The network consists of three input layers and an intermediate layer.
The three input layers – an eye-centered layer, an eye position layer
and a head-centered layer – are also output layers; the final
estimates of the network are read from these layers after relaxation.
The three input layers consist of three topographic layers of N units
indexed by their position, i(or j,k), where i(or
j,k) = 1…N. Similarly,
the intermediate layer is a topographic 2D map of N´N units indexed by their position l,
m, where l = 1…N and m=1…N.
The input layers are symmetrically interconnected with the intermediate
layer (hidden layer), and the corresponding matrices of connection
weights are denoted by Wr, We, Wa for,
respectively, the eye-centered, eye position and head-centered layers.
The connection strengths between unit i
(j,k) in each input
layer and unit (l,m) in the
intermediate layer are given by
The variable sw
represents lateral spread: unit i is strongly connected if
. Note that with these connection matrices, unit (l,m)
in the intermediate layer is most strongly interconnected with unit i=l
in the eye-centered layer, j=m
in the eye position layer and k=l+m
in the head-centered layer. Unit (l,m)
is connected more weakly to neighboring units in each layers, with the
spatial extent of these connections is controlled by sw
Rri(t), Rej(t), and Rak(t)
are denoted as the activity of unit i
(j,k) in the eye-centered,
eye-position and head-centered layer at time t.
For eye-centered position, the probability distribution for the initial
activity, denoted Rri(0),
is given by
The expressions for P(Rej(0)|xe)
and P(Rak(0)|xa) are identical to P(Rri(0)|xr),
except that r is replaced by e
or a. The expressions for f (xa)
and f (xe) are
identical to fi(xr)
except that r is replaced by a or e.
The activity in the intermediate layer, Alm(0),
is initialized to 0: Alm(0)
= 0 for all l,m.
Recurrent network evolution
The evolution of the activities in the recurrent network is described by
a set of coupled nonlinear equations. Denoting Alm(t)
as the activity of unit (l,m)
in the intermediate layer at time t, the evolution equations are written
Llm(t) represents a
linear pooling of activities from the three input layers. The activation
functions Alm or Rri, Rej,Rak
implement a quadratic nonlinearity coupled with a divisive
Parameters used in the simulation
= 20 Hz, n
= 1 Hz, s
= 0.40 radians, Kw =
= 0.002 s and S = 0.1 Hz.
sw = 0.37 radians, Cr=Ce=Ca=1 s.