[PDF] - CSE/NB 528 Lecture 10: Recurrent Networks (Chapter 7) Lecture PDF Document

SLIDE 1

1

R. Rao, CSE528: Lecture 10

CSE/NB 528 Lecture 10: Recurrent Networks

(Chapter 7)

Lecture figures are from Dayan & Abbott’s book http://people.brandeis.edu/~abbott/book/index.html 2

R. Rao, CSE528: Lecture 10

What’s on our smörgåsbord today?

F Computation in Linear Recurrent Networks Eigenvalue analysis F Non-linear Recurrent Networks Eigenvalue analysis F Covered in: Chapter 7 in Dayan & Abbott

SLIDE 2

3

R. Rao, CSE528: Lecture 10

Linear Recurrent Networks

v u v v M W     dt d 

Output Decay Input Feedback

4

R. Rao, CSE528: Lecture 10

What can a Linear Recurrent Network do?

On-Board analysis based on eigenvectors of recurrent weight matrix M

v u v v M W     dt d 

SLIDE 3

5

R. Rao, CSE528: Lecture 10

Example of a Linear Recurrent Network

Recurrent connections M = cosine function of relative angle ( - ’) +

Excitation nearby,

Inhibition further away

) ' cos( ) ' , (       M

Is M symmetric? M(, ’)= M(’, ’)? Each neuron codes for an angle between -180 to +180 degrees

6

R. Rao, CSE528: Lecture 10

Example of a Linear Recurrent Network

Each neuron has a preferred angle between -180 to +180 degrees

) ' cos( ) ' , (       M

All eigenvalues = 0 except

10

1

1 9 .

1 1

     fication i.e. ampli

(See section 7.4 in Dayan & Abbott)

SLIDE 4

7

R. Rao, CSE528: Lecture 10

Example: Amplification in a Linear Recurrent Network Input Output (noisy cosine)

Preferred angle of neuron

8

R. Rao, CSE528: Lecture 10

Example: Memory for Maintaining Eye Position

Input: Bursts of spikes from brain stem oculomotor neurons Output: Memory of eye position in medial vestibular nucleus

SLIDE 5

9

R. Rao, CSE528: Lecture 10

Nonlinear Recurrent Networks

) M W ( v u v v     F dt d 

Output Decay Input Feedback (Convenient to use Wu = h)

10

R. Rao, CSE528: Lecture 10

Amplification in a Nonlinear Recurrent Network

Input Output 9 . 1

1 

 (but stable due to rectification)

(F = rectification nonlinearity: F(x) = x if x > 0 and 0 o.w.)

SLIDE 6

11

R. Rao, CSE528: Lecture 10

Selective “Attention” in a Nonlinear Recurrent Network

Network performs “winner-takes-all” input selection Input Output

12

R. Rao, CSE528: Lecture 10

Gain Modulation in a Nonlinear Recurrent Network

Inputs Outputs Changing the level of input multiplies the output

SLIDE 7

13

R. Rao, CSE528: Lecture 10

Gain Modulation in Parietal Cortex Neurons

Gaze 1 Gaze 2 Responses of Area 7a neuron Example of a gain- modulated tuning curve

14

R. Rao, CSE528: Lecture 10

Short-Term Memory Storage in a Nonlinear Recurrent Network

Local Input + Output Background Turn off input Output

Network maintains a memory of previous activity when input is turned off. Similar to “short- term memory” or “working memory” in prefrontal cortex Memory is maintained by recurrent activity

SLIDE 8

15

R. Rao, CSE528: Lecture 10

What about Non-Symmetric Recurrent Networks?

F Example: Network of Excitatory (E) and Inhibitory (I)

Neurons

Connections can’t be symmetric: Why?

   

          

I E IE I II I I I E I EI E EE E E E

v M v M v dt dv v M v M v dt dv    

Simple 2-neuron network representing interacting populations One excitatory neuron and one inhibitory neuron

16

R. Rao, CSE528: Lecture 10

Stability Analysis of Nonlinear Recurrent Networks

dt d t J t dt d e i t t dt d dt d t t dt d ε ε ε v f v ε v f v f v f ε v ε v v v v f v v f v

v v

              

 

    

) ( ) ( . . ) ( ) ( ) ) ( ( : expansion Taylor ) (i.e., ) ( ) ( , Near 0) ) ( (i.e., point fixed a is Suppose ) ( : case General

(see Mathematical Appendix A.3 in textbook)

Derive solution for v(t) based on eigen-analysis of J Eigenvalues of J determine stability of network J is the “Jacobian matrix”

SLIDE 9

17

R. Rao, CSE528: Lecture 10

Example: Non-Symmetric Recurrent Networks

F Specific Network of Excitatory (E) and Inhibitory (I)

Neurons:

   

          

I E IE I II I I I E I EI E EE E E E

v M v M v dt dv v M v M v dt dv    

1.25

1 -10

1 10 10 ms Parameter we will vary to study the network

18

R. Rao, CSE528: Lecture 10

Linear Stability Analysis

F Matrix of derivatives (the “Jacobian Matrix”):

              

I II I IE E EI E EE

M M M M J     ) 1 ( ) 1 (

   

I I E IE I II I I E E I EI E EE E E

v M v M v dt dv v M v M v dt dv              

Take derivatives of right hand side with respect to both vE and vI

SLIDE 10

19

R. Rao, CSE528: Lecture 10

Compute the Eigenvalues

F Jacobian Matrix: F Its two eigenvalues (obtained by solving det(J – I) = 0):

              

I II I IE E EI E EE

M M M M J     ) 1 ( ) 1 (

                          

I E IE EI I II E EE I II E EE

M M M M M M        4 1 1 ) 1 ( ) 1 ( 2 1

2

Different dynamics depending on real and imaginary parts of 

(see pages 410-412 of Appendix in Text)

20

R. Rao, CSE528: Lecture 10

Phase Plane and Eigenvalue Analysis

   

            10 10 25 . 1 10

E I I I I I E E E

v v v dt dv v v v dt dv 

I = 30 ms 50ms

SLIDE 11

21

R. Rao, CSE528: Lecture 10

Damped Oscillations in the Network

I = 30 ms (negative real eigenvalue) Stable Fixed Point

22

R. Rao, CSE528: Lecture 10

Unstable Behavior and Limit Cycle

I = 50 ms (positive real eigenvalue) Limit cycle

SLIDE 12

23

R. Rao, CSE528: Lecture 10

Oscillatory Activity in Real Networks

Activity in rabbit (or wabbit)

lfactory bulb during 3 sniffs

Sniff Sniff Sniff

(see Chapter 7 in textbook for details)

24

R. Rao, CSE528: Lecture 10