Possible Neural Implementation
We now sketch a possible neuronal mechanism for the creation and merging
of contour sheets in scale space. As basic computing units pools of densely
interconnected excitatory and inhibitory spiking neurons are used. Single
neurons are described by their mean firing rate, which is given by the
internal field h via a sigmoid transfer function 
. Connecting all neurons within a pool in an all-to-all fashion leads to
the following mean-field equations for the averaged firing rates of the
excitatory (E) and inhibitory (I) neuron populations:
Figure 3: Mean firing rate (a) and slow modulation frequency of
the mean firing rate (b) for pools of coupled excitatory and inhibitory
neurons. Simulation parameters were 
, 
, 
, 
, 
, and 
.
These equations have been derived in a variety of contexts [14, 12, 9] and display the following basic behavior: as the applied stimulus S gets stronger, the mean firing rate of the pool increases (Fig. 3.a). In addition, the system displays limit-cycle behavior, resulting in a slow modulation of the mean firing rate. The frequency of this slow modulation is a monotonic function of the input stimulus (fig. 3.b) until saturation effects take over.
It is well known that systems of coupled limit-cycle oscillators display frequency and phase locking for a variety of connection schemes [5, 4, 10, 8]. In figure 4.a a prototypical network is displayed. Four oscillators are receiving fixed stimuli, whereas a a fifth oscillator is tuned through the available stimulus range. The oscillators with fixed input have small synaptic links from their excitatory pools to the inhibitory pool of the oscillator which is tuned through.
Figure 4: a) In a prototypical network, one of the oscillators is
tuned through the stimulus range, while four other oscillators are kept
fixed at stimulus values of 0.2, 0.4, 0.6, and 0.8. In b) the correlation
measure indicates synchronization within a small 
-range.
The correlation measure 
(where 
) between the various oscillator-pairs shows four pronounced peaks over
the whole stimulus range (figure 4.b).
These peaks indicate frequency locking of the central oscillator with one
of the four driving oscillators. Generally, depending on the coupling strength
between them, oscillators close in frequency will tend to group together.
They will form clusters oscillating at a common frequency, while oscillators
further away from the principal cluster frequency stay unsynchronized with
that cluster.
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Comments are welcome! © 1997 by Rolf Henkel - all rights reserved.