Fuzzy BSB-neuro-model

«Brain-State-in-a-Box Model» (BSB-model)

W

)(kx

)1( kx

1z

)(ky

)(kx

Dynamic of BSB-model:

),()1

),)

)y(k,(k,x

(k,Wxx(k,)y(k,

(1)

Activation function:

.,...,2,1

,1),(если ,1

,1),(1-если ),,(

,1),(если ,1

)),(()1,(

ni

ky

kyky

ky

kykxi

ii

i

ii

(2)

2

Artificial neural network of associative memory

)(1 kx

)(2 kx

)(3 kx

)(kxn

)(1 ky

)(2 ky

)(kym

3

Fuzzy clustering

Mapping: nn RkxRx )( (3)

Absolute capacity oflinear auto-associative memory :

1l/n nn 2 (4)

Membership function:

n

xkxdkx q

q 2

)),,((1)),((

* (5)

Hamming distance:

n

iiqiq xkxxkxd

1

*,

* ),()),,(( (6)

4

Adjustment of synaptic weights

Learning of correlation matrix-memory:

.0)0(),1()1()()1( IWkxkykWkW T (7)

Widrow-Hoff autoassociative rule:

).1())1()()1()(()()1( kxkxkWkxkkWkW T (8)

Orthogonal projection:

)()( lXlXW (9)

5

Adjustment of synaptic weights

Recurrent form of projection algorithm:

)()())1(()1()( kbkxkWkWkW (10)

)( nn

2

2

f ( ) ( ) ( ) , if f(k) x(k)-X(k-1)d(k) 0,

b(k) d ( ) ( 1), if

1 ( )

k f k f k

k X kf(k) 0,

d k

-+ T

T +

ìï = = ¹ïïïï=í -ï =ïï +ïïî

(11)

)()1()( kxkXkd (12)

where - unit matrix

6

2

( ) ( )( ( ) ( ) ( )) , z(r)x(r) 1,

1 ( ) ( )W(l,r) X(l,r)X ( , )

( ) ( )( ( ) ( ) ( )) , ( ) ( ) 1,

( )

x r z rW l x r z r if

z r x rl r

z r z rW l x r z r if z r x r

z r

+

T

ì æ öï ÷ï ç- I + ÷ ¹ï ç ÷çï ÷ç -è øïï= =í æ öï ÷çï ÷çï - I - =÷çï ÷çï ÷çè øïî

(14)

)(rz

)(

)()(

rz

rSlX (15)

Delete of the pattern from matrix-memory:

))(),(()( rxrlXlX (13)

Adjustment of synaptic weights

Learning algorithm:

7

where - last row of matrix