Symmetries,non-Euclidean metrics,and patterns in a Swift–Hohenberg model of the visual cortex

The aim of this work is to investigate the effect of the shift-twist symmetry on pattern formation processes in the visual cortex. First, we describe a generic set of Riemannian metrics of the feature space of orientation preference that obeys properties of the shift-twist, translation, and reflection symmetries. Second, these metrics are embedded in a modified Swift-Hohenberg model. As a result we get a pattern formation process that resembles the pattern formation process in the visual cortex. We focus on the final stable patterns that are regular and periodic. In a third step we analyze the influences on pattern formation using weakly nonlinear theory and mode analysis. We compare the results of the present approach with earlier models.     

A theory of the symmetries of filamentous bacteriophages.

A mathematical model is presented which explains the symmetries observed for the protein coats of filamentous bacterial viruses. Three viruses (Ff, IKe, and If1) all have five-start helices with rotation angles of 36 degrees and axial translations of 16 A (Type I symmetry), and three other viruses (Pf1, Xf, and Pf3) all have one-start helices with rotation angles of approximately equal to 67 degrees and translations of approximately 3 A (Type II symmetry). The coat protein subunits in each group diverge from each other in amino acid sequence, and Type II viruses differ dramatically in DNA structure. Regardless of the differences, both Type I and Type II symmetry can be understood as direct, natural consequences of the close-packing of alpha-helical protein subunits. In our treatment, an alpha-helical subunit is modeled as consisting of two interconnected, flexible tubular segments that follow helical paths around the DNA, one in an inner layer and the other in an outer layer. The mathematical model is a set of algebraic equations describing the disposition of the flexible segments. Solutions are described by newly introduced symmetry indices and other parameters. An exhaustive survey over the range of indices has produced a library of all structures that are geometrically feasible within our modeling scheme. Solutions which correspond in their rotation angles to Type I and Type II viruses occur over large ranges of the parameter space. A few solutions with other symmetries are also allowed, and viruses with these symmetries may exist in nature. One solution to the set of equations, obtained without any recourse to the x-ray data, yields a calculated x-ray diffraction pattern for Pf1 which compares reasonably with experimental patterns. The close-packing geometry we have used helps explain the near constant linear mass density of known filamentous phages. Helicoid, rigid cylinder, and maximum entropy structure models proposed by others for Pf1 are reconciled with the flexible tube models and with one another.     

Morphology and the gradient of a symmetric potential predict gait transitions of dogs

Gaits and gait transitions play a central role in the movement of animals. Symmetry is thought to govern the structure of the nervous system, and constrain the limb motions of quadrupeds. We quantify the symmetry of dog gaits with respect to combinations of bilateral, fore–aft, and spatio-temporal symmetry groups. We tested the ability of symmetries to model motion capture data of dogs walking, trotting and transitioning between those gaits. Fully symmetric models performed comparably to asymmetric with only a \(22\%\) increase in the residual sum of squares and only one-quarter of the parameters. This required adding a spatio-temporal shift representing a lag between fore and hind limbs. Without this shift, the symmetric model residual sum of squares was \(1700\%\) larger. This shift is related to (linear regression, \(n=5\), \(p=0.0328\)) dog morphology. That this symmetry is respected throughout the gaits and transitions indicates that it generalizes outside a single gait. We propose that relative phasing of limb motions can be described by an interaction potential with a symmetric structure. This approach can be extended to the study of interaction of neurodynamic and kinematic variables, providing a system-level model that couples neuronal central pattern generator networks and mechanical models.     

Optimality of human contour integration

For processing and segmenting visual scenes, the brain is required to combine a multitude of features and sensory channels. It is neither known if these complex tasks involve optimal integration of information, nor according to which objectives computations might be performed. Here, we investigate if optimal inference can explain contour integration in human subjects. We performed experiments where observers detected contours of curvilinearly aligned edge configurations embedded into randomly oriented distractors. The key feature of our framework is to use a generative process for creating the contours, for which it is possible to derive a class of ideal detection models. This allowed us to compare human detection for contours with different statistical properties to the corresponding ideal detection models for the same stimuli. We then subjected the detection models to realistic constraints and required them to reproduce human decisions for every stimulus as well as possible. By independently varying the four model parameters, we identify a single detection model which quantitatively captures all correlations of human decision behaviour for more than 2000 stimuli from 42 contour ensembles with greatly varying statistical properties. This model reveals specific interactions between edges closely matching independent findings from physiology and psychophysics. These interactions imply a statistics of contours for which edge stimuli are indeed optimally integrated by the visual system, with the objective of inferring the presence of contours in cluttered scenes. The recurrent algorithm of our model makes testable predictions about the temporal dynamics of neuronal populations engaged in contour integration, and it suggests a strong directionality of the underlying functional anatomy.     

Posterior Probability Matching and Human Perceptual Decision Making

Probability matching is a classic theory of decision making that was first developed in models of cognition. Posterior probability matching, a variant in which observers match their response probabilities to the posterior probability of each response being correct, is being used increasingly often in models of perception. However, little is known about whether posterior probability matching is consistent with the vast literature on vision and hearing that has developed within signal detection theory. Here we test posterior probability matching models using two tools from detection theory. First, we examine the models’ performance in a two-pass experiment, where each block of trials is presented twice, and we measure the proportion of times that the model gives the same response twice to repeated stimuli. We show that at low performance levels, posterior probability matching models give highly inconsistent responses across repeated presentations of identical trials. We find that practised human observers are more consistent across repeated trials than these models predict, and we find some evidence that less practised observers more consistent as well. Second, we compare the performance of posterior probability matching models on a discrimination task to the performance of a theoretical ideal observer that achieves the best possible performance. We find that posterior probability matching is very inefficient at low-to-moderate performance levels, and that human observers can be more efficient than is ever possible according to posterior probability matching models. These findings support classic signal detection models, and rule out a broad class of posterior probability matching models for expert performance on perceptual tasks that range in complexity from contrast discrimination to symmetry detection. However, our findings leave open the possibility that inexperienced observers may show posterior probability matching behaviour, and our methods provide new tools for testing for such a strategy.     

The common patterns of nature

We typically observe large‐scale outcomes that arise from the interactions of many hidden, small‐scale processes. Examples include age of disease onset, rates of amino acid substitutions and composition of ecological communities. The macroscopic patterns in each problem often vary around a characteristic shape that can be generated by neutral processes. A neutral generative model assumes that each microscopic process follows unbiased or random stochastic fluctuations: random connections of network nodes; amino acid substitutions with no effect on fitness; species that arise or disappear from communities randomly. These neutral generative models often match common patterns of nature. In this paper, I present the theoretical background by which we can understand why these neutral generative models are so successful. I show where the classic patterns come from, such as the Poisson pattern, the normal or Gaussian pattern and many others. Each classic pattern was often discovered by a simple neutral generative model. The neutral patterns share a special characteristic: they describe the patterns of nature that follow from simple constraints on information. For example, any aggregation of processes that preserves information only about the mean and variance attracts to the Gaussian pattern; any aggregation that preserves information only about the mean attracts to the exponential pattern; any aggregation that preserves information only about the geometric mean attracts to the power law pattern. I present a simple and consistent informational framework of the common patterns of nature based on the method of maximum entropy. This framework shows that each neutral generative model is a special case that helps to discover a particular set of informational constraints; those informational constraints define a much wider domain of non‐neutral generative processes that attract to the same neutral pattern.     

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

The symmetries of filamentous phage particles

The helical symmetries of two classes of filamentous bacteriophage particles are distinctly different. The symmetry of the class I particles is²C₅S_~2.0 (a 5-fold rotation axis combined with an approximately 2-fold screw axis). The symmetry of the class II particles is C₁S_5.4 (a one-start helix with 27 subunits equally spaced along five turns). The same basic α-helical interlocking arrangement of the largely α-helical coat protein subunits can be accommodated by the symmetry of the two classes of phage particles. The conservation of this structural pattern reflects intrinsic packing properties of α-helices. The difference between the symmetries of the class I and class II particles suggests that different assembly processes may have evolved to form these structures with very similar protein packing architectures.     

Sex-specific meiotic drive and selection at an imprinted locus

We present a one-locus model that breaks two symmetries of Mendelian genetics. Whereas symmetry of transmission is breached by allowing sex-specific segregation distortion, symmetry of expression is breached by allowing genomic imprinting. Simple conditions for the existence of at least one polymorphic stable equilibrium are provided. In general, population mean fitness is not maximized at polymorphic equilibria. However, mean fitness at a polymorphic equilibrium with segregation distortion may be higher than mean fitness at the corresponding equilibrium with Mendelian segregation if one (or both) of the heterozygote classes has higher fitness than both homozygote classes. In this case, mean fitness is maximized by complete, but opposite, drive in the two sexes. We undertook an extensive numerical analysis of the parameter space, finding, for the first time in this class of models, parameter sets yielding two stable polymorphic equilibria. Multiple equilibria exist both with and without genomic imprinting, although they occurred in a greater proportion of parameter sets with genomic imprinting.     

Theoretical morphology of tetrapod skull networks

Network models of the tetrapod skull in which nodes represent bones and links represent sutures have recently offered new insights into the structural constraints underlying the evolutionary reduction of bone number in the tetrapod skull, known as Williston's Law. Here, we have built null network model-derived generative morphospaces of the tetrapod skull using random, preferential attachment, and geometric proximity growth rules. Our results indicate that geometric proximity is the best null model to explain the disparity of skull structures under two structural constraints: bilateral symmetry and presence of unpaired bones. The analysis of the temporal occupation of this morphospace, concomitant with Williston's Law, indicates that the tetrapod skull has followed an evolutionary path toward more constrained morphological organizations.     

An electrostatic spatial resonance model for coaxial helical structures with applications to the filamentous bacteriophages.

A model is presented that treats the symmetry matching problem in structures made of two interacting coaxial helices of point charges. The charges are sources of a potential field that mediates a non-specific attractive interaction between the helices. The problem is represented in Fourier space, which affords the most generality. It is found that coaxial helices with optimally mated symmetries can lock into spatial resonance configurations that maximize their interaction. The resonances are represented as vectors in a discrete three-dimensional space. Two algebraic relations are given for the four symmetry parameters of two helices in resonance. One-start inner helices interacting with coaxial one-start or NR-start outer helices are considered. Applications are made to the filamentous bacteriophages Ff, Pf1, Xf, and Pf3. The interaction given by the linearized Poisson-Boltzmann equation is calculated in this formalism to allow comparison of the electrostatic free energy of interaction of different resonance structures. Experimental nucleotide/subunit ratios are accounted for, and models for the DNA-protein interfaces are presented, with particular emphasis on Pf1.     

Perceptual Anthropology: The Cultural Salience of Symmetry

In this paper, I advance the position that knowledge about the universals of form perceived by the visual system is fundamental to a theory of how art communicates. I focus on how the perceptual system uses the universal property of symmetry to recognize and classify form. I propose that the symmetries that structure design parts in non-representational geometric patterns metaphorically encode a culture's fundamental relationships about the world. This metaphorical use of the property of symmetry is illustrated by showing how bifold symmetries in ceramic design embody Puebloan concepts of life. [symmetry of pattern, perceptual universals, worldview in design structure, metaphorical readings]     

An excitable cortex and memory model successfully predicts new pseudopod dynamics

Motile eukaryotic cells migrate with directional persistence by alternating left and right turns, even in the absence of external cues. For example, Dictyostelium discoideum cells crawl by extending distinct pseudopods in an alternating right-left pattern. The mechanisms underlying this zig-zag behavior, however, remain unknown. Here we propose a new Excitable Cortex and Memory (EC&M) model for understanding the alternating, zig-zag extension of pseudopods. Incorporating elements of previous models, we consider the cell cortex as an excitable system and include global inhibition of new pseudopods while a pseudopod is active. With the novel hypothesis that pseudopod activity makes the local cortex temporarily more excitable--thus creating a memory of previous pseudopod locations--the model reproduces experimentally observed zig-zag behavior. Furthermore, the EC&M model makes four new predictions concerning pseudopod dynamics. To test these predictions we develop an algorithm that detects pseudopods via hierarchical clustering of individual membrane extensions. Data from cell-tracking experiments agrees with all four predictions of the model, revealing that pseudopod placement is a non-Markovian process affected by the dynamics of previous pseudopods. The model is also compatible with known limits of chemotactic sensitivity. In addition to providing a predictive approach to studying eukaryotic cell motion, the EC&M model provides a general framework for future models, and suggests directions for new research regarding the molecular mechanisms underlying directional persistence.     

The effects of temperature, inlet pollutant concentration, and microorganism concentration on the rate of aerobic biological treatment

Systematic errors obtained in using traditional models of aerobic waste treatment processes are discussed. These errors are observed to arise due to the lack of matching of traditional models to experimental data in the case of broad variations of process parameters, for instance, organic loading. It is shown that the generalized models of waste treatment proposed earlier by the author permit these errors to be eliminated.     

Generalization and the evolution of symmetry preferences

Biological displays are often symmetrical, and there is growing evidence that receivers are sensitive to these symmetries. One explanation for the evolution of such sensitivity is that symmetry reflects the quality of the signaller. An alternative is that the sensitivity may arise as a by-product of general properties of biological recognition systems. In line with the latter idea, simulations of the recognition process based on simple, artificial neural networks have suggested that generalization can give rise to preferences for particular symmetrical stimuli. However, it is not clear from these studies exactly how the preferences emerge, and to what extent the results are relevant to biological recognition systems. Here, we employ a different class of recognition models (gradient interaction models) to demonstrate more clearly how generalization can generate a preference for symmetrical variants of a display. We also point out that the predictions of the gradient interaction and network-based models regarding the effects of generalization closely match the results from empirical studies of stimulus control. Our analysis demonstrates that the effects of generalization cannot be ignored when studying the evolution of symmetry preferences and symmetric signals.