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.
Highlights
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 1991, Felleman and Van Essen formalized the idea of a visual cortical hierarchy using a large number of tract tracing results obtained from macaque monkeys
Since the publication of this article, the question emerged: what is the optimal visual cortical hierarchy? Using the same set of criteria and notion of optimality, Hilgetag et al (1996) demonstrated that there are at least 100,000 hierarchies which are even more optimal than that introduced by Felleman and Van Essen
Summary
In 1991, Felleman and Van Essen formalized the idea of a visual cortical hierarchy using a large number of tract tracing results obtained from macaque monkeys. Their general premise was that the laminar source and termination patterns of corticocortical projections contained information about their hierarchical directionality, which allowed projections to be labelled as ascending, descending, or lateral. Our approach utilized a continuous, rather than discrete scale for describing hierarchical level, and introduced the measurement of deviation from a constraint as a cost function for optimization, rather than a count of discrete violations This method does not produce a unique optimal hierarchy – an indeterminacy problem similar to that reported by Hilgetag et al (1996) – it does always produce an optimal hierarchy. The method is implemented, such that an optimal hierarchy can be calculated for any arbitrary set of cortical areas with tract tracing information, and updated if new data are produced
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