ELECTRONIC SIMULATION OF THE HODGKIN-HUXLEY MODEL
Figure 7—System diagram for electronic simulation of the Hodgkin-Huxley model
Given a suitable means of generating the conductance functions, GNa(v,t) and GK(v,t), one can readily stimulate the essential aspects of the Modern Ionic Hypothesis. If we wish to do this electronically, we have two problems. First, we must synthesize a network whose input is the membrane potential and whose output is a voltage or current proportional to the desired conductance function. Second, we must transform the output from a voltage or current to an effective electronic conductance. The former implies the need for nonlinear, active filters, while the latter implies the need for multipliers. The basic block diagram is shown in [Figure 7]. Several distinct realizations of this system have been developed in our laboratory, and in each case the results were the same. With parameters adjusted to closely match the data of Hodgkin and Huxley, the electronic model exhibits all of the important properties of the axon. It produces spikes of 1 to 2 msec duration with a threshold of approximately 5% to 10% of the spike amplitude. The applied stimulus is generally followed by a prepotential, then an active rise of less than 1 msec, followed by an active recovery. The after-depolarization generally lasts several msec, followed by a prolonged after-hyperpolarization. The model exhibits the typical strength-duration curve, with rheobase of 5% to 10% of the spike amplitude. For sufficiently prolonged sodium inactivation (long time constant of recovery from inactivation), the model also exhibits an effect identical to classical Wedensky inhibition [(18)]. Thus, as would be expected, the electronic model simulates very well the electrical properties of the axon.
In addition to the axon properties, however, the electronic model is able to reproduce all of the somatic and dendritic activity outlined in the section on subthreshold activity. Simulation of the pacemaker and graded-response potentials is accomplished without additional circuitry. In the case of synaptically induced potentials, however, auxiliary networks are required. These networks provide additive terms to the variable conductances in accordance with current notions on synaptic transmission [(19)]. Two types of networks have been used. In both, the inputs are simulated presynaptic spikes, and in both the outputs are the resulting simulated chemical transmitter concentration. In both, the transmitter substance was assumed to be injected at a constant rate during a presynaptic spike and subsequently inactivated in the presence of an enzyme. One network simulates a first-order chemical reaction, where the enzyme concentration is effectively constant. The other simulates a second-order chemical reaction, where the enzyme concentration is assumed to be reduced during the inactivation process. For simulation of an excitatory synapse, the output of the auxiliary network is added directly to GNa in the electronic model. For inhibition, it is added to GK. With the parameters of the electronic membrane model set at the values measured by Hodgkin and Huxley, we have attempted to simulate synaptic activity with the aid of the two types of auxiliary networks. In the case of the simulated first-order reaction, the excitatory synapse exhibits facilitation, antifacilitation, or neither—depending on the setting of a single parameter, the transmitter inactivation rate (i.e., the effective enzyme concentration). This parameter would appear, in passing, to be one of the most probable synaptic variables. In this case, the mechanisms for facilitation and antifacilitation are contained in the simulated postsynaptic membrane. Facilitation is due to the nonlinear dependence of GNa on membrane potential, while antifacilitation is due to inactivation of GNa. The occurrence of one form of response or the other is determined by the relative importance of the two mechanisms [(18)]. Grundfest [(20)] has mentioned both of these mechanisms as potentially facilitory and antifacilitory, respectively. The simulated inhibitory synapse with the first order input is capable of facilitation [(18)], but no antifacilitation has been observed. Again, the presence or absence of facilitation is determined by the inactivation rate.
With the simulated second-order reaction, both excitatory and inhibitory synapses exhibit facilitation. In this case, two facilitory mechanisms are present—one in the postsynaptic membrane and one in the nonconstant transmitter inactivation reaction. The active membrane currents can, in fact, be removed; and this system will still exhibit facilitation. With the second-order auxiliary network, the presence of excitatory facilitation, antifacilitation, or neither depends on the initial, or resting, transmitter inactivation rate. The synaptic behavior also depends parametrically on the simulated enzyme reactivation rate. Inhibitory antifacilitation can be introduced with either type of auxiliary network by limiting the simulated presynaptic transmitter supply.
Certain classes of aftereffects are inherent in the mechanisms of the Ionic Hypothesis. In the electronic model, aftereffects are observed following presynaptic volleys with either type of auxiliary network. Following a volley of spikes into the simulated excitatory synapse, for example, rebound hyperpolarization may or may not occur depending on the simulated transmitter inactivation rate. If the inactivation rate is sufficiently high, rebound will occur. This rebound can be monophasic (inhibitory phase only) or polyphasic (successive cycles of excitation and inhibition). Following a volley of spikes into the simulated inhibitory synapse, rebound depolarization may or may not occur depending on the simulated transmitter inactivation rate. This rebound can also be monophasic or polyphasic. Sustained postexcitatory depolarization and sustained postinhibitory hyperpolarization [(2)] have been achieved in the model by making the transmitter inactivation rate sufficiently low.
The general forms of the postsynaptic potentials simulated with the electronic model are strikingly similar to those published in the literature for real neurons. The first-order auxiliary network produces facilitation of a form almost identical to that shown by Otani and Bullock [(8)] while the second-order auxiliary network produces facilitation of the type shown by Chalazonitis and Arvanitake [(2)]. The excitatory antifacilitation is almost identical to that shown by Hagiwara and Bullock [(1)] in both form and dependence on presynaptic spike frequency. In every case, the synaptic behavior is determined by the effective rate of transmitter inactivation, which in real neurons would presumably be directly proportional to the effective concentration of inactivating enzyme at the synapse.
Pacemaker potentials are easily simulated with the electronic model without the use of auxiliary networks. This is achieved either by inserting a large, variable shunt resistor across the simulated membrane ([see Figure 5]) or by allowing a small sodium current leakage at the resting potential. With the remaining parameters of the model set as close as possible to the values determined by Hodgkin and Huxley, the leakage current induces low-frequency, spontaneous spiking. The spike frequency increases monotonically with increasing leakage current. In addition, if the sodium conductance inactivation is allowed to accumulate over several spikes, periodic spike pairs and spike bursts will result. Subthreshold pacemaker potentials have also been observed in the model, but with parameter values set close to the Hodgkin-Huxley data these are generally higher in frequency than pacemaker potentials in real neurons. It is interesting that a pacemaker mode may exist in the absence of the simulated sodium conductance. It is a very high-frequency mode (50 cps or more) and results from the alternating dominance of potassium current and chloride (or leakage ion) current in determining the membrane potential. The significance of this mode cannot be assessed until better data is available for the potassium conductance at low levels of depolarization in real neurons. In general, as far as the model is concerned, pacemaker potentials are possible because the potassium conductance is delayed in both its rise with depolarization and its fall with repolarization.
Rate sensitive graded response has also been observed in the electronic model. The rate sensitivity—or accommodation—is due to the sodium conductance inactivation. The response of the model to an imposed ramp depolarization was discussed in [Reference 18]. At this time, several alternative model parameters could be altered to bring about reduced electrical excitability. None of the parameter changes was very satisfying, however, because none of them was in any way justified by physiological data. We have since found that the membrane capacitance, a plausible parameter in view of recent physiological findings, can completely determine the electrical excitability. Thus, with the capacitance determined by Hodgkin and Huxley (1 microfarad per cm²), the model exhibits excitability characteristic of the axon. As the capacitance is increased, the model becomes less excitable until, with 10 or 12 μμf, it is effectively inexcitable. Thus, with an increased capacitance—but with all the remaining parameters set as close as possible to the Hodgkin-Huxley values—the electronic model exhibits the characteristics of Bullock’s graded-response regions.