Simple models of
neurons!
!
Lecture 4!
1!
Recap: Phase Plane Analysis!
2!
FitzHugh-Nagumo Model!
Membrane potential
K activation variable
Notes from Zillmer, INFN
3!
FitzHugh Nagumo Model Demo!
4!
Phase Plane Analysis - Summary !
Scholarpedia, FitzHugh Nagumo Model
5!
Recap – HH model!
6!
Reducing HH models(2)!
Original
Reduced
7!
Reducing HH models(3)!
Dynamical Systems Neuroscience, Izhikevich8!
Reducing HH to a 2-D equation!
g
g
For all three variables!
First approximation:
Replace m by its
asymptoic value!
Abbott and Kepler, Model Neurons: From HH to Hopfield
9!
Reducing HH to a 2-D equation!
g
g
2-d model!
We want the time-dependence of U in f in
reduced model to approximate the timedependence of F in the full model by
changing h and n!
10!
Two-dimensional version of HH!
11!
Two-dimensional version of HH!
Limit cycle
due to
constant current injection
(train of action potentials)
ï30
0
ï40
V - nullcline
0
ï50
0
0
0
ï60
0
20
ï100
40
20
ï70
20
U
0
ï50
0
50
V
12!
General HH Model Reduction Strategy!
g
g
Start with one equation for V and one for
recovery variable (lets call it R)!
To match the models in phase space, the first
equation has to be a cubic polynomial in V!
[Wilson, Spikes, decisions, and actions, 1998]!
13!
General HH Model Reduction Strategy!
g
Start with one equation for V and one for
recovery variable (lets call it R)!
!
g
Null clines of this system of equations are:!
14!
[Wilson, Spikes, decisions, and actions, 1999]!
General HH Model Reduction Strategy!
Direct !
reduction!
Phase plane!
reduction (eqn below)!
!
g
Fitting to nullclines we get:!
15!
[Wilson, Spikes, decisions, and actions, 1999]!
General HH Model Reduction Strategy!
[Wilson, Spikes, decisions, and actions, 1999]!
16!
What is the code here?!
g
Intracellular recording from a locust
projection neuron to 1s odor puffs!
60 mV
17!
One-Dimensional Reductions!
g
Perfect Integrate and Fire Model!
$" (t # t )
V
I(t)
C
i
Whatʼs missing!
In this model!
compared to HH?!
! t
ref
dV (t)
linear
C
= I(t)
dt
V ( t ) = VThr " Fire+reset threshold
18!
One-dimensional Reductions!
g
Perfect Integrate and Fire Model!
Spike emission
$" (t # t )
V
I(t)
C
!
i
V
VThr
reset
tref
!
!
dV (t)
linear
C
= I(t)
dt
V ( t ) = VThr " Fire+reset threshold
19!
One dimensional reductions!
g
The successive times, ti, of spike occurrence:!
t i +1
!
g
!
g
" I(t)dt = CV
th
ti
Firing rate vs. input current of the perfect
integrator:!
I
!
f =
CVThr
If you force a refractory period Tref following a
spike, such that V = 0mV for Tref period
following
! a spike, then:!
I
f =
CVThr + t ref I
20!
One-dimensional Reductions!
g
I(t)
C
Leaky Integrate and Fire Model!
$" (t # t )
V
R
Spike emission
i
V
VThr
reset
! t
ref
!
!
dV (t) V (t)
linear
C
+
= I(t)
dt
R
V ( t ) = VThr " Fire+reset threshold
21!
Comparison of the two models!
10
Perfect Integrate & Fire
Leaky Integrate & Fire
Model Input
9
8
7
V(mV)
6
C = 1nF
R = 10MOhm
VSPK = 70mV
VTHR = 15 mV
I = 1 nA
5
4
3
What is the main !
difference?!
2
1
0
0
50
100
150
200
250
300
Time (ms)
350
400
450
500
22!
One-dimensional Reductions!
g
I(t)
Adapting Leaky Integrate and Fire Model!
$" (t # ti )
V
C R
! t
gadapt
ref
dV (t) V (t)
C
+
+ gadaptV (t) = I(t)
dt
R
" adapt
dgadapt (t)
= #gadapt (t)
dt
linear
adaptation
V ( t ) = VThr " Fire+reset threshold gadapt (t) = gadapt (t) + Ginc
23!
Comparison of the three models!
300
Perfect Integrate & Fire
Leaky Integrate & Fire
Input
Adapting Integrate & Fire
250
200
C = 1nF
R = 10MOhm
VSPK = 70mV
VTHR = 15 mV
τG= 50 ms
Ginc = 0.2 nS
I = 1nA
150
100
50
0
0
50
100
150
200
250
300
350
400
450
500
24!
Comparison of the three models!
250
Perfect I&F
Leaky I&F
Adapt I&F
Firing rate (hertz)
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
Input Amplitude (nA)
3.5
4
4.5
5
25!
Low pass filter!
26!
One Dimensional Models!
g
A more principled approach (again based on
2-d phase plane dynamics!!!)!
3
a = 0.7
b = 0.8
x ’ = x ï x /3 ï y + I
y ’ = p (x + a ï b y)
I=0
p = 1/13
1
y
Input current moves!
V nullcline!
and!
therefore alters stability
of the!
equilibrium point!
Slow Variable (W)
0.5
W-nullcline meets!
left or right segments of!
V-nullcline then the!
equilibrium point is stable!
0
ï0.5
ï1
W-nullcline meets!
Center segments of!
V-nullcline then the!
equilibrium point is unstable!
ï1.5
FitzHugh-Nagumo Model
ï2
ï2
ï1.5
ï1
ï0.5
0
x
0.5
Fast Variable (V)
1
1.5
2
27!
One Dimensional Models!
g
A more principled approach (again based on
2-d phase plane dynamics!!!)!
3
a = 0.7
b = 0.8
x ’ = x ï x /3 ï y + I
y ’ = p (x + a ï b y)
I=0
p = 1/13
1
y
Input current moves!
V nullcline!
and!
therefore alters stability
of the!
equilibrium point!
Slow Variable (W)
0.5
0
W-nullcline meets!
left or right segments of!
V-nullcline then the!
equilibrium point is stable!
Threshold behavior
once the current alters!
the stability of the
fixed-point (causes
spike)!
ï0.5
ï1
W-nullcline meets!
Center segments of!
V-nullcline then the!
equilibrium point is unstable!
ï1.5
FitzHugh-Nagumo Model
ï2
ï2
ï1.5
ï1
ï0.5
0
x
0.5
Fast Variable (V)
1
1.5
2
28!
One Dimensional Models!
g
A more principled approach (again based on
2-d phase plane dynamics!!!)!
If I want to construct a
integrate and fire
model only this region
is important!!!
29!
Sub-threshold dynamics
Dynamical Systems Neuroscience, Izhikevich
captured by this highlighted region
Quadratic Integrate and Fire Model!
g
A more principled approach (again based on
2-d phase plane dynamics!!!)!
Notice: in this highlighted region
V-nullcline is a parbola
U-nullcline is still a line
30!
Sub-threshold dynamics
Dynamical Systems Neuroscience, Izhikevich
captured by this highlighted region
Izhikevich Model!
g
A simple model that captures the subthreshold behavior in a small neighborhood
of the left knee (confined to the shaded
square) and the initial segment of the upstroke of an action potential is given by:!
where a, b, c, d are dimensionless parameters !
31!
Izhikevich Model!
32!
Izhikevich Model!
33!
Firing rate models!
g
I(t)
C
The potential in a continuous firing rate unit
has same dynamics as in a LIF neuron!
f= g(V)
V
R
dV (t) V (t)
C
+
= I(t)
dt
R
f = g(V )
Sigmoidal function
34!