Criteria for optimizing cortical hierarchies with continuous ranges

Abstract

In a recent paper (Reid et al., 2009) we introduced a method to calculate optimal hierarchies in the visual network that utilizes continuous, rather than discrete, hierarchical levels, and permits a range of acceptable values rather than attempting to fit fixed hierarchical distances. There, to obtain a hierarchy, the sum of deviations from the constraints that define the hierarchy was minimized using linear optimization. In the short time since publication of that paper we noticed that many colleagues misinterpreted the meaning of the term "optimal hierarchy". In particular, a majority of them were under the impression that there was perhaps only one optimal hierarchy, but a substantial difficulty in finding that one. However, there is not only more than one optimal hierarchy but also more than one option for defining optimality. Continuing the line of this work we look at additional options for optimizing the visual hierarchy: minimizing the number of violated constraints and minimizing the maximal size of a constraint violation using linear optimization and mixed integer programming. The implementation of both optimization criteria is explained in detail. In addition, using constraint sets based on the data from Felleman and Van Essen (1991), optimal hierarchies for the visual network are calculated for both optimization methods.

Keywords: connectivity; hierarchy; linear programming; macaque; mixed integer programming; optimality; visual system.

Figure 1

A directed graph with vertices v₁,v₂,v₃,v₄,v₅,v₆,v₇ and edge ranges.

Figure 2

Vertices u and v with the hierarchical distance [x,y].

Figure 3

Left: The linear program to calculate an optimal hierarchy for the graph from Figure 1. The objective is to minimize the sum of all deviations (defined by constraint c20). Right: The resulting hierarchy. Red numbers are the hierarchy levels of nodes, blue numbers the actual distances in the hierarchy.

Figure 4

(A) A graph with edge values. (B) One possible hierarchy obtained by optimizing for the number of constraint violations. (C) Another possible hierarchy obtained by optimizing for the maximum deviation. Both (B) and (C) red numbers are the hierarchy levels of nodes, blue numbers the actual distances in the hierarchy.

Figure 5

(A) An optimal hierarchy for the graph from Figure 1 given the objective to minimize primarily the sum of all deviations and in addition the maximal deviation. Red numbers are the hierarchy levels of nodes, blue numbers the actual distances in the hierarchy. (B) An optimal hierarchy for the graph from Figure 1 given the objective is to minimize primarily the sum of all deviations and in addition the number of deviations. Again, red numbers are the hierarchy levels of nodes, blue numbers the actual distances in the hierarchy.

Figure 6

Geometrical distribution of hierarchy levels in the FV91 visual network of the macaque, both as three-dimensional cortical surface renderings (left), and as a two-dimensional “flat map” representation of the cortical sheet (right). Regions are coloured by their mean normalized hierarchical position, obtained by the first optimization method over the 10 constraint sets described in Table 1. Directed edges in the flat map illustration represent interregional connections, and are coloured according to projection class: A+ (purple, opaque), A (purple, transparent), L (black), D (green, transparent), and D+ (green, opaque). Compare this representation to Figures 2 and 4 in Felleman and Van Essen (1991)..