Transcript Representation of movement in near extra

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:
• 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
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 electronically
by a threshold discriminator
circuit
Neuronal Codes – Probabilistic response and Bayes’ rule
stimulus
conditional probability:
P{ti } | s(t )
spike
trains
stimulus
Neuronal Codes – Probabilistic response and Bayes’ rule
natural situation:
ensembles of signals
joint probability:
Ps(t )
P{ti }, s(t )
experimental situation:
• we choose s(t)
P{ti }, s(t )  P{ti } | s(t ) Ps(t )
joint
probability
conditional
probability
prior
distribution
Neuronal Codes – Probabilistic response and Bayes’ rule
experimental situation:
P{ti }, s(t )  P{ti } | s(t ) Ps(t )
• 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
P{ti }, s(t )  Ps(t ) | {ti } P{ti }
response-conditional ensemble
Neuronal Codes – Probabilistic response and Bayes’ rule
what we see:
P{ti }, s(t )  P{ti } | s(t ) Ps(t )
what our
brain “sees”:
P{ti }, s(t )  Ps(t ) | {ti } P{ti }
Ps(t ) | {ti } P{ti }  P{ti } | s(t ) Ps(t )
Bayes’ rule:
Ps(t )
Ps(t ) | {ti }  P{ti } | s(t )
P{ti }
Neuronal Codes – Probabilistic response and Bayes’ rule
motion sensitive neuron H1 in the fly’s brain:
determined by
the experimenter
property of the
neuron
average angular velocity
of motion across the VF
spike count
in a 200ms window
Pn, v   P(n) P(v)
correlation
Neuronal Codes – Probabilistic response and Bayes’ rule
determine the probability of a
stimulus from given spike train
Ps(t ) | {ti }
stimuli
spikes
Neuronal Codes – Probabilistic response and Bayes’ rule
determine the probability of a
stimulus from given spike train
Ps(t ) | {ti }
Neuronal Codes – Probabilistic response and Bayes’ rule
determine probability of
a spike train
from a given stimulus
P{ti } | s(t )
Neuronal Codes – Probabilistic response and Bayes’ rule
determine probability of
a spike train
from a given stimulus
P{ti } | s(t )
r (t )
Neuronal Codes – Probabilistic response and Bayes’ rule
How do we measure this time dependent firing rate?
r (t )
Neuronal Codes – Probabilistic response and Bayes’ rule
Nice probabilistic stuff, but
SO, WHAT?
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
P{ti } | s(t )
Ps(t ) | {ti }
(traditional approach)
(how the brain “sees” it)
Bayes’ rule:
Ps(t )
Ps(t ) | {ti }  P{ti } | s(t )
P{ti }
-> 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
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
Neuronal Codes – Probabilistic response and Bayes’ rule
average stimulus
depending on
spike count
average number
of spikes
depending on
stimulus amplitude
Neuronal Codes – Probabilistic response and Bayes’ rule
average number
of spikes
depending on
stimulus amplitude
average stimulus
depending on
spike count
almost perfectly linear
relation
non-linear
relation
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:
charging
current
Ion
channels
I inj (t )  I C (t )   I k (t )
k
with
Q
C
u
and
du
dV
IC  C
C
dt
dt
dVm
C
  I k (t )  I inj (t )
dt
k
3
4
I

g
m
h
(
V

V
)

g
n
(Vm  VK )  g L (Vm  VL )
 k Na
m
Na
K
k
dVm
C
  g Na m3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
Hodgkin Huxley Model:
dVm
C
  g Na m3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
• voltage dependent gating variables
m   m (u )(1  m)   m (u )m
n   n (u )(1  n)   n (u )n
h   (u )(1  h)   (u )h
h
asymptotic
value
1
x  
[ x  x0 (u )]
 x (u )
time
constant
h
with
x0 (u ) 
(for the giant squid axon)
x
[ x (u )   x (u )]
 x (u )  [ x (u )   x (u )]1
dVm
C
  g Na m3 h(Vm  VNa )  g K n 4 (Vm  VK )  g L (Vm  VL )  I inj
dt
1
x  
[ x  x0 (u )]
 x (u )
action potential
• 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
General reduction of the Hodgkin-Huxley Model
I Na
IK
stimulus
I leak
du
C
 g Na m3 h(u  VNa )  g K n 4 (u  VK )  g l (u  VL )  I (t )
dt
1) dynamics of m are fast
2) dynamics of h and n are similar
General Reduction the of Hodgkin-Huxley Model:
2 dimensional Neuron Models
stimulus
du

 F (u, w)  I (t )
dt
dw
w
 G (u , w)
dt
FitzHugh-Nagumo Model
du
u3
 u   w I
dt
3
dw
  (u    w)
dt
u: membran potential
w: recovery variable
I: stimulus
dw
 0.08(u  0.7  0.8w)
dt
FitzHugh-Nagumo Model
nullclines
3
du
u
 u   w I
dt
3
dw
  (u    w)
dt
du
0
dt
dw
0
dt
FitzHugh-Nagumo Model
nullclines
stimulus
du
u3
 u   w I
dt
3
dw
  (u    w)
dt
w
dw
0
dt
I(t)=I0
u
du
0
dt
FitzHugh-Nagumo Model
nullclines
du
u3
 u   w I
dt
3
w
dw
  (u    w)
dt
For I=0:
• convergence to a stable fixed point
dw
0
dt
I(t)=0
u
du
0
dt
FitzHugh-Nagumo Model
nullclines
stimulus
du
u3
 u   w I
dt
3
w
dw
0
dt
dw
  (u    w)
dt
I(t)=I0
u
- unstable fixed point
limit cycle
du
0
dt
limit cycle
FitzHugh-Nagumo Model
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
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
Integrate and Fire model
Spike emission
j
i
ui
Spike reception
models two key aspects of neuronal excitability:
• passive integrating response for small inputs
• stereotype impulse, once the input exceeds a particular amplitude

Integrate and Fire model
Spike emission
j
i
ui
I

du
m
 u (t )  RI (t )
dt
ui t     Fire+reset
Spike reception: EPSP
reset
threshold

Integrate and Fire model
I(t)
Time-dependent input
i
u

I(t)

-spikes are events
-threshold
-spike/reset/refractoriness
Integrate and Fire model (linear)
du

 u (t )  RI (t )
dt
I=0
du
dt
If firing:
du
dt
u  u reset
I>0
0
-40
-80
u

u
repetitive
u
resting
t

u

t
Integrate and Fire model
du

 u (t )  RI (t )
dt
du

 F (u )  RI (t )
dt
If firing:
linear
non-linear
u  u reset
Integrate and Fire model (non-linear)
du
dt
I=0
du
dt
u

d
 u  F (u )  RI (t ) non-linear
dt
ut     Fire+reset threshold
I>0
u

Quadratic I&F:
F (u)  c2u  c0
2
Integrate and Fire model (non-linear)
du
dt
I=0
u

critical voltage
for spike initiation
(by a short current pulse)
Integrate and Fire model (non-linear)
d
u
dt
I=0
d
u
dt
u

d
  u  F (u )  RI (t ) non-linear
dt
ut     Fire+reset threshold
I>0
u

Quadratic I&F:
F (u)  c2u  c0
2
exponential I&F:
F (u)  u  c0 exp(u   )
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
Comparison: detailed vs non-linear I&F
I
C
gNa gKv1 gKv3 gl
du
dt
I(t)


du
 F (u )  RI (t )
dt
u
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:
I&F:
• parameters depend on the time since
the last output spike
• voltage dependent parameters
• integral over the past
• 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
Spike response model
(for details see Gerstner and Kistler, 2002)
Spike emission
t  ti^ 
j
i
Spike reception: EPSP
 t  t
f
j
ui


Spike reception: EPSP
 t  t
Spike emission: AP
t  t
^
i


Last spike of i
form of the AP
and the after-potential
ui t   t  t 
^
i
ui t    
f
j
All spikes, all neurons
 w  t  t 
ij
j
Firing:
f
f
j
time course of the
response to an
^
incoming spike
i
synaptic efficacy
t t
Spike response model
(for details see Gerstner and Kistler, 2002)
j
i
ui
ui t   t  t 
^
i

 w  t  t 
ij
 j
f
f
j
external driving
current
  k (t  ti^ , s ) I ext (t  s ) ds
0
Spike response model – dynamic threshold
   (t  t ' )
ti^
threshold
u t   
du(t )
0
dt
Firing:
t'  t
Comparison: detailed vs SRM
<2ms
80% of spikes
correct (+/-2ms)
I(t)
I
C
Spike

gNa gKv1 gKv3 g
l
detailed model
threshold model (SRM)
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