simple models of neurons lecture 4

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Simple models of neurons Lecture 4

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Page 1: Simple models of neurons Lecture 4

1!

Simple models of neurons!

!Lecture 4!

Page 2: Simple models of neurons Lecture 4

2!

Recap: Phase Plane Analysis!

Page 3: Simple models of neurons Lecture 4

3!

FitzHugh-Nagumo Model!

Membrane potential

K activation variable

Notes from Zillmer, INFN

Page 4: Simple models of neurons Lecture 4

4!

FitzHugh Nagumo Model Demo!

Page 5: Simple models of neurons Lecture 4

5!

Phase Plane Analysis - Summary !

Scholarpedia, FitzHugh Nagumo Model

Page 6: Simple models of neurons Lecture 4

6!

Recap – HH model!

Page 7: Simple models of neurons Lecture 4

7!

Reducing HH models(2)!

Original

Reduced

Page 8: Simple models of neurons Lecture 4

8!

Reducing HH models(3)!

Dynamical Systems Neuroscience, Izhikevich

Page 9: Simple models of neurons Lecture 4

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

Page 10: Simple models of neurons Lecture 4

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!

Page 11: Simple models of neurons Lecture 4

11!

Two-dimensional version of HH!

Page 12: Simple models of neurons Lecture 4

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

Page 13: Simple models of neurons Lecture 4

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]!

Page 14: Simple models of neurons Lecture 4

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]!

Page 15: Simple models of neurons Lecture 4

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]!

Page 16: Simple models of neurons Lecture 4

16!

General HH Model Reduction Strategy!

[Wilson, Spikes, decisions, and actions, 1999]!

Page 17: Simple models of neurons Lecture 4

17!

What is the code here?!g  Intracellular recording from a locust

projection neuron to 1s odor puffs!

60 mV

Page 18: Simple models of neurons Lecture 4

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?!

Page 19: Simple models of neurons Lecture 4

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

Page 20: Simple models of neurons Lecture 4

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

Page 21: Simple models of neurons Lecture 4

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

Page 22: Simple models of neurons Lecture 4

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

Page 23: Simple models of neurons Lecture 4

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

Page 24: Simple models of neurons Lecture 4

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

Page 25: Simple models of neurons Lecture 4

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

Page 26: Simple models of neurons Lecture 4

26!

Low pass filter!

Page 27: Simple models of neurons Lecture 4

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

Page 28: Simple models of neurons Lecture 4

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

Page 29: Simple models of neurons Lecture 4

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!!!

Page 30: Simple models of neurons Lecture 4

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

Page 31: Simple models of neurons Lecture 4

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 !

Page 32: Simple models of neurons Lecture 4

32!

Izhikevich Model!

Page 33: Simple models of neurons Lecture 4

33!

Izhikevich Model!

Page 34: Simple models of neurons Lecture 4

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