RULE-BASED GROWTH OF FFT BU~ERFLY NETWORKS. Georae McKee, College of Computer Science, Northeastern University, 360 Huntington Avenue, 161CN,Boston, MA 02115 USA
In biological organisms, the mode by which the genetic program is transformed into a fullscale multtcellular system is by the developmental process of growth. Growth is governed by processes that controt the sequence in which cells in different locations of the organism multiply their numbers, change their characteristics, and affect the properties and behavior of their neighbors. In biological systems, these processes have their origin in regulatory genes, i.e. DNA sequences that c o d e for enzymes that control the timing and degree to which other DNA sequences are translated into structural proteins. This abstract describes a realization of these processes in a biologically-derived set of abstract developmental rules, and their expression in a developmental program that leads to the same connectivity as the "butterfly' network that describes the flow of computation in the Fast Fourier Transform. It should be noted that simply interpreted as living neurons, this network would not compute Fourier transforms, since the transform involves transmission and multiplication of complex numbers. While biologically plausible systems carrying complex values can be hypothesized, it is unclear what neuroanatomical or physiological techniques could d e t e c t them. The set of rules we are currently working with have as their subject both migrating cells and migrating neuritic growth cones which trail neural processes attached to their parent cell bodies. The rules operate in a morphogenetic field defined by sources and sinks, and consist of two classes, one set of migration rules: •Centrifugal migration away from a morphogen source, • Centripetal migration towards a morphogen sink, • Unear migration along a morphogen gradient; and a second set of growth rules: • Mitosis without differentiation (normal cell division), • Mitosis with differentiation (stem cells), • Differentiation without mitosis, • Deletion (cell death). As a special mode of differentiation, mitoses can lose, dupllcate or modify the connectivity of their parent cell. A Iisl:>-based parallel programming language suitable for precise expression of large-scale configurations generated by these rules is under development. The growth of a second-order FFT network is shown in the diagrams at left. It takes place on a field of two morphogenetic gradients, linear from top to bottom, and linear from left to right. The growth of each FFTstage consists of a two-step initialization, followed by a series of four-step replication phases. Each step involves either a mitosis or a migration. The sequence of steps in the replication phase determines whether the replication creates several simple bulfferflies or a single higher-order but'iertly. The "developmental networks" that include the buffrerfly configuration form a class of net that gives an additional measure on network complexity in comparison with the fullyc o n n e c t e d connectionist nets of the perceptron family and the matrix-based genetic networks of Mjolsness et al. The well-known scaling problems of coarse-grained parallel computers are becoming apparent in synthetic neural networks as well; rule-governed connectivity offers a principled way to approach this problem. A neural connectivity pattern recognizable as a modified butterfly network has been described in the nematode C. elegans by J.G.White et al.; heterochronic mutants created by changes in the sequence of mitoses and migrations are a classic topic in developmental ontogeny and phylogeny, and have been described in detail tn this organism. The butterfly network in C. elegans connects the left side of its nervous system with the right. It is intriguing to speculate that a similar structure might connect the left and right sides of the mammalian brain, providing a means for the sequential thought processes of the left hemisphere to communicate with the "appositional' thought processes of the right hemisphere via an abstract "Conceptual Fourier Transform' definable on the lattice of symbols created as the history of an inferential derivation.