Proyectos a Analizar

To rec ap itu late , given c active correct cells and s active spurious cells, to tallin g

w = c + s active cells, G ardner-M edw in (1976) estim ated th a t th e probability th a t any one correct cell receives exactly r effective in p u ts is:

=

. ) = ( =

( I )

® (I- O

(31)

F u rth e r, he approxim ates th e probability using a Poisson d istrib u tio n , which is reasonable to num erically com pute th e probability.

Sim ilarly, th e probability th a t a spurious cell receives exactly r effective in p u ts was proposed to be:

P{dspur = r ) = B c, (3.2) w here p is th e p ro po rtio n of modified synapses, p can be calculated as:

=

(3 3)

(For m ore d etailed explanation, see A ppendix B)

F u rth er, w hen c{t) correct cells and s{t) spurious cells are active a t tim e

t, th e expected num ber of active correct cells and active spurious cells a t th e n e x t tim e step w hen threshold T ( i -b 1) is applied are:

{c{t + 1)) = VFP(4orr > T { t + 1)) (3.4)

{s{t + 1)) = { N - W ) P { d , ^ r > T { t -f 1)) (3.5) Perform ance analysis using th e above form alism was carried out by G ardner- M edw in (1976) (See figure 3.3).

T h e above analysis does not accurately m odel th e behaviour of th e n e t­ work, because, as poin ted out by B uckingham (B uckingham 1991), it relies on

dendritic input distribution

200

estim ated by Gardner-Medwin 1976 --- p resent a n a ly s is ... simulation ... 150 spurious correct 10 15 effective input 20 25

F igure 3.2: D endritic in p u t distrib u tio n [ N = 5000,i? = 500,M = 200,PF = 200,c = 100,s = 0) T he new analysis closely m atches w ith th e sim ulated resu lt. T h e p rio r theory u nderestim ates th e d en d ritic in p u t of th e spurious cells. (A fter Buckingham (1991), using different p a ra m ete r sets.)

th e assu m p tion th a t every principal cell p a rtic ip a tes in th e same num ber of events (hence, use of th e con stan t p is ju stified). In reality, u n it usage of a p rin cip al cell varies from cell to cell an d th e p ro p o rtio n of m odified synapses to th e cell depends on th e level of u n it usage. In fact, u n it usage is binom ially d is trib u te d w ith th e principal probability W / N . Therefore, d istrib u tio n s in­ volving p m u st be modified to acquire more accu rate estim ates. T he modified version of form ula 3.2 is:

M _{/ H \}

P ( D ^ = r) = ^ P^{m)B ( X W j ÿ , c , r ) (3-6) w here is th e probability th a t a principal cell p a rtic ip a te s in m events and

p{m) is th e pro b ab ility th a t a synapse im pinging onto th e principal cell w ith u n it usage m is modified to be effective. T hey are B i W / N ^ M , m ) an d 1 — (1 —

respectively.

rh o = 0 .2 rh o = 0 .3 g o 1 0.8 0.6 0 .4 P rior --- M odified ... 0.2 0 0.6 0 0.2 0 .4 0.8 1 03 3 o 1 0.8 0.6 0 .4 P rior M odified 0.2 0 0 0.2 0 .4 0.6 0.8 1

Initial s e e d s iz e (wo/W ) Initial s e e d s iz e (wo/W )

Figure 3,3: A nalysis of th e expected value of recall perform ance (R eproduced and m odified from G ardner-M edw in (1976). These graphs com pare th e p e r­ form ance estim ated by G ardner-M edw in (1976) [Prior] w ith th e new analysis of th e p rio r strateg y based on B uckingham (1991) [Modified]. N ote th a t th e prior estim ates were overestim ates. (See figure 3.7.)

and has a longer tail, although th e m ean of th e d istrib u tio n is th e same. The im p act,o n th e original theory of th e corrections to th e p ro b ab ility of r effective in p u ts can be seen in figure 3.2, which com pares th e to ta l dendritic in p u t expected for correct and spurious cells in an exam ple netw ork configuration. In th is figure th e prior and present theories b o th p red ict th e same correct cell d istrib u tio n as th a t m easured in a sim ulation of th e netw ork. However th e prior m odel u n d erestim ates th e spurious cell d en d ritic in p u t, while th e present m odel m atches well w ith th e sim ulated network.

U sing th e corrected formalism, the capacity analysis is reassessed in figure 3.3. T h e horizontal axis of th e graph is th e size of in itial seed p a tte rn which contains only correct cells. In th e initial m odel (G ardner-M edw in 1976), th e vertical axis was percentile m atch of correct cells. In th is thesis, th e previously

defined q u ality m easure is used instead.

T h e increase in dendritic in p u t for spurious cells due to th e corrected den­ d ritic d istrib u tio n has a significant im pact on th e expected perform ance of progressive recall, as shown in Figure 3.3. In th is figure th e change in ex­ p ected qu ality is shown for one configuration of a netw ork. Note th e decrease in th e expected perform ance of th e prior thresh o ld strategy, th a t resu lts from th e incorrect estim ation of th e dendritic in p u t in spurious cells.