Читать книгу EEG Signal Processing and Machine Learning - Saeid Sanei - Страница 45

The objective in this section is to introduce some established models for generating normal and some abnormal EEGs. These models are generally nonlinear and some have been proposed [15] for modelling a normal EEG signal and some others for the abnormal EEGs.

A simple distributed model consisting of a set of simulated neurons, thalamocortical relay cells, and interneurons was proposed [16, 17] that incorporates the limited physiological and histological data available at that time. The basic assumptions were sufficient to explain the generation of the alpha rhythm, i.e. the EEGs within the frequency range of 8–13 Hz.

A general nonlinear lumped model may take the form shown in Figure 3.8. Although the model is analogue in nature all the blocks are implemented in discrete form. This model can take into account the major characteristics of a distributed model and it is easy to investigate the result of changing the range of excitatory and inhibitory influences of thalamocortical relay cells and interneurons.

Figure 3.8 A nonlinear lumped model for generating the rhythmic activity of the EEG signals; h_e (t) and h_i (t) are the excitatory and inhibitory post synaptic potentials, f(v) is normally a simplified nonlinear function, and the C_i s are respectively the interaction parameters representing the interneurons and thalamocortical neurons [16].

In this model [16] there is a feedback loop including the inhibitory post‐synaptic potentials, the nonlinear function, and the interaction parameters C₃ and C₄. The other feedback includes mainly the excitatory potentials, nonlinear function, and the interaction parameters C₁ and C₂. The role of the excitatory neurons is to excite one or two inhibitory neurons. The latter, in turn, serve to inhibit a collection of excitatory neurons. Thus, the neural circuit forms a feedback system. The input p(t) is considered as a white noise signal. This is a general model; more assumptions are often needed to enable generation of the EEGs for the abnormal cases. Therefore, the function f(v) may change to generate the EEG signals for different brain abnormalities. Accordingly, the C_i coefficients can be varied. In addition, the output is subject to environment and measurement noise. In some models, such as the local EEG model (LEM) [16] the noise has been considered as an additive component in the output.

Figure 3.9 shows the LEM model. This model uses the formulation by Wilson and Cowan [18] who provided a set of equations to describe the overall activity (not specifically the EGG) in a cartel of excitatory and inhibitory neurons having a large number of interconnections [19]. Similarly, in the LEM the EEG it is assumed that the rhythms are generated by distinct neuronal populations, which possess frequency selective properties. These populations are formed by the interconnection of the individual neurons and are assumed to be driven by a random input. The model characteristics, such as the neural interconnectivity, synapse pulse response, and threshold of excitation are presented by the LEM parameters. The changes in these parameters produce the relevant EEG rhythms.

The input p(t) is assumed to result from the summation of a randomly distributed series of random potentials which drive the excitatory cells of the circuit, producing the ongoing background EEG signal. Such signals originate from other deeper brain sources within the thalamus and brain stem and constitute part of the ongoing or spontaneous firing of the central nerve system (CNS). In the model, the average number of inputs to an inhibitory neuron from the excitatory neurons is designated by C_e and the corresponding average number from inhibitory neurons to each individual excitatory neuron is C_i . The difference of two decaying exponentials are used for modelling each post‐synaptic potential h_e or h_i :

Figure 3.9 The local EEG model (LEM). The thalamocortical relay neurons are represented by two linear systems having impulse responses h_e (t), on the upper branch, and the inhibitory post‐synaptic potential by h_i (t). The nonlinearity of this system is denoted by f_e (v) representing the spike generating process. The interneuron activity is represented by another linear filter h_e (t) in the lower branch, which generally can be different from the first linear system, and a nonlinearity function f_i (v). C_e and C_i represent respectively, the number of interneuron cells and the thalamocortical neurons.

(3.35)

(3.36)

where A, B, a_k , and b_k are constant parameters, which control the shape of the pulse waveforms. The membrane potentials are related to the axonal pulse densities via the static threshold functions f_e and f_i . These functions are generally nonlinear; however, to ease the manipulations they are considered linear for each short time interval. Using this model, the normal brain rhythms such as alpha wave is considered as filtered noise.

A more simplified model is the one presented by Jansen and Rit [20]. This model is shown in Figure 3.10. The model is a neurophysiologically inspired model simulating electrical brain activity (including EEG or evoked potentials (EPs)). A previously developed lumped‐parameter model [16] of a single cortical column has been implemented in their work. The model could produce a large variety of EEG‐like waveforms and rhythms. Coupling two models, with delays in the interconnections to simulate the synaptic connections within and between cortical areas, made it possible to replicate the spatial distribution of alpha and beta activity.

EPs were simulated by presenting pulses to the input of the coupled models. In general, the responses were more realistic than those produced using a single model. The proposed model is based on a nonlinear model of a cortical column described by Jansen and Rit [20] and also based upon Lopes da Silva's lumped‐parameter model [16, 18]. The cortical column is modelled by a population of ‘feed forward’ pyramidal cells, receiving inhibitory and excitatory feedback from local interneurons (i.e. other pyramidal, stellate or basket cells residing in the same column) and excitatory input from neighbouring or more distant columns. The input can be a pulse, arbitrary function, or noise.

Each of the neuron populations is modelled by two blocks. The first block transforms the average pulse density of APs coming to the population of neurons into an average post‐synaptic membrane potential which can either be excitatory or inhibitory. This block is referred to as the post‐synaptic potential (PSP) block and represents a linear transformation with 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]:

(3.39)

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 a_d = 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.