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5m
úQ¯ ¥(¯ õ2gf¢©c®Û÷ <g ®Úe3ª¨eQ« e ¦ ó e ñ2ªÜi
ð
II
ï
1
alpha 1.0
alpha 0.5
alpha 0.1
alpha 0.05
alpha 0.01
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
nporqjsLtvu1dx 2Lx }M^
7/#_1X#*%*#¨1/#±*1#%=%:
7^2#¦£»2X#1#
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c=*1#_/ec_X#_/d*®1#1T¨_/%-
α ¢8X%%_%)¤-9
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k4/mn/.1KG
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.:35*. _<.(
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_#!1"+!
D T
!!D!d(/N,!1/.D
.1
(
[*(2"!#qd.1 cc
f!4[" [
:!M
.1º ≈ N 1.52 / log(N ) K
1.48
/ log(N
mn!1A/.1q(2*<! N≈3/2N/ log(N
P) )
©
,.1:!(&
:! !A3f!A+?+c(/[(/0!#q¯(
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, [!1"*!!°:q)[
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35*.+!1"*MH Q °=35,"q(2*
(2!_9!!.%(2!.-2! T.13 N ,0!% c/.1_(%9q)[
(
û/(/) m0!% 335.1+!1"*!&+?, *,!335.1+!!1!+(/
mn:!*._9 kI = kN 2 /2 P a " [
¥!1!c!1.1
:!c[m"º),±m
. z 3
[!:!1/.1x(2**!%max
I m"!1!1!D(/mnT.12!1./*
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2
.d.%¼[!:!1/.1d(/:! 3 H = N ρ c/.1 0 < ρ < 1 )q)[n/.%(2max
, c/2I
(%
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T
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d:
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Iρ = N ρ log2 (N 1−ρ ) = (1 − ρ)N ρ log2 (N )
S:
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I@
Æ
2
# of retrieved patterns
10
1
10
1000
400
100
50
10
Counter
0
10
−6
10
−5
−4
10
10
−3
10
−2
10
α
−1
0
10
10
1
10
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+*1#_N
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nporqjsLtvuLx052x
:c(%&_
j
zmax =
N 2−ρ
kN 2
k
=
2(1 − ρ)N ρ log2 (N )
2(1 − ρ) log2 (N )
: :
:
W: >: d:
.
!9(
[*(2"!#qp.1 dº 3/2
K c(%N>!1 -
" c"!!1+2?)W
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ρ!1=x.11/2
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(
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!p[[:_#! /jP a p(2*,!
. )/+
(2Nmn,!°*+ (/"
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á ¼ / log 2 ≈ 0.693 m0!% i :q)[: =¯!1/.1 3m"!1 i q[ c9!1
[" [
:!¬ / *.qº
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!1=log
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c!
(
Amn(%/*
c0!1p(2**!1"),*,:Wj,_°q)[
jCm"!1 i q[<35.
(2
1/2 log 2 ≈ 0.721 35*.d:[.%:º(2 º.1/._K»G
UY)O ^P
S!1D!!¬!17!1/*./!(/+? *(
[*(2"!#q35*.¬c>?)_=S M+?)+*35. (2*D!1
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.q.1c)q)[
.1(/, 7*.4+)q/.1q±+)q)[n/.%ρ(2, !1 "
:!º35
*:m)q)[n/.%(2, =. ρ = 1 − log 2/ log N K(2*..1
[*°!
! T&,"!1º
*(%MP a "
:!º!.1max
/! -*.*d! T&q)[n/.%(2*", K»["q)
P
ρ≥ρ
= log 2/ log N
a
*:
2: :
Z
:
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>:
min
Â
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PEÁ
Z Û
>:
>:
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ð
I*O
ï
15000
Counter1.48
model
0.79 (N −N) / log (N)
2
Retrieved Patterns
Palimpsest
model
0.42 (N1.52−N) / log2(N)
10000
5000
0
0
1000
2000
N
3000
4000
í=*#0%
n%
)/%j
¥º#0
+
jx/#X½%d/T#~c1_#1
*1#%=%:
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x/X^_ /:d#
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.<.1 k(N 2 − N 2−2ρ )/2 K !1 < ?"+n),mn/.
3.12!1./m"<[!:!
.¥m_(2 _
zmax =
¸FP CZQ
k
N 2−ρ − N 2−2ρ
2(1 − ρ)
log2 (N )
c: .!9(/[(/0!#qp*.:d<"
.1q
a (
[*(2"!#qp*º+(/D+?), !
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.
/)!°35.d+ P
a "D!-(/[(/0!#qf.1d: `3¸:!
.»35.¬2!>T: .1;c: "!=.1/.¬<35-: .)q)[
.1(/, á
!1/q. *./:"!=!19*!35.1 3D :WX(^!1"!
v(2, 2¸ "(/!1/.1
.º35:-
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.
(/, 7 *.c.1m,#!_P a 4(
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ρmax
ρ
N
ρ
ρmin
N →∞
α
ρ
ρ
IY
b Ã× fg@®WúT¯ ò ó giÚª e¿bdce S¦Cg@eÜVg Û® c 3× « f¢ªcePõWg@¡5® õWg@fE©c®Û÷*it¬ ò b õ9õ{°
Æ
600
zmax
α=0.01, No noise
α=0.01, 6 disturbed hypercolumns
α=0.1, No noise
α=0.1, 6 disturbed hypercolumns
500
Capacity
400
300
200
100
0
0.1
0.2
0.3
0.4
0.5
ρ
0.6
0.7
0.8
0.9
T* 1x
`#1X#%¨_^±/X#1#*4/M@
"!º*71¡x
#¢
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y *r½ : Ö{: : :
>:
r:
úQ¯ ¥(¯ õ2gf¢©c®Û÷ <g ®Úe3ª¨eQ« e ¦ ó e ñ2ªÜi
ð
IU
ï
2.5
Median convergence time
2
1.5
1
0.5
0
0
10
20
30
40
Pattern position
50
60
70
nporqjsLtvuGLxzy#HLx6| ^*0
°%_¨#_%*%<#=º/)e:
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>:
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>: >: S:
C Kw
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n
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@*E
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
Median convergence time
Median convergence time
Æ
1
0.8
0.6
1
0.8
0.6
0.4
0.4
0.2
0.2
0
0
20
40
60
# patterns
80
0
100
0
20
40
60
# patterns
80
nporqjsLtvuJLxzy{y{x| ^0
%_¨1#
%1-#=¥/)e:
)*/:*%¨0/#

71#%
X
%=/X0__
α = 0.1 ¢81¤7
α = 0.01 ¢#
~¤¦x}x~djx/#±7X:
%
=
7#~*±#~!º/X#1#*%¶ *x
α = 0.1 ~)/D^

%¡±
4_*±2#_C"D=6)#¦
Median convergence time
0.5
0.4
0.3
0.2
10
8
10
6
8
6
4
4
2
List length
0
2
0
List position
^*0
&%_¨1#
%1<#=f/0*1#^/X0* =%=/X+_&
)e
¦
α = 0.4
nporqjsLtvuKdxûyÞwLx|
_X#_±

:
Q !(
MK -
/._K#!1"»mn=./!.1"
°
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100
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1
0.9
0.8
0.7
0.6
0.5
0
0.1
0.2
0.3
0.4
0.5
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0.6
0.7
0.8
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1
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100
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80
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40
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20
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0.8
0.7
0.7
0.6
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0.9
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
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0
5
10
15
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20
25
30
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0
5
10
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0.8
0.7
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Mean overlap
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Kappa=1.0
0.9
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
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2
2.5
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4
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250
500
200
400
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Frequency
150
300
100
200
50
100
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0
5
10
15
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25
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30
35
40
45
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120
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100
80
60
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30
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140
120
100
80
60
40
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0
0
2
4
6
8
10
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14
16
18
20
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200
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π̂i (t) = P
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X
k
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k
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κ(t)τL
dt
βi (t) = log(Λi (t))
κ(t)τL
wij (t) =
(1 − λ0 )Λij (t) + λ0
((1 − λ0 )Λi (t) + λ0 )((1 − λ0 )Λj (t) + λ0 )
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dt
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τA
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dt
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τA
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:
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((1 − λ0 )µi (t) + λ0 )((1 − λ0 )µj (t) + λ0 )
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1
2
3
5
Support
Pattern
4
6
7
8
9
10
1
2
3
4
5
Time (s)
6
7
8
9
0
ÙzÉQÝmwOxQÏÕÑÊfÉ_Í
1
2
3
4
Time (s)
5
6
7
8
9
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Activity
1
0.5
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
Time (msec)
700
800
900
1000
0
−20
−40
−60
Effective bias
Membrane potential
0
20
3
2
1
0
Effective weight
−1
3
2
1
0
−1
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40
Support
20
0
−20
−40
−60
−2.5
−2
−1.5
−1
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Effective bias
0
0.5
1
1.5
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Mutual information
Markov information
2.5
Mutual information (bits)
W
2
1.5
1
0.5
0
0
500
Delay (msec)
1000
1500

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20
20
18
18
16
16
14
14
12
12
Patterns reached
Patterns reached
KW
10
8
10
8
6
6
4
4
2
2
0
0
500
1000
1500
τL
Ù°ÉQÝ
2000
2500
0
3000
0
500
1000
1500
τA
Ùzã{Ý
τL
2000
2500
3000
τA
160
160
140
140
120
120
100
100
Time (msec)
Time (msec)
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80
80
60
60
40
40
20
20
0
0
2
4
6
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8
10
Pattern
τL = 360
12
ÌÎÏØÛ
14
16
18
20
0
0
2
4
6
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8
10
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12
14
16
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1200
1800
Other patterns
Isolate
1600
1000
1400
800
1000
Dwell time (msec)
Mean dwell time (msec)
1200
800
600
600
400
400
200
200
0
0
300
600
900
1200
1500
τA
1800
2100
2400
2700
0
3000
0
1
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2
3
4
5
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6
7
8
9
10
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15
140
120
Mean stay time (timesteps)
Timesteps in attractor
100
80
60
10
5
40
20
0
0
0.05
0.1
0.15
Sum of overlaps
0.2
0.25
0
0
0.02
0.04
Ù°ÉQÝ
0.06
0.08
0.1
Sum of overlaps
0.12
0.14
0.16
0.18
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Adaptation gain modulation
10
0.5
Input on
Input off
1
9
8
gA
6
5
1.5
4
3
Proportion of time in input states
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0.4
0.2
2
2
5
10
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Time (s)
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25
30
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10
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40
50
60
10
20
30
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40
50
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2
4
6
8
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Bayesian Attractor Neural Network Models of Memory

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