstable fixed point

limit cycle


Iwu

udt

du

3

3

)( wudt

dw

stimulus


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:





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


ui

-spikes are events-threshold-spike/reset/refractoriness

I(t)

I(t)

Time-dependent input


)()( 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

)()( tRIuFdt

du

)()( tRItudt

du linear

non-linear

resetuu If firing:


)()( 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


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:


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:

CglgKv1gNa

I

gKv3

I(t)

dt

du

u

)()( tRIuFdt

du

Comparison: detailed vs non-linear I&F

Neuronal codes

Spiking models:





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


fjtt


îtt

îtt

Spike emission: AP

fjtt ^

itt tui j f

ijw

tui Firing: tti ^

Spike emission

Last spike of i

All spikes, all neurons


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


0

^ )(),( dsstIsttk exti

external driving current

)'( tt

0)(

dt

tdu

tu Firing: tt '

threshold

ît

Spike response model – dynamic threshold

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

Neural codes and spiking models. Neuronal codes Spiking models: Hodgkin Huxley Model (small regeneration) Reduction of the HH-Model to two dimensions.

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