therefore have a significant effect on the electrical behavior of neurons within short time. The charge propagation is motivated by the fact that there exist traveling waves in the retina (Jacobs and Werblin, 1998).

6.2.2Method of Latencies

The second transformation that we suggest uses the idea that the input magnitude determines the latency of the first spike, a transformation called time-to-first spike by Thorpe (Thorpe, 1990), or generally called a latency code (section 3.3, see specifically figure 13). To implement this, one would continuously feed the neuron with a specific intensity value - it charges up the neuron at a rate determined by this value: hence, high pixel values will trigger early firing, low pixel values will trigger late firing. That will result in signaling bright areas first, followed by signaling of darker areas. Hence, contours are separated in time.

Before demonstrating the operation of the above processes on grayscale images, we firstly introduce the idea of charge propagation in a two-dimensional map and subsequently introduce the above contour detection process.

6.3 The Propagation Map

The propagation map is a two-dimensional array of connected neuronal units (figure 30). Each neuron, depicted as circle, is connected with its eight neighbors via a ‘horizontal’ resistance. The neuron model is a perfect integrate-and-fire unit (chapter 3, section 3.7), but the exact neuron model does not matter: a different model would just require an adjustment of certain parameters. The map is in some sense the compartmental modeling approach used for approximating dendrites, laid out in two dimensions, with each compartment having a spiking unit.

Figure 30: Connectivity of the propagation map. A circle represents a neuron, modeled as an integrate-and-fire type.

If the activity level of a neuron is raised above the spiking threshold of the map, then the generated spike will contribute significantly to the

Figure 31: Behavior of the propagation map in response to simple stimuli (Matlab simulation). a. Block source. b. Diagonal line. c. Horizontal line with gap. The snapshots are taken at times t=2, t=10 and t=18, with time equal the number of time steps. Grey: subthreshold voltages. Black: Spikes.

activity level of its neighbors and cause them to fire, thus triggering a traveling spike wave. In order to avoid the bounce back of a wave, the neuronal unit requires a short refractory period. Figure 31 illustrates this propagation process for three different stimuli. In figure 31a the stimulus is a point source: four neurons are activated by raising their activity threshold above the spiking threshold. The spikes are shown in black, the gray values ahead of the spike front represent subthreshold activity. The traveling wave is an annular outward growing wave. For a straight line, the traveling wave is of oval shape (figure 31b).

The propagation map has a characteristic which is extremely useful for encoding space: when the two traveling waves meet, they merge. This is shown in figure 31c for two short lines triggering traveling waves, which eventually merge into a single one. Thus, contour propagation seals gaps.

When a luminance landscape is placed into such a map - that does not raise the activity level above the spiking threshold -, then a smoothening process takes place by means of the lateral connections: the landscape starts to even out, but in particularly fast for ‘noisy’ values that pop out of an area or a local plateau. This subthreshold