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.
Whether membrane capacitance is the determining factor in real neurons is, of course, a matter of speculation. Quite a controversy is raging over membrane capacity measurements ([see Rall (21)]), but the evidence indicates that the capacity in the soma is considerably greater than that in the axon [(6)], [(22)].
It should be added that increasing the capacitance until the membrane model becomes inexcitable has little effect on the variety of available simulated synaptic responses. Facilitation, antifacilitation, and rebound are still present and still depend on the transmitter inactivation rate. Thus, in the model, we can have a truly inexcitable membrane which nevertheless utilizes the active membrane conductances to provide facilitation or antifacilitation, and rebound. The simulated subthreshold pacemaker potentials are much more realistic with the increased capacitance, being lower in frequency and more natural in form.
In one case, the electronic model predicted behavior which was subsequently reported in real neurons. This was in respect to the interaction of synaptic potentials and pacemaker potential. It was noted in early experiments that when the model was set in a pacemaker mode, and periodic spikes were applied to the simulated inhibitory synapse, the pacemaker frequency could be modified; and, in fact, it would tend to lock on to the stimulus frequency. This produced a paradoxical effect whereby the frequency of spontaneous spikes was actually increased by increasing the frequency of inhibitory synaptic stimuli. At very low stimulus frequencies, the spontaneous pacemaker frequency was not appreciably perturbed. As the stimulus frequency was increased, and approached the basic pacemaker frequency, the latter tended to lock on and follow further increases in the stimulus frequency. When the stimulus frequency became too high for the pacemaker to follow, the latter decreased abruptly in frequency and locked on to the first subharmonic. As the stimulus frequency was further increased, the pacemaker frequency would increase, then skip to the next harmonic, then increase again, etc. This type of behavior was observed by Moore et al. [(23)] in Aplysia and reported at the San Diego Symposium for Biomedical Electronics shortly after it was observed by the author in the electronic model.
Thus, we have shown that an electronic analog with all parameters except membrane capacitance fixed at values close to those of Hodgkin and Huxley, can provide all of the normal threshold or axonal behavior and also all of the subthreshold somatic and dendritic behavior outlined on [page 7]. Whether or not this is of physiological significance, it certainly provides a unifying basis for construction of electronic neural analogs. Simple circuits, based on the Hodgkin-Huxley model and providing all of the aforementioned behavior, have been constructed with ten or fewer inexpensive transistors with a normal complement of associated circuitry [(18)]. In the near future we hope to utilize several models of this type to help assess the information-processing capabilities not only of individual neurons but also of small groups or networks of neurons.
REFERENCES
| 1. | Hagiwara, S., and Bullock, T. H. |
| “Intracellular Potentials in Pacemaker and Integrative Neurons of the Lobster Cardiac Ganglion,” | |
| J. Cell and Comp. Physiol. 50 (No. 1):25-48 (1957) | |
| 2. | Chalazonitis, N., and Arvanitaki, A., |
| “Slow Changes during and following Repetitive Synaptic Activation in Ganglion Nerve Cells,” | |
| Bull. Inst. Oceanogr. Monaco No. 1225:1-23 (1961) | |
| 3. | Hodgkin, A. L., Huxley, A. F., and Katz, B., |
| “Measurement of Current-Voltage Relations in the Membrane of the Giant Axon of Loligo,” | |
| J. Physiol. 116:424-448 (1952) | |
| 4. | Hagiwara, S., and Saito, N., |
| “Voltage-Current Relations in Nerve Cell Membrane of Onchidium verruculatum,” | |
| J. Physiol. 148:161-179 (1959) | |
| 5. | Hagiwara, S., and Saito, N., |
| “Membrane Potential Change and Membrane Current in Supramedullary Nerve Cell of Puffer,” | |
| J. Neurophysiol. 22:204-221 (1959) | |
| 6. | Hagiwara, S., |
| “Current-Voltage Relations of Nerve Cell Membrane,” | |
| “Electrical Activity of Single Cells,” | |
| Igakushoin, Hongo, Tokyo (1960) | |
| 7. | Bullock, T. H., |
| “Parameters of Integrative Action of the Nervous System at the Neuronal Level,” | |
| Experimental Cell Research Suppl. 5:323-337 (1958) | |
| 8. | Otani, T., and Bullock, T. H., |
| “Effects of Presetting the Membrane Potential of the Soma of Spontaneous and Integrating Ganglion Cells,” | |
| Physiological Zoology 32 (No. 2):104-114 (1959) | |
| 9. | Bullock, T. H., and Terzuolo, C. A., |
| “Diverse Forms of Activity in the Somata of Spontaneous and Integrating Ganglion Cells,” | |
| J. Physiol. 138:343-364 (1957) | |
| 10. | Bullock, T. H., |
| “Neuron Doctrine and Electrophysiology,” | |
| Science 129 (No. 3355):997-1002 (1959) | |
| 11. | Chalazonitis, N., and Arvanitaki, A., |
| “Slow Waves and Associated Spiking in Nerve Cells of Aplysia,” | |
| Bull. Inst. Oceanogr. Monaco No. 1224:1-15 (1961) | |
| 12. | Bullock, T. H., |
| “Properties of a Single Synapse in the Stellate Ganglion of Squid,” | |
| J. Neurophysiol. 11:343-364 (1948) | |
| 13. | Bullock, T. H., |
| “Neuronal Integrative Mechanisms,” | |
| “Recent Advances in Invertebrate Physiology,” | |
| Scheer, B. T., ed., Eugene, Oregon:Univ. Oregon Press 1957 | |
| 14. | Hodgkin, A. L., and Huxley, A. F., |
| “Currents Carried by Sodium and Potassium Ions through the Membrane of the Giant Axon of Loligo,” | |
| J. Physiol. 116:449-472 (1952) | |
| 15. | Hodgkin, A. L., and Huxley, A. F., |
| “The Components of Membrane Conductance in the Giant Axon of Loligo,” | |
| J. Physiol. 116:473-496 (1952) | |
| 16. | Hodgkin, A. L., and Huxley, A. F., |
| “The Dual Effect of Membrane Potential on Sodium Conductance in the Giant Axon of Loligo,” | |
| J. Physiol. 116:497-506 (1952) | |
| 17. | Hodgkin, A. L., and Huxley, A. F., |
| “A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve,” | |
| J. Physiol. 117:500-544 (1952) | |
| 18. | Lewis, E. R., |
| “An Electronic Analog of the Neuron Based on the Dynamics of Potassium and Sodium Ion Fluxes,” | |
| “Neural Theory and Modeling,” | |
| R. F. Reiss, ed., Palo Alto, California:Stanford University Press, 1964 | |
| 19. | Eccles, J. C., |
| Physiology of Synapses, | |
| Berlin:Springer-Verlag, 1963 | |
| 20. | Grundfest, H., |
| “Excitation Triggers in Post-Junctional Cells,” | |
| “Physiological Triggers,” | |
| T. H. Bullock, ed., Washington, D.C.:American Physiological Society, 1955 | |
| 21. | Rall, W., |
| “Membrane Potential Transients and Membrane Time Constants of Motoneurons,” | |
| Exp. Neurol. 2:503-532 (1960) | |
| 22. | Araki, T., and Otani, T., |
| “The Response of Single Motoneurones to Direct Stimulation,” | |
| J. Neurophysiol. 18:472-485 (1955) | |
| 23. | Moore, G. P., Perkel, D. H., and Segundo, J. P., |
| “Stability Patterns in Interneuronal Pacemaker Regulation,” | |
| Proceedings of the San Diego Symposium for Biomedical Engineering, | |
| San Diego, California, 1963 | |
| 24. | Eccles, J. C., |
| The Neurophysiological Basis of Mind, | |
| Oxford:Clarendon Press, 1952 |
Fields and Waves in Excitable
Cellular Structures
R. M. STEWART
Space General Corporation
El Monte, California
“Study of living processes by the physiological method only proceeded laboriously behind the study of non-living systems. Knowledge about respiration, for instance, began to become well organized as the study of combustion proceeded, since this is an analogous operation....”