Nonparametric density estimation and regression achieved with topographic maps maximizing the information-theoretic entropy of their outputs

We introduce an unsupervised competitive learning rule, called the extended Maximum Entropy learning Rule (eMER), for topographic map formation. Unlike Kohonen's Self-Organizing Map (SOM) algorithm, the presence of a neighborhood function is not a prerequisite for achieving topology-preserving mappings, but instead it is intended: (1) to speed up the learning process and (2) to perform nonparametric regression. We show that, when the neighborhood function vanishes, the neural weigh t density at convergence approaches a linear function of the input density so that the map can be regarded as a nonparametric model of the input density. We apply eMER to density estimation and compare its performance with that of the SOM algorithm and the variable kernel method. Finally, we apply the ‘batch’ version of eMER to nonparametric projection pursuit regression and compare its performance with that of back-propagation learning, projection pursuit learning, constrained topolog ical mapping, and the Heskes and Kappen approach. Received: 12 August 1996 / Accepted in revised form: 9 April 1997     

Emergence of invariant-feature detectors in the adaptive-subspace self-organizing map

 A new self-organizing map (SOM) architecture called the ASSOM (adaptive-subspace SOM) is shown to create sets of translation-invariant filters when randomly displaced or moving input patterns are used as training data. No analytical functional forms for these filters are thereby postulated. Different kinds of filters are formed by the ASSOM when pictures are rotated during learning, or when they are zoomed. The ASSOM can thus act as a learning feature-extraction stage for pattern recognizers, being able to adapt to many sensory environments and to many different transformation groups of patterns. Received: 14 September 1995 / Accepted in revised form: 8 May 1996     

Classification of protein families and detection of the determinant residues with an improved self-organizing map

 Using a SOM (self-organizing map) we can classify sequences within a protein family into subgroups that generally correspond to biological subcategories. These maps tend to show sequence similarity as proximity in the map. Combining maps generated at different levels of resolution, the structure of relations in protein families can be captured that could not otherwise be represented in a single map. The underlying representation of maps enables us to retrieve characteristic sequence patterns for individual subgroups of sequences. Such patterns tend to correspond to functionally important regions. We present a modified SOM algorithm that includes a convergence test that dynamically controls the learning parameters to adapt them to the learning set instead of being fixed and externally optimized by trial and error. Given the variability of protein family size and distribution, the addition of this feature is necessary. The method is successfully tested with a number of families. The rab family of small GTPases is used to illustrate the performance of the method. Received: 25 July 1996 / Accepted in revised form: 13 February 1997     

A SOM projection technique with the growing structure for visualizing high-dimensional data

The Self-Organizing Map (SOM) is an efficient tool for visualizing high-dimensional data. In this paper, an intuitive and effective SOM projection method is proposed for mapping high-dimensional data onto the two-dimensional grid structure with a growing self-organizing mechanism. In the learning phase, a growing SOM is trained and the growing cell structure is used as the baseline framework. In the ordination phase, the new projection method is used to map the input vector so that the input data is mapped to the structure of the SOM without having to plot the weight values, resulting in easy visualization of the data. The projection method is demonstrated on four different data sets, including a 118 patent data set and a 399 checical abstract data set related to polymer cements, with promising results and a significantly reduced network size.     

Classifying habitat characteristics of wetlands using a self-organizing map

Wetlands are nutrient-rich and biodiverse ecosystems that provide habitats for various animals and plants and protect against flooding. Classification of wetlands provides information to conservation planners and resource managers for ecosystem service determination. Many ecological case studies illuminate the self-organizing map (SOM) as a robust and powerful data classification and visualization tool. In this study, we use the SOM to analyze the habitat characteristics of inland wetlands in South Korea. We surveyed the plants, benthic macroinvertebrates, and bird species inhabiting 530 nationwide wetlands for four years from 2016 to 2019. Nine environmental features, including the proportion of urban area, farmland, grassland, a forest within a 1 km buffer zone, distance from the river and nearest wetland, area, perimeter, and average slope of wetland polygons, were used to train the SOM and examine the habitat characteristics of the surveyed living components. A map size of 10 × 11 pixels was considered for SOM training, and the output data were classified into eight clusters. Based on the occurrence frequency of the surveyed species group, most species were distributed in all clusters, whereas some dominated in specific clusters. We believe that our study contributes significantly to the literature because it highlights the significance of the SOM approach to cluster wetlands with dependent habitats and provides ecological information to build sustainable wetland conservation policies.     

Manifold learning for robot navigation

In this paper we introduce methods to build a SOM that can be used as an isometric map for mobile robots. That is, given a dataset of sensor readings collected at points uniformly distributed with respect to the ground, we wish to build a SOM whose neurons (prototype vectors in sensor space) correspond to points uniformly distributed on the ground. Manifold learning techniques have already been used for dimensionality reduction of sensor space in navigation systems. Our focus is on the isometric property of the SOM. For reliable path-planning and information sharing between several robots, it is desirable that the robots build an internal representation of the sensor manifold, a map, that is isometric with the environment. We show experimentally that standard Non-Linear Dimensionality Reduction (NLDR) algorithms do not provide isometric maps for range data and bearing data. However, the auxiliary low dimensional manifolds created can be used to improve the distribution of the neurons of a SOM (that is, make the neurons more evenly distributed with respect to the ground). We also describe a method to create an isometric map from a sensor readings collected along a polygonal line random walk.     

Diffusion-based learning theory for organizing visuo-motor coordination

A diffusion-based learning theory is presented and applied to organize the visuomotor coordination of an eye-hand system which has redundant motion degree of freedom (dof). This theory considers the spatial optimality of the coordination: to minimize the end-effector position error of the eye-hand system as well as the differentiation of the joint angles with respect to the end-effector positions over all the bounded work space. By introducing variational methods with respect to the space, we derive a partial differential equation (PDE) of the joint angles with respect to the work space. The equation includes a diffusion term. For the given boundary conditions and the initial conditions, it can be solved uniquely, and the solution is a well organized map. From the motor learning point of view, our approach contains both the aspects of supervised learning as well as self-organization. Firstly, we assume that the forward relation from the hand system's joint angles to its end-effector positions can be obtained using supervised learning, and at the boundary of the work space, the supervisor can provide correct joint information. Then, by evolving the diffusion equation, we organize the visuomotor coordination. We show the effectiveness of this approach using a 3-dof scale manipulator. The problems of how to realize the visuomotor map; how to utilize the resultant map in several motions; and what are the influences of the initial conditions on the map formation and the relation to the boundary conditions are also discussed using computer simulations. Our approach has three advantages: (1) it does not require too many trial motions for the eye-hand system; (2) during the map formation process, it requires only the local interactions between each node; and (3) it guarantees the final map's spatial optimality over all the bounded work space. Received: 8 May 1997 / Accepted in revised form: 12 June 1998     

Applications of self-organizing map for patterning macrofungal diversity of xerothermic swards

Macrofungal communities were investigated in four associations of xerothermic swards: Festucetum pallentis, Origano-Brachypodietum, Adonido-Brachypodietum pinnati and Diantho-Armerietum elongatae in a Jurassic area of the Częstochowa Upland (southern Poland). A total of 47 species were recorded. The self-organising map (SOM)—an unsupervised algorithm for artificial neural networks—was used to recognise patterns in the macrofungal communities of diverse xerothermic swards. Only two associations were mycologically similar: Origano-Brachypodietum and Adonido-Brachypodietum pinnati. Species with high and significant IndVal (the species indicator value) for each investigated phytocoenoses are presented. The presence of macrofungal species and the participation of indicator species were connected with habitat factors of plant associations, as documented by the IndVal application. In the least fertile phytocoenoses, macrofungal communities were poor with few indicator species. The more fertile phytocoenoses had richer and more varied communities of macrofungi with higher numbers of indicator species. The ordering methods applied in this study were very effective for analyzing the macrofungal communities existing in plant associations.     

Spatial representation of the glomerular map in the Drosophila protocerebrum

In the fruit fly, Drosophila, olfactory sensory neurons expressing a given receptor project to spatially invariant loci in the antennal lobe to create a topographic map of receptor activation. We have asked how the map in the antennal lobe is represented in higher sensory centers in the brain. Random labeling of individual projection neurons using the FLP-out technique reveals that projection neurons that innervate the same glomerulus exhibit strikingly similar axonal topography, whereas neurons from different glomeruli display very different patterns of projection in the protocerebrum. These results demonstrate that a topographic map of olfactory information is retained in higher brain centers, but the character of the map differs from that of the antennal lobe, affording an opportunity for integration of olfactory sensory input.     

A brain region-specific predictive gene map for autism derived by profiling a reference gene set

Molecular underpinnings of complex psychiatric disorders such as autism spectrum disorders (ASD) remain largely unresolved. Increasingly, structural variations in discrete chromosomal loci are implicated in ASD, expanding the search space for its disease etiology. We exploited the high genetic heterogeneity of ASD to derive a predictive map of candidate genes by an integrated bioinformatics approach. Using a reference set of 84 Rare and Syndromic candidate ASD genes (AutRef84), we built a composite reference profile based on both functional and expression analyses. First, we created a functional profile of AutRef84 by performing Gene Ontology (GO) enrichment analysis which encompassed three main areas: 1) neurogenesis/projection, 2) cell adhesion, and 3) ion channel activity. Second, we constructed an expression profile of AutRef84 by conducting DAVID analysis which found enrichment in brain regions critical for sensory information processing (olfactory bulb, occipital lobe), executive function (prefrontal cortex), and hormone secretion (pituitary). Disease specificity of this dual AutRef84 profile was demonstrated by comparative analysis with control, diabetes, and non-specific gene sets. We then screened the human genome with the dual AutRef84 profile to derive a set of 460 potential ASD candidate genes. Importantly, the power of our predictive gene map was demonstrated by capturing 18 existing ASD-associated genes which were not part of the AutRef84 input dataset. The remaining 442 genes are entirely novel putative ASD risk genes. Together, we used a composite ASD reference profile to generate a predictive map of novel ASD candidate genes which should be prioritized for future research.     

Iterative cooperation between parallel pathways for object and background motion

In a typical visual scene, one or more objects move relative to a larger background, which can itself be in motion as a result of the observer’s eyes moving with respect to the outside world. Here we show that accurate estimation of the background motion from an image velocity field can be accomplished through an iterative cooperation between two modules: one that specializes in calculating a weighted average velocity and another one calculating a velocity contrast map. We build on our analysis to provide a model for the tectum-pretectum loop in the nonmammalian midbrain. Our model accounts for some of the known properties of the tectal neurons (sensitivity to relative motion) and pretectal neurons (sensitivity to whole-field motion). It also agrees with our knowledge of the pretectotectal projection (divergent and inhibitory), and with the results of lesion studies in which the pretectal input to the tectum was removed, leading to hyperactivity of the tectal neurons and the animal. Our model also makes a testable prediction regarding the tectopretectal projection, i.e., that the presence of a larger object and a bigger discrepancy between the directions of motion for the object and the background lead to a larger error by the pretectum in estimating the background motion when the tectal input is abolished.     

Partition decoupling for multi-gene analysis of gene expression profiling data

Background

Molecular dynamics (MD) simulations are powerful tools to investigate the conformational dynamics of proteins that is often a critical element of their function. Identification of functionally relevant conformations is generally done clustering the large ensemble of structures that are generated. Recently, Self-Organising Maps (SOMs) were reported performing more accurately and providing more consistent results than traditional clustering algorithms in various data mining problems. We present a novel strategy to analyse and compare conformational ensembles of protein domains using a two-level approach that combines SOMs and hierarchical clustering.

Results

The conformational dynamics of the α-spectrin SH3 protein domain and six single mutants were analysed by MD simulations. The Cα's Cartesian coordinates of conformations sampled in the essential space were used as input data vectors for SOM training, then complete linkage clustering was performed on the SOM prototype vectors. A specific protocol to optimize a SOM for structural ensembles was proposed: the optimal SOM was selected by means of a Taguchi experimental design plan applied to different data sets, and the optimal sampling rate of the MD trajectory was selected. The proposed two-level approach was applied to single trajectories of the SH3 domain independently as well as to groups of them at the same time. The results demonstrated the potential of this approach in the analysis of large ensembles of molecular structures: the possibility of producing a topological mapping of the conformational space in a simple 2D visualisation, as well as of effectively highlighting differences in the conformational dynamics directly related to biological functions.

Conclusions

The use of a two-level approach combining SOMs and hierarchical clustering for conformational analysis of structural ensembles of proteins was proposed. It can easily be extended to other study cases and to conformational ensembles from other sources.     

Representation of the glomerular olfactory map in the Drosophila brain

We explored how the odor map in the Drosophila antennal lobe is represented in higher olfactory centers, the mushroom body and lateral horn. Systematic single-cell tracing of projection neurons (PNs) that send dendrites to specific glomeruli in the antennal lobe revealed their stereotypical axon branching patterns and terminal fields in the lateral horn. PNs with similar axon terminal fields tend to receive input from neighboring glomeruli. The glomerular classes of individual PNs could be accurately predicted based solely on their axon projection patterns. The sum of these patterns defines an "axon map" in higher olfactory centers reflecting which olfactory receptors provide input. This map is characterized by spatial convergence and divergence of PN axons, allowing integration of olfactory information.     

An integrated pan‐tropical biomass map using multiple reference datasets

We combined two existing datasets of vegetation aboveground biomass (AGB) (Proceedings of the National Academy of Sciences of the United States of America, 108 , 2011, 9899; Nature Climate Change, 2 , 2012, 182) into a pan‐tropical AGB map at 1‐km resolution using an independent reference dataset of field observations and locally calibrated high‐resolution biomass maps, harmonized and upscaled to 14 477 1‐km AGB estimates. Our data fusion approach uses bias removal and weighted linear averaging that incorporates and spatializes the biomass patterns indicated by the reference data. The method was applied independently in areas (strata) with homogeneous error patterns of the input (Saatchi and Baccini) maps, which were estimated from the reference data and additional covariates. Based on the fused map, we estimated AGB stock for the tropics (23.4 N–23.4 S) of 375 Pg dry mass, 9–18% lower than the Saatchi and Baccini estimates. The fused map also showed differing spatial patterns of AGB over large areas, with higher AGB density in the dense forest areas in the Congo basin, Eastern Amazon and South‐East Asia, and lower values in Central America and in most dry vegetation areas of Africa than either of the input maps. The validation exercise, based on 2118 estimates from the reference dataset not used in the fusion process, showed that the fused map had a RMSE 15–21% lower than that of the input maps and, most importantly, nearly unbiased estimates (mean bias 5 Mg dry mass ha^?1 vs. 21 and 28 Mg ha^?1 for the input maps). The fusion method can be applied at any scale including the policy‐relevant national level, where it can provide improved biomass estimates by integrating existing regional biomass maps as input maps and additional, country‐specific reference datasets.     

Spatial cognition and neuro-mimetic navigation: a model of hippocampal place cell activity

 A computational model of hippocampal activity during spatial cognition and navigation tasks is presented. The spatial representation in our model of the rat hippocampus is built on-line during exploration via two processing streams. An allothetic vision-based representation is built by unsupervised Hebbian learning extracting spatio-temporal properties of the environment from visual input. An idiothetic representation is learned based on internal movement-related information provided by path integration. On the level of the hippocampus, allothetic and idiothetic representations are integrated to yield a stable representation of the environment by a population of localized overlapping CA3-CA1 place fields. The hippocampal spatial representation is used as a basis for goal-oriented spatial behavior. We focus on the neural pathway connecting the hippocampus to the nucleus accumbens. Place cells drive a population of locomotor action neurons in the nucleus accumbens. Reward-based learning is applied to map place cell activity into action cell activity. The ensemble action cell activity provides navigational maps to support spatial behavior. We present experimental results obtained with a mobile Khepera robot. Received: 02 July 1999 / Accepted in revised form: 20 March 2000     

Development and regeneration of the retinotectal map in goldfish: a computational study

We present a simple computational model to study the interplay of activity-dependent and intrinsic processes thought to be involved in the formation of topographic neural projections. Our model consists of two input layers which project to one target layer. The connections between layers are described by a set of synaptic weights. These weights develop according to three interacting developmental rules: (i) an intrinsic fibre-target interaction which generates chemospecific adhesion between afferent fibres and target cells; (ii) an intrinsic fibre-fibre interaction which generates mutual selective adhesion between the afferent fibres; and (iii) an activity-dependent fibre-fibre interaction which implements Hebbian learning. Additionally, constraints are imposed to keep synaptic weights finite. The model is applied to a set of eleven experiments on the regeneration of the retinotectal projection in goldfish. We find that the model is able to reproduce the outcome of an unprecedented range of experiments with the same set of model parameters, including details of the size of receptive and projective fields. We expect this mathematical framework to be a useful tool for the analysis of developmental processes in general. <br>     

Visualizing the Topical Structure of the Medical Sciences: A Self-Organizing Map Approach

Background

We implement a high-resolution visualization of the medical knowledge domain using the self-organizing map (SOM) method, based on a corpus of over two million publications. While self-organizing maps have been used for document visualization for some time, (1) little is known about how to deal with truly large document collections in conjunction with a large number of SOM neurons, (2) post-training geometric and semiotic transformations of the SOM tend to be limited, and (3) no user studies have been conducted with domain experts to validate the utility and readability of the resulting visualizations. Our study makes key contributions to all of these issues.

Methodology

Documents extracted from Medline and Scopus are analyzed on the basis of indexer-assigned MeSH terms. Initial dimensionality is reduced to include only the top 10% most frequent terms and the resulting document vectors are then used to train a large SOM consisting of over 75,000 neurons. The resulting two-dimensional model of the high-dimensional input space is then transformed into a large-format map by using geographic information system (GIS) techniques and cartographic design principles. This map is then annotated and evaluated by ten experts stemming from the biomedical and other domains.

Conclusions

Study results demonstrate that it is possible to transform a very large document corpus into a map that is visually engaging and conceptually stimulating to subject experts from both inside and outside of the particular knowledge domain. The challenges of dealing with a truly large corpus come to the fore and require embracing parallelization and use of supercomputing resources to solve otherwise intractable computational tasks. Among the envisaged future efforts are the creation of a highly interactive interface and the elaboration of the notion of this map of medicine acting as a base map, onto which other knowledge artifacts could be overlaid.     

The reference genetic linkage map for the multinational Brassica rapa genome sequencing project

We describe the construction of a reference genetic linkage map for the Brassica A genome, which will form the backbone for anchoring sequence contigs for the Multinational Brassica rapa Genome Sequencing Project. Seventy-eight doubled haploid lines derived from anther culture of the F₁ of a cross between two diverse Chinese cabbage (B. rapa ssp. pekinensis) inbred lines, ‘Chiifu-401-42’ (C) and ‘Kenshin-402-43’ (K) were used to construct the map. The map comprises a total of 556 markers, including 278 AFLP, 235 SSR, 25 RAPD and 18 ESTP, STS and CAPS markers. Ten linkage groups were identified and designated as R1–R10 through alignment and orientation using SSR markers in common with existing B. napus reference linkage maps. The total length of the linkage map was 1,182 cM with an average interval of 2.83 cM between adjacent loci. The length of linkage groups ranged from 81 to 161 cM for R04 and R06, respectively. The use of 235 SSR markers allowed us to align the A-genome chromosomes of B. napus with those of B. rapa ssp. pekinensis. The development of this map is vital to the integration of genome sequence and genetic information and will enable the international research community to share resources and data for the improvement of B. rapa and other cultivated Brassica species. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

 We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold. Received: 19 May 1997 / Revised version: 6 April 1998