Plato's Cave: Orientational Harmonic Model

The Orientational Harmonic Model

An extension to the [Directed Diffusion model], the Orientational Harmonic model [Lehar 1994] accounts for all of the illusory vertex phenomena discussed above, and many more by way of a single simple mechanism. This model proposes that the cells in the cooperative layer representing different orientations at a single spatial location are arranged in rings, as shown below, the cell at three o'clock representing a horizontal edge to the right, nine o'clock a horizontal edge to the left, twelve o'clock a vertical edge above, etc.

This arrangement is consistent with the pinwheel model of cortical organization proposed by Braitenberg [2]. Each cell in the cooperative ring receives oriented input directly from the corresponding oriented cell at the same location, as well as indirectly from neighboring regions of the cooperative layer by way of monopolar receptive fields which receive oriented activation from like-oriented cooperative cells at adjacent locations in the oriented direction. For example the cell at three o'clock receives activation directly from the horizontal oriented cell at the same location (suggested by the shaded oriented cell in the figure) and indirectly from horizontal oriented cells displaced in the three o'clock direction by way of a cooperative receptive field which receives cooperative activation from that direction.

According to the Orientational Harmonic model, the cells in the cooperative ring are coupled so as to support standing waves of circular harmonic resonance within the cooperative ring. Harmonic resonance, whether mechanical, acoustical, or electrical, is a fundamental property of all physical systems, and has the property of sub-dividing the resonating system into integer numbers of equal intervals of alternating active and inactive regions. For example A in the figure below illustrates the first four harmonics of acoustical vibration in a linear tube, like a flute, where the grey shading denotes regions of high amplitude oscillation. C illustrates the first four harmonics of oscillation in a circular resonant system, like a closed circular tube. In an orientational representation such as the one proposed by this model, the patterns of standing waves depicted in B represent patterns of edges intersecting at a vertex, as shown in C. For example the fourth harmonic represents a four-way, or "+" vertex, the third harmonic represents a three-way "Y" vertex, the second harmonic represents a straight-through or colinear feature, while the first harmonic represents a single edge which terminates at the center, or an end-stop feature. There is also a zeroth harmonic which represents edge signals at all orientations equally, which is the pattern seen in response to a small circular dot.

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