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Segmentation in Scale Space


Rolf D. Henkel

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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 tex2html_wrap_inline403 . 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:

eqnarray56

   figure62

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 tex2html_wrap_inline409 , tex2html_wrap_inline411 , tex2html_wrap_inline413 , tex2html_wrap_inline415 , tex2html_wrap_inline417 , and tex2html_wrap_inline419 .

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.

   figure73

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 tex2html_wrap_inline423 -range.

The correlation measure tex2html_wrap_inline425 (where tex2html_wrap_inline427 ) 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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