Diseño Un Sistema De Riego Por Aspersión En Jardines

G ibson an d R obinson(G ibson and R obinson 1992) calculated th e statistic a l dynam ics of activ ity of th e netw ork for progressive recall. T h ey are also aware of th e fact th a t wij and Wik are n o t independently d istrib u ted . T h eir m eth o d of calcu latin g th e statistic a l dynam ics, /e m progressive recall equation is re­ viewed as below (slightly modified from th e original derivation to fit th e con­ ventions defined sor far).

T h ey approxim ated th e dendritic sum d istrib u tio n using th e norm al d istri­ b u tio n s. G iven w active cells which consists of c correct cells and s spurious

cells:

P ( ^corr ) *A/*( fJ-corr j ^corr )

-P ( ^ apu r ) ” ■A/'( f^apur j ^ a p u r )

where is th e norm al d istrib u tio n w ith m ean ji and variance cr^. ficoTT is th e m ean num ber of dendritic in p u t received by a correct cell. Likewise, fiapur is th e m ean num ber of d en d ritic in p u t received by a spurious cell. T h ey are:

R R R

f^corr — C ^ fJ-apur ~ P~^

where w = c + s

^^corr is th e variance of th e dendritic in p u t d istrib u tio n to a correct cell and is th e variance of th e dendritic in p u t to a spurious cell. T his can be calculated by tre a tin g each of th e incom ing syn ap tic in p u t as a random variable so th a t th e dendritic in p u t is th e sum of th e ran d o m variables.

For a correct cell, th e probability d istrib u tio n for receiving an effective in p u t from a n o th er active correct cell is equivallent to a B ernoulli tria l w ith m ean ^ an d variance § ( l ~ Similarly, th e p ro b ab ility d istrib u tio n for receiving an effective in p u t from an o th er active spurious cell is a B ernoulli tria l w ith m ean and variance . As discussed in th e previous section, th e covariance for th e random variables Wij and Wik m ust also be taken in to consideration. T he covariance is denoted as cov{wij,W ik) for convenience,

is calcu lated as:

R / R \ R r R

^ "

]v)

Sim ilarly, is th e variance of th e sum of th e ran do m variable w*,*, where * ind icates th e indecies to th e active cell (th ere are w of them ).

R r R

cov{wij,W ik, th e covariance between wij an d Wik m u st be calculated. E labora- tive work by G ibson and Robinson (G ibson a n d R obinson 1992) d e m o n strated th a t th e covariance can be com puted as follows.

M 2M

=e)i-s)-æj -i‘-æ

Consequentially, given threshold value T { t + 1), th e expected activities of

correct cells an d spurious cell after th e th resh o ld is applied are:

fj>c — T { t + 1) \

— T { t + 1)

O', J

w here 0 (») is th e norm al d istrib u tio n function (for th e definition, see ap p en d ix

A).

T h e above is so called th e level 1 equations. (T he level 0 equation is th e

above w ith o u t th e consideration of th e covariance am ong th e modified sy n ap ­ tic connections — this is essentially th e same as G ardner-M edw in’s analysis (G ardner-M edw in 1976).) T he level 1 analysis is equivalent to approxim ating

B uckingham ’s dendritic sum d istrib u tio n to a no rm al d istrib u tio n . For sim ple recall (single step recall) th e formalism applies as c correct cells an d s spurious cells are a rb itra rily chosen. In progressive recall, th e level 1 equation p redicts

th e netw ork behaviour when th e netw ork loading is low. (i.e. sm all W and

M ) . For high netw ork loadings, it fails to p red ict th e behaviour of th e netw ork. As G ibson and Robinson (G ibson and R obinson 1992) inspected, th is is b e­ cause th e re is (yet again) a correlation betw een p re-th resh o ld netw ork activ ity (X (t)) an d po st-th resh o ld netw ork activ ity (X (i -h 1)). T h ey have worked

ou t th e level 2 equation which incorporates th e correlation betw een X (t) and X (( 4- 1). It tu rn s out th a t this can be w ritte n w ith 4 coupled non-linear

If we define x{t) as th e expected p ro b ab ility of a correct cell become active a t tim e t and y{t) as th e expected p ro b ability of a spurious cell become active a t tim e t (i.e. x {i) = { c { t))/W and x(t) = (s(i))/(A ^ — W) ) , th e level 2

equations are sum m arised as follows:

w here x (t)', y { ty , an d associated variables w ith ’ (dash) are subsidiary vari­ ables th a t need be com puted to find Xt+i an d yt+i^ x \ t ) and y'{t) can explicitly be w ritte n as:

_

w ’ N - W

Now, fJ>corr{t),fJ>3puT{t),(^corT{t), and u,pur(0 ^^.ve to be defined. G ibson and

R obinson calculates th a t they are:

l^corrit) = +

(l

^ )

= ^ ^ ' ( < ) + ( i - ^ ) § p y ' W

Mcorr( 0 rnu'gj^j.{t) can be calculated by replacing p by p' where pf is defined

below.

[ri(Ti{t)f — n a c {l — c)xt + n ( l - a)cpy[ 4- n ^ (l - aY c^^[y'yf

[n(Tn{t)Ÿ = nacpx[ ^ 1 — -f n ( l — a)cpy[ — c p ^ ^ +

n^a^c^^{x[Y 4- 2n^a(a - a)c^^x[y[ -f 4- ( 1 — o)^c^((?/{)^

Let us define a function pk which op erates on a an d M as: A = [1 - a ( l — (1 — a )^ )]^

T h e n , p' an d can com pactly expressed as: 1 — 2/3i + /?2 P = 1 - A H _ 1 — 3/3i + 3/32 — A _ /2 ^ ~ 1 - A