1 spiking models. 2 neuronal codes spiking models: hodgkin huxley model (brief repetition) reduction...
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Spiking models
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Neuronal codes
Spiking models:
• Hodgkin Huxley Model (brief repetition)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
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Varying firing properties
???
Influence of steady hyperpolarization
Rhythmic burst in the absence of synaptic inputs
Influence of the neurotransmitter Acetylcholin
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Action Potential / Shapes:
Squid Giant Axon Rat - Muscle Cat - Heart
5
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Your neurons surely don‘t like this guy!
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Voltage clamp method
• developed 1949 by Kenneth Cole• used in the 1950s by Alan Hodgkin and Andrew Huxley to measure ion current while maintaining specific membrane potentials
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Voltage clamp method
Small depolarization
Ic: capacity currentIl: leakage current
Large depolarization
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The sodium channel (patch clamp)
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The sodium channel
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Function of the sodium channel
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Hodgkin and Huxley
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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
)( xmxx VVgI
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Hodgkin Huxley Model:
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
huhuh
nunun
mumum
hh
nn
mm
)()1)((
)()1)((
)()1)((
)]([
)(
10 uxx
ux
x
1
0
)]()([)(
)]()([)(
uuu
uuux
xxx
xx
x
with
• voltage dependent gating variables
time constant
asymptotic value
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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
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General Reduction of the Hodgkin-Huxley Model: 2 dimensional Neuron Models
)(),( tIwuFdt
du
stimulus
),( wuGdt
dww
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Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
)8.07.0(08.0 wudt
dw
u: membran potentialw: recovery variableI: stimulus
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Introduction to dynamical systems
Simple Example: Harmonic Oscillator
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Introduction to dynamical systems
Simple Example: Harmonic Oscillator
Force is proportional to displacementof spring F = à kx =mx�
..
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Introduction to dynamical systems
Simple Example: Harmonic Oscillator
Description as differential equation:
One second order equation
Force is proportional to displacementof spring
dt2d2x =à í x
F = à kx =mx�..
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Introduction to dynamical systems
Simple Example: Harmonic Oscillator
Description as differential equation:
One second order equation
Two first order equations
Force is proportional to displacementof spring F = à kx =mx�
dt2d2x =à í x
dtdx
= v
dtdv
= à í x
..
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Analysis of the Dynamics
In the time domain: x v
t
x and v have the same sinusoidaltime course but have a phase shiftof π/2, i.e. when x is in equillibrium position velocity is maximal and when velocity is zero x reaches theamplitude.
xç= vvç= à í x
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Analysis of the Dynamics
In the time domain: x v
t
x and v have the same sinusoidaltime course but have a phase shiftof π/2, i.e. when x is in equillibrium position velocity is maximal and when velocity is zero x reaches theamplitude.
Phase space: Abstract representationof all dynamical parameters
Energy defines shape => Ellipse
v
x
xç= vvç= à í x
ax2+bv2= constant
E = Ekin+Epot= 21mv2+ 2
1kx2= constant
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Analysis of the Dynamics
In the time domain: x v
t
x and v have the same sinusoidaltime course but have a phase shiftof π/2, i.e. when x is in equillibrium position velocity is maximal and when velocity is zero x reaches theamplitude.
Phase space: Abstract representationof all dynamical parameters v
x
Points of return: v = 0
xç= vvç= à í x
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Analysis of the Dynamics
In the time domain: x v
t
x and v have the same sinusoidaltime course but have a phase shiftof π/2, i.e. when x is in equillibrium position velocity is maximal and when velocity is zero x reaches theamplitude.
Phase space: Abstract representationof all dynamical parameters v
x
Equilibrium Position x=0: v = v
max and v = -v
max
xç= vvç= à í x
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Analysis of the Dynamics
In the time domain: x v
t
x and v have the same sinusoidaltime course but have a phase shiftof π/2, i.e. when x is in equillibrium position velocity is maximal and when velocity is zero x reaches theamplitude.
Phase space: Abstract representationof all dynamical parameters v
x
Equilibrium Position x=0: v = v
max and v = -v
max
!!! Important: Same position x = 0, but two different velocities. Happens often that observable (position) has one value, but system is in different STATES (moving upor moving down).
xç= vvç= à í x
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
We have four variables:
xç= vvç= à í xx;v;xç;vç
(xç;vç) (x;v)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
xç= vvç= à í x
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
Arrow up (or down) means x=0, velocity at the location (x,v) does not change AND v=large (in this case even maximal).
This is evident for these two locations as they are the reversal points of the pendulum, where there is no velocity but a maximal acceleration.
.
.
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
xç= vvç= à í x
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
... and others.
xç= vvç= à í x
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
This is special and idealized case: Trajectoryin phase space (ellipse) remains the same.
xç= vvç= à í x
... and others.
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
This is special and idealized case: Trajectoryin phase space (ellipse) remains the same.
It depends only on the the initial conditions(how far we strain the spring in the beginning).
xç= vvç= à í x
... and others.
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
We have four variables:
Examples:
A denser picture of a circular vector field.
... and others.
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
xç= vvç= à í x
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
We have four variables:
Examples:
Here is an example for an oscillator with damping: The phase space trajectory alwaysreturns to the resting position
and
xç= vvç= à í x
... and others.
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
(x;v) = (0;0) (xç;vç) = (0;0)
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Vector Field
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
We have four variables:
Examples:
And vector fields can be really crazy...
xç= vvç= à í x
... and others.
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Fixed Points
For analysis of dynamical systems there are points of specialinterest where the system or a particular variable does notchange, i.e. the vector (xç;vç) = (0;0)
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Fixed Points
For analysis of dynamical systems there are points of specialinterest where the system or a particular variable does notchange, i.e. the vector
Look at our simple example:
For which values (x,v) is the upper equation true and what does this solution mean?
xç= vvç= à í x
(xç;vç) = (0;0)
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Fixed Points
For analysis of dynamical systems there are points of specialinterest where the system or a particular variable does notchange, i.e. the vector
For which values (x,v) is the upper equation true and what does this solution mean?
Look at our simple example:xç= vvç= à í x
(xç;vç) = (0;0)
Only for one single point , the FIXED POINT(x;v) = (0;0)
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Fixed Points
For analysis of dynamical systems there are points of specialinterest where the system or a particular variable does notchange, i.e. the vector
For which values (x,v) is the upper equation true and what does this solution mean?
This is also true for the damped oscillator, but there thefixed point is also an ATTRACTOR, i.e. no matter wherewe start in phase space we will always end up at theresting position. No surprise!
Look at our simple example:xç= vvç= à í x
Only for one single point , the FIXED POINT
In the simplest case: a resting pendulum!
(x;v) = (0;0)
(xç;vç) = (0;0)
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Fixed Points II and Other Stuff
Still interesting where the system doesn't change:
Consider a more general case:xç= f (x;v)vç= g(x;v)
xç= f (x;v) = 0vç= g(x;v) = 0
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Fixed Points II and Other Stuff
Still interesting where the system doesn't change:
Consider a more general case:
The geometric functions that we get from these equations are called
NULLCLINES.
They signify where the vectors of the vector fields are only horizontalor vertical. At the intersection of the the nullclines we find fixed points.
xç= f (x;v)vç= g(x;v)
xç= f (x;v) = 0vç= g(x;v) = 0
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Fixed Points II and Other Stuff
Still interesting where the system doesn't change:
Consider a more general case:
The geometric functions that we get from these equations are called
NULLCLINES.
They signify where the vectors of the vector fields are only horizontalor vertical. At the intersection of the the nullclines we find fixed points.
For the simple spring example we find the trivialnullclines:
xç= f (x;v)vç= g(x;v)
xç= f (x;v) = 0vç= g(x;v) = 0
They intersect at which is the fixed point.(x;v) = (0;0)
xç= 0) v(x) = 0vç= 0) x(v) = 0
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Nullclines
We have four variables:
The change of a point in phase space can be depicted by drawing
the vector of change at each point
v
x
We have four variables:
Examples:
xç= vvç= à í x
(xç;vç) (x;v)
x;v;xç;vç
(x;v) = (æxmax;0) , (xç;vç) = (0;ç í x)
(x;v) = (0;ævmax) , (xç;vç) = (ævmax;0)
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Limit cycles
We know already a simple example of an attractor: A fixed point (red cross).
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Limit cycles
In more complex systems we find other kinds of attractors such as
LIMIT CYCLES
Attracting circular trajectories approached by all other trajectories no matter whether they start from the outside (blue spirals) or from the inside (green spirals).
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Limit cycles
In more complex systems we find other kinds of attractors such as
LIMIT CYCLES
Attracting circular trajectories approached by all other trajectories no matter whether they start from the outside (blue spirals) or from the inside (green spirals).
Attracting trajectory is called limit cycle because mathematically it is reached for all other trajectories in the limit of infinite time – they are getting closer and closer but in theory never quite reach the limit cycle.
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Limit cycles II
Another famous example is the Van-der-Pol-oscillator:
The shape of the limit cycle depends on parameters in the differential equations.
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General Reduction of the Hodgkin-Huxley Model: 2 dimensional Neuron Models
)(),( tIwuFdt
du
stimulus
),( wuGdt
dww
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
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0
0
u
w
u = u + ww = u - w
u = -w
u = w
Phase space
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Hodgkin Huxley Model:
injLmLKmKNamNam IVVgVVngVVhmg
dt
dVC )()()( 43
huhuh
nunun
mumum
hh
nn
mm
)()1)((
)()1)((
)()1)((
• voltage dependent gating variables
m -> 0h = const.
Assumputions that leadto the FH-Model
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Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
)8.07.0(08.0 wudt
dw
u: membran potentialw: recovery variableI: stimulus
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FitzHugh-Nagumo Model
0dt
du
0dt
dw
Iwu
udt
du
3
3
)( wudt
dw
nullclines
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dt
du
0dt
dw
w
u
The vector field always points towards the stable fixed points!
Thus, at every stable fixed point vectors must
turn around.
Unstable fixpoint(vectors point away)
Stable fixpoint(vectors point towards)
Ball in a bowl Ball on a bowl
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dt
du
0dt
dw
w
u
The vector direction at a nullcline is always either horizontal (blue) or vertical (red).
Thus, we can often approximate the behavior of the system by treating the whole vector field as being either horizontal or vertical.
From every given point we just follow the initial vector (which we must know) until we hit a nullcline, then turn as appropriate and move on to the next nullcline, turn 90 deg now and so on!
Some startingpoint
etc.
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0dt
du
0dt
dww
uI(t)=I0
Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
nullclines
stimulus
Adding a constat term to an equation shifts the curve up- (or down-)wards!
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0dt
du
0dt
dw
w
u
Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
nullclines
We had been here (and stable)
I(t)=I0
The system got shifted a bit by current input
We receive a contraction to a new fixed point!
We get a new stable fixed point as soon as the minimum of the new red nullcline is lower and to the right of the old fixed point.
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0dt
du
0dt
dw
w
u
Iwu
udt
du
3
3
)( wudt
dw
FitzHugh-Nagumo Model
nullclines
We had been here (and stable)
I(t)=I0
The system got shifted a lot by current input
We would receive an expansion (divergence) in this case.
up, up!
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0dt
du
0dt
dww
uI(t)=I0
limit cycle represents spiking
FitzHugh-Nagumo Model
Iwu
udt
du
3
3
)( wudt
dw
stimulus
FitzHugh-Nagumo Model
nullclines
Things are not that bad!As the vector field has a curl it will not diverge. Rather we get a limit cycle!
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FitzHugh-Nagumo Model
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FitzHugh-Nagumo ModelGreen area: Passive repolarization, no spike!
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FitzHugh-Nagumo ModelGreen area: Active process, spike!
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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)
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The Fitzhugh-Nagumo model – Anodal break excitation
Post-inhibitory (rebound) spiking:transient spike after hyperpolarization
Shows the effect of the quasi-threshold!
t
V
Original threshold
New „threshold“
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Even with a lot of current we will always hit the left branch of the red (now green!) nullcline and then travel done again. No new spike can be elicted!
Being absolute refractory
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Being relative refractory
With a little current (green) we will again hit the downward branch of the nullcline (as before) with a lot of current (grey) we will, however, hit the blue (other) nullcline and a
new spike will be elicited!etc.
etc.
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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
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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 (brief repetition)
• Reduction of the HH-Model to two dimensions (general)
• FitzHugh-Nagumo Model • Integrate and Fire Model
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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
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iui
Spike reception: EPSP
)()( tRItudt
dum
tui Fire+reset threshold
Spike emission
resetI
j
Integrate and Fire model
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)()( tRItudt
du 0uIf firing:
I=0
dt
du
u
I>0
dt
du
u
resting
t
u
repetitive
t
Integrate and Fire model (linear)
u0
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)()( tRIuFdt
du
)()( tRItudt
du linear
non-linear
0uIf firing:
Integrate and Fire model
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)()( 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)
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)()( 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)
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I=0
dt
du
u
Integrate and Fire model (non-linear)
critical voltagefor spike initiation
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CglgKv1gNa
I
gKv3
I(t)
dt
du
u
)()( tRIuFdt
du
Comparison: detailed vs non-linear I&F
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neuronalfeatures
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79E.M. Izhikevich, IEEE Transactions on Neural Networks, 15:1063-1070, 2004
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)(
140504.0 2
wubaw
Iwuuu
E.M. Izhikevich, 2003
mVu 30if threshold
dww
cu
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)(
140504.0 2
wubaw
Iwuuu
mVu 30
dww
cu
u
w
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End here
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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 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
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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
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iuij
fjtt ^
itt tui j f
ijw
Spike response model (for details see Gerstner and Kistler, 2002)
0
^ )(),( dsstIsttk exti
external driving current
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)'( tt
0)(
dt
tdu
tu Firing: tt '
threshold
^it
Spike response model – dynamic threshold
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CglgKv1gNa
I
gKv3
Comparison: detailed vs SRM
I(t)
detailed model
Spike
threshold model (SRM)
<2ms
80% of spikescorrect (+/-2ms)
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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. 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