Electroencephalography (EEG) provides a non-invasive measure of brain electrical activity. Neural population models, where large numbers of interacting neurons are considered collectively as a macroscopic system, have long been used to understand features in EEG signals. By tuning dozens of input parameters describing the excitatory and inhibitory neuron populations, these models can reproduce prominent features of the EEG such as the alpha-rhythm. However, the inverse problem, of directly estimating the parameters from fits to EEG data, remains unsolved. Solving this multi-parameter non-linear fitting problem will potentially provide a real-time method for characterizing average neuronal properties in human subjects. Here we perform unbiased fits of a 22-parameter neural population model to EEG data from 82 individuals, using both particle swarm optimization and Markov chain Monte Carlo sampling. We estimate how much is learned about individual parameters by computing Kullback-Leibler divergences between posterior and prior distributions for each parameter. Results indicate that only a single parameter, that determining the dynamics of inhibitory synaptic activity, is directly identifiable, while other parameters have large, though correlated, uncertainties. We show that the eigenvalues of the Fisher information matrix are roughly uniformly spaced over a log scale, indicating that the model is sloppy, like many of the regulatory network models in systems biology. These eigenvalues indicate that the system can be modeled with a low effective dimensionality, with inhibitory synaptic activity being prominent in driving system behavior.
Alpha blocking, a phenomenon where the alpha rhythm is reduced by attention to a visual, auditory, tactile or cognitive stimulus, is one of the most prominent features of human electroencephalography (EEG) signals. Here we identify a simple physiological mechanism by which opening of the eyes causes attenuation of the alpha rhythm. We fit a neural population model to EEG spectra from 82 subjects, each showing a different degree of alpha blocking upon opening of their eyes. Though it has been notoriously difficult to estimate parameters by fitting such models, we show how, by regularizing the differences in parameter estimates between eyes-closed and eyes-open states, we can reduce the uncertainties in these differences without significantly compromising fit quality. From this emerges a parsimonious explanation for the spectral differences between states: Changes to just a single parameter, p ei , corresponding to the strength of a tonic excitatory input to the inhibitory cortical population, are sufficient to explain the reduction in alpha rhythm upon opening of the eyes. We detect this by comparing the shift in each model parameter between eyes-closed and eyes-open states. Whereas changes in most parameters are weak or negligible and do not scale with the degree of alpha attenuation across subjects, the change in p ei increases monotonically with the degree of alpha blocking observed. These results indicate that opening of the eyes reduces alpha activity by increasing external input to the inhibitory cortical population.
Electroencephalography (EEG) provides a non-invasive measure of brain electrical activity. Neural population models, where large numbers of interacting neurons are considered collectively as a macroscopic system, have long been used to understand features in EEG signals. By tuning dozens of input parameters describing the excitatory and inhibitory neuron populations, these models can reproduce prominent features of the EEG such as the alpha-rhythm. However, the inverse problem, of directly estimating the parameters from fits to EEG data, remains unsolved. Solving this multi-parameter non-linear fitting problem will potentially provide a real-time method for characterizing average neuronal properties in human subjects. Here we perform unbiased fits of a 22-parameter neural population model to EEG data from 82 individuals, using both particle swarm optimization and Markov chain Monte Carlo sampling. We estimate how much is learned about individual parameters by computing Kullback-Leibler divergences between posterior and prior distributions for each parameter. Results indicate that only a single parameter, that determining the dynamics of inhibition, is directly identifiable, while other parameters have large, though correlated, uncertainties. We show that the eigenvalues of the Fisher information matrix are roughly uniformly spaced over a log scale, indicating that the model is sloppy, like many of the regulatory network models in systems biology. These eigenvalues indicate that the system can be modeled with a low effective dimensionality, with inhibition being prominent in driving system behavior. Author summaryElectroencephalography (EEG), where electrodes are used to measure electric potential on the outside of the scalp, provides a simple, non-invasive way to study brain activity. Physiological interpretation of features in EEG signals has often involved use of collective models of neural populations. These neural population models have dozens of input parameters to describe the properties of inhibitory and excitatory neurons. Being able to estimate these parameters by direct fits to EEG data holds the promise of providing a real-time non-invasive method of inferring neuronal properties in different December 5, 2018 1/27 individuals. However, it has long been impossible to fit these nonlinear, multi-parameter models effectively. Here we describe fits of a 22-parameter neural population model to EEG spectra from 82 different subjects, all exhibiting alpha-oscillations. We show how only one parameter, that describing inhibitory dynamics, is constrained by the data, although all parameters are correlated. These results indicate that inhibition plays a central role in the generation and modulation of the alpha-rhythm in humans. Introduction1 The classical alpha rhythm is one of the most remarkable features observed in 2 electroencephalogram (EEG) recordings from humans [1, 2]. First discovered by Berger 3 in the 1920s [3, 4] these waxing and waning oscillations of 8 -13 Hz, that are prominent 4 during eyes-closed...
In the modern environment, people are more and more dependent from software-controlled systems. If it comes, however, to the certification of those systems, frequently used methods like fault tree analysis or Markov models have some crucial disadvantages, especially in modeling the hierarchical decomposition as well as the temporal behavior of the underlying system. To deal with these weak points, state/event fault trees (SEFTs) [9] were developed. All analysis methods of SEFTs - qualitative and quantitative - based on their reachability graphs which are traditionally developed by translation into equivalent Petri-nets. However, this Petri-net translation and its subsequent reachability graph development is a computationally expensive procedure that comes mostly with a state-space explosion. In this paper, we propose an approach that is able to develop an SEFT's reachability graph directly without an intermediate Petri-net model. In the evaluation we have carrie out some case studies in order to show that the direct approach increases the efficiency w. r. t. memory and time consumption while maintaining the same results as the traditional approach
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