Theoretical Neuroscience Computational and Mathematical Modeling of Neural Systems-Dayan,Abbott

. ,

, , 1991. , . . ,

1992 : ,

. , , . , , 1992

, , 1993 ,

. , 1994 :

, , , . , 1995

, ., . , . , 1995

: , , , , , 1997

, , , .. , , . , . , 1997

: , , , 1998

: , . , . , . , 1998

: , . , 1998

: , . , 1998

, . , .. , . , 1999

: , . , 2000

: , .. , 2001

..

, ,

2001 . ( , , ) .

2. .

, . :

/ .. .. . ^ ( )

. 0262041995 (. : . )1. () ^ . 2.

^ . 3. .. , .. . . .

363.3 .39 2001573.8a01a1321

2001044005

This page intentionally left blank

1

1 : 31.1 31.2 81.3 ? 171.4 241.5 341.6 391.7 401.8 43

2 : 452.1 452.2 452.3 512.4 : 602.5 : 742.6 772.7 1 792.8 812.9 812.10 84

3 873.1 873.2 893.3 973.4 1133.5 118

3.6 1193.7 122

4 1234.1 1234.2 1304.3 1454.4 1494.5 1504.6 150

151

5 : 1535.1 1535.2 1535.3 1615.4 1625.5 1665.6 1735.7 1755.8 1785.9 1885.10 1915.11 1915.12 193

6 : 1956.1 1956.2 1956.3 2036.4 2176.5 2246.6 2246.7 228

7 2297.1 2297.2 2317.3 241

Ani Desu

Ani Desu

7.4 2447.5 2657.6 2737.7 2767.8 2767.9 277

279

8 2818.1 2818.2 2848.3 2938.4 3138.5 3268.6 3278.7 328

9 3319.1 3319.2 3329.3 3409.4 3469.5 3549.6 3559.7 357

10 35910.1 35910.2 36810.3 37310.4 38910.5 39410.6 39510.7 396

399.1 399.2 408.3 410

.4 413.5 415.6 418

419

439

http://mitpress.mit.edu/dayan-abbott

, , , . . . , , , , , , , .

.

, , . . , , . , .

, , , , , . , . ,, , , . , , , , . . , .

. . , .

. , , ( 1^4)

. , (5^7). , ( 810). 5 6, , .

, , , & (2000); , , & (1992);, , & (1996); (1997); . (1998); . (2000).

, , . , . . , . , , .

, , . , http://mitpress.mit.edu/dayan-abbott. . , .

, . ( ), , . . . . . .

, , , , , . , , , , , , . , , : , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , a, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

, , . .. . . , .

Ani Desu

1 :

1.1

. , , . . , , , , , , .

. , 1 2, . , , . , , , , . 3. 4, . , . 5 6.

, . , 1.1, .

4 :

. 1.1 1.1 , 1.1 100,000. 1.1

. , , . , 40 10 . 180 , , 2 . 10 50.

, . , (+), (+), (2+), (), . .

. , 70 ( 0 ), .

. , + , + . . ( ) , .

. .

, , . 100 1 ( 1.2). . , . . , , .

1.1 5

dendrite

dendrite

apicaldendrite

soma

soma

soma

axon

axon

axon

basaldendrite

axoncollaterals

A B

C

1.1 . () . . , , .() . . () . . 150. ( ,1911; , 1992.)

. 1 . , , .

, 2+ ( 1.2). ,

6 :

microtubules

mitochondrion

synapticvesicles

cleft

dendrite

dendriticspecialization

axon terminalof presynapticneuron

dendritic spineof postsynapticneuron

-60

-40

-20

0

20

mem

bran

e po

tent

ial (

)

2001000

time ()

A B

1.2 () . () . . . , 1 . . , , . ( . . ; ., 1991.)

. , , , , , .

1.3 ( ). , . ,

. , , . 1.3 . . .

1.1 7

100

15

0.1

100

dendrite

axon

1.3 . . . . . . . 0.1 . 1000 0.1 . ( 1.1.)

, ; , , . , . , , , . , , .

1.3 , . .

, . , . , . , . .

8 :

. . , . . , , , . . , .

, , . , .

, , . , . , , , . 2, . , () , , .

1.2

. , , , . ( 1 ), . , = 1,2, . . . ,.

1.2 9

0 , 0 . , ( ),

() =

=1( ) . (1.1)

() () . ,

(),

=1( ) =

()( ) , (1.2)

. ,

( )() = () , (1.3)

( , 0).

, . , , . , m n . , . ,

= =

1

0

() . (1.4)

() = .

, . , . , +%, %, %. , %, , 0 1, . . ,

10 :

( ) +%, .

+% . . , , , , . (), r()

r() = 1%

+%

() . (1.5)

r() ( ), m n , r(). , % 0 , , , % r() .

%, r()% +% . r() . % , +% . r()% ., r()% . ; r()% % .

1.2, . , 1.5, . , r() , ,

() ( ) =

()r( ) (1.6)

. ; . r() .

1.2 11

() r(), , .

= =

1

0

() = 1

0 r() . (1.7)

. ( 1.6). r() . , , .

m n , r(), , . , m n, mn, m n r(), , , , , . , r() ( r() ). , .

r() . , r(). . , .

1.4 r() . 1.4 3 . , . %, , %. 1.4 100 . , , r() .

12 :

A

E

B

D

C

100

50

0

rate

()

3.02.52.01.51.00.50.0time ()

100

50

0ra

te (

)100

50

0ra

te (

)

100

50

0ra

te (

)sp

ikes

1.4 . () . () % = 100 . () % = 100 . () = 100 . () 1.12 1/ = 100 . ( .,1997.)

1.4 , . , 1/%, . % , . . .

. , % ,

1.2 13

. 1.4 100 . = 1,2, . . . , ,

r() =

=1w( ) , (1.8)

w() ={

1/% %/2 < %/20 . (1.9)

., .

1.8 ( 1.2):

r() =

w()( ) . (1.10)

1.10 , w, ,

.

1.4 . w() 0 = 0, 1. , 1.4,w() :

w() = 12w

(

2

22w

). (1.11)

, w , %. 1.8 ( 1.4).

, . .

14 :

. .

w() = [2 ()]+ (1.12) 1/ . []+

[ ]+ ,

[]+ ={

00 . (1.13)

1.4 . 1.4 .

. , . . 2, .

. . ( , ) , , , 1.7. , = (), .

() . . , .

1.5 1 (1) . ,

. . . 1.5 , .

() = (12

(

)2), (1.14)

1.2 15

A60

50

40

30

20

10

0-40 -20 0 20 40

(orientation angle in degrees)

(

)

B

1.5 () . . . . . () 1 . 1.14 = 52.14 , = 0, = 14.73. ( , 1995, ,1968; ., 1974).)

, ( 90 +90), . = , .

. , . 1.6

1 . . , . , , , .

1.6 1 .

() = 0 + ( 0) ( ) , (1.15) ,

16 :

A 6050

40

30

20

10

0350300250200150100500

(movement direction in degrees)

(

)

B

1.6 () . , . .() . 1.15 = 54.69 , 0 = 32.34 , = 161.25. ( ., 1982, .)

, 0 . 1.15 20 . 1.6, , 1 20 < 0, 1.15 . , ( 1.13),

() = [0 + ( 0) ( )]+ . (1.16)

1.7 1 . ( 1.7). 1 , . 1.7,

,

() = 1 + ((1/2 )/%) . (1.17)

, , 1/2 , % . % , 1.7.

,

1.3 ? 17

A

40

30

20

10

0-1.0 -0.5 0.0 0.5 1.0

(retinal disparity in degrees)

(

)

B

1.7 () . . , . . , . . () 1 , . , . 1.17 = 36.03 , 1/2 = 0.036, % = 0.029. ( , 1995; , 1981.)

= () . , . , (). () , . (), , (). ().

. , . , r(), . , .

1.3 ?

.

18 :

. , mn . ., , m, , ?n , , . () . , .

. , . , . , . . , m n %. , , % , %/ . a . [ . a , . a .[

, , . , . ,() 0,

0 ()/ = 0

0 ()/ = 0. . : , , . , . , . ,

.

1.3 ? 19

() , 2 .

0 (( + )) =

+

(()) =

0 (()) . (1.18)

, ( + ) = () , .

, (), . , , ( ), , = 1,2, . . . ,, . , . ,

()() =

1

=1

( ) 1

=1( )

. (1.19)

, , . ( ). 1.8 . , . 1.19 , . , () 0 . , () 0 < 0, . , , 1.8. .

1.1. , 1.2, () 1.19,

() = 1

0 () ( ) = 1

0

r()( ) . (1.20)

20 :

time

spike-triggered average

1.8 . . , . . ( .,1997.)

() r() . 1.20 .

. ,

r

+ . , .

r() = 1

0 r()( + ) . (1.21)

1.20 1.21,

() = 1r() , (1.22)

= / . 1.22 , .

.

1.3 ? 21

(ms)

C()

(mV)

200

50

600

300 200 100

100

-100

-200

-300

()

()

1.9 . . . . ( ., 1996.)

. () , . , . .

1.9 . . . . , 1.9, , . , , , , .

. ? 2, , . , (.., ), . , , . 1.9 .

22 :

. .

. , , 1.21

() = 1

0

()( + ) . (1.23)

, . , 0 /2 < < /2 = 0,

() = 2 () . (1.24) , , . , 1.24 .

. , . , %, , = % = 1,2, . . . , % = . () . , 1.8, . ,

1

=1

+ ={2 /% = 00 . (1.25)

1/% 1.24 % 0. , 0 = 0 %. , % 1/(2%).

1.3 ? 23

C50

0

B 50

0

velo

city

(degs

/ ) 50

0

A

10 25

velo

city

(degs

/ ) ve

loci

ty(de

gs/ )

5

1.10 1 . () . () 10 1 . () 5 1 . ( , 1988; ., 1997.)

0 2 /%. . , , . 1/% . .

, . 1.10 . 1 . 1 . . , () . 1.10, , 15 . 1.10 10 1 . , 10 . , 10 , . . 1.10

24 :

5 1 . 5 .

, , . , .

1.4

. , , , . , , , +%. %, , %. . +%, % ( , % 0), []%, [] .

, [ ] [ ] . [ ] . , ( ) , , .

, [1, 2, . . . , ]. , [1, 2, . . . , ] +% = 1,2, . . . , [1, 2, . . . , ] = [1, 2, . . . , ](%)., . , , . r() , r() , , . , ,

1.4 25

. , , (.., ) .

, , . , . , , . , , . . , , , , , .

r() = r . , . , [1, 2, . . . , ] [], . 0 1 2 . . . ,

[1, 2, . . . , ] = ![](%

). (1.26)

1.37 .

[], % = /. % , , % 0. [] : , , . r%, (r%). , (1 r%), (1 r%). ,

26 :

1.0

0.8

0.6

0.4

0.2

0.0

[

]

6543210r

= 0 = 1 = 2 = 5

0.15

0.10

0.05

0.00

[

]

20151050

A B

1.11 () = 0, 1, 2, 5. , r, . () r = 10 () 10 ().

!/( )!!. ,

[] = %0

!( )!! (r%)

(1 r%) . (1.27) , % 0, % = . , = /%. = r%,

%0

(1 r%) = 0

((1 + )1/)r = r = (r) (1.28)

0(1 + )1/ , , = (1). ,!/( )! = (/%),

[] = (r)

! (r) . (1.29) . [],

, r 1.11. , . 1.11 10. r, , r. 1.11 r = 10.

1.29. , ( )

2 = 2 2 = r . (1.30)

1.4 27

. , 2/, 1 , .

,

. . + +1 < + +%, %, , , r%, %. 1.29, = 0, (r), +%

[ +1 < +%] = r% (r) . (1.31) , , % . , . , .

, ,

=

0 r (r) = 1

r

, (1.32)

,

2 =

0 2r (r) 2 = 1

r2 . (1.33)

,

= , (1.34)

1 . , , . 1. , 2 .

.

28 :

. , , . 1.1 . , 1.4, ,

= /. ,

() = 1

0 (() ) (( + ) ) . (1.35)

, , , a.

. ( 1/2)% ( + 1/2)% % . , . . 2 0 , 2%/ . 2%/ . , =/ 2%/2. , = 0 , = % 0, 0 = / . %, = 0 . /% % 0, . , = 0 , 0/% (). , ( 1.31). , ( ) . , =0 =0 . = r

() = r() . (1.36)

,

1.4 29

time ()time ()

B

0 0 +80+80-80 -80

A

1.12 . () () () 40 . . () .( ., 1991.)

. , ()= (), . . 0 , . . 1.12, 40 .

, , [1, 2, . . . , ] . , [1, 2, . . . , ] r(). , , 0 1 2 . . . , ( )

[1, 2, . . . , ] = (

0 r()

) =1

r() . (1.37)

30 :

. , , , 1, , ! .

, r(), , . % r()%. % , , 0 1. r()% > , ; .

, =1,2, . . . ( 1.31). 0 1, . , +1 = ()/r. , . , . r r() r . r +1 = ()/r. r()/r < . r()/r , . .

1.13 1 . , 1.5,

r() = (()) = (12

(()

)2). (1.38)

, . 2. 1.13, . , .

1.4 31

-60-40-20

0

204060

0 50 100 150 200 250 300 350 400 450 500time ()

firin

g ra

te(

)st

imul

us o

rient

atio

n (de

grees

)sp

ikes

0

20

40

60

1.13 . () 40 20 100 . ( ) ( ) . .

, . , . , . 0 , .

, ? , , , .

32 :

1000

0.1 1 10 1000.1

1

10

100

mean (spikes)3 10 30 100 300 1000

1

1.1

exp

onen

tm

ulti

plie

r

1

1.1

count duration ()

A B

Cva

rianc

e (sp

ikes2

)

0.9

0.9

1.14 . () 256 . . 94 . () . () . ( a ., 1997.)

. 2 2 = , . 1.0 1.5. = = 1, . , , .

1.14 . ( ) . 1.14, . , 1, ( 1.14). , , . , , .

. 1.15

1.4 33

interspike interval ()

perc

ent o

f isi

00

20 40 60 80 100

3.2

6.4

interspike interval ()0 20 40 60 80 100

0

3.2

A B6.4

1.15 () , . , , .() . ( ., 1994.)

( 3 ). 10 , , 1.31. , . 1.31 , . .

,

[] = r(r) (r)

! (1.39)

> 0, , = 0. 1.15 . , . 5 2 ( ). 1.15 .

(1) 1.16. , , 1/. , 1, 1, . . .

,

34 :

30201000

0.2

0.4

0.6

0.8

1.0

1.2

average isi ()

1.16 1 . . ( , 1992.)

. , . , , . , , . 1.17 1 . ( ). ( ) (). , . 5.

. . ( ).

1.5

. , .

1.5 35

in vitro in vivo in vivo

20

100

current injection current injection visual stimulation

1.17 1 . . () (). ( ., 1996.)

, . .

, ,

, , . , r(), 1.37. , r() , ., . ,

. . , , . , , . , , .

. , , , . . .

36 :

, . , , 10% . , , mn , . , , , .

, . , , . ,

( 3). . , . , .

.

, . . , , . , .

. . , , . , , , 712 .

1.5 37

1.41.31.21.1position ()

20

0

0

180

360sp

ikes

phas

e (de

grees

)

1.18 . . . . ( a , 1993.)

. 1.18 , , a , . , .

. . , . , . , .

38 :

. . ( ) , . .

. , r(). r() , , , . 1.19 ( ). , . , r() .

, . , , . , . , , . , , .

, , . , . r() . .

. , .

1.6 39

2000150010005000time ()

140

spike

s/

1.19 . , . , , . . , ( 3 ). ( , 1996.)

.

1.6

, . , , : r(), ,

40 :

. r() , . , , , . , , . . , , , , , . , .

1.7

:

( ),

() = 1 /2/2

()() , (1.40)

. , 2/. () = 2 () ,

() = 2

/2/2

()() = 2

. (1.41)

; .

,

() = 1

0 () 1

/2/2

( + )() (1.42)

= 1

0 ()() 1

/2/2

( + )(( + )) .

1.7 41

,

() = 1

0 ()() . (1.43)

, ( 2/) (). ,

() = |()|2 , (1.44) . .

1.40 1.44 , . , (%) = 1,2, . . . ,/%, ,

%

/%=1

()(%)2

,

(). . , , (, .., , 1993).

:

r

=

=0[] =

=0

(r)! (r) , (1.45)

2() =

=02[] 2 =

=0

2(r)! (r)

2 . (1.46)

, .

() =

=0

(r) ()! (r) . (1.47)

42 :

, = 0,

=0

=

=0

(r)! (r) , (1.48)

, . , () = /!,

() = (r)

=0

(r ()

)! = (r) (r

) . (1.49)

= r

(r)(r) (1.50)

22 = (r

)2 (r)(r)+ r (r)(r) . (1.51)

= 0 1.45 1.46 = r 2() = (r)2 + r (r)2 = r.

:

= 1,2, . . . , . +1 . % % = +1 . % 0. r( + %). r( + %)%, 1 r( + %)%. , , ,

[ ] =

=1(1 r( + %)%) . (1.52)

,

[ ] =

=1 (1 r( + %)%) , (1.53)

1.8 43

. (1 r( +%)%)r( + %)%, %,

[ ] =

=1r( + %)% . (1.54)

% 0, r() +1,

[ ] = +1

r() . (1.55)

,

[ ] = ( +1

r()

). (1.56)

[1, 2, . . . , ] , 0 , :

[1, 2, . . . , ] = ( 1

0 r()

)

(

r()

)

r()1=1

r()( +1

r()

). (1.57)

, :

( 1

0 r()

)

(

r()

) 1=1

( +1

r()

)

= (( 1

0 r()+

1=1

+1

r()+

r()

))

= (

0 r()

). (1.58)

1.57 1.37.

1.8

& (1991) . . (1997)

44 :

, 1^4. & (1998) . , , (1962) & (1980), . (1992), ( 2) & (1968), & (1978), (1958). & (1998) .

1 3, ; 1 , . (1997), , & (1998). (1995) , .

2 :

2.1

1 . , . , .

1, 17

( ), (), (1 17). , ( , ) . ( , ) . 2, 18, , . , , , , , . , , , , 7.

2.2

1, ,

46

. . . r() r() (). . , mn

r() ,

r() = r0 +

0 ()( ) . (2.1)

r0 = 0.() , , . 2.1 r() 1.

1, . . () . () 0.

2.1, , . , , . ,

r() = r0 +

()( )+

12 2(1, 2)( 1)( 2)+123 3(1, 2, 3)( 1)( 2)( 3)+ . . . . (2.2)

. 2.1. , , , , ( ) , .

2.1,

2.2 47

, ,

= 1

0 (r() r())2 . (2.3)

0 ( ). 1, , r() =

r() ( + )/,

, () =

() ( + )/: 0

( )() = r() . (2.4)

^ . 2.4 , ( ). () = 2 () ( 1), 2.4

2

0

( )() = 2 () . (2.5)

,

() = r()2

= ()2

, (2.6)

() . , r()= () 1. , .

1, 1 , . 2.1 . 2.1, 2.1. , , 2.1. , , . , . .

48

A

B

C

time ()

firin

g ra

te(

)sp

ikes

(de

grees

/ )

60

40

20

0

-20

-40

-60

100

50

0100 200 300 400 5000

2.1 1 . () . () 100 . () . . . ( .,1997.)

. , . 2.1 . . . 2.1 , . , , . , .

2.2 49

2.1. , 2.1 . , . .

. , , . : , , . . :

() =

0()( ) . (2.7)

r() = r0 + () r()

r() = r0 + (()) , (2.8)

. . , .

2.2. , 2.4. ((), r()) , r() . . , , , (). . . () , .

.

50

100

80

60

40

20

0

r (

)

543210

100

80

60

40

20

0

(

)

543210

Eq. 2.9

Eq. 2.10

Eq. 2.11

A B

2.2 () . r ( ) () ( ). () . , 2.9, 2.10, 2.11 = 25 , 0 = 1, 1/2 = 3, = 100 , 1 = 2, 2 = 1/2.

() = [ 0]+ , (2.9) 0 . , , . 0 =0. . , , ,

() = r1 + (1(1/2 )) . (2.10)

, 1/2 , 1 . ,

() = r[(2( 0)

)]+ , (2.11)

2 2.10, 0 . 2.2 .

, 2.8 . , , 2.4 r() = r0 + (),

2.3 51

stimulusLinear Filter Static Nonlinearity Spike Generator

response

2.3 . . r0 ., .

r() = r0 + (()). ( ) 2.6 , , .

, . , . , , 2.7 ,

() =

0() (( )) . (2.12)

. , , = (), , 1. , 2.12 .

2.8 ( 1). 2.3 , , . , r()% , 1 . , 2.3, (), .

2.3

,

52

A B

rod and conereceptors (R)

bipolar (B)horizontal (H)

amacrine (A)

retinalganglion (G)

R R

H

B B

G1

G2

A

lightto opticnerve

2.4 () . . , , . () 1 . , 4 , 8 . , . ( ., 1992; , 1911. 1969; , 1992.)

. . 2.4 , 2.4 . , . 2.4 ( ). 2.4. , 2.4, . . , , . . , ., .

, . 2.4,

2.3 53

primaryvisualcortex

lateralgeniculatenucleus

lateralgeniculatenucleus

retina

retina

opticchiasm

2.5 () . ( ., 1992.)

, . .2 , 1 . , .

, ( 2.5). , . , ( 2.5).

, , . , . , . , . , , . . m n , .

, , ,

54

. , , .

, , , , , . .

. . , . .

, . . , . , , ( ) . ; .

, , . , . , , 1 , , , . , ; 10.

, , . . .

2.3 55

F

A B

F

2.6 () . , . . , , . . . . () a , . . 2.7.

. . , m n , . ( 2.6), , , , ,

, . , , 90 +90 0 70 ( ) , .

, ( ) . , , , ( 2.6). ( 2.6). . , 180/ . , , .

56

0

0

1

1

2

2

-1

3 ()

(

)

A B

-21

2.7 () . 2.6. 1, 2.3, 5.4, ( ) 90,45, 0, 45, 90. . () 2.15 2.17. ( ., 1982.)

2.7 . 2.7 , 2.6. , . , . , .

2.7 2.7. , , . , ().

. +% . %. %. , % = ()%

2.3 57

, % % 0,

= () . (2.13)

, 2.7,

() = 0 + , (2.14)

12 0 1. 2.13 = 0 = 0,

= (1 + /0) . (2.15) . % %/180. , , /180 . , %, . % 0,

=

180 () . (2.16)

. 2.16

= (0 + )180 . (2.17)

2.7 2.7.

1, 2.15 2.17 (/0) /180. = + = (/0)(/180) ( 1) = (). , . ,

. + () . . , 2.15 2.17. , , .

58

, () . . , , , , . , 2.6. , . , , . , (, , ) (, ) . (, , ) , . .

, . , ,

(, , ) = ( -+ -.) () . (2.18)

2.8 ( ) , -,., .

- .

( ), - , . , . . , - , 2/ ( 2.8). , 2/ ( 2.8). , - , -= 0 , , . . . . %. %./2 . . , /2 . . , /, /2 .

. , ,

1

0

(, , )(, , + ) = 2 ()( )( ) . (2.19)

2.3 59

0

A B

2.8 . () 2.18. >0, < 0 . -, . , ., . () 2/.

, . , ( , ) .

, . . , . , % % = 1,2, . . . . ()

= % . (2.20)

, 2, < . = () = ((2 )). , ((2 )%) = (2 %) = (%) = (%) ( 2.9). , . < 2

60

-1.0

-0.5

0.0

0.5

1.0

(/

6)

(or

/

2)

129630

2.9 . (/6) (/2) , % = 3. /3, 2/3 /6 = /2.

, . , , . 120 , /(2) = 1/(2%) = 60 . , 2.20, , 2 .

2.4 :

, 1, , = 1,2, . . . , . , ,

(, , ) = 1

=1(, , )

. (2.21)

, 1, , 1/ 1/. (, , ) (, ) ., + , ,

r(, , ) = 1

0 r()(, , + ) . (2.22)

2.4 : 61

, 1,

(, , ) = r(, ,) , (2.23) , , , =/. , r0, 2.8, r() = r0 + (()). 2.7, () . , ().

, () :

() =

0

(, , )(, , ) . (2.24) (, , ) , , (, ) . 2.6, , , 2

(, , ) = r(, ,)2

= (, , )2

. (2.25)

(, , ) . (, , ) ,

. , , , ,

(, , ) = (, )() . (2.26) .

. (, , ) , .

2.8 ( ), 1, . , . , , 1.

62

2.10 ( 17) . . , , , . () () ( 2.11) . (>0) (< 0) . , . , 2.10 , ( 2.11). 2.10 2.11 . . , .

, . . , . , , , . . ,

(, ) = 12 (

2

22

2

22

)( ) . (2.27)

: , , ; , ,

( 2/); ,

. , 2.18 = , . = , - = 0. 2.10 2.10 2.10 2.10. 2.12 .

2.4 : 63

-4-2 0 2 4 -5

05

0

A B

C D

-4-2 0 2 4 -5

05

0

2.10 . () () 50 100 . , (, ). . (, )> 0, (, ) < 0. () () ( 2.27) = 1, = 2, 1/ = 0.56, = 1 /2 () = 1 (), . ( , 1987.)

64

B CA

2.11 2.10. , 0. () , , . () , .() .

2.12, , ( ) . . = 2(+/), +> < = . , . , .

. (2( )2/2) ( 2.34). + (2( )2/2) = 1/2. = (2 (2))1/2/,

= 2(

+

2 (2)

2 (2)

) =

2 (2) 2

+ 12 1 . (2.28)

> (2 (2))1/2, . 0.5 2.5, 1.7 6.9.

2.27 , 0.

2.4 : 65

1.0

0.5

0.0

-0.5

-2 0 2 (degrees)

1.00.80.60.40.20.0

-4 0 4 (degrees)

-0.8

-0.4

0.0

0.4

0.8

-2 0 2 (degrees)

0.8

0.4

0.0

-0.4

-2 0 2 (degrees)

A B C D

2

0

2

0

2.12 2.27. 2 . () = 1,1/ = 0.5, = 0 = 0. = 0. () = 1, 1/ = 0.5, = /2 = 0. = 0 2.27. () = 1, 1/ = 0.33, = /4 = 0. = 0. () 2.27 = 2 = 0. .

, ()+ () () () 2.27.

- = . , (0, 0) 0 0. 0, 0

2.13 . 300 , . 210 ( = 210 ), , 2.10, . ( ), , 75 = 210 . , < 75. .

2.13 . 2.13 , . , , . , () 0, , ,

66

30 75 120 165 210 = 255

(de

grees

)0

5

0 5 (degrees)

2.13 . (, , ) . 2.10, (, , )>0 (, , )

2.4 : 67

0.4

0.2

0.03210

/

0.4

0.2

0.0

-1.0 0.0 1.0

A B

-0.5

0.0

0.5

-2 0 2

C

2.15 = = 0, = = , = 2 = 1. () - , = . = 0. () ,/, - = 0 .= 0. () . , = - = 0.

=

(, ) ( (-)+ (-).) (2.31)

() =

0 () (( )) . (2.32)

= = . , ( 2.15) -, , . , , . . , 0 (. = = 0),

= (

2(2 + 2)2

)

(2 (-)

), (2.33)

. , - = 0 , (2) 0 ( , = = 2,(2) = 0.02). ,

= 2 (

2( )22

)(.) , (2.34)

.

68

0.5

0.4

0.3

0.2

0.1

0.0

am

plitu

de

20151050frequency ()

2.16 2.29. () , /2.

250

0

60

BA

-2-1 0 1 2 0

50100

150200

0

2.17 . () . . . () ( = 0) = 1,1/ = 0.56, = /2 2.29 1/ = 15. ( ., 1995.)

(/2 ) 2.16. 4 10 1 .

(, , ) ( 2.13), . 2.17 . , , 2.13. 2.17

2.4 : 69

A B

2.18 . () . () . , .

, 2.17, 2.29.

, . , 2.18 . 2.18, . 2.18 . .

, . 2.17 , 2.19 2.20. 2.19 . , 2.17. 2.19, , . . 2.19 () . , () = 0. , . 2.19 , , (). 2.19 , (). , 2.19.

2.19 2.19 .

70

- location of bar

A B

()

C D

E F

()

2.19 . . . () . () () ()> 0. () () < 0. () () . , () . () . , (). () , .

, (). 2.19 .

2.20 . 2.20. 2.20, ., (). , , 2.20, , 2.20. () (2.20). 2.20 . 2.20, (2.20), , (2.20), , (2.20). , .

2.4 : 71

A

B C

D

E F

()

()

2.20 . 2.19. () . , , . , , () . () () . () , . () . .

. . 2.21. , ( ). . 2.29 , . , 2.21, ,

(, , ) = (, )() (2.35)

72

200

0

60

A B

-2-1 0 1 2 0

50100

150200

0

-3

2.21 . () . . . () 2.35 2.37. ( = 0) = 1, 1/ = 0.5, = 0. 2.29 = 20 , , , . = /9, = 0.02 /. ( ., 1995.)

= () () (2.36)

= ()+ () . (2.37) () (), . . .

2.22 . 2.22 , 2.22, . (). , (2.22), , () (2.22). 2.22 , (2.22). , , , 2.36, ().

2.4 : 73

D

E F

B C

()

()

A

2.22 . , . () . , , . , , () . () () . , () . () . , ().

:

() , 2.8, r() = r0 + (()), . , = []+, . , , . , r /(1/2 + ) 2, 1/2 . (1992)

() = []2+

21/2 + []2+, (2.38)

74

. , , , . mn . .

2.5 :

. , , , , . . , . , , . , , . , .,

. .

. , 2.20 2.22. 2, .

, , , . 2.34, . , 1 2, /2. , , ,

1 = (,) (.) ( ) , (2.39)

2.5 : 75

0.3

0.2

0.1

0.0-2 0 2

0.2

0.1

0.03210

/

A B

0.2

0.1

0.0 12

+

22

-1.0 0.0 1.0

C

2.23 . = 2. () ,21 + 22, - = . () 21 + 22 , /, - = 0. () 21 + 22 . , = - = 0.

(,) . (,) , . /2,

2 = (,) (.) ( ) (2.40) ( /2.) = (.). , .,

21 + 22 = 22(,) 2( ) , (2.41) 2(.)+ 2(.)= 1. ,

r() = r0 + (21 + 22

), (2.42)

. , , 2.23. , , 2.15, . 2.23, 1 2 2.31 2.32, 2.34 . , . .

2.41 2.42 2( ) 2.

2( ) = 12 (2( ))+12 . (2.43)

76

-1

0

1

0204060

400 500 600 70020010000

204060

()

r (

)

simple

complex

stimulusA

B

C

r (

)

2.24 . () (, , ) (, ) . () 2.29 = 1/(15 ). () . r0 = 5 . .

, . 2.24 , .

mn . , /2, . , 21 22 . , . , ,

r() = r0 + ([1]2+ + [2]2+ + [3]2+ + [4]2+

), (2.44)

[]+ , + /2, + , + 3/2. , , . , . 7.

2.6 77

2.6

. , . , ( ) . , . , . , . 2.25 .

(, ) = ( 1

22

(

2 + 222

) 22

(

2 + 222

)).

(2.45)

= = 0. 2.45 , , . , , , . . (+) () . 2.25 2.25.

2.25 , , . , , . , , .

(, , ) = (

()22

(

2 + 222

)

()

22

(

2 + 222

)).

(2.46)

78

0 3

0

200

00 3

3

-2-1 0 1 2 0

100200

0

-2-1 0 1 2 -2

02

0

B

D

A

C

2.25 . () . ( ) ( ). () ( 2.45) = 0.3, = 1.5, = 5. () . . () 2.46 , 2.47 1/ = 16 , 1/ = 32 , 1/ = 1/ = 64 . ( ., 1995.)

, , ,

, () = 2, (,) 2, (,) . (2.47) , , . 2.29, . 2.25 2.46 2.25.

2.26 . . ,

2.7 1 79

time ()

firin

g ra

te(

)

2.26 . . . . ( ., 1996.)

, . 2.26 , . 0.5, . .

2.7 1

, , , . , (1962) . , 2.27,

. 2.27

80

LGN receptive fields

simple cellOn OffOff

complexcell

A B

simple cells

2.27 () . , . , . , . . () . ( ), (1,2, 3, 4), . , . , .

, , . . . , 7, .

, . , . . (1962), 2.27.

, . , , . . , . 7.

2.8 81

2.8

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

2.9

:

2.1 , 2.3

= 1

0(r0 +

0

()( ) r())2

. (2.48)

0. , , ,

, , . . % %. r = r(%), = (%), = (( )%). % , 2.48 ,

= %/%=0

(r0 +%

=0

r)2

. (2.49)

0,

= 0 = 2%/%=0

(r0 +%

=0

r)

% . (2.50)

82

%

=0

(%

/%=0

)= %

/%=0

(r r0) . (2.51)

% 0 % , % , % , 2.51 , ,

0 ()

( 1

0

( )( ))= 1

0

(r() r0) ( ) .(2.52)

r0 0. , r(). 2.52 . + , ( ), 2.52 2.4. 2.6 2.4 . , 2.4 ,

r() = r0 +

()( ) . (2.53)

2.4,

( )() = r() . (2.54)

( )

() =

()() () =

()() ,(2.55)

r() ().

2.54 . 2.54 . , ( ),

()

( )() = ()() . (2.56)

2.9 83

, 2.54

()() = r() , (2.57)

() = r()/(). () ()

() = 12

()() , (2.58)

() = 12

r()()

() . (2.59)

:

0

(()

)2 = . (2.60) ( ),

r()+

0 2() = r0 +

0

()( )+

0

(()

)2,

(2.61)

. 0 ( )

() = 2( ) . (2.62) ( 0) 2.60 , . ().

: `

(1952, 1975) (

84

) . r = r0 + (()) 2.6, . ,

() = 12

0

r()( ) = 12

0

(())( ) . (2.63)

r0 0. , 2.63 , , 2.63 . , ,

12

0

(())( ) = ()

0 (()) . (2.64)

(), , , 1. . 2.64 . .

2.10

& (1978), . (1997), & (1998) . . (1996). (1958) &(1978).

& (1962,1977), (1984), (1988), (1995), & (1990). (1987) , (1977), . (1984), & (1984) . & (1979). . (1978, 1978) , (1985) & (1987) (, 1946) , . (1995) . & (1985), & (1982), & (1984), & (1985), (1961) & (1965). a(1992, 1993) .

2.10 85

(1996), 10 & (1999). (1965) (1966). & (1991). (1994).

3

3.1

1 2, . , . , . , . 7. . , , 4.

1, , , . , . , . .

, . . , =(1, 2, . . . , ) , , , . .

, . [|]. :

88

[], ,

[],

[, ], .

[|], ,

[|], , .

[|] , , [] . [] [|] ,

[] =

[|][] [] =

[|][] . (3.1)

[, ] [|] , [|] ,

[, ] = [|][] = [|][] . (3.2) [|] [|]:

[|] = [|][][] , (3.3)

[] = 0. [|] . , , [|]. , [|] [|], [] . , .

, , , . , . , . , , [], [|], . , .

3.2 89

0% coherence 50% coherence 100% coherence

3.1 . 45 . 0% . 50% , ( ). 100% . ( ., 1992.)

, . , . . , , .

3.2

, . (1992). , . 45 . , ( 3.1). , . . 0% , . , , 100%, . , .

. ,

90

0 10 20 30 40 50 60

20

20

20

0

0

0

firing rate ()

num

ber o

f tria

ls

1.0

0.8

0.6

0.51.00.1 10 100

fra

ctio

n co

rrect

A B

behaviorneuronal

3.2 . () . , . () . , , . . ( ., 1992.)

. , . ; , . a . 3.2 . 1% , ( = 0.5), ( = 1) 10% .

3.2 . 2 , 2 . , ( +) ( ), , ( ) ( ). , , , . , . ,

3.2 91

, 2 , ,+ .

= +

, (3.4)

. , .

, . . , mn; mn. 3.2 , , . 3.2 , .

( ) , [ |+]. mn ( ) [ |]. ( mn mn, mn mn) 1[ |+] 1[ |], . , , , , mn :

() = [ |]

() = [ |+] . (3.5)

.

+ 1 1

. ,

92

0 1. , ; . , (()+ 1 ())/2 . , . , .

() . .

, , , , . , . =0, , mn. , = 0, ==1, . , , , mn, ==0. . , .

3.3 . . , , ()= 0,()= 0 , () mn 1 (). , mn mn , (). , ; , , =, . 3.3.

3.3 . 3.3 , 1/2, ( ). 3.3, 1, ( ). ,

3.2 93

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.012.8%

6.4%3.2%

1.6%

0.8%

3.3 . ((),()) . 3.2. . ( ., 1992.)

. , . ,

, . , , 1 2. 1 1 2 2. .

, . , , , 1 = 2, [1 |+] = () = 2. , 2 . %, 2 +% [|]%, [|] = . ,

[] =

0 [|]() . (3.6)

,

94

.

, (), [|],

() =

[|] . (3.7)

,

= [|] . (3.8)

[|] 3.6 . =1 =0 =0 =,

[] = 1

0 . (3.9)

, . , .

[|+] [|] + , 2 . , ,

[] = 12( +

2

)= 12

(

2

), (3.10)

3.4 () (

)

() = 2

(2) . (3.11)

, . [] 3.10.

, . , , mn

3.2 95

. , , . . .

3.2 . 3.2 . , / . , . , . .

. ? ? ( )

(, , ),

() = [|+][|] , (3.12)

. . , , ., . , , .

. 3.7 3.8,

() =

[|+] = [|+] . (3.13)

96

3.8,

=

=

[|+][|] = () , (3.14)

.

. mn mn . , mn mn +. , . mn mn, , [+|] [|] . ,

[+|] = [|+][+][] [|] =[|][]

[] . (3.15)

mn ,+ = +[|]. , mn = [+|]. , mn + mn . 3.15, mn

() = [|+][|] +

[][+] . (3.16)

. , . . . , .

[|+] [|] + 2 , [+] = [] = 1/2, [+|] ,

[+|] = 11 + (( )/) , (3.17)

= (+ + )/2. ; [+|].

3.3 97

, . +% . ,

[|+%][|]

[|] +%[|]/[|]

= 1 +% [|] . (3.18)

%,

() = [|] (3.19)

( 1)/% . () . ()

3.3

. , . , , , . , . , [|] [|], . , , . . . , .

, , .

98

3602701809000.0

0.5

1.0

(degrees)

/

3.4 . . 90 , 40 . . ( , 1991.)

. . . a . , . .

(1991) . , . 3.4 . . , . . ( = 1,2,3,4) , . , = (), , (

()

)= [(( )]+ , (3.20)

. , , . , .

3.3 99

, . , v |v|= 1 ( ). , . , () = v . , ,(

()

)= [v ]+ . (3.21)

. , , v v. , v, ,v = v v = v . , 3.21. , , , ( 3.5). , , . , ! , (1998) .

v v = v+v . , , , , . ,

v =4

=1

(

)

. (3.22)

, . . 3.5 3.22 . 6 .

100

1

23

4

1

2

A B

-180 90 180-90 00

2

4

6

8

wind direction (degrees)

err

or

(degre

es)

3.5 () a . v . 1, 2, 3, 4 90, .() . , . 3.22 . ( , 1994.)

; , .

1, (1) . , 1. 1 , 0, . , v, 1, ( ), ( 0

)=(

() 0

)= v , (3.23)

. , , , . , 1 , . , ; ,

3.3 101

. .

, 1 . , . , 3.22 v. , ,

v =

=1

( 0

)

(3.24)

, , v. 3.24 3.23,

v =

=1(v ) . (3.25)

v v , . , . 3.24 , , 3.6.

, . , , , . () . () . . , , ( ), . +%, .

102

3.6 . . 45. . . . ( ., 1991, ., 1983.)

, [|], [|] ( 3.3),

[|] = [|][][] . (3.26)

[|] . , ,

, (, ) mn . . , ,

(, )[|].

3.3 103

-180 -90 0 90 1800

2

4

6

8

-180 -90 0 90 1800

2

4

6

8

err

or

(degre

es)


maximum likelihood Bayesian


3.7 . . 3.5. . ( , 1994.)

, (, )= ()2,

=

[|] . (3.27)

, (, ) =| |, [|]. () , , , [|]. . [] , [|] [|] , []/[] 3.26 . , , , [|], () .

. 3.7 , . , [|] 3.4. 3.7 . [], . , .

104

. 3.7, 3.5, ( 3.4). 3.5, , . , ( 3.11).

3.5 . , . 90 . .

, . . . ,

() = (12

(

)2), (3.28)

3.8. , ( 3.8 ). ,

() .

3.8 . .

. , . , , , [|] . ( = /), 1. , = , = (), ( 1)

[|] = ( ())

()!( ()) . (3.29)

,

3.3 105

1.00.8

0.60.40.20.0

/

-5 -4 -3 -2 -1 0 1 2 3 4 5

3.8 5 5. = 1. , . 0.5 1/5 , .

,

[|] =

=1

( ())()!

( ()) . (3.30)

.

3.9 3.8 =0. : . 3.9 =0. | |, =5,4, . . . ,5 =0. , [|] . 3.30 ,

[|] =

=1

(()

)+ . . . , (3.31) , , ,

().

, .

, , 3.31. 0,

106

1.0

0.5

0.0

/

-5 -4 -3 -2 -1 0 1 2 3 4 5

3.9 11 3.8 0. , , ( ). = 0. . > ( = 0) , .

=1

()()

= 0 , (3.32)

. 3.28, ()/() = ( )/2 ,

=

/2/2

. (3.33)

,

=

, (3.34)

. .

3.33 , . . , 0. , . , , , .

3.3 107

, , , [] . . , [] , . , , , . [|] . , [|] = [|] + [] []. , , 3.31,

[|] =

=1

(()

)+ [] + . . . . (3.35) ,

=1

()()

+ []

[]= 0 . (3.36)

, , 3.36

=

/2 + /2

/2 + 1/2. (3.37)

3.10 [|] ( ) =2 =1, 3.9. ,[|] 0. 2, , . , . . 3.10 3.28 3.35 1/(

/2 ) (

) 1/(

/2 + 1/2) ( ). , () 2(). (.., ),

() = . (3.38)

108

1.5

1.0

0.5

0.0-1.0 -0.5 0.0 0.5 1.0

3.10 , 3.9 3.8. [|] = 0. 2 1, = 0. =0.0385 =0.107, .

. () = 0 . , ,

2() = ( )2 . (3.39) ,

( )2

.

. = (),

( )2 = ( + ())2 = 2()+ 2() . (3.40)

, . , .

. , , , . . , .

3.3 109

( )

2() (1 + ()

)2()

, (3.41)

() (). [|], () 3.41 , [|] ( )

() =

2 [|]2

=

[|](

2 [|]2

). (3.42)

() =(

[|]

)2=

[|]( [|]

)2. (3.43)

. () = 0, 3.40 2 , 3.41, 2 1/(). , , . , . , 3.40 . , . , , 1/().

, . , , . , ( ) 2() = 1/(). . , () , , 1/2( ) . , , ,

[||>]0 >0. 3.42 , .

110

. , , . , . , , . , . ., . a , 4, .

( 3.8, ) 3.31. , 3.42,

() =

2 [|]2

=

=1

(( ()()

)2

()()

). (3.44)

, 3.8, 0. = (),

() =

=1

( ()

)2()

. (3.45)

, . . 3.11 3.45 , = 0 3.8. , . , ()= 0 . . , .

.

3.3 111

1.0

0.8

0.6

0.4

0.2

0.0-4 -3 -2 -1 0 1 2 3 4

/

/ ( )

3.11 =0 =1, . () , . . , =0, .

, 3.8. 3.45 ( ), ( ). , . , , . , . , , = . 3.45,

() =

=1

( )24

(12

(

)2). (3.46)

. , . ( ),

()

( )2

4

(12

(

)2)

=

22

. (3.47)

,

112

. , , .

. . , , 3.47,

= (2)/2

2= (2)/22 . (3.48)

, = 1, . . ( ) = 1, > 2, = 2.

, . . +% %. , % ( ) ( %) 1/(). , , 3.4,

=%() . (3.49)

, . %, () 3.19.

, . , , , %, 3.19 ,

=

=1

()()

. (3.50)

3.4 113

0 1000 2000 30000

5

10

orie

ntat

ion

thre

shol

d o

r(

)-1/2 (de

grees

)

3.12 . , () . . , 2. ( , 1988.)

3.12 . , , .

3.4

. . , = 1,2, . . . , . 2, 1. , . , 2, 0.

, . . () , < , , ,

114

7

time

4 8

200 -200

400

B

C

A

3.13 . () ( ). 0. , 7. . . 0 , 8 (.., ). () . . () , . , 1. , . 0 = 160 .

> . , , , . , . , , 0, , 0 , 0 ( 3.13). ( 0) ( 0 < < ; 7 3.13), ( 0) ( < 0; 16 3.13), . 0 , . 0.

.

3.4 115

( ), ,

( 0) =

=1( )

() . (3.51)

, = / , 0, . 3.51 , ( 17 3.13) ( ) = 0 0. . , . 3.13 , 3.13.

3.51 () = ( ) 1,

( 0) =

(( ) )() . (3.52)

, 2. , ,

1

0

(

(( ) )() ( 0)

)2, (3.53)

. 2,

( )() = r( 0) , (3.54)

,

( ) = 1

0(( ) )(( ) ) , (3.55)

1. r , , 1,

r( 0) = (0 ) = 1

=1( + 0)

. (3.56)

2,

116

3.54 . . , , . , ,

() = () , (3.57) 3.54

() = 1r( 0) = (0 ) =1

=1( + 0)

. (3.58)

0 . 0 , , , . : , , 0 ( 3.13).

3.58. , , . , , () 0 > 0. , () = 0 < 0., 0 = 0 , () = 0 .

, ,

() = 12

()() , (3.59)

, ,

() = r()(0)()

. (3.60)

r r . r( 0) 3.58. . , 3.60 .

, , , () 3.58 0. , 0 .

3.4 117

0 , . < 0 () -()(), -() -() = 1 > 0 -() = 0 < 0. 3.13. -(), , .

() ( ). . 0. () 0 . , , , ().

3.14 1 2. , , . 3.14 . 1 , , . , . , 1, . (1991) 1 , () , (). . 1 , / . ( ) 1 ( ) . 1 , ( 3.14). , , . , 1 . , ,

118

time ()

time ()re

spon

seve

loci

ty (de

grees

/)

kern

el a

mpl

itude

(degre

es/

)-20

20

0

0 100

-20

-40

-60

20

40

60

0

0 100 200 300 400 500

3.14 1 . . , . , 0 = 40 . 1 () ( ) () ( ). , . ( ., 1997.)

1 , .

3.5

, , , . ,

3.6 119

, . ; , , ; ; . .

3.6

:

% . (), (). () ,

() =

[|]-(() ) () =

[|+]-(() ) .(3.61)

( ) ( ) () . - , 1 0 , . ,

% = ( )() =

[|+]-(() )

[|+]-(()) .(3.62)

() () , , ,

% =

[|+] (-(() )-(())-(())-(())) .(3.63)

() = [|+]/[|], [|+] ()[|] ,

% =

()[|](-(() )-(())-(())-(())

).

(3.64)

, () - , () , .

%

[|] (-(() )-(())-(())-(())) .(3.65)

120

( () () ),

% [

[|]-(() )

[|]-(())]. (3.66)

, , . % 0, (), .

:

, ,

22 2 . (3.67) , (2 )2 0 (3.68) .

222 22 0 , (3.69) .

3.67 = / = . 3.43 3.39, 2 = 2 = 2.

2() [|]

( )2

. (3.70)

3.70, ( 3.38),

+ () = =

[|] , (3.71)

1 + () =

[|]

=

[|] [|]

=

[|] [|] ( ) . (3.72)

3.6 121

[|] [|]

=

[|] = 0 (3.73)

[|] = 1. 3.72 3.70,

2() (1 + ())2 , (3.74), , 3.41.

:

3.54. , , () ,

()

( )() =

()r( 0) .(3.75)

+ 0,

(0)

()r() = (0)r() , (3.76)

r() r(). 3.54 3.75 . ( ),

()

( )() = ()() , (3.77)

() () () ():

() =

()() () =

()() .(3.78)

3.75 ,

()() = (0)r() . (3.79) 3.60 , 3.59 () ().

122

3.7

, , , , & (1974). & (1966) (1989). . (1997). (1994). . (1989) .(1992) , . (1996) .

, .(1970), . (1986 & 1988), . (1987), . (1988). , , (1988), (1988), (1990), & (1992), (1992), & (1993), . (1993), & (1994), (1994, 1996), (1996), . (1998). (, 1981). , (1999), & (1992) (1984).

4

4.1

n ?n n ?n . , . 8 10.

( ) . ( ), . , , , , . . . , , . .

, , . , , . . .

124

. , mn (.., ) . . . , .

, , mn mn . . , , . , . ?

(.., ), . [] . , , mn , ([]), ([]) . ([]), , . , ([]) [] . , . . 1 2 . , , [1][2],

([1][2]) = ([1])+ ([2]) . (4.1) ., . , 2 . 2 , mn,

([]) = 2 [] . (4.2)

4.1 125

, . . , , .

(4.2) . a ,

=

[] 2 [] . (4.3)

, = 2 [] . , , . [] 0, , 0 2 0 2 0, 0. 0. , .

. , : . , [] 0, , 1. (4.3) 0 [] = 0 2 1 = 0., . , , + . , (4.3), , [] = 1 [+],

= (1 [+]) 2(1 [+]) [+] 2 [+] . (4.4) , 4.1, 1 [] = [+] = 1/2. , .

, . , . , .

126

1.0

0.8

0.6

0.4

0.2

0.0

(b

its)

1.00.80.60.40.20.0 [+]

1.0

0.8

0.6

0.4

0.2

0.0

(bi

ts)

0.50.40.30.20.10.0

A B

4.1 () . [+] +, [] = 1 [+] , . [] = [+] = 1/2. () . . 1/2 > 1/2 1 .

. . .

. . , 4.3. [|], , [] 4.3.

=

[|] 2 [|] . (4.5)

,

=

[] =

,

[][|] 2 [|] . (4.6)

, .

4.1 127

, 4.3 4.6

= =

[] 2 [] +

,

[][|] 2 [|] . (4.7)

[|] [] ( 3),

[] =

[][|] . (4.8)

, 4.7 ,

=,

[][|] 2(

[|][]

). (4.9)

3

[, ] = [][|] = [][|] , (4.10) [, ] . 4.10 ,

=,

[, ] 2(

[, ][][]

). (4.11)

, . , 4.10

=

[] 2 [] +

,

[][|] 2 [|] . (4.12)

4.7, . 4.12 . , , , .

, . , . ,[|] = [], 4.9 = 0. ,

128

. , []= [] [|] 1 = 0 . 4.9 ,

=

[] 2

( 1[]

)=

[] 2 [] . (4.13)

, [] = [], . , , .

, , + , , + . + + , , . , [+|+] =[|] = 1 , [+|] = [|+] = . [+] = [] = 1/2, , 4.4, 1 . (1 ) 2(1 ) 2 .,

= 1 + (1 ) 2(1 )+ 2 . (4.14) 4.1. ( = 0), 1 , . ( =1/2), 0.

. . , + , + . , [+|+] = [+|+][+]/[+] = 1 . , + . + , 1 . , , .

() . [] []

(,) =

[] 2

([][]

). (4.15)

, (,) 0 = ( ). , ,

4.1 129

. 4.15 4.11, [, ] [][]. , [, ] [][]. , [, ] . 0 . , .

. 3, . . , . , .

, . . , . , . [] [], .

, +%, %, []%. %, , (4.3),

=

[]% 2([]%)=

[]% 2 [] 2% . (4.16)

1. % 0 , 2% . . , (.., )[],

130

% 0. ,

%0

{ + 2%

} = [] 2 [] . (4.17)% . , . , , % , . 4.17 . 4.17 .

, , 4.17, [|] :

%0

{ + 2%

} = [][|] 2 [|] . (4.18) 4.17 4.18,

=

[][|] 2(

[|][]

). (4.19)

2% .

3, . [|] ( ) . [|], . [|] , . , , , 3. , , , .

4.2

. , ,

4.2 131

. , . .

, . . , , , . . , .

. . , . , . .

, [] 4.17 . , % , 2% , . , . 1. . , 0 . [] ,

0 [] 2 [] , (4.20)

0

[] = 1 . (4.21)

, ( ),

132

,

[] = 1 , (4.22)

. , %,

= 2 2% = 2( %

). (4.23)

4.22 . ,

[] = 1/ . = (). %, +% []%. ( +%) (). , [] = 1/, | ( +%) ()|/. , | ( +%) ()|/ = []%. ( +%) > () %. , % 0,

= [] , (4.24)

() =

[] , (4.25)

, . , .

(1981) () . , . 4.2 . 4.2 . , , . 4.2 , , .

4.2 133

0

1

0.5re

lativ

e re

spon

se

0 1-1contrast

4.2 ( ) ( ). . . ( , 1981.)

, . , , . , , . .

, . . (.., ). = 1,2, . . . , , %. [] ,

=

[] 2 [] 2% . (4.26)

4.26,

134

. [] =

= []

,

=

[] 2 [] 2% =

[] 2 [] 2% .(4.27)

,

. (4.28)

,

=

[] 2(

[] []

) 0 . (4.29)

[]

[],

.

[] =

[] , (4.30)

, . , , 4.30 . . , . 4.29 .

, . , , [] = [] . , (.., , , ). , . , [] , .

. , [|] . , .

4.2 135

, . , , . . , , = ( )2= 2 . , ,

=

[]( )( ) = 2 . (4.31)

4.31 . , . , , .

, , . . , . . .

. , , . , , , . , .

2, (, , ) 0. ,

136

= (, ), =(, ). , 2

() =

0

(, )(, ) . (4.32)

(, ) , (, )= ()(), () . , , (, ) = ()(). , () = ()

=

()() (4.33)

() =

0 ()( ) . (4.34)

, , , . , . , .

, . ( ) . ( ), , . , . , ( ).

() =

( )() . (4.35)

. . . , . ,

4.2 137

. , , . () . , , , .

. , ( 4.31). , , , ,

(, ) = ()() =

( )( )()() .(4.36)

, , . 4.31, , , ,

(, ) = 2( ) . (4.37) ( 4.31). .

()() 4.36 . (, , | |),

( ) = ()() . (4.38) , 4.37 . ,

( ) = 142

( ( ))() (4.39)

( ) = 142

( ( ))() . (4.40)

138

, , ( 1). , 4.37

|()|2() = 2 , (4.41)

|()| = ()

. (4.42)

4.42 , ()|()|2 . . , . , 4.42 . , .

|()|, () . , . 4.42 .

, () 1/||2 . = 0 , 1/||2 = 0 () 1/(||2 + 20) 0 . , . () ( ).

,

() (||)||2 + 20, (4.43)

. 4.42 |()|, , ||. . , , . . ,

4.2 139

. . , , , . .

. , , . , , . , , .

() () , , . , () : , (), ; , (), 4.42. () = () (). , ()+ (), , (), . 2 ( ) 3 ( ) ( ). . , 4.35 , . , , .

140

1.0

0.8

0.6

0.4

0.2

0.0

(degrees)

A B1.0

0.8

0.6

0.4

0.2

0.01.00.80.60.40.2

(cycles/degree)1.00.5 1.5 2.0 2.5 3.0

4.3 . () ( ) ( ) . |()| . () () ( ) . . () , (||) . 1/ = 0.16 /, 0 = 0.16/, /(0) = 0.05 1 .

, ()() = 0. ,

(()+ ())() = ( ) , (4.44)

(()+ ())(()+ ()) = ( )+ ( ) , (4.45) , , . , ,

() = ()()+ () . (4.46)

, ( 4.42). ,

|()|

()()+ () . (4.47)

4.47, 4.43 , 4.3. , . ( 4.3)

4.2 141

() ( 4.3) . 4.3 () 4.47 , . () , || , 4.3. ( 4.3), , 2. : .

( 4.3), . , , . . , , . ( ) . , .

. , . . (), 4.34. 4.37

()() = 2( ) . (4.48) 4.37 . , , . () ,

() = 12

Theoretical Neuroscience Computational and Mathematical Modeling of Neural Systems-Dayan,Abbott

Documents

Transcript of Theoretical Neuroscience Computational and Mathematical Modeling of Neural Systems-Dayan,Abbott