Читать книгу EEG Signal Processing and Machine Learning - Saeid Sanei - Страница 44

A simpler model than that of Hodgkin–Huxley for simulating spiking neurons is the Morris–Lecar model [14]. This model is a minimal biophysical model, which generally exhibits single AP. This model considers that the oscillation of a slow calcium wave depolarizing the membrane leads to a bursting state. The Morris–Lecar model was initially developed to describe the behaviour of barnacle muscle cells. The governing equations relating the membrane potential (E) and potassium activation w_k to the activation parameters are given as:

(3.28)

(3.29)

where I_i is the combination of three ionic currents, calcium (Ca), potassium (K) and leak (l) and similar to the Hodgkin–Huxley model, are products of a maximal conductance , activation components (in such as a_Ca , w_k ), and the driving force E. The changes in the potassium activation variable w_k is proportional to a steady‐state activation function w _∞(E) (a sigmoid curve) and a time‐constant function τ_k (E) (a bell‐shaped curve). These functions are respectively defined as:

(3.30)

(3.31)

The steady‐state activation function a_ca (E), involved in calculation of the calcium current, is defined as:

(3.32)

Similar to the sodium current in the Hodgkin–Huxley model, the calcium current is an inward current. Since the calcium activation current is a fast process in comparison with the potassium current, it is modelled as an instantaneous function. This means that for each voltage E, the steady‐state function a_Ca (E) is calculated. The calcium current does not incorporate any inactivation process. The activation variable w_k here is similar to a_k in the Hodgkin–Huxley model, and finally the leak currents for both models are the same [12]. A simulation of the Morris–Lecar model is presented in Figure 3.6.

Calcium‐dependent potassium channels are activated by intracellular calcium, the higher the calcium concentration the higher the channel activation [12]. For the Morris–Lecar model to exhibit bursting behaviour, the two parameters of maximal time constant and the input current have to be changed [12]. Figure 3.7 shows the bursting behaviour of the Morris–Lecar model. The basic characteristics of a bursting neuron are the duration of the spiky activity, the frequency of the APs during a burst, and the duration of the quiescence period. The period of an entire bursting event is the sum of both active and quiescence duration [12].

Figure 3.6 Simulation of an AP within the Morris–Lecar model. The model parameters are: C_m = 22 μF cm⁻², =3.8 mS cm⁻², =8.0 mS cm⁻², =1.6 mS cm⁻², E_Ca = 125 mV, E_k = −80 mV, E_l = −60 mV, λ_t = 0.06, h_Ca = −1.2, S_m = 8.8.

Figure 3.7 An illustration of the bursting behaviour that can be generated by the Morris–Lecar model.

Neurons communicate with each other across synapses through axon‐dendrites or dendrites‐dendrites connections, which can be excitatory, inhibitory, or electric [12]. By combining a number of the above models, a neuronal network can be constructed. The network exhibits oscillatory behaviour due to the synaptic connection between the neurons. It is commonly assumed that excitatory synaptic coupling tends to synchronize neural firing, while inhibitory coupling pushes neurons towards anti‐synchrony. Such behaviour has been seen in models of neuronal circuits [1].

A synaptic current is produced as soon as a neuron fires an AP. This current stimulates the connected neuron and may be modelled by an alpha function multiplied by a maximal conductance and a driving force as:

(3.33)

where:

(3.34)

and t is the latency or time since the trigger of the synaptic current, u is the time to reach to the peak amplitude, E_syn is the synaptic reversal potential, and is the maximal synaptic conductance. The parameter u alters the duration of the current while changes the strength of the current. This concludes the treatment of the modelling of APs.

As the nature of the EEG sources cannot be determined from the electrode signals directly, many researchers have tried to model these processes on the basis of information extracted using signal processing techniques. The method of linear prediction (LP) described in the later sections of this chapter is frequently used to extract a parametric description.

A tutorial on realistic neural modelling using the Hodgkin–Huxley excitation model by David Beeman has been documented at the first annual meeting of the World Association of Modelers (WAM) Biologically Accurate Modeling Meeting (BAMM) in 2005 in Texas, USA. This can be viewed at http://www.brains‐minds‐media.org/archive/218.