simple models of neurons lecture 4
TRANSCRIPT
1!
Simple models of neurons!
!Lecture 4!
2!
Recap: Phase Plane Analysis!
3!
FitzHugh-Nagumo Model!
Membrane potential
K activation variable
Notes from Zillmer, INFN
4!
FitzHugh Nagumo Model Demo!
5!
Phase Plane Analysis - Summary !
Scholarpedia, FitzHugh Nagumo Model
6!
Recap – HH model!
7!
Reducing HH models(2)!
Original
Reduced
8!
Reducing HH models(3)!
Dynamical Systems Neuroscience, Izhikevich
9!
Reducing HH to a 2-D equation!
g For all three variables!
g First approximation: Replace m by its asymptoic value!
Abbott and Kepler, Model Neurons: From HH to Hopfield
10!
Reducing HH to a 2-D equation!g 2-d model!
g We want the time-dependence of U in f in reduced model to approximate the time-dependence of F in the full model by changing h and n!
11!
Two-dimensional version of HH!
12!
Two-dimensional version of HH!
00
0
0
00
0
20
20
20
40
100 50 0 50
70
60
50
40
30
V - nullcline
Limit cycle due to
constant current injection (train of action potentials)
V
U
13!
General HH Model Reduction Strategy!g Start with one equation for V and one for
recovery variable (lets call it R)!
g To match the models in phase space, the first equation has to be a cubic polynomial in V!
[Wilson, Spikes, decisions, and actions, 1998]!
14!
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:!
[Wilson, Spikes, decisions, and actions, 1999]!
15!
General HH Model Reduction Strategy!
!g Fitting to nullclines we get:!
Direct !reduction!
Phase plane!reduction (eqn below)!
[Wilson, Spikes, decisions, and actions, 1999]!
16!
General HH Model Reduction Strategy!
[Wilson, Spikes, decisions, and actions, 1999]!
17!
What is the code here?!g Intracellular recording from a locust
projection neuron to 1s odor puffs!
60 mV
18!
One-Dimensional Reductions!g Perfect Integrate and Fire Model!
!
C dV (t)dt
= I(t)
!
V t( ) =VThr "Fire+reset
linear
threshold
I(t)
C !
"(t # ti)$
tref
V
Whatʼs missing!In this model!
compared to HH?!
19!
One-dimensional Reductions!g Perfect Integrate and Fire Model!
!
V
!
VThr
Spike emission
reset
I(t)
C
I(t)
C !
"(t # ti)$
tref
!
C dV (t)dt
= I(t)
!
V t( ) =VThr "Fire+reset
linear
threshold
V
20!
One dimensional reductions!g The successive times, ti, of spike occurrence:!
!g Firing rate vs. input current of the perfect
integrator:!
!g If you force a refractory period Tref following a
spike, such that V = 0mV for Tref period following a spike, then:!
!
I(t)dt = CVthti
ti+1
"
!
f =I
CVThr
!
f =I
CVThr + tref I
21!
One-dimensional Reductions!g Leaky Integrate and Fire Model!
!
V
!
VThr
Spike emission
reset
I(t)
C
I(t)
C !
"(t # ti)$
tref
!
C dV (t)dt
+V (t)R
= I(t)
!
V t( ) =VThr "Fire+reset
linear
threshold
R
V
22!
Comparison of the two models!
0 50 100 150 200 250 300 350 400 450 5000
1
2
3
4
5
6
7
8
9
10
Time (ms)
V(m
V)
Perfect Integrate & FireLeaky Integrate & FireModel Input
What is the main !difference?!
C = 1nF R = 10MOhm VSPK = 70mV VTHR = 15 mV
I = 1 nA
23!
One-dimensional Reductions!g Adapting Leaky Integrate and Fire Model!
I(t) I(t)
C !
"(t # ti)$
tref
!
C dV (t)dt
+V (t)R
+ gadaptV (t) = I(t)
!
V t( ) =VThr " Fire+reset
linear
threshold
gadapt
V
R
!
"adaptdgadapt (t)
dt= #gadapt (t) adaptation
!
gadapt (t) = gadapt (t) +Ginc
24!
Comparison of the three models!
0 50 100 150 200 250 300 350 400 450 5000
50
100
150
200
250
300
Perfect Integrate & FireLeaky Integrate & FireInputAdapting Integrate & Fire
C = 1nF R = 10MOhm VSPK = 70mV VTHR = 15 mV τG= 50 ms
Ginc = 0.2 nS I = 1nA
25!
Comparison of the three models!
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
50
100
150
200
250
Input Amplitude (nA)
Firin
g ra
te (h
ertz
)
Perfect I&FLeaky I&FAdapt I&F
26!
Low pass filter!
27!
One Dimensional Models!g A more principled approach (again based on
2-d phase plane dynamics!!!)!
W-nullcline meets!left or right segments of!
V-nullcline then the!equilibrium point is stable!
x ’ = x x3/3 y + Iy ’ = p (x + a b y)
I = 0p = 1/13
a = 0.7b = 0.8
2 1.5 1 0.5 0 0.5 1 1.5 22
1.5
1
0.5
0
0.5
1
x
y
W-nullcline meets!Center segments of! V-nullcline then the!
equilibrium point is unstable!
Input current moves !V nullcline!
and !therefore alters stability
of the !equilibrium point!
Fast Variable (V)
Slo
w V
aria
ble
(W)
FitzHugh-Nagumo Model
28!
One Dimensional Models!g A more principled approach (again based on
2-d phase plane dynamics!!!)!
W-nullcline meets!left or right segments of!
V-nullcline then the!equilibrium point is stable!
x ’ = x x3/3 y + Iy ’ = p (x + a b y)
I = 0p = 1/13
a = 0.7b = 0.8
2 1.5 1 0.5 0 0.5 1 1.5 22
1.5
1
0.5
0
0.5
1
x
y
W-nullcline meets!Center segments of! V-nullcline then the!
equilibrium point is unstable!
Input current moves !V nullcline!
and !therefore alters stability
of the !equilibrium point!
Threshold behavior once the current alters!
the stability of the fixed-point (causes
spike)!
Fast Variable (V)
Slo
w V
aria
ble
(W)
FitzHugh-Nagumo Model
29!
One Dimensional Models!g A more principled approach (again based on
2-d phase plane dynamics!!!)!
Dynamical Systems Neuroscience, Izhikevich Sub-threshold dynamics captured by this highlighted region
If I want to construct a integrate and fire
model only this region is important!!!
30!
Quadratic Integrate and Fire Model!g A more principled approach (again based on
2-d phase plane dynamics!!!)!
Dynamical Systems Neuroscience, Izhikevich Sub-threshold dynamics captured by this highlighted region
Notice: in this highlighted region V-nullcline is a parbola U-nullcline is still a line
31!
Izhikevich Model!g A simple model that captures the sub-
threshold behavior in a small neighborhood of the left knee (confined to the shaded square) and the initial segment of the up-stroke of an action potential is given by:!
where a, b, c, d are dimensionless parameters !
32!
Izhikevich Model!
33!
Izhikevich Model!
34!
Firing rate models!g The potential in a continuous firing rate unit
has same dynamics as in a LIF neuron!
I(t)
C
I(t)
C R
V f= g(V)
!
C dV (t)dt
+V (t)R
= I(t)
!
f = g(V ) Sigmoidal function