P. Valdés

Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex

Information processing in the visual cortex depends on complex and context sensitive patterns of interactions between neuronal groups in many different cortical areas. Methods used to date for disentangling this functional connectivity presuppose either linearity or instantaneous interactions, assumptions that are not necessarily valid. In this paper a general framework that encompasses both linear and non-linear modelling of neurophysiological time series data by means of Local Linear Non-linear Autoregressive models (LLNAR) is described. Within this framework a new test for non-linearity of time series and for non-linearity of directedness of neural interactions based on LLNAR is presented. These tests assess the relative goodness of fit of linear versus non-linear models via the bootstrap technique. Additionally, a generalised definition of Granger causality is presented based on LLNAR that is valid for both linear and non-linear systems. Finally, the use of LLNAR for measuring non-linearity and directional influences is illustrated using artificial data, reference data as well as local field potentials (LFPs) from macaque area TE. LFP data is well described by the linear variant of LLNAR. Models of this sort, including lagged values of the preceding 25 to 60 ms, revealed the existence of both uni- and bi-directional influences between recording sites.

Nonlinear EEG analysis based on a neural mass model.

Abstract. The well-known neural mass model described by Lopes da Silva et al. (1976) and Zetterberg et al. (1978) is fitted to actual EEG data. This is achieved by reformulating the original set of integral equations as a continuous-discrete state space model. The local linearization approach is then used to discretize the state equation and to construct a nonlinear Kalman filter. On this basis, a maximum likelihood procedure is used for estimating the model parameters for several EEG recordings. The analysis of the noise-free differential equations of the estimated models suggests that there are two different types of alpha rhythms: those with a point attractor and others with a limit cycle attractor. These attractors are also found by means of a nonlinear time series analysis of the EEG recordings. We conclude that the Hopf bifurcation described by Zetterberg et al. (1978) is present in actual brain dynamics.

Frequency domain models of the EEG

SummaryThe structure of the normal resting EEG crosspectrum Svv(ω) is analyzed using complex multivariate statistics. Exploratory data analysis with Principal Component Analysis (PCA) is followed by hypothesis testing and computer simulations related to possible neural generators. The Svv(ω) of 211 normal individuals (ages 5 to 97) may be decomposed into two types of processes: the ξ process with spatial isotropicity reflecting diffuse, correlated cortical generators with radial symmetry, and processes that seem to be generated by more spatially concentrated, correlated sources. The latter are reflected as spectral peaks such as the process. The eigenvectors of the ξ process are the Spherical Harmonic Functions which explains the recurring pattern of maps characteristic of the spatial PCA of qEEG data. A new method for estimating sources in the frequency domain which fits dipoles to the whole crosspectrum is applied to explain the characteristics of the localized sources.

High resolution quantitative EEG analysis

SummaryHigh resolution spectral methods are explored as an alternative to broad band spectral parameters (BBSP) in quantitative EEG analysis. In a previous paper (Valdes et al. 1990b) regression equations (“Developmental surfaces”) were introduced to characterize the age-frequency distribution of the mean and standard deviation of the log spectral EEG power in a normative sample. These normative surfaces allow the calculation of z transformed spectra for all derivations of the 10/20 system and z maps for each frequency. Clinical material is presented that illustrates how these procedures may pinpoint frequencies of abnormal brain activity and their topographic distribution, avoiding the frequency and spatial “smearing” that may occur using BBSP. The increased diagnostic accuracy of high resolution spectral methods is demonstrated by means of receiver operator characteristic (ROC) curve analysis. Procedures are introduced to avoid type I error inflation due to the use of more variables in this type of procedure.

Brain-stem auditory evoked potentials and brain death

BAEP records were obtained from 30 brain-dead patients. Three BAEP patterns were observed: (1) no identifiable waves (73.34%), (2) an isolated bilateral wave I (16.66%), and (3) an isolated unilateral wave I (10%). When wave I was present, it was always significantly delayed. Significant augmentation of wave I amplitude was present bilaterally in one case and unilaterally in another. On the other hand, in serial records from 3 cases wave I latency tended to increase progressively until this component disappeared. During the same period, wave I amplitude fluctuations were observed. A significant negative correlation was found for wave I latency with heart rate and body temperature in 1 case. Two facts might explain the progressive delay and disappearance of wave I in brain-dead patients: a progressive hypoxic-ischaemic dysfunction of the cochlea and the eighth nerve plus hypothermia, often present in brain-dead patients. Then the incidence of wave I preservation reported by different authors in single BAEP records from brain-dead patients might depend on the moment at which the evoked potential study was done in relation to the onset of the clinical state. It is suggested that, although BAEPs provide an objective electrophysiological assessment of brain-stem function, essential for BD diagnosis, this technique could be of no value for this purpose when used in isolation.

EEG-distributed inverse solutions for a spherical head model

The theoretical study of the minimum norm solution to the MEG inverse problem has been carried out in previous papers for the particular case of spherical symmetry. However, a similar study for the EEG is remarkably more difficult due to the very complicated nature of the expression relating the voltage differences on the scalp to the primary current density (PCD) even for this simple symmetry. This paper introduces the use of the electric lead field (ELF) on the dyadic formalism in the spherical coordinate system to overcome such a drawback using an expansion of the ELF in terms of longitudinal and orthogonal vector fields. This approach allows us to represent EEG Fourier coefficients on a 2-sphere in terms of a current multipole expansion. The choice of a suitable basis for the Hilbert space of the PCDs on the brain region allows the current multipole moments to be related by spatial transfer functions to the PCD spectral coefficients. Properties of the most used distributed inverse solutions are explored on the basis of these results. Also, a part of the ELF null space is completely characterized and those spherical components of the PCD which are possible silent candidates are discussed.

Projective Methods for the Magnetic Direct Problem

The prediction of magnetic fields due to prescribed current sources in realistically shaped body models is usually approached by approximating the body as a set of piecewise homogeneous conductors with arbitrary geometry, separated by surfaces S l (l = 1, ..., M). One of the regions contains the primary current sources \(\vec j\left( {\vec r} \right)\) and S 1 encloses all other surfaces; \(\vec r\) is a arbitrary position vector.

EEG predictability: adequacy of non-linear forecasting methods

The predictive properties of EEG segments were analyzed. The sample included alpha, delta as well as spike and wave EEG activity recordings. Most of these segments are better described with non-linear autoregressive models, and a non-linear forecasting algorithm is routinely required. In terms of their predictive properties, segments can be divided into unpredictable, predictable and very predictable, these three groups being similarly represented among the alpha activity EEG segments. In EEG segments with alpha activity, poor predictability is associated with poor organization of the rhythmic pattern. Concerning dynamic properties, it was found that cyclic skeletons were highly represented among the very predictable segments, which reflect a contribution of the deterministic component of the autoregressive model to the predictability of the segments. Notable contributions of the noise component may explain the properties of unpredictable segments. These results point to a great diversity of predictive patterns among EEG recordings. Other factors besides the existence of chaotic dynamics must be regarded.

Maximum a posteriori estimation of change points in the EEG

A new approach for EEG segmentation is introduced. This is based on a methodology for optimal segmentation of non-stationary signals derived from the maximum a posteriori estimation principle. It is a model-based, not sequential approach that allows for segmentation at different resolution levels. The features of the methodology are illustrated by its application to EEG recordings containing several types of spectral changes due to normal and pathological variations of spontaneous brain rhythmic activities, as well as physiological artifacts.

Multivariate statistical brain electromagnetic mapping

SummaryBrain Electromagnetic Topography (BET) has attained widespread use. The representation of EEG or MEG parameters as scalp maps (BETm) aids its clinical interpretation. However, some critical issues limit the usefulness of BETm. In particular, the conventional statistical assessment of BETm with respect to normative data is based upon marginal significance probability scales which involve multiple univariate comparisons (one at each recording site). As a consequence, the probability of false positive findings (type I error) is increased above its nominal level. The use of conservative levels avoids this phenomenon but results in a considerable increase of the probability of not detecting real abnormality (type II error). Furthermore, BETm are constructed without taking into consideration the patterns of correlations characteristic of electromagnetic data under normal states of brain functioning. This limits the capability of BETm of representing multivariate aspects of abnormality. This paper introduces some techniques to approach these difficulties. Multivariate Brain Electromagnetic Topographic maps (MBETm) are defined, which retain the attractive features of mapping but also take advantage of multivariate characteristics (in the spatial and frequency domains) to highlight aspects of neuropathology. Moreover, simultaneous significance probability (SSP) scales, valid for both BETm and MBETm, are introduced for the global control of the probability of a type I error. The use of these techniques is illustrated with data from patients with cortical tumours and with epilepsy. ROC analysis shows that in some cases there is a significant improvement in both detection and localization accuracy.