Semantic geodesic maps: a unifying geometrical approach for studying the structure and dynamics of single trial evoked responses

Abstract

Objectives: A general framework for identifying and describing structure in a given sample of evoked response single-trial signals (STs) is introduced. The approach is based on conceptually simple geometrical ideas and enables the convergence of pattern analysis and non-linear time series analysis.

Methods: Classical steps for analyzing the STs by waveform are first employed and the ST-analysis is transferred to a multidimensional space, the feature space, the geometry of which is systematically studied via multidimensional scaling (MDS) techniques giving rise to semantic maps. The structure in the feature space characterizes the trial-to-trial variability and this is utilized to probe functional connectivity between two brain areas. The underlying dynamic process responsible for the emerged structure can be described by a multidimensional trajectory in the feature space. This in turn enables the detection of dynamical interareal coupling as similarity between the corresponding trajectories.

Results and conclusions: The utility of semantic maps was demonstrated using magnetoencephalographic data from a simple auditory paradigm. The coupling of ongoing activity and evoked response is vividly demonstrated and contrasted with the apparent deflection from zero baseline that survives averaging. Prototypes are easily identified as the end points of distinct paths in the semantic map representation, and their neighborhood is populated by STs with distinct properties not only in the latencies where the evoked response is expected to be strong, but also and very significantly in the prestimulus period. Finally our results provide evidence for interhemispheric binding in the (4–8 Hz) range and dynamical coupling at faster time scales.

Introduction

The traditional approach to characterize the evoked response is to perform ensemble averaging and record for each site, the polarity, latency and amplitude of the major deflections of the average waveform. The ensemble average usually includes a complex of partially overlapping components reflecting different processing stages along the neural pathways. However, due to the non-stationary character of the response generation mechanism, these components are usually smeared after averaging, and the final estimation will be only a coarse summary, or worse distortion, of the events that took place during the recording session (Liu and Ioannides, 1996).

To advance beyond ensemble averaging it is necessary to deal with the low signal-to-noise ratio (SNR) of the evoked responses which are always embedded within (and quite often coupled with) the ongoing brain activity. An evoked response does not depend solely on the stimulus physical characteristics, but also on many other factors like subject's performance and psychophysiological state. Improvement in the analysis of single-trial signals (STs) is also required for the study of interactions between brain areas (Gray, 1999, Ioannides et al., 2000, Leocani et al., 2001).

The analysis of evoked responses has been traditionally supported by pattern analysis methodologies (Donchin and Heffley, 1978, Gath et al., 1985). The main objective is the identification of structure in the data (Geva and Pratt, 1994, Zouridakis et al., 1997a, Zouridakis et al., 1997b, Lange et al., 2000, Hoppe et al., 2000). To this end, features are extracted from the individual ST-patterns (i.e. ST-waveforms) and classification/clustering techniques are employed to split the record of evoked responses to homogeneous groups of ST patterns sharing similar morphological characteristics (Geva, 1998). However, the low SNR makes the feature-selection and feature-extraction tasks non-trivial. The feature-extraction step can be guided by the pattern of the averaged response (Lange et al., 1997) or performed, in a fully unsupervised manner, via principal component analysis (PCA) (Geva and Pratt, 1994).

Techniques for analyzing ST-responses have been biased by the transient waveshape seen in the averaged response. Possible interdependence between successive ST-responses, non-linear interaction of the response and the ongoing brain waves and any causal relationship between prestimulus activity and the response are often overlooked (Arieli et al., 1996, Ioannides et al., 1998, Karakas et al., 2000). Powerful techniques to characterize the evolution of brain activity from a continuous-mode time series recorded at a single site have been developed in non-linear dynamics (Pritchard and Duke, 1995, Burioka et al., 2001, Kowalik et al., 2001, Lee et al., 2001), where also efficient numerical methods, well suited to the oscillatory nature of the measured signals, have been introduced to study the dynamical interdependence between brain areas (Arnhold et al., 1999). Our work brings together ideas from pattern analysis and non-linear dynamics in a unified geometrical approach for ST-analysis, which is both model-free and general.

The STs are analyzed based on their waveforms and this endows the approach with a great generality since, by modifying the feature extraction step, a rich exploration can be achieved. The approach is multivariate in nature and therefore can cope with the realistic case of interrelate features (Mardia et al., 1979). With the feature-extraction step the ST-patterns, from one recording site, can be represented by points in a (possibly) high dimensional vector space, the feature-space. The identification of structure in the data is then formulated as detection of the deterministic-skeleton of the point sample or as delineation of the signal related manifold (i.e. constrained-surface) of the point distribution in the feature-space. The first objective of this work is to introduce efficient ways to represent the structure of the point sample in a parsimonious, but yet meaningful way. Dimensionality reduction techniques, with a structure-preserving character, are suggested for producing an accessible image of the point-sample. The second objective is to suggest a general statistical scheme that quantifies the coherence in structure between two feature-spaces and can be used to detect (upon proper feature-selection) functional connectivity between brain areas. The final objective is to highlight the common geometrical ground shared by both pattern analysis and non-linear time series analysis: the multidimensional trajectories describing the brain's activity evolution trace paths through the feature space. This observation suggests a straightforward way to trace back the history of the observed structure in feature space, in order to understand better the dynamics of the response generation.

The paper has been divided into two parts. Section 2 introduces the different aspects of the unifying geometrical approach. Section 3 presents the results from analysing M100 evoked responses. The Appendices include all the necessary algorithmic steps for understanding the technical details of the approach and facilitating a direct implementation.