To understand evolutionary paths connecting diverse biological forms, we defined a three-dimensional genotypic space separating two flower color morphs of Antirrhinum. A hybrid zone between morphs showed a steep cline specifically at genes controlling flower color differences, indicating that these loci are under selection. Antirrhinum species with diverse floral phenotypes formed a U-shaped cloud within the genotypic space. We propose that this cloud defines an evolutionary path that allows flower color to evolve while circumventing less-adaptive regions. Hybridization between morphs located in different arms of the U-shaped path yields low-fitness genotypes, accounting for the observed steep clines at hybrid zones.

Understanding evolutionary change requires phenotypic differences between organisms to be placed in a genetic context. However, there are few cases where it has been possible to define an appropriate genotypic space for a range of species. Here we address this problem by defining a genetically controlled space that captures variation in shape and size between closely related species of Antirrhinum. The axes of the space are based on an allometric model of leaves from an F 2 of an interspecific cross between Antirrhinum majus and Antirrhinum charidemi. Three principal components were found to capture most of the genetic variation in shape and size, allowing a three-dimensional allometric space to be defined. The contribution of individual genetic loci was determined from QTL analysis, allowing each locus to be represented as a vector in the allometric space. Leaf shapes and sizes of 18 different Antirrhinum taxa, encompassing a broad range of leaf morphologies, could be accurately represented as clouds within the space. Most taxa overlapped with, or were near to, at least one other species in the space, so that together they defined a largely interconnected domain of viable forms. It is likely that the pattern of evolution within this domain reflects a combination of directional selection and evolutionary tradeoffs within a high dimensional space.leaf ͉ morphometry ͉ QTL analysis ͉ shape variation ͉ species

A combination of experimental analysis and mathematical modelling shows how the genetic control of tissue polarity plays a fundamental role in the development and evolution of form.

A key approach to understanding how genes control growth and form is to analyze mutants in which shape and size have been perturbed. Although many mutants of this kind have been described in plants and animals, a general quantitative framework for describing them has yet to be established. Here we describe an approach based on Principal Component Analysis of organ landmarks and outlines. Applying this method to a collection of leaf shape mutants in Arabidopsis and Antirrhinum allows low-dimensional spaces to be constructed that capture the key variations in shape and size. Mutant phenotypes can be represented as vectors in these allometric spaces, allowing additive gene interactions to be readily described. The principal axis of each allometric space reflects size variation and an associated shape change. The shape change is similar to that observed during the later stages of normal development, suggesting that many phenotypic differences involve modulations in the timing of growth arrest. Comparison between allometric mutant spaces from different species reveals a similar range of phenotypic possibilities. The spaces therefore provide a general quantitative framework for exploring and comparing the development and evolution of form

A fully adaptive normalized nonlinear complex-valued gradient descent (FANNCGD) learning algorithm for training nonlinear (neural) adaptive finite impulse response (FIR) filters is derived. First, a normalized nonlinear complex-valued gradient descent (NNCGD) algorithm is introduced. For rigour, the remainder of the Taylor series expansion of the instantaneous output error in the derivation of NNCGD is made adaptive at every discrete time instant using a gradient-based approach. This results in the fully adaptive normalized nonlinear complex-valued gradient descent learning algorithm that is suitable for nonlinear complex adaptive filtering with a general holomorphic activation function and is robust to the initial conditions. Convergence analysis of the proposed algorithm is provided both analytically and experimentally. Experimental results on the prediction of colored and nonlinear inputs show the FANNCGD outperforming other algorithms of this kind.

Abstract-A fully adaptive normalized nonlinear gradient descent (FANNGD) algorithm for online adaptation of nonlinear neural filters is proposed. An adaptive stepsize that minimizes the instantaneous output error of the filter is derived using a linearization performed by a Taylor series expansion of the output error. For rigor, the remainder of the truncated Taylor series expansion within the expression for the adaptive learning rate is made adaptive and is updated using gradient descent. The FANNGD algorithm is shown to converge faster than previously introduced algorithms of this kind.Index Terms-Adaptive step size, gradient descent algorithms, neural networks, nonlinear adaptive prediction.

Analysis of a normalised backpropagation (NBP) algorithm employed in feed-forward multilayer nonlinear adaptive filters trained by backpropagation is provided. It is first shown that a degree of freedom in training of a nonlinear adaptive filter can be removed according to the relationship between the gain of the activation function, learning rate and weight matrix. The derivation of the NBP algorithm for a multilayer feed-forward neural adaptive filter is then provided based upon the minimisation of the instantaneous output error of the filter. Simulation results show that the NBP algorithm converges faster than a standard backpropagation algorithm and achieves better prediction gain when applied to nonlinear and non-stationary signal

A nonlinear gradient descent (NGD) learning algorithm with an adaptive amplitude of the nonlinearity is derived for the class of nonlinear finite impulse response (FIR) adaptive filters (dynamical perceptron). This is based on the adaptive amplitude backpropagation (AABP) algorithm for large-scale neural networks. The amplitude of the nonlinear activation function is made gradient adaptive to give the adaptive amplitude nonlinear gradient descent (AANGD) algorithm, making the AANGD suitable for processing nonlinear and nonstationary input signals with a large dynamical range. Experimental results show the AANGD algorithm outperforming the standard NGD algorithm on both colored and nonlinear input with large dynamics. Despite its simplicity, the considered algorithm proves suitable for adaptive filtering of nonlinear and nonstationary signals.Index Terms-Adaptive amplitude of the nonlinearity, gradient descent algorithms, neural networks, nonlinear adaptive filters.