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.
min max S ( x, y )
x
!
y
Multiple goals
• Analogy to game theory
– zero-sum game
• Equilibrium
• Oscillations
• Complex non-periodic behavior
Synaptic interactions
excitatory
population
inhibitory
population
T
B
A
y
x
!C
!B
• A and C symmetric
• excitatory-inhibitory interpretation
– A, B, C nonnegative matrices
Matrix-vector notation
" x x˙ + x = f ( u + Ax # By )
T
˙
" y y + y = g(v + B x # Cy )
!
Saddle function
• Excitatory neurons try to minimize
• Inhibitory neurons try to maximize
T
1
2
T
T
1
2
T
S = "u x " x Ax + v y " y Cy
T
T
T
T
+1 F ( x ) + y B x "1 G ( y )
• Platt & Barr (1987)
• Mjolness & Garrett (1990)
!
Saddle function gradients
#S
"1
"
= u + Ax " By " f ( x )
#x
"1
"1
˙
= f ($ x x + x ) " f ( x )
#S
T
"1
= v + B x " Cy " g ( y )
#y
=g
"1
($
y
y˙ + y ) " g
"1
( y)
Pseudo gradient ascent-
descent
%S
" x x˙ # $
%x
%S
" y y˙ #
%y
descent
ascent
• The components of these vectors have
the same sign.
!
True gradient ascent-descent
#S
x˙ = "
#x
#S
y˙ =
#y
descent
ascent
• When does this dynamics converge to
the solution of the minimax problem?
!
min max S ( x, y )
x
y
It depends
!
x2 y2
S=
"
2
2
S = xy
steady state
oscillations
!
Steady state
2
2
x
y
S=
"
2
2
#S
x˙ = " = "x
#x
#S
y˙ =
= "y
#y
Periodic behavior
S = xy
#S
x˙ = " = "y
#x
#S
y˙ =
=x
#y
Kinetic energy
2
2
#
S
#
S
T
T
˙
T = " x˙
x˙ + y˙
y˙
2
2
#x
#y
x˙ 2 y˙ 2
T=
+
2
2
!
• lower bounded
• nonincreasing if
!
" 2S
positive definite
2
"x
" 2S
negative definite
2
"y
Proof
# 2S
# 2S
x˙˙ = " 2 x˙ "
y˙
#x
#x#y
#S
˙x = "
#x
#S
y˙ =
#y
# 2S
# 2S
y˙˙ =
x˙ + 2 y˙
#x#y
#y
T˙ = x˙x˙˙ + y˙ y˙˙
2
! # S
2
#
S 2
2
= " 2 x˙ + 2 y˙
#x
#y
!
!
The saddle function could
either increase or decrease
dS
T "S
T "S
T
T
= x˙
+ y˙
= # x˙ x˙ + y˙ y˙
dt
"x
"y
!
Lyapunov function
L = T + rS
2
2
$
'
$
'
#
S
#
S
T
T
L˙ = " x˙ & 2 + rI ) x˙ + y˙ & 2 + rI ) y˙
% #x
(
% #y
(
!
!
" 2S
+ rI positive definite
2
"x
" 2S
+ rI negative definite
2
"y
choose r to satisfy these conditions
and keep L lower bounded
!
Legendre transform pairs
Legendre transformation
F !######" F
F !( x ) = f ( x ) F !( x ) = f
{
"1
(x )
}
F ( x ) = max px ! F (p)
p
!(p, x ) = 1T F (p)" p T x + 1T F ( x )
Generalized kinetic energy
! x x& + x = f (u + Ax " By )
1
2
! x x& 2 "
"# ! x$1%(u + Ax $ By , x )
!(p, x ) = 1T F (p)" p T x + 1T F ( x )
!(p, x ) # 0
!(p, x ) = 0 for f (p) = x
likewise, ! (q , x ) = 1T G(q )" q T x + 1T G ( x )
Lyapunov function
F "( x ) = f ( x ) F "( x ) = f #1 ( x ) G"( x ) = g( x ) G "( x ) = g#1 ( x )
!
kinetic
energy
saddle
function
!
"( p, x ) = 1T F ( p) # pT x + 1T F ( x )
$(q, x ) = 1T G(q) # qT x + 1T G ( x )
S = "uT x " 12 x T Ax + v T y " 12 y T Cy
+1T F ( x ) + y T BT x "1T G ( y )
1
1
Lyapunov
L = #( u + Ax $ By, x ) + %(v + BT x $ Cy, y ) + rS
function
"x
"y
!
Need to verify that L is lower bounded
!
Sufficient conditions for
stability
T
"1
"1
˙
˙
L˙ = x˙ T Ax˙ " y˙ T Cy˙ " (# "1
+
r
x
f
#
x
+
x
"
f
(
)
( x )]
) [ x
x
[
T
"1
"1
˙
˙
+( r " # "1
y
g
#
y
+
y
"
g
( y)
(
)
)
y
y
sufficient condition for L˙ " 0
!
T
max
a,b
(a # b) A(a # b)
(a # b)
T
( f (a) # f (b))
#1
#1
" 1+ r$ x
T
min
a,b
!
(a # b) C (a # b)
(a # b)
T
(g (a) # g (b))
#1
#1
% r$ y #1
]
Excitatory-inhibitory pair
• inhibitory feedback causes oscillations
• self-excitation required to sustain them
"
#
y
x
!"
Competitive network
local excitation
!
x
1
global inhibition
x
!
y
2
x˙ i + x i = f ( ui " y + #x i )
x
3
!
&
)
$y˙ + y = g( % x i +
' i *
Sufficient conditions
T = $ [ F ( ui + "x i # y ) # ( ui + "x i # y ) x i + F ( x i )]
i
%
1 2
V = $'#ui x i # "x i + F ( x i ) + G
&
2
i
(
(
$i x i *)
)
L =T +V +
L˙ = ${"x˙ i2 # (+ #1 + 1) x˙ i [ f #1 ( x˙ i + x i ) # f #1 ( x i )]}
i
!
Conclusion
• excitatory-inhibitory network
• dynamics on a saddle
– gradient ascent/descent
– shape of saddle determines behavior