The Synergy between Complex Channel-Specific FIR Filter and Spatial Filter for Single-Trial EEG Classification

The common spatial pattern analysis (CSP), a frequently utilized feature extraction method in brain-computer-interface applications, is believed to be time-invariant and sensitive to noises, mainly due to an inherent shortcoming of purely relying on spatial filtering. Therefore, temporal/spectral filtering which can be very effective to counteract the unfavorable influence of noises is usually used as a supplement. This work integrates the CSP spatial filters with complex channel-specific finite impulse response (FIR) filters in a natural and intuitive manner. Each hybrid spatial-FIR filter is of high-order, data-driven and is unique to its corresponding channel. They are derived by introducing multiple time delays and regularization into conventional CSP. The general framework of the method follows that of CSP but performs better, as proven in single-trial classification tasks like event-related potential detection and motor imagery.     

Regularized common spatial patterns with subject-to-subject transfer of EEG signals

In the context of brain-computer interface (BCI) system, the common spatial patterns (CSP) method has been used to extract discriminative spatial filters for the classification of electroencephalogram (EEG) signals. However, the classification performance of CSP typically deteriorates when a few training samples are collected from a new BCI user. In this paper, we propose an approach that maintains or improves the recognition accuracy of the system with only a small size of training data set. The proposed approach is formulated by regularizing the classical CSP technique with the strategy of transfer learning. Specifically, we incorporate into the CSP analysis inter-subject information involving the same task, by minimizing the difference between the inter-subject features. Experimental results on two data sets from BCI competitions show that the proposed approach greatly improves the classification performance over that of the conventional CSP method; the transformed variant proved to be successful in almost every case, based on a small number of available training samples.     

The Essential Complexity of Auditory Receptive Fields

Encoding properties of sensory neurons are commonly modeled using linear finite impulse response (FIR) filters. For the auditory system, the FIR filter is instantiated in the spectro-temporal receptive field (STRF), often in the framework of the generalized linear model. Despite widespread use of the FIR STRF, numerous formulations for linear filters are possible that require many fewer parameters, potentially permitting more efficient and accurate model estimates. To explore these alternative STRF architectures, we recorded single-unit neural activity from auditory cortex of awake ferrets during presentation of natural sound stimuli. We compared performance of > 1000 linear STRF architectures, evaluating their ability to predict neural responses to a novel natural stimulus. Many were able to outperform the FIR filter. Two basic constraints on the architecture lead to the improved performance: (1) factorization of the STRF matrix into a small number of spectral and temporal filters and (2) low-dimensional parameterization of the factorized filters. The best parameterized model was able to outperform the full FIR filter in both primary and secondary auditory cortex, despite requiring fewer than 30 parameters, about 10% of the number required by the FIR filter. After accounting for noise from finite data sampling, these STRFs were able to explain an average of 40% of A1 response variance. The simpler models permitted more straightforward interpretation of sensory tuning properties. They also showed greater benefit from incorporating nonlinear terms, such as short term plasticity, that provide theoretical advances over the linear model. Architectures that minimize parameter count while maintaining maximum predictive power provide insight into the essential degrees of freedom governing auditory cortical function. They also maximize statistical power available for characterizing additional nonlinear properties that limit current auditory models.     

Processing visual signals with nonlinear spatial filters

Summary The properties of nonlinear spatial filters as a major part of the peripheral nervous system are investigated with respect to processing of two types of signals: a deterministic stepfunction and Gaussian noise. The nonlinearity of the nerve cell is treated as a threshold element. It is shown that certain unfavorable characteristics of linear filters do not occur in such nonlinear filters. The basic nonlinear effects of combined space-time-filtering are demonstrated by using bandpass filters.Time- and space-dependent noise is applied to the nonlinear bandpass. The two subsystems formed by the linear bandpass section and the threshold are treated separately.Two examples for the application of nonlinear spatial filtering are given.This paper contains results of a thesis submitted to the Fachbereich Biologie der Johannes Gutenberg-Universität, Mainz, in partial fulfillment of the requirements for the degree of a Dr. rer. nat.The author is grateful to Prof. Dr.-Ing. W. v. Seelen for his continual guidance and support.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

Sparse spatial filter via a novel objective function minimization with smooth ℓ1 regularization

Common spatial pattern (CSP) method is widely used in brain machine interface (BMI) applications to extract features from the multichannel neural activity through a set of spatial projections. These spatial projections minimize the Rayleigh quotient (RQ) as the objective function, which is the variance ratio of the classes. The CSP method easily overfits the data when the number of training trials is not sufficiently large and it is sensitive to daily variation of multichannel electrode placement, which limits its applicability for everyday use in BMI systems. To overcome these problems, the amount of channels that is used in projections, should be limited to some adequate number. We introduce a spatially sparse projection (SSP) method that exploits the unconstrained minimization of a new objective function with approximated ?₁ penalty. Unlike the RQ, this new objective function depends on the magnitude of the sparse filter. The SSP method is employed to classify the multiclass ECoG and two class EEG data sets. We compared our results with a recently introduced sparse CSP solution based on ?₀ norm. Our method outperforms the standard CSP method and provides comparable results to ?₀ norm based solution and it is associated with less computational complexity. We also conducted several simulation studies on the effect of noisy channel and intersession variability on the performance of the CSP and sparse filters.     

Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Low-Rank Regularization for Learning Gene Expression Programs

Learning gene expression programs directly from a set of observations is challenging due to the complexity of gene regulation, high noise of experimental measurements, and insufficient number of experimental measurements. Imposing additional constraints with strong and biologically motivated regularizations is critical in developing reliable and effective algorithms for inferring gene expression programs. Here we propose a new form of regulation that constrains the number of independent connectivity patterns between regulators and targets, motivated by the modular design of gene regulatory programs and the belief that the total number of independent regulatory modules should be small. We formulate a multi-target linear regression framework to incorporate this type of regulation, in which the number of independent connectivity patterns is expressed as the rank of the connectivity matrix between regulators and targets. We then generalize the linear framework to nonlinear cases, and prove that the generalized low-rank regularization model is still convex. Efficient algorithms are derived to solve both the linear and nonlinear low-rank regularized problems. Finally, we test the algorithms on three gene expression datasets, and show that the low-rank regularization improves the accuracy of gene expression prediction in these three datasets.     

A Neural Network-Based Optimal Spatial Filter Design Method for Motor Imagery Classification

In this study, a novel spatial filter design method is introduced. Spatial filtering is an important processing step for feature extraction in motor imagery-based brain-computer interfaces. This paper introduces a new motor imagery signal classification method combined with spatial filter optimization. We simultaneously train the spatial filter and the classifier using a neural network approach. The proposed spatial filter network (SFN) is composed of two layers: a spatial filtering layer and a classifier layer. These two layers are linked to each other with non-linear mapping functions. The proposed method addresses two shortcomings of the common spatial patterns (CSP) algorithm. First, CSP aims to maximize the between-classes variance while ignoring the minimization of within-classes variances. Consequently, the features obtained using the CSP method may have large within-classes variances. Second, the maximizing optimization function of CSP increases the classification accuracy indirectly because an independent classifier is used after the CSP method. With SFN, we aimed to maximize the between-classes variance while minimizing within-classes variances and simultaneously optimizing the spatial filter and the classifier. To classify motor imagery EEG signals, we modified the well-known feed-forward structure and derived forward and backward equations that correspond to the proposed structure. We tested our algorithm on simple toy data. Then, we compared the SFN with conventional CSP and its multi-class version, called one-versus-rest CSP, on two data sets from BCI competition III. The evaluation results demonstrate that SFN is a good alternative for classifying motor imagery EEG signals with increased classification accuracy.     

Online Classification of Multiple Motor Imagery Tasks Using Filter Bank Based Maximum-a-Posteriori Common Spatial Pattern Filters

ObjectiveThe main objective of this paper is to propose a novel technique, called filter bank maximum a-posteriori common spatial pattern (FB-MAP-CSP) algorithm, for online classification of multiple motor imagery activities using electroencephalography (EEG) signals. The proposed technique addresses the overfitting issue of CSP in addition to utilizing the spectral information of EEG signals inside the framework of filter banks while extending it to more than two conditions.Materials and methodsThe classification of motor imagery signals is based upon the detection of event-related de-synchronization (ERD) phenomena in the μ and β rhythms of EEG signals. Accordingly, two modifications in the existing MAP-CSP technique are presented: (i) The (pre-processed) EEG signals are spectrally filtered by a bank of filters lying in the μ and β brainwave frequency range, (ii) the framework of MAP-CSP is extended to deal with multiple (more than two) motor imagery tasks classification and the spatial filters thus obtained are calculated for each sub-band, separately. Subsequently, the most imperative features over all sub-bands are selected and un-regularized linear discriminant analysis is employed for classification of multiple motor imagery tasks.ResultsPublicly available dataset (BCI Competition IV Dataset I) is used to validate the proposed method i.e. FB-MAP-CSP. The results show that the proposed method yields superior classification results, in addition to be computationally more efficient in the case of online implementation, as compared to the conventional CSP based techniques and its variants for multiclass motor imagery classification.ConclusionThe proposed FB-MAP-CSP algorithm is found to be a potential / superior method for classifying multi-condition motor imagery EEG signals in comparison to FBCSP based techniques.     

Cluster detection based on spatial associations and iterated residuals in generalized linear mixed models

Summary .  Spatial clustering is commonly modeled by a Bayesian method under the framework of generalized linear mixed effect models (GLMMs). Spatial clusters are commonly detected by a frequentist method through hypothesis testing. In this article, we provide a frequentist method for assessing spatial properties of GLMMs. We propose a strategy that detects spatial clusters through parameter estimates of spatial associations, and assesses spatial aspects of model improvement through iterated residuals. Simulations and a case study show that the proposed method is able to consistently and efficiently detect the locations and magnitudes of spatial clusters.     

A Hierarchical Bayesian Model for Spatial Prediction of Multivariate Non‐Gaussian Random Fields

Summary As most georeferenced data sets are multivariate and concern variables of different types, spatial mapping methods must be able to deal with such data. The main difficulties are the prediction of non‐Gaussian variables and the modeling of the dependence between processes. The aim of this article is to present a new hierarchical Bayesian approach that permits simultaneous modeling of dependent Gaussian, count, and ordinal spatial fields. This approach is based on spatial generalized linear mixed models. We use a moving average approach to model the spatial dependence between the processes. The method is first validated through a simulation study. We show that the multivariate model has better predictive abilities than the univariate one. Then the multivariate spatial hierarchical model is applied to a real data set collected in French Guiana to predict topsoil patterns.     

Using eigenstructure decompositions of time-varying autoregressions in common spatial patterns-based EEG signal classification

Brain–computer interfaces based on common spatial patterns (CSP) depend on the operational frequency bands of the events to be discriminated. This problem has been addressed through sub-band decompositions of the electroencephalographic signals using filter banks, then the performance relies on the number of filters that are stacked and the criteria to select their bandwidths. Here, we propose an alternative approach based on an eigenstructure decomposition of the signals’ time-varying autoregressions (TVAR). The eigen-based decomposition of the TVAR allows for subject-specific estimation of the principal time-varying frequencies, then such principal eigencomponents can be used in the traditional CSP-based classification. We show through a series of numerical experiments that the proposed classification scheme can achieve a performance which is comparable with the one obtained through the filter bank-based approach. However, our method does not rely on a preliminary selection of a frequency band, yet good performance is achieved under realistic conditions (such as reduced number of sensors and small amount of training data) independently of the time interval selected.     

Modelling geographical patterns in species richness using eigenvector-based spatial filters

Aim To test the mechanisms driving bird species richness at broad spatial scales using eigenvector‐based spatial filtering. Location South America. Methods An eigenvector‐based spatial filtering was applied to evaluate spatial patterns in South American bird species richness, taking into account spatial autocorrelation in the data. The method consists of using the geographical coordinates of a region, based on eigenanalyses of geographical distances, to establish a set of spatial filters (eigenvectors) expressing the spatial structure of the region at different spatial scales. These filters can then be used as predictors in multiple and partial regression analyses, taking into account spatial autocorrelation. Autocorrelation in filters and in the regression residuals can be used as stopping rules to define which filters will be used in the analyses. Results Environmental component alone explained 8% of variation in richness, whereas 77% of the variation could be attributed to an interaction between environment and geography expressed by the filters (which include mainly broad‐scale climatic factors). Regression coefficients of environmental component were highest for AET. These results were unbiased by short‐scale spatial autocorrelation. Also, there was a significant interaction between topographic heterogeneity and minimum temperature. Conclusion Eigenvector‐based spatial filtering is a simple and suitable statistical protocol that can be used to analyse patterns in species richness taking into account spatial autocorrelation at different spatial scales. The results for South American birds are consistent with the climatic hypothesis, in general, and energy hypothesis, in particular. Habitat heterogeneity also has a significant effect on variation in species richness in warm tropical regions.     

Simulation of the effect of local adaption with nonlinear spatial filters

The visual system of vertebrates is capable of processing pattern signals over a wide range of intensity reaching from nearly absolute darkness to very bright sunlight. Typically the visual system of humans extracts fine contours of patterns of sufficiently high intensity or at high background intensity level, showing signal processing properties which can be explained by a bandpass system. Conversely, at very low intensity levels that system shows low-pass response: only coarse contours of patterns are recognized, however, the amplification of the signals has increased. The effect is called local adaption. A model is shown on the basis of a one-stage nonlinear spatial filter which, controlled by the local distribution of pattern intensity, can alter its frequency characteristic between low-pass response and bandpass response. Results are stated for computer-modelled filters. The investigation is restricted to one-dimensional filters, however, the results can be used to explain the function of two-dimensional filters qualitatively.     

Linear and Weakly Nonlinear Stability Analyses of Turing Patterns for Diffusive Predator–Prey Systems in Freshwater Marsh Landscapes

We study a diffusive predator–prey model describing the interactions of small fishes and their resource base (small invertebrates) in the fluctuating freshwater marsh landscapes of the Florida Everglades. The spatial model is described by a reaction–diffusion system with Beddington–DeAngelis functional response. Uniform bound, local and global asymptotic stability of the steady state of the PDE model under the no-flux boundary conditions are discussed in details. Sufficient conditions on the Turing (diffusion-driven) instability which induces spatial patterns in the model are derived via linear analysis. Existence of one-dimensional and two-dimensional spatial Turing patterns, including rhombic and hexagonal patterns, are established by weakly nonlinear analyses. These results provide theoretical explanations and numerical simulations of spatial dynamical behaviors of the wetland ecosystems of the Florida Everglades.     

A model of human pattern perception: association fields for adaptive spatial filters.

Visual neurons in the primary visual cortex 'look' at the retinal image through a four-dimensional array of spatial receptive fields (filter-elements): two spatial dimensions and, at each spatial location, two Fourier dimensions of spatial frequency and orientation. In general, visual objects activate filter-elements along each of these dimensions, suggesting a need for some kind of linking mechanism that determines whether two or more filter-elements are responding to the same or different contours or objects. In the spatial domain, a (spatial) association field between filter-elements, arranged to form first-order curves, has been inferred as a flexible method by which different parts of extended (luminance) contours become associated (Field et al., 1993). Linking has also been explored between filters selective for different regions in Fourier space (e.g. Georgeson and Meese, 1997). Perceived structure of stationary plaids suggests that spatial filtering is adaptive: synthetic filters can be created by the linear summation of basis-filters across orientation or spatial frequency in a stimulus-dependent way. For example, a plaid with a pair of sine-wave components at +/-45 deg looks like a blurred checkerboard; a structure that can be understood if features are derived after linear summation of spatial filters at different orientations. However, the addition of an oblique third-harmonic component causes the plaid to perceptually segment into overlapping oblique contours. This result can be understood if filters are summed across spatial frequency, but, in this case, treated independently across orientation. In the present paper, the architecture of an association field is proposed to permit linking and segmentation of filter-elements across spatial frequency and orientation. Three types of link are proposed: (1) A chain of constructive links around sites of common spatial frequency but different orientation, to promote binding of filters across orientation; (2) Constructive links between sites with common orientation but different spatial frequency, to promote binding of filters across spatial frequency; (3) Long-range links between sites of common spatial frequency but different orientation, whose activation and role are determined by activity in a higher spatial frequency band. A model employing the proposed network of links is consistent with at least six previously reported effects on the perception of briefly presented stationary plaids.     

Integrated framework of nonlinear prediction and process monitoring for complex biological processes

Bioprocesses and biosystems have nonlinear and multiple operation patterns depending on the influent loads, temperatures, the activity of microorganisms, and other factors. In this paper, an integrated framework of nonlinear modeling and process monitoring methods is developed for a complex biological process. The proposed method is based on modeling by fuzzy partial least squares (FPLS) and on process monitoring by a statistical decomposition, which is suitable for predicting and supervising a nonlinear biological process. Case studies in the bio-simulated process and industrial biological plant show that the proposed method can give superior prediction and monitoring performance in complex biological plants compared to other linear and nonlinear methods, since it can effectively capture the nonlinear causal relationship within the biosystem. This gives us the integrated framework that is able to both model and monitor the nonlinear bioprocess simultaneously.     

Intermolecular electrostatics via molecular fragmentation: an application to crystal structure prediction

Molecular fragmentation is an attractive approach to the simulation of large molecules, in which calculations are carried out on small segments of the molecule, achieving linear scaling but reduced accuracy. Its application to crystal structure prediction (CSP) is challenged by high accuracy requirements. In this study, the applicability of a fragmentation scheme is tested for distributed multipoles, which are used in CSP to model intermolecular electrostatic interactions. Four test systems are investigated: a molecular salt, the highly conjugated molecule retinal, a model pharmaceutical molecule and the nonlinear molecule nitrotriacetanilide. It is demonstrated that fragment-based electrostatics reproduce, to an acceptable degree, a set of crystal structures generated using whole-molecule electrostatics. Inclusion of the molecular environment of each fragment out to four bonds separation is found to provide a sufficiently accurate set of distributed multipoles for the purposes of CSP.     

System analysis of Phycomyces light-growth response with gaussian white-noise test stimuli

The light-growth response of the Phycomyces sporangiophore is a transient change of elongation rate in response to changes in ambient blue-light intensity. The white-noise method of nonlinear system identification (Wiener-Lee-Schetzen theory) has been applied to this response, and the results have been interpreted by system analysis methods in the frequency domain. Experiments were performed on the Phycomyces tracking machine. Gaussian white-noise stimulus patterns were applied to the logarithm of the light intensity. The log-mean intensity of the broadband blue illumination was 0.1 W m^-2 and the standard deviation of the Gaussian white-noise was 0.58 decades. The results, in the form of temporal functions called Wiener kernels, represent the input-output relation of the light-growth response system. The transfer function, which was obtained as the Fourier transform of the first-order kernel, was analyzed in the frequency domain in terms of a dynamic model that consisted of a first-order high-pass filter, two secondorder low-pass filters, a delay element, and a gain factor. Parameters in the model (cutoff frequencies, damping coefficients, latency, and gain constant) were evaluated by nonlinear least-squares methods applied to the complex-valued transfer function. Analysis of the second-order kernel in the frequency domain suggests that the residual nonlinearity of the system lies close to the input.