A parametric method to measure time-varying linear and nonlinear causality with applications to EEG data.

A linear and nonlinear causality detection method called the error-reduction-ratio causality (ERRC) test is introduced in this paper to investigate if linear or nonlinear models should be considered in the study of human electroencephalograph (EEG) data. In comparison to the traditional Granger methods, one significant advantage of the ERRC approach is that it can effectively detect the time-varying linear and nonlinear causalities between two signals without fitting a complete nonlinear model. Two numerical simulation examples are employed to compare the performance of the new method with other widely used methods in the presence of noise and in tracking time-varying causality. Finally, an application to measure the linear and nonlinear relationships between two EEG signals from different cortical sites for patients with childhood absence epilepsy is discussed.

A new NARX-based Granger linear and nonlinear casual influence detection method with applications to EEG data.

A new NARX-based Granger linear and nonlinear casual influence detection method is presented in this paper to address the potential for linear and nonlinear models in data with applications to human EEG data analysis. Considering two signals initially, the paper introduces four indexes to measure the linearity and nonlinearity of a single signal, and one signal influencing the second signal. This method is then extended to the time-varying and multivariate cases. An adaptation of an Orthogonal Least Squares routine is employed to select the significant terms in the models. A numerical example is provided to illustrate the effectiveness of the new algorithms together with the application to real EEG data collected from 4 patients.

Tracking time-varying causality and directionality of information flow using an error reduction ratio test with applications to electroencephalography data.

This paper introduces an error reduction ratio-causality (ERR-causality) test that can be used to detect and track causal relationships between two signals. In comparison to the traditional Granger method, one significant advantage of the new ERR-causality test is that it can effectively detect the time-varying direction of linear or nonlinear causality between two signals without fitting a complete model. Another important advantage is that the ERR-causality test can detect both the direction of interactions and estimate the relative time shift between the two signals. Numerical examples are provided to illustrate the effectiveness of the new method together with the determination of the causality between electroencephalograph signals from different cortical sites for patients during an epileptic seizure.

Balanced excitation and inhibition: model based analysis of local field potentials.

We have developed a model of the local field potential (LFP) based on the conservation of charge, the independence principle of ionic flows and the classical Hodgkin-Huxley (HH) type intracellular model of synaptic activity. Insights were gained through the simulation of the HH intracellular model on the nonlinear relationship between the balance of synaptic conductances and that of post-synaptic currents. The latter is dependent not only on the former, but also on the temporal lag between the excitatory and inhibitory conductances, as well as the strength of the afferent signal. The proposed LFP model provides a method for decomposing the LFP recordings near the soma of layer IV pyramidal neurons in the barrel cortex of anaesthetised rats into two highly correlated components with opposite polarity. The temporal dynamics and the proportional balance of the two components are comparable to the excitatory and inhibitory post-synaptic currents computed from the HH model. This suggests that the two components of the LFP reflect the underlying excitatory and inhibitory post-synaptic currents of the local neural population. We further used the model to decompose a sequence of evoked LFP responses under repetitive electrical stimulation (5Hz) of the whisker pad. We found that as neural responses adapted, the excitatory and inhibitory components also adapted proportionately, while the temporal lag between the onsets of the two components increased during frequency adaptation. Our results demonstrated that the balance between neural excitation and inhibition can be investigated using extracellular recordings. Extension of the model to incorporate multiple compartments should allow more quantitative interpretations of surface Electroencephalography (EEG) recordings into components reflecting the excitatory, inhibitory and passive ionic current flows generated by local neural populations.

Time-varying linear and nonlinear parametric model for Granger causality analysis.

Statistical measures such as coherence, mutual information, or correlation are usually applied to evaluate the interactions between two or more signals. However, these methods cannot distinguish directions of flow between two signals. The capability to detect causalities is highly desirable for understanding the cooperative nature of complex systems. The main objective of this work is to present a linear and nonlinear time-varying parametric modeling and identification approach that can be used to detect Granger causality, which may change with time and may not be detected by traditional methods. A numerical example, in which the exact causal influences relationships, is presented to illustrate the performance of the method for time-varying Granger causality detection. The approach is applied to EEG signals to track and detect hidden potential causalities. One advantage of the proposed model, compared with traditional Granger causality, is that the results are easier to interpret and yield additional insights into the transient directed dynamical Granger causality interactions.

Nonlinear fisher discriminant analysis using a minimum squared error cost function and the orthogonal least squares algorithm.

The nonlinear discriminant function obtained using a minimum squared error cost function can be shown to be directly related to the nonlinear Fisher discriminant (NFD). With the squared error cost function, the orthogonal least squares (OLS) algorithm can be used to find a parsimonious description of the nonlinear discriminant function. Two simple classification techniques will be introduced and tested on a number of real and artificial data sets. The results show that the new classification technique can often perform favourably compared with other state of the art classification techniques.

On-line identification of nonlinear systems using Volterra polynomial basis function neural networks.

An on-line identification scheme using Volterra polynomial basis function (VPBF) neural networks is considered for nonlinear control systems. This comprises a structure selection procedure and a recursive weight learning algorithm. The orthogonal least-squares algorithm is introduced for off-line structure selection and the growing network technique is used for on-line structure selection. An on-line recursive weight learning algorithm is developed to adjust the weights so that the identified model can adapt to variations of the characteristics and operating points in nonlinear systems. The convergence of both the weights and the estimation errors is established using a Lyapunov technique. The identification procedure is illustrated using simulated examples.

Dual-orthogonal radial basis function networks for nonlinear time series prediction.

A new structure of Radial Basis Function (RBF) neural network called the Dual-orthogonal RBF Network (DRBF) is introduced for nonlinear time series prediction. The hidden nodes of a conventional RBF network compare the Euclidean distance between the network input vector and the centres, and the node responses are radially symmetrical. But in time series prediction where the system input vectors are lagged system outputs, which are usually highly correlated, the Euclidean distance measure may not be appropriate. The DRBF network modifies the distance metric by introducing a classification function which is based on the estimation data set. Training the DRBF networks consists of two stages. Learning the classification related basis functions and the important input nodes, followed by selecting the regressors and learning the weights of the hidden nodes. In both cases, a forward Orthogonal Least Squares (OLS) selection procedure is applied, initially to select the important input nodes and then to select the important centres. Simulation results of single-step and multi-step ahead predictions over a test data set are included to demonstrate the effectiveness of the new approach.

On-line Supervised Adaptive Training Using Radial Basis Function Networks.

A new recursive supervised training algorithm is derived for the radial basis neural network architecture. The new algorithm combines the procedures of on-line candidate regressor selection with the conventional Givens QR based recursive parameter estimator to provide efficient adaptive supervised network training. A new concise on-line correlation based performance monitoring scheme is also introduced as an auxiliary device to detect structural changes in temporal data processing applications. Practical and simulated examples are included to demonstrate the effectiveness of the new procedures. Copyright 1996 Elsevier Science Ltd.

Radial Basis Function Network Configuration Using Mutual Information and the Orthogonal Least Squares Algorithm.

Input nodes of neural networks are usually predetermined by using a priori knowledge or selected by trial and error. For example, in pattern recognition applications the input nodes are usually the given pattern features and in system identification applications the past input and output data are often used as inputs to the network. Some of the input variables may be irrelevant to the task in hand and therefore may cause a deterioration in network performance. Some may be redundant and may increase the complexity of the network and consume expensive computation time. In the present study, the mutual information between the input variables and the output of the network is used to select a suboptimal set of input variables for the network. The variables are selected according to the information content relevant to the output. Variables which have a higher mutual information with the output and lower dependence on other selected variables are used as network inputs. The algorithms are derived based on heuristics and performance is assessed by using radial basis function (RBF) networks trained with the orthogonal least squares algorithm (OLS), which selects the hidden layer nodes of the network according to the error reduction ratios on the network output. Both real and simulated data sets are used to demonstrate the effectiveness of the new algorithms. Copyright 1996 Elsevier Science Ltd.