Consciousness, cognition, and the hierarchy of context: extending the global neuronal workspace model

The ui , vi , i = 1, 2 are functions of KR, JR, but not explic-
itly of R itself.

We expand these equations about the critical value KR =
KC and about JR = 0, obtaining

Solving this for RC and substituting into the first of equa-
tion (2) gives, as a first iteration of a far more general proce-
dure (e.g. Shirkov and Kovalev, 2001)

dKR/dR = (KR-Kr)ydlog(f)/dR+(KR-Kr)z/R
dJR/dR = wJRdlog(f)/dR + xJR/R.

(6)

The terms y =     du1 /dKR |K_R =K_C , z     =

du2/dKR|K_R=K_C, w = v1(Kr, 0), x = v2(Kr, 0) are
constants.

Solving the first of these equations gives

χ(K, 0) ≈ χ(Kc/2, 0)Rr = XoRr

(10)

which are the essential relationships.

Note that a power law of the form f(R) = R^m, m = 3,
which is the direct physical analog, may not be biologically
reasonable, since it says that ‘language richness’ can grow
very rapidly as a function of increased network size. Such
rapid growth is simply not observed.

If we take the biologically realistic example of non-integral
‘fractal’ exponential growth,

KR = Kr + (K - Kr)R^zf(R)^y,

(11)

again remembering that K₁ = K, J₁ = J, f(1) = 1.

Wilson’s essential trick is to iterate on this relation, which
is supposed to converge rapidly (Binney, 1986), assuming that
for KR near Kr , we have

where δ > 0 is a real number which may be quite small, we
can solve equation (8) for R_r, obtaining

[KC∕(KC - K)][V(^δy+^z)]
2V(^δy+^z)

Kr/2 ≈ Kr + (K - Kr)R^zf (R)^y.

(8)

We iterate in two steps, first solving this for f(R) in terms
of known values, and then solving for R, finding a value Rr
that we then substitute into the first of equations (2) to obtain
an expression for H [K, 0] in terms of known functions and
parameter values.

The first step gives the general result

for K near Kr . Note that, for a given value of y , we might
want to characterize the relation α ≡ δy + z = constant as
a “tunable universality class relation” in the sense of Albert
and Barabasi (2002).

Substituting this value for Rr back into equation (9) gives
a somewhat more complex expression for H than equation
(2), having three parameters, i.e. δ, y, z .

A more biologically interesting choice for f (R) is a loga-
rithmic curve that ‘tops out’, for example

<<   5   6   7   8   9   10   11   >>

More intriguing information