Biological Computation

188    ◾    Biological Computation

where the sign() function returns 1 for positive values and –1 for nega-

tive values.

By inserting the expression for T

i,j

:

i

v

i j

j

v

j

i

k

j

k

k

U

T

U

N

U U

= =

∑ ∑

sign sign

,

1

∑

j

v

j

U

Isolating the k = v term and manipulating the equation gives

U

N

U U U U U

i

v

i

k

j

k

i

v

j

v

k v

j

j

= +

≠

∑∑

sign

1

ν

U

N

U U U

N

U U U

i

v

i

k

j

k

k v

j

i

v

j

v

j

v

= +

( )

≠

∑

sign

1 1

ν

jjj

∑∑

U U

N

U U U

i

v

i

v

i

k

j

k

k v

j

j

= +

≠

∑∑

sign

1

ν

(4.13)

If the second term is 0, U

v

is clearly stable. It will also be stable if the

second term is small enough: if

U

i

v

is 1, then if the second term is greater

than or equal to –1 it cannot ip the sign of

U

i

v

. Similarly, if

U

i

v

is –1 then

the second term will ha ...