Neural codes and spiking models

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Neural codes and spiking models
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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

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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

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Neuronal Codes Action potentials as the
elementary units
voltage clamp from a brain cell of a fly
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Neuronal Codes Action potentials as the
elementary units
voltage clamp from a brain cell of a fly
after band pass filtering
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Neuronal Codes Action potentials as the
elementary units
voltage clamp from a brain cell of a fly
after band pass filtering
generated electronically by a threshold
discriminator circuit
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Neuronal Codes Probabilistic response and
Bayes rule
stimulus
conditional probability
stimulus
spike trains
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Neuronal Codes Probabilistic response and
Bayes rule
natural situation
ensembles of signals
joint probability

• experimental situation
• we choose s(t)

conditional probability
prior distribution
joint probability
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Neuronal Codes Probabilistic response and
Bayes rule
experimental situation

• 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

response-conditional ensemble
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Neuronal Codes Probabilistic response and
Bayes rule
what we see
what our brain sees
Bayes rule
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Neuronal Codes Probabilistic response and
Bayes rule
motion sensitive neuron H1 in the flys brain
determined by the experimenter
property of the neuron
average angular velocity of motion across the VF
spike count
in a 200ms window
correlation
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Neuronal Codes Probabilistic response and
Bayes rule
determine the probability of a stimulus from
given spike train
stimuli
spikes
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Neuronal Codes Probabilistic response and
Bayes rule
determine the probability of a stimulus from
given spike train
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Neuronal Codes Probabilistic response and
Bayes rule
determine probability of a spike train from a
given stimulus
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Neuronal Codes Probabilistic response and
Bayes rule
determine probability of a spike train from a
given stimulus
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Neuronal Codes Probabilistic response and
Bayes rule
How do we measure this time dependent firing
rate?
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Neuronal Codes Probabilistic response and
Bayes rule
Nice probabilistic stuff, but SO, WHAT?
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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
(traditional approach)
(how the brain sees it)
Bayes rule

• -gt 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

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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!

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Neuronal Codes Probabilistic response and
Bayes rule
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Neuronal Codes Probabilistic response and
Bayes rule
average number of spikes depending on stimulus
amplitude
average stimulus depending on spike count
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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 linear relation
Thats interesting, isnt it?
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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.
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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

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Hodgkin Huxley Model
charging current
Ionchannels
with
and
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Hodgkin Huxley Model
asymptotic value

• voltage dependent gating variables

time constant
with
(for the giant squid axon)
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action potential

• If u increases, m increases -gt 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

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General reduction of the Hodgkin-Huxley Model
1) dynamics of m are fast 2) dynamics of h and n
are similar
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General Reduction of the Hodgkin-Huxley Model
2 dimensional Neuron Models
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FitzHugh-Nagumo Model
u membran potential w recovery variable I
stimulus
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FitzHugh-Nagumo Model
nullclines
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FitzHugh-Nagumo Model nullclines
w
I(t)I0
u
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FitzHugh-Nagumo Model nullclines
w
I(t)0
u

• For I0
• convergence to a stable fixed point

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FitzHugh-Nagumo Model
FitzHugh-Nagumo Model nullclines
w
I(t)I0
u
- unstable fixed point
limit cycle
limit cycle
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FitzHugh-Nagumo Model
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The FitzHugh-Nagumo model Absence of
all-or-none spikes
(java applet)

• 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

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The FitzHugh-Nagumo model Excitation block and
periodic spiking
Increasing I shifts the V-nullcline upward-gt
periodic spiking as long as equilibrium is on the
unstable middle branch-gt Oscillations can be
blocked (by excitation) when I increases further
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The Fitzhugh-Nagumo model Anodal break
excitation
Post-inhibitory (rebound) spikingtransient
spike after hyperpolarization
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The Fitzhugh-Nagumo model Spike accommodation

• no spikes when slowly depolarized
• transient spikes at fast depolarization

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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

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Integrate and Fire model
i
Spike reception

• models two key aspects of neuronal excitability
• passive integrating response for small inputs
• stereotype impulse, once the input exceeds a
  particular amplitude

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Integrate and Fire model
Spike emission
i
reset
I
Firereset
threshold
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Integrate and Fire model
I(t)
Time-dependent input
i
I(t)
-spikes are events -threshold -spike/reset/refract
oriness
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Integrate and Fire model (linear)
I0
0
u
-40
-80
u
resting
t
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Integrate and Fire model
linear
non-linear
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Integrate and Fire model (non-linear)
I0
u
Quadratic IF
non-linear
Firereset
threshold
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Integrate and Fire model (non-linear)
I0
u
critical voltage for spike initiation
(by a short current pulse)
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Integrate and Fire model (non-linear)
I0
u
Quadratic IF
non-linear
exponential IF
Firereset
threshold
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Linear integrate-and-fire
Strict voltage threshold - by construction -
spike threshold reset condition
Non-linear integrate-and-fire
There is no strict firing threshold - firing
depends on input - exact reset condition of
minor relevance
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Comparison detailed vs non-linear IF
I
C
gKv1
gKv3
gl
gNa
I(t)
u
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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

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Spike response model (for details see Gerstner
and Kistler, 2002)
generalization of the IF model

• SRM
• parameters depend on the time since the last
  output spike
• integral over the past

• IF
• voltage dependent parameters
• differential equations

• allows to model refractoriness as a combination
  of three components
• reduced responsiveness after an output spike
• increase in threshold after firing
• hyperpolarizing spike after-potential

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Spike response model (for details see Gerstner
and Kistler, 2002)
Spike emission
i
Spike emission AP
Last spike of i
All spikes, all neurons
form of the AP and the after-potential
time course of the response to an incoming spike
synaptic efficacy
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Spike response model (for details see Gerstner
and Kistler, 2002)
i
external driving current
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Spike response model dynamic threshold
threshold
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Comparison detailed vs SRM
80 of spikes correct (/-2ms)
I(t)
I
C
gNa
gKv1
gKv3
gl
detailed model
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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. 1445-466
• Nagumo J. et al. (1962) An active pulse
  transmission line simulating nerve axon. Proc
  IRE. 5020612070
• Gerstner, W. and Kistler, W. M. (2002) Spiking
  Neuron Models. Cambridge University Press. online
  at http//diwww.epfl.ch/gerstner/SPNM/SPNM.html