Electronic Neuron Models
The Nernst equation (Equation 3.21), which expresses the required membrane voltage to equilibrate the ion flux through the membrane for an existing concentration ratio of a particular ion species. Because the Nernst equation evaluates the ion moving force due to a concentration gradient as a voltage [V], this may be represented in equivalent electric circuits as a battery.
The cable model of an axon, which is composed of external and internal resistances as well as the electric properties of the membrane. This equivalent circuit may be used to calculate the general cable equation of the axon (Equation 3.45) describing the subthreshold transmembrane voltage response to a constant current stimulation. The time-varying equations describing the behavior of the transmembrane voltage due to a step-impulse stimulation are also of interest (though more complicated). Their solutions were illustrated in Figure 3.11. The equivalent circuit for (approximate) derivation of the strength-duration equation, Equation 3.58, was shown in Figure 3.12.
The equivalent electric circuits describing the behavior of the axon under conditions of nerve propagation, or under space-clamp and voltage-clamp conditions, are shown in Figures 4.1, 4.2, and 4.3; the corresponding equations are 4.1, 4.2, and 4.3, respectively.
The electric circuit for the parallel-conductance model of the membrane, which contains pathways for sodium, potassium, and chloride ion currents, is illustrated in Figure 4.10, and its behavior described by Equation 4.10. This equation includes the following passive electric parameters (electronic components): membrane capacitance, Nernst voltages for sodium, potassium, and chloride ions, as well as the leakage conductance. Further, the circuit includes the behavior of the active parameters, the sodium and potassium conductances, as described by the Hodgkin-Huxley equations (Equations 4.12-4.24).
It provides us with an opportunity to verify that the models we have constructed really behave the same as the excitable tissue that they should model - that is, that the model is correct. If the behavior of the model is not completely correct, we may be able to adjust the properties of the model and thereby improve our understanding of the behavior of the tissue. This analysis of the behavior of the excitable tissue is one general purpose of modeling work.
There exists also the possibility of constructing, or synthesizing electronic circuits, whose behavior is similar to neural tissue, and which perform information processing in a way that also is similar to nature. In its most advanced form, it is called neural computing.
Figure 10.11A and 10.11B compare the potassium and sodium ion currents of the Lewis model to those in the Hodgkin-Huxley model, respectively. Figure 10.12 illustrates the action pulse generated by the Lewis model. The peak magnitude of the simulated sodium current is 10 mA. This magnitude is equivalent to approximately 450 µA/cm2 in the membrane, which is about half of the value calculated by Hodgkin and Huxley from their model. The maximum potassium current in the circuit is 3 mA, corresponding to 135 µA/cm2 in the membrane. The author gave no calibration for the membrane potential or for the time axis.
The electronic realizations of the Hodgkin-Huxley model are very accurate in simulating the function of a single neuron. However, when one is trying to simulate the function of neural networks, they become very complicated. Many scientists feel that when simulating large neural networks, the internal construction of its element may not be too important. It may be satisfactory simply to ensure that the elements produce an action pulse in response to the stimuli in a manner similar to an actual neuron. On this basis, Leon D. Harmon constructed a neuron model having a very simple circuit. With this model he performed experiments in which he simulated many functions characteristic of the neuron (Harmon, 1961).
The circuit of the Harmon neuron model is given in Figure 10.13. Figures 10.13A and 10.13B show the preliminary and more advanced versions of the circuit, respectively. The model is equipped with five excitatory inputs which can be adjusted. These include diode circuits representing various synaptic functions. The signal introduced at excitatory inputs charges the 0.02 µF capacitor which, after reaching a voltage of about 1.5 V, allows the monostable multivibrator, formed by transistors T1 and T2, to generate the action pulse. This impulse is amplified by transistors T3 and T4. The output of one neuron model may drive the inputs of perhaps 100 neighboring neuron models. The basic model also includes an inhibitory input. A pulse introduced at this input has the effect of decreasing the sensitivity to the excitatory inputs..
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