MIT Department of Brain and Cognitive Sciences 9.641J, Spring 2005 - Introduction to Neural Networks Instructor: Professor Sebastian Seung

Excitatory-inhibitory networks

Sebastian Seung

Two neural populations

• “excitatory” and “inhibitory” • interactions

– within populations: symmetric – between populations: antisymmetric

The two populations of an excitatory-inhibitory network

behave as if they have opposing goals.

Minimax

• An excitatory-inhibitory network is a method of solving a minimax problem.

!

minxmax

yS x,y( )

Multiple goals

• Analogy to game theory – zero-sum game

• Equilibrium • Oscillations • Complex non-periodic behavior

Synaptic interactionsexcitatory inhibitorypopulation population

B

!B

A !Cx y

T

• A and C symmetric • excitatory-inhibitory interpretation

– A, B, C nonnegative matrices

Matrix-vector notation

!

" x˙ x + x = f u + Ax # By( )

" y˙ y + y = g v + B

Tx #Cy( )

Saddle function

• Excitatory neurons try to minimize• Inhibitory neurons try to maximize

!

S = "uT

x " 1

2x

TAx + v

Ty " 1

2y

TCy

+1TF x( ) + y

TB

Tx "1

TG y( )

• Platt & Barr (1987) • Mjolness & Garrett (1990)

Saddle function gradients

!

"#S

#x= u + Ax " By " f

"1x( )

= f"1 $ x

˙ x + x( ) " f"1

x( )

#S

#y= v + B

Tx "Cy " g

"1y( )

= g"1 $ y

˙ y + y( ) " g"1

y( )

Pseudo gradient ascent-descent

!

" x˙ x # $

%S

%x

" y˙ y #

%S

%y

descent

ascent

• The components of these vectors have the same sign.

True gradient ascent-descent

!

˙ x = "#S

#x

˙ y =#S

#y

descent

ascent

• When does this dynamics converge tothe solution of the minimax problem?

!

minxmax

yS x,y( )

It depends

!

S =x2

2"y2

2

!

S = xy

steady state oscillations

Steady state

!

S =x

2

2"

y2

2

˙ x = "#S

#x= "x

˙ y =#S

#y= "y

Periodic behavior

!

S = xy

˙ x = "#S

#x= "y

˙ y =#S

#y= x

Kinetic energy

!

T =˙ x

2

2+

˙ y 2

2

!

˙ T = " ˙ x T #

2S

#x2

˙ x + ˙ y T #

2S

#y2

˙ y

• lower bounded

• nonincreasing if

!

" 2S

"x 2 positive definite

" 2S

"y 2 negative definite

Proof

!

˙ ̇ x = "# 2

S

#x2

˙ x "# 2

S

#x#y˙ y

˙ ̇ y =# 2

S

#x#y˙ x +

# 2S

#y2

˙ y

!

˙ x = "#S

#x

˙ y =#S

#y

!

˙ T = ˙ x ̇ ̇ x + ˙ y ̇ ̇ y

= "# 2

S

#x2

˙ x 2

+# 2

S

#y2

˙ y 2

The saddle function couldeither increase or decrease

!

dS

dt= ˙ x

T "S

"x+ ˙ y

T "S

"y= # ˙ x

T˙ x + ˙ y

T˙ y

Lyapunov function

!

˙ L = " ˙ x T # 2

S

#x2

+ rI$

% &

'

( ) ̇ x + ˙ y

T # 2S

#y2

+ rI$

% &

'

( ) ̇ y

!

L = T + rS

!

" 2S

"x 2+ rI positive definite

" 2S

"y 2+ rI negative definite

choose r to satisfy these conditions and keep L lower bounded

Legendre transform pairs

( ) ( ){ }F x px F pp

= !max

( ) ( ) ( ) ( )! = ! = "F x f x F x f x1

F FLegendre transformation

! "######

( ) ( ) ( )! p x F p p x F xT T T

, = " +1 1

Generalized kinetic energy

( ) ( ) ( )

( )

( ) ( )

!

!

!

p x F p p x F x

p x

p x f p x

T T T,

,

,

= " +

#

= =

1 1

0

0 for

( )1

2

2 1! !x xx u Ax By x& ,"#" + $$%

( )!x x x f u Ax By& + = + "

( ) ( ) ( )likewise, ! q x G q q x G xT T T

, = " +1 1

Lyapunov function

!

" F x( ) = f x( ) " F x( ) = f#1

x( ) " G x( ) = g x( ) " G x( ) = g#1

x( )

kineticenergy

saddlefunction

Lyapunovfunction

!

" p,x( ) = 1TF p( ) # p

Tx + 1T

F x( )

$ q,x( ) = 1TG q( ) # q

Tx + 1T

G x( )

!

S = "uTx " 1

2x

TAx + v

Ty " 1

2y

TCy

+1TF x( ) + y

TB

Tx "1

TG y( )

!

L =1

" x# u + Ax $ By,x( ) +

1

" y% v + BT

x $Cy,y( ) + rS

Need to verify that L is lower bounded

Sufficient conditions forstability

!

˙ L = ˙ x T

A˙ x " ˙ y TC˙ y " # x

"1 + r( ) ˙ x T

f"1 # x

˙ x + x( ) " f"1

x( )[ ]

+ r " # y

"1( ) ˙ y T

g"1 # y

˙ y + y( ) " g"1

y( )[ ]

!

sufficient condition for ˙ L " 0

maxa,b

a # b( )T

A a # b( )

a # b( )T

f#1

a( ) # f#1

b( )( )"1+ r$ x

mina,b

a # b( )TC a # b( )

a # b( )T

g#1

a( ) # g#1

b( )( )% r$ y #1

Excitatory-inhibitory pair• inhibitory feedback causes oscillations

• self-excitation required to sustain them

!"

# x y

"

Competitive networklocal excitation

global inhibition

y! x2

!

x3

!

x1

!

˙ x i + xi = f ui " y +#xi( )

$˙ y + y = g xi

i

%&

' (

)

* +

Sufficient conditions

!

T = F ui +"xi # y( ) # ui +"xi # y( )xi + F xi( )[ ]i

$

V = #uixi #1

2"xi

2 + F xi( ) + G xii

$( )%

& ' (

) * i

$

L = T + V +

˙ L = "˙ x i2 # +#1 +1( ) ˙ x i f

#1˙ x i + xi( ) # f

#1xi( )[ ]{ }

i

$

Conclusion

• excitatory-inhibitory network • dynamics on a saddle

– gradient ascent/descent – shape of saddle determines behavior