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