Fetsje Bijma

In vivo measurement of the brain and skull resistivities using an EIT-based method and realistic models for the head

In vivo measurements of equivalent resistivities of skull (/spl rho//sub skull/) and brain (/spl rho//sub brain/) are performed for six subjects using an electric impedance tomography (EIT)-based method and realistic models for the head. The classical boundary element method (BEM) formulation for EIT is very time consuming. However, the application of the Sherman-Morrison formula reduces the computation time by a factor of 5. Using an optimal point distribution in the BEM model to optimize its accuracy, decreasing systematic errors of numerical origin, is important because cost functions are shallow. Results demonstrate that /spl rho//sub skull///spl rho//sub brain/ is more likely to be within 20 and 50 rather than equal to the commonly accepted value of 80. The variation in /spl rho//sub brain/ (average = 301 /spl Omega/ /spl middot/ cm, SD = 13%) and /spl rho//sub skull/ (average = 12230 /spl Omega/ /spl middot/ cm, SD = 18%) is decreased by half, when compared with the results using the sphere model, showing that the correction for geometry errors is essential to obtain realistic estimations. However, a factor of 2.4 may still exist between values of /spl rho//sub skull///spl rho//sub brain/ corresponding to different subjects. Earlier results show the necessity of calibrating /spl rho//sub brain/ and /spl rho//sub skull/ by measuring them in vivo for each subject, in order to decrease errors associated with the electroencephalogram inverse problem. We show that the proposed method is suited to this goal.

A maximum-likelihood estimator for trial-to-trial variations in noisy MEG/EEG data sets

The standard procedure to determine the brain response from a multitrial evoked magnetoencephalography (MEG) or electroencephalography (EEG) data set is to average the individual trials of these data, time locked to the stimulus onset. When the brain responses vary from trial-to-trial this approach is false. In this paper, a maximum-likelihood estimator is derived for the case that the recorded data contain amplitude variations. The estimator accounts for spatially and temporally correlated background noise that is superimposed on the brain response. The model is applied to a series of 17 MEG data sets of normal subjects, obtained during median nerve stimulation. It appears that the amplitude of late component (30-120 ms) shows a systematic negative trend indicating a weakening response during stimulation time. For the early components (20-35 ms) no such a systematic effect was found. The model is furthermore applied on a MEG data set consisting of epileptic spikes of constant spatial distribution but varying polarity. For these data, the advantage of applying the model is that positive and negative spikes can be processed with a single model, thereby reducing the number of degrees of freedom and increasing the signal-to-noise ratio.

The spatiotemporal MEG covariance matrix modeled as a sum of Kronecker products.

The single Kronecker product (KP) model for the spatiotemporal covariance of MEG residuals is extended to a sum of Kronecker products. This sum of KP is estimated such that it approximates the spatiotemporal sample covariance best in matrix norm. Contrary to the single KP, this extension allows for describing multiple, independent phenomena in the ongoing background activity. Whereas the single KP model can be interpreted by assuming that background activity is generated by randomly distributed dipoles with certain spatial and temporal characteristics, the sum model can be physiologically interpreted by assuming a composite of such processes. Taking enough terms into account, the spatiotemporal sample covariance matrix can be described exactly by this extended model. In the estimation of the sum of KP model, it appears that the sum of the first 2 KP describes between 67% and 93%. Moreover, these first two terms describe two physiological processes in the background activity: focal, frequency-specific alpha activity, and more widespread non-frequency-specific activity. Furthermore, temporal nonstationarities due to trial-to-trial variations are not clearly visible in the first two terms, and, hence, play only a minor role in the sample covariance matrix in terms of matrix power. Considering the dipole localization, the single KP model appears to describe around 80% of the noise and seems therefore adequate. The emphasis of further improvement of localization accuracy should be on improving the source model rather than the covariance model.

A mathematical approach to the temporal stationarity of background noise in MEG/EEG measurements

The general spatiotemporal covariance matrix of the background noise in MEG/EEG signals is huge. To reduce the dimensionality of this matrix it is modeled as a Kronecker product of a spatial and a temporal covariance matrix. When the number of time samples is larger than, say, J = 500, the iterative Maximum Likelihood estimation of these two matrices is still too time-consuming to be useful on a routine basis. In this study we looked for methods to circumvent this computationally expensive procedure by using a parametric model with subject-dependent parameters. Such a model would additionally help with interpreting MEG/EEG signals. For the spatial covariance, models have been derived already and it has been shown that measured MEG/EEG signals can be understood spatially as random processes, generated by random dipoles. The temporal covariance, however, has not been modeled yet, therefore we studied the temporal covariance matrix in several subjects. For all subjects the temporal covariance shows an alpha oscillation and vanishes for large time lag. This gives rise to a temporal noise model consisting of two components: alpha activity and additional random noise. The alpha activity is modeled as randomly occurring waves with random phase and the covariance of the additional noise decreases exponentially with lag. This model requires only six parameters instead of 12 J(J + 1). Theoretically, this model is stationary but in practice the stationarity of the matrix is highly influenced by the baseline correction. It appears that very good agreement between the data and the parametric model can be obtained when the baseline correction window is taken into account properly. This finding implies that the background noise is in principle a stationary process and that nonstationarities are mainly caused by the nature of the preprocessing method. When analyzing events at a fixed sample after the stimulus (e.g., the SEF N20 response) one can take advantage of this nonstationarity by optimizing the baseline window to obtain a low noise variance at this particular sample.

Independently outgrowing neurons and geometry-based synapse formation produce networks with realistic synaptic connectivity.

Neuronal signal integration and information processing in cortical networks critically depend on the organization of synaptic connectivity. During development, neurons can form synaptic connections when their axonal and dendritic arborizations come within close proximity of each other. Although many signaling cues are thought to be involved in guiding neuronal extensions, the extent to which accidental appositions between axons and dendrites can already account for synaptic connectivity remains unclear. To investigate this, we generated a local network of cortical L2/3 neurons that grew out independently of each other and that were not guided by any extracellular cues. Synapses were formed when axonal and dendritic branches came by chance within a threshold distance of each other. Despite the absence of guidance cues, we found that the emerging synaptic connectivity showed a good agreement with available experimental data on spatial locations of synapses on dendrites and axons, number of synapses by which neurons are connected, connection probability between neurons, distance between connected neurons, and pattern of synaptic connectivity. The connectivity pattern had a small-world topology but was not scale free. Together, our results suggest that baseline synaptic connectivity in local cortical circuits may largely result from accidentally overlapping axonal and dendritic branches of independently outgrowing neurons.

Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

This paper deals with multivariate Gaussian models for which the covariance matrix is a Kronecker product of two matrices. We consider maximum likelihood estimation of the model parameters, in particular of the covariance matrix. There is no explicit expression for the maximum likelihood estimator of a Kronecker product covariance matrix. We investigate whether the maximum likelihood estimator of the covariance matrix exists and whether it is unique. We consider models with general, with double diagonal, and with one diagonal Kronecker product covariance matrices, and find different results.

Dynamics underlying spontaneous human alpha oscillations: A data-driven approach

Although the cognitive and clinical correlates of spontaneous human alpha oscillations as recorded with electroencephalography (EEG) or magnetoencephalography (MEG) are well documented, the dynamics underlying these oscillations is still a matter of debate. This study proposes a data-driven method to reveal the dynamics of these oscillations. It demonstrates that spontaneous human alpha oscillations as recorded with MEG can be viewed as noise-perturbed damped harmonic oscillations. This provides evidence for the hypothesis that these oscillations reflect filtered noise and hence do not possess limit-cycle dynamics. To illustrate the use of the model, we apply it to two data-sets in which a decrease in alpha power can be observed across conditions. The associated differences in the estimated model parameters show that observed decreases in alpha power are associated with different kinds of changes in the dynamics. Thus, the model parameters are useful dynamical biomarkers for spontaneous human alpha oscillations.

A Morpho-Density Approach to Estimating Neural Connectivity

Neuronal signal integration and information processing in cortical neuronal networks critically depend on the organization of synaptic connectivity. Because of the challenges involved in measuring a large number of neurons, synaptic connectivity is difficult to determine experimentally. Current computational methods for estimating connectivity typically rely on the juxtaposition of experimentally available neurons and applying mathematical techniques to compute estimates of neural connectivity. However, since the number of available neurons is very limited, these connectivity estimates may be subject to large uncertainties. We use a morpho-density field approach applied to a vast ensemble of model-generated neurons. A morpho-density field (MDF) describes the distribution of neural mass in the space around the neural soma. The estimated axonal and dendritic MDFs are derived from 100,000 model neurons that are generated by a stochastic phenomenological model of neurite outgrowth. These MDFs are then used to estimate the connectivity between pairs of neurons as a function of their inter-soma displacement. Compared with other density-field methods, our approach to estimating synaptic connectivity uses fewer restricting assumptions and produces connectivity estimates with a lower standard deviation. An important requirement is that the model-generated neurons reflect accurately the morphology and variation in morphology of the experimental neurons used for optimizing the model parameters. As such, the method remains subject to the uncertainties caused by the limited number of neurons in the experimental data set and by the quality of the model and the assumptions used in creating the MDFs and in calculating estimating connectivity. In summary, MDFs are a powerful tool for visualizing the spatial distribution of axonal and dendritic densities, for estimating the number of potential synapses between neurons with low standard deviation, and for obtaining a greater understanding of the relationship between neural morphology and network connectivity.

How are evoked responses generated? The need for a unified mathematical framework

When the EEG is recorded during multiple events of the same type (e.g., the same stimuli), and the EEG (or MEG) is averaged, time locked by the events, a stable signal arises which is commonly called event related potential (ERP), respectively event related field (ERF). Despite its widespread use in both the scientific and clinical domain, the precise mechanism by which ERPs and ERFs are generated is still a matter of debate. Up to the publication of Nikulin et al. (2007) the main controversy was between the ‘‘signal plus noise” (SPN)-model and the ‘‘phase reset” (PR)-model. In the SPN model the response to the ‘event’ is considered to be a signal that is the same from trial to trial which is superimposed onto the ongoing EEG, often considered to be correlated Gaussian noise. Within the framework of the SPN-model the ERP is a maximum likelihood estimator of the event specific brain response. This point of view is not altered by the fact that the ongoing EEG may consist of frequency bursts in the 10 Hz alpha range instead of Gaussian noise or ongoing alpha oscillations. Since in the SPN paradigm these bursts are not affected by the external stimulus, the only contribution of the bursts to the model fitting procedure is an alteration of the spatio-temporal characteristics of the noise covariance (Bijma et al., 2003). In the PR-model, introduced by Sayers et al. (1974) the role of the alpha waves is fundamentally different. Based on the argument that the spectral power of the EEG is (hardly) altered by stimulation, these authors proposed that the ERP is generated by a mechanism where the external events somehow result in a phase synchronization of the alpha bursts. The validity of the phase synchronization argument for the PR-model, as proposed by Makeig et al. (2002), was questioned by Mäkinen et al. (2005) because simulations show that this argument does not truly discriminate between the SPN and the PR model. As an alternative, Mäkinen et al. (2005) proposed to consider the standard deviation of the measured responses as function of time, because the PR-model implies that these standard deviations show a dip at the time of phase resetting, whereas the SPNmodel results in a standard deviation that is constant over time. Based on these arguments, Mäkinen et al. (2005) found experimental evidence for the SPN-model. More generally, Bijma et al. (2003) considered the stationarity of the full temporal covariance matrix instead of only its diagonal part (i.e., the squared standard deviations) and found highly stationary background noise, which also supports the SPN-model. Further evidence

Three-way matrix analysis, the MUSIC algorithm and the coupled dipole model

The inverse problem of multi-channel MEG/EEG data is considered as a parameter estimation problem. The stability of the solution of the inverse problem, which decreases with the number of included dipoles, can be improved by either adding constraints to the model parameters, or by adding more data of related data sets. The latter approach was taken by Bijma et al. [Bijma F, de Munck JC, Böcker KBE, Huizenga HM, Heethaar RM. The coupled dipole model: an integrated model for multiple MEG/EEG data sets. NeuroImage 2004;23(3):890-904; Bijma F, de Munck JC, Huizenga HM, Heethaar RM, Nehorai A. Simultaneous estimation and testing in multiple MEG data sets. IEEE Trans SP 2005;53(9):3449-60] by introducing coupling matrices that link dipole parameters and source time functions of different data sets. Here, the theoretical foundations of the coupled dipole model are explored and the MUSIC algorithm is generalised to the analysis of multiple related data sets. Similar to the MUSIC algorithm, the number of sources and the number of constraints are derived from the data by considering the minimum possible residual error as a function of the number of sources and constraints. However, contrary to the MUSIC algorithm, where the minimum residual error can be obtained from an SVD analysis of a two-way data matrix, here we deal with multiple data sets and therefore three-way matrix analysis is used. From a simulation study it appears that the number of sources and constraints can be clearly determined from a generalised SVD analysis. The generalisation of the MUSIC algorithm to three-way data gives reasonable estimates of the dipole parameters. These results can be used in the simultaneous analysis of MEG/EEG data of multiple subjects, multiples data sets of the same subject or models where subsequent trials of data show habituation effects.