EEG Signal Processing and Machine Learning. Saeid Sanei
an impulse response given by [20]:
Figure 3.10 Simplified model for brain cortical alpha generation. The input is a pulse‐shaped waveform.
(3.37)
for the excitatory case, and the following for the inhibitory case:
(3.38)
A and B determine the maximum amplitudes of the excitatory and inhibitory PSPs respectively, and a and b are the lumped representation of the sum of reciprocal of the time constant of passive membrane and all other spatially distributed delays in the neuron dendric network. The Sigm function is defined as [21]:
where e 0 indicates the maximum firing rate of the neural population, v 0 the PSP for which a 50% firing rate is achieved, and r the steepness of sigmoidal transformation. The connectivity constants C1 – C4 in the figure characterize the interaction between the pyramidal cells and the excitatory and inhibitory interneurons which account for the total number of synapses established between the neurons.
As an empirical value, considering C = C1 = 135, typical values suggested for the rest of the parameters are C2 = 0.8C, C3 = 0.25C, C4 = 0.25C, A = 3.25, B = 22, v 0 = 6, a = 100 s−1, b = 50 s−1 and ad = 30.
The above system has been extended to modelling the visual evoked potentials (VEPs) based on the fact that VEPs are the results of interaction of two or more of so‐called cortical columns. A proposed two‐column model for VEP can be seen in Figure 3.11 where each block represents the model in Figure 3.10.
The main problem with such a model is due to the fact that only a single channel EEG is generated and unlike the phase‐coupling model explained in subsection 3.2.2, there is no modelling of interchannel relationships and the inherent connectivity of the brain zones. Therefore, a more accurate model has to be defined to enable simulation of a multichannel EEG generation system. This is still an open question and remains an area of research.
Figure 3.11 A two‐column model for generation of VEP. Two connectivity constants K1 and K2 attenuate the output of a column before it is fed to the other.
3.4 Mathematical Models Derived Directly from the EEG Signals
These models are basically the description of single and multichannel signals in terms of a limited number of statistical parameters which not only represent the morphology of the waveforms but also their temporal or spatial sample correlations. In these models the internal and external noise is often considered as an uncorrelated temporal signal independent of the brain generated signals.
3.4.1 Linear Models
3.4.1.1 Prediction Method
The main objective of using prediction methods is to find a set of model parameters which best describe the signal generation system. Such models generally require a noise type input. In autoregressive (AR) modelling of signals each sample of a single channel EEG measurement is defined to be linearly related with respect to a number of its previous samples, i.e.:
(3.40)
where ak , k = 1,2,…,p, are the linear parameters, n denotes the discrete sample time normalized to unity, p is called the model or prediction order and x(n) is the noise input. In an autoregressive moving average (ARMA) linear predictive model each sample is obtained based on a number of its previous input and output sample values, i.e.:
(3.41)
where bk , k = 1, 2,…, q are the additional linear parameters. The parameters p and q are the model orders. The Akaike criterion can be used to determine the order of the appropriate model of a measurement signal by maximizing the log-likelihood equation [22] with respect to the model order:
(3.42)
where p and q represent respectively, the assumed AR and MA model prediction orders, N is the number of signal samples, and
In a multivariate AR (MVAR) approach a multichannel scheme is considered. Therefore, each signal sample is defined versus both its previous samples and the previous samples of the signals from other channels, i.e. for channel i we have:
(3.43)
where m represents the number of channels and xi (n) represents the noise input to channel i. Similarly, the model parameters can be calculated iteratively in order to minimize the error between the actual and predicted values [23].
Figure 3.12 A linear model for the generation of EEG signals.
There are numerous applications for linear models. These applications are discussed in other chapters of this book. Different algorithms have been developed to find efficiently the model coefficients. In the maximum likelihood estimation (MLE) method [23–25] the likelihood function is maximized over the system parameters formulated from the assumed real, Gaussian distributed, and sufficiently long input signals of approximately 10–20 seconds (consider a sampling frequency of fs = 250 Hz as often used for EEG recordings). Using Akaike's method the gradient of the squared error is minimized using the Newton–Raphson approach applied to the resultant nonlinear equations [23, 26]. This is considered as an approximation to the MLE approach. In the Durbin method [27] the Yule–Walker equations, which relate the model coefficients to the autocorrelation of the signals, are iteratively solved. The approach and the results are equivalent to those using a least‐squared‐based scheme [28]. The MVAR coefficients are often calculated using the Levinson–Wiggins–Robinson