Dynamics and topographic organization of recursive self-organizing maps

Recently there has been an outburst of interest in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, there is no general consensus as to how best to process sequences using topographicmaps, and this topic remains an active focus of neurocomputational research. The representational capabilities and internal representations of the models are not well understood. Here, we rigorously analyze a generalization of the self-organizingmap (SOM) for processing sequential data, recursive SOM (RecSOM) (Voegtlin, 2002), as a nonautonomous dynamical system consisting of a set of fixed input maps. We argue that contractive fixed-input maps are likely to produce Markovian organizations of receptive fields on the RecSOM map. We derive bounds on parameter β (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed-input maps is guaranteed. Some generalizations of SOM contain a dynamic module responsible for processing temporal contexts as an integral part of the model. We show that Markovian topographic maps of sequential data can be produced using a simple fixed (nonadaptable) dynamic module externally feeding a standard topographic model designed to process static vectorial data of fixed dimensionality (e.g., SOM). However, by allowing trainable feedback connections, one can obtain Markovian maps with superior memory depth and topography preservation. We elaborate on the importance of non-Markovian organizations in topographic maps of sequential data. © 2006 Massachusetts Institute of Technology.

The topographic distribution of the magnetic P100M to full- and half-field stimulation

Visual evoked magnetic responses were recorded to full-field and left and right half-field stimulation with three check sizes (70′, 34′ and 22′) in five normal subjects. Recordings were made sequentially on a 20-position grid (4 × 5) based on the inion, by means of a single-channel direct current-Superconducting Quantum Interference Device second-order gradiometer. The topographic maps were consistent on the same subjects recorded 2 months apart. The half-field responses produced the strongest signals in the contralateral hemisphere and were consistent with the cruciform model of the calcarine fissure. Right half fields produced upper-left-quadrant outgoing fields and lower-left-quadrant ingoing fields, while the left half field produced the opposite response. The topographic maps also varied with check size, with the larger checks producing positive or negative maximum position more anteriorly than small checks. In addition, with large checks the full-field responses could be explained as the summation of the two half fields, whereas full-field responses to smaller checks were more unpredictable and may be due to sources located at the occipital pole or lateral surface. In addition, dipole sources were located as appropriate with the use of inverse problem solutions. Topographic data will be vital to the clinical use of the visual evoked field but, in addition, provides complementary information to visual evoked potentials, allowing detailed studies of the visual cortex. © 1992 Kluwer Academic Publishers.

Topographic mapping and source localization of the pattern reversal visual evoked magnetic response

The topography of the visual evoked magnetic response (VEMR) to pattern reversal stimulation was studied in four normal subjects using a single channel BTI magnetometer. VEMRs were recorded from 20 locations over the occipital scalp and the topographic distribution of the most consistent component (P100M) studied. A single dipole in a sphere model was fitted to the data. Topographic maps were similar when recorded two months apart on the same subject to the same stimulus. Half field (HF) stimulation elicited responses from sources on the medial surface of the calcarine fissure mainly in the contralateral hemisphere as predicted by the cruciform model. The full field (FF) responses to large checks were approximately the sum of the HF responses. However, with small checks, FF stimulation appeared to activate a different combination of sources than the two HFs. In addition, HF topography was more consistent between subjects than FF for small check sizes. Topographic studies of the VEMR may help to explain the analogous visual evoked electrical response and will be essential to define optimal recording positions for clinical applications.

Recursive self-organizing map as a contractive iterative function system

Recently, there has been a considerable research activity in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, the representational capabilities and internal representations of the models are not well understood. We rigorously analyze a generalization of the Self-Organizing Map (SOM) for processing sequential data, Recursive SOM (RecSOM [1]), as a non-autonomous dynamical system consisting off a set of fixed input maps. We show that contractive fixed input maps are likely to produce Markovian organizations of receptive fields o the RecSOM map. We derive bounds on parameter $\beta$ (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed input maps is guaranteed.

Topography of visual evoked magnetic responses to pattern shift, pattern onset and flash stimuli

Different visual stimuli may activate separate channels in the visual system and produce magnetic responses from the human bran which originate from distinct regions of the visual cortex. To test this hypothesis, we have investigated the distribution of visual evoked magnetic responses to three distinct visual stimuli over the occipital region of the scalp with a DC-SQUID second-order gradiometer in an ubshielded environment. Patterned stimuli were presented full field and to the right half field, while a flash stimulus was presented full field only, in five normal subjects. Magnetic responses were recorded from 20 to 42 positions over the occipital scalp. Topographic maps were prepared of the major positive component within the first 150ms to the three stimuli, i.e., the P100m (pattern shift), C11m (pattern onset) and P2m (flash). For the pattern shift stimulus the data suggested the source of the P100m was close to the midline with the current directed towards the medial surface. The data for the pattern onset C11m suggested a source at a similar depth but with the current directed away from the midline towards the lateral surface. The flash P2m appeared to originate closer to the surface of the occipital pole than both the patterned stimuli. Hence the pattern shift (which may represent movement), and the pattern onset C11m (representing contrast and contour) appear to originate in similar areas of brain but to represent different asepcts of cortical processing. By contrast, the flash P2m (representing luminance change) appears to originate in a distinct area of visual cortex closer to the occipital pole.

Analysis of magnetoencephaoographic data as a nonlinear dynamical system

This thesis presents the results from an investigation into the merits of analysing Magnetoencephalographic (MEG) data in the context of dynamical systems theory. MEG is the study of both the methods for the measurement of minute magnetic flux variations at the scalp, resulting from neuro-electric activity in the neocortex, as well as the techniques required to process and extract useful information from these measurements. As a result of its unique mode of action - by directly measuring neuronal activity via the resulting magnetic field fluctuations - MEG possesses a number of useful qualities which could potentially make it a powerful addition to any brain researcher's arsenal. Unfortunately, MEG research has so far failed to fulfil its early promise, being hindered in its progress by a variety of factors. Conventionally, the analysis of MEG has been dominated by the search for activity in certain spectral bands - the so-called alpha, delta, beta, etc that are commonly referred to in both academic and lay publications. Other efforts have centred upon generating optimal fits of "equivalent current dipoles" that best explain the observed field distribution. Many of these approaches carry the implicit assumption that the dynamics which result in the observed time series are linear. This is despite a variety of reasons which suggest that nonlinearity might be present in MEG recordings. By using methods that allow for nonlinear dynamics, the research described in this thesis avoids these restrictive linearity assumptions. A crucial concept underpinning this project is the belief that MEG recordings are mere observations of the evolution of the true underlying state, which is unobservable and is assumed to reflect some abstract brain cognitive state. Further, we maintain that it is unreasonable to expect these processes to be adequately described in the traditional way: as a linear sum of a large number of frequency generators. One of the main objectives of this thesis will be to prove that much more effective and powerful analysis of MEG can be achieved if one were to assume the presence of both linear and nonlinear characteristics from the outset. Our position is that the combined action of a relatively small number of these generators, coupled with external and dynamic noise sources, is more than sufficient to account for the complexity observed in the MEG recordings. Another problem that has plagued MEG researchers is the extremely low signal to noise ratios that are obtained. As the magnetic flux variations resulting from actual cortical processes can be extremely minute, the measuring devices used in MEG are, necessarily, extremely sensitive. The unfortunate side-effect of this is that even commonplace phenomena such as the earth's geomagnetic field can easily swamp signals of interest. This problem is commonly addressed by averaging over a large number of recordings. However, this has a number of notable drawbacks. In particular, it is difficult to synchronise high frequency activity which might be of interest, and often these signals will be cancelled out by the averaging process. Other problems that have been encountered are high costs and low portability of state-of-the- art multichannel machines. The result of this is that the use of MEG has, hitherto, been restricted to large institutions which are able to afford the high costs associated with the procurement and maintenance of these machines. In this project, we seek to address these issues by working almost exclusively with single channel, unaveraged MEG data. We demonstrate the applicability of a variety of methods originating from the fields of signal processing, dynamical systems, information theory and neural networks, to the analysis of MEG data. It is noteworthy that while modern signal processing tools such as independent component analysis, topographic maps and latent variable modelling have enjoyed extensive success in a variety of research areas from financial time series modelling to the analysis of sun spot activity, their use in MEG analysis has thus far been extremely limited. It is hoped that this work will help to remedy this oversight.

Remote sensing applications to flood hydrology in Belize

The research compares the usefullness of four remote sensing information sources, these being LANDSAT photographic prints, LANDSAT computer compatible tapes, Metric Camera and SIR-A photographic prints. These sources provide evaluations of the catchment characteristics of the Belize and Sibun river basins in Central America. Map evaluations at 1:250,000 scale are compared to the results of the same scale, remotely sensed information sources. The values of catchment characteristics for both maps and LANDSAT prints are used in multiple regression analysis, providing flood flow formulae, after investigations to provide a suitable dependent variable discharge series are made for short term records. The use of all remotely sensed information sources in providing evaluations of catchment characteristics is discussed. LANDSAT prints and computer compatible tapes of a post flood scene are used to estimate flood distributions and volumes. These are compared to values obtained from unit hydrograph analysis, using the dependent discharge series and evaluate the probable losses from the Belize river to the floodplain, thereby assessing the accuracy of LANDSAT estimates. Information relating to flood behaviour is discussed in terms of basic image presentation as well as image processing. A cost analysis of the purchase and use of all materials is provided. Conclusions of the research indicate that LANDSAT print material may provide information suitable for regression analysis at levels of accuracy as great as those of topographic maps, that the differing information sources are uniquely applicable and that accurate estimates of flood volumes may be determined even by post flood imagery.

Predictive gene lists for breast cancer prognosis: a topographic visualisation study

Background: The controversy surrounding the non-uniqueness of predictive gene lists (PGL) of small selected subsets of genes from very large potential candidates as available in DNA microarray experiments is now widely acknowledged 1. Many of these studies have focused on constructing discriminative semi-parametric models and as such are also subject to the issue of random correlations of sparse model selection in high dimensional spaces. In this work we outline a different approach based around an unsupervised patient-specific nonlinear topographic projection in predictive gene lists. Methods: We construct nonlinear topographic projection maps based on inter-patient gene-list relative dissimilarities. The Neuroscale, the Stochastic Neighbor Embedding(SNE) and the Locally Linear Embedding(LLE) techniques have been used to construct two-dimensional projective visualisation plots of 70 dimensional PGLs per patient, classifiers are also constructed to identify the prognosis indicator of each patient using the resulting projections from those visualisation techniques and investigate whether a-posteriori two prognosis groups are separable on the evidence of the gene lists. A literature-proposed predictive gene list for breast cancer is benchmarked against a separate gene list using the above methods. Generalisation ability is investigated by using the mapping capability of Neuroscale to visualise the follow-up study, but based on the projections derived from the original dataset. Results: The results indicate that small subsets of patient-specific PGLs have insufficient prognostic dissimilarity to permit a distinction between two prognosis patients. Uncertainty and diversity across multiple gene expressions prevents unambiguous or even confident patient grouping. Comparative projections across different PGLs provide similar results. Conclusion: The random correlation effect to an arbitrary outcome induced by small subset selection from very high dimensional interrelated gene expression profiles leads to an outcome with associated uncertainty. This continuum and uncertainty precludes any attempts at constructing discriminative classifiers. However a patient's gene expression profile could possibly be used in treatment planning, based on knowledge of other patients' responses. We conclude that many of the patients involved in such medical studies are intrinsically unclassifiable on the basis of provided PGL evidence. This additional category of 'unclassifiable' should be accommodated within medical decision support systems if serious errors and unnecessary adjuvant therapy are to be avoided.