The Ionic Hypothesis and Neuron Models

E. R. Lewis

Librascope Group, General Precision, Inc.Research and Systems CenterGlendale, California

The measurements of Hodgkin and Huxley were aimed at revealing the mechanism of generation and propagation of the all-or-none spike. Their results led to the Modern Ionic Hypothesis. Since the publication of their papers in 1952, advanced techniques with microelectrodes have led to the discovery of many modes of subthreshold activity not only in the axon but also in the somata and dendrites of neurons. This activity includes synaptic potentials, local response potentials, and pacemaker potentials.We considered the question, “Can this activity also be explained in terms of the Hodgkin-Huxley Model?” To seek an answer, we have constructed an electronic analog based on the ionic hypothesis and designed around the data of Hodgkin and Huxley. Synaptic inputs were simulated by simple first-order or second-order networks connected directly to simulated conductances (potassium or sodium). The analog has, with slight parameter adjustments, produced all modes of threshold and subthreshold activity.

We considered the question, “Can this activity also be explained in terms of the Hodgkin-Huxley Model?” To seek an answer, we have constructed an electronic analog based on the ionic hypothesis and designed around the data of Hodgkin and Huxley. Synaptic inputs were simulated by simple first-order or second-order networks connected directly to simulated conductances (potassium or sodium). The analog has, with slight parameter adjustments, produced all modes of threshold and subthreshold activity.

In recent years physiologists have become quite adept at probing into neurons with intracellular microelectrodes. They are now able, in fact, to measure (a) the voltage change across the postsynaptic membrane elicited by a single presynaptic impulse (see, for examples, references1and2) and (b) the voltage-current characteristics across a localized region of the nerve cell membrane(3),(4),(5),(6). With microelectrodes, physiologists have been able to examine not only the all-or-none spike generating and propagating properties of axons but also the electrical properties of somatic and dendritic structures in individual neurons. The resulting observations have led many physiologists to believe that the individual nerve cell is a potentially complex information-processing system far removed from the simple two-state device envisioned by many early modelers. This new concept of the neuron is well summarized by Bullock in his 1959Sciencearticle(10). In the light of recent physiological literature, one cannot justifiably omit the diverse forms of somatic and dendritic behavior when assessing the information-processing capabilities of single neurons. This is true regardless of the means of assessment—whether one uses mathematical idealizations, electrochemical models, or electronic analogs. We have been interested specifically in electronic analogs of the neuron; and in view of the widely diversified behavior which we must simulate, our first goal has been to find a unifying concept about which to design our analogs. We believe we have found such a concept in the Modern Ionic Hypothesis, and in this paper we will discuss an electronic analog of the neuron which was based on this hypothesis and which simulated not only the properties of the axon but also the various subthreshold properties of the somata and dendrites of neurons.

We begin with a brief summary of the various types of subthreshold activity which have been observed in the somatic and dendritic structures of neurons. This is followed by a brief discussion of the Hodgkin-Huxley data and of the Modern Ionic Hypothesis. An electronic analog based on the Hodgkin-Huxley data is then introduced, and we show how this analog can be used to provide all of the various types of somatic and dendritic activity.

In studying the recent literature in neurophysiology, one is immediately struck by the diversity in form of both elicited and spontaneous electrical activity in the single nerve cell. This applies not only to the temporal patterns of all-or-none action potentials but also to the graded somatic and dendritic potentials. The synaptic membrane of a neuron, for example, is often found to be electrically inexcitable and thus incapable of producing an action potential; yet the graded, synaptically induced potentials show an amazing diversity in form. In response to a presynaptic impulse, the postsynaptic membrane may become hyperpolarized (inhibitory postsynaptic potential), depolarized (excitatory postsynaptic potential), or remain at the resting potential but with an increased permeability to certain ions (a form of inhibition). The form of the postsynaptic potential in response to an isolated presynaptic spike may vary from synapse to synapse in several ways, as shown inFigure 1. Following a presynaptic spike, the postsynaptic potential typically rises with some delay to a peak value and then falls back toward the equilibrium or resting potential. Three potentially important factors are the delay time (synaptic delay), the peak amplitude (spatial weighting of synapse), and the rate of fall toward the equilibrium potential (temporal weighting of synapse). The responses of a synapse to individual spikes in a volley may be progressively enhanced (facilitation), diminished (antifacilitation), or neither(1),(2),(7),(8). Facilitation may be in the form of progressively increased peak amplitude, or in the form of progressively decreased rate of fall (see Figure 2). The time course and magnitude of facilitation or antifacilitation may very well be important synaptic parameters. In addition, the postsynaptic membrane sometimes exhibits excitatory or inhibitory aftereffects (or both) on cessation of a volley of presynaptic spikes(2),(7); and the time course and magnitude of the aftereffects may be important parameters. Clearly, even if one considers the synaptic potentials alone, he is faced with an impressive variety of responses. Examples of the various types of postsynaptic responses may be found in the literature, but for purposes of the present discussion the idealized wave forms inFigure 2will demonstrate the diversity of electrical behavior with which one is faced.

Figure 1—Excitatory postsynaptic potentials in response to a single presynaptic spike

Figure 2—Idealized postsynaptic potentials

In addition to synaptically induced potentials, low-frequency, spontaneous potential fluctuations have been observed in many neurons(2),(7),(9),(10),(11). These fluctuations, generally referred to as pacemaker potentials, are usually rhythmic and may be undulatory or more nearly saw-toothed in form. The depolarizing phase may be accompanied by a spike, a volley of spikes, or no spikes at all. Pacemaker frequencies have been noted from ten or more cycles per second down to one cycle every ten seconds or more. Some idealized pacemaker wave forms are shown inFigure 3.

Figure 3—Idealized pacemaker potentials

Figure 4—Graded response

Bullock(7),(10),(12),(13)has demonstrated the existence of a third type of subthreshold response, which he calls the graded response. While the postsynaptic membrane is quite often electrically inexcitable, other regions of the somatic and dendritic membranes appear to be moderately excitable. It is in these regions that Bullock observes the graded response. If one applies a series of pulsed voltage stimuli to the graded-response region, the observed responses would be similar to those shown inFigure 4A. Plotting the peak response voltage as a function of the stimulus voltage would result in a curve similar to that inFigure 4B(see Ref. 3, page 4).For small values of input voltage, the response curve is linear; the membrane is passive. As the stimulus voltage is increased, however, the response becomes more and more disproportionate. The membrane is actively amplifying the stimulus potential. At even higher values of stimulus potential, the system becomes regenerative; and a full action potential results. The peak amplitude of the response depends on the duration of the stimulus as well as on the amplitude. It also depends on the rate of application of the stimulus voltage. If the stimulus potential is a voltage ramp, for example, the response will depend on the slope of the ramp. If the rate of rise is sufficiently low, the membrane will respond in a passive manner to voltages much greater than the spike threshold for suddenly applied voltages. In other words, the graded-response regions appear to accommodate to slowly varying potentials.

In terms of functional operation, we can think of the synapse as a transducer. The input to this transducer is a spike or series of spikes in the presynaptic axon. The output is an accumulative, long-lasting potential which in some way (perhaps not uniquely) represents the pattern of presynaptic spikes. The pacemaker appears to perform the function of a clock, producing periodic spikes or spike bursts or producing periodic changes in the over-all excitability of the neuron. The graded-response regions appear to act as nonlinear amplifiers and, occasionally, spike initiators. The net result of this electrical activity is transformed into a series of spikes which originate at spike initiation sites and are propagated along axons to other neurons. The electrical activity in the neuron described above is summarized in the following outline (taken in part fromBullock (7)):

Hodgkin, Huxley, and Katz(3)and Hodgkin and Huxley(14),(15),(16), in 1952, published a series of papers describing detailed measurements of voltage, current, and time relationships in the giant axon of the squid (Loligo). Hodgkin and Huxley(17)consolidated and formalized these data into a set of simultaneous differential equations describing the hypothetical time course of events during spike generation and propagation. The hypothetical system which these equations describe is the basis of the Modern Ionic Hypothesis.

The system proposed by Hodgkin and Huxley is basically one of dynamic opposition of ionic fluxes across the axon membrane. The membrane itself forms the boundary between two liquid phases—the intracellular fluid and the extracellular fluid. The intracellular fluid is rich in potassium ions and immobile organic anions, while the extracellular fluid contains an abundance of sodium ions and chloride ions. The membrane is slightly permeable to the potassium, sodium, and chloride ions; so these ions tend to diffuse across the membrane. When the axon is inactive (not propagating a spike), the membrane is much more permeable to chloride and potassium ions than it is to sodium ions. In this state, in fact, sodium ions are actively transported from the inside of the membrane to the outside at a rate just sufficient to balance the inward leakage. The relative sodium ion concentrations on both sides of the membrane are thus fixed by the active transport rate, and the net sodium flux across the membrane is effectively zero. The potassium ions, on the other hand, tend to move out of the cell; while chloride ions tend to move into it. The inside of the cell thus becomes negative with respect to the outside. When the potential across the membrane is sufficient to balance the inward diffusion of chloride with an equal outward drift, and the outward diffusion of potassium with an inward drift (and possibly an inward active exchange), equilibrium is established. The equilibrium potential is normally in the range of 60 to 65 millivolts.

The resting neural membrane is thus polarized, with the inside approximately 60 millivolts negative with respect to the outside. Most of the Hodgkin-Huxley data is based on measurements of the transmembrane current in response to an imposed stepwise reduction (depolarization) of membrane potential. By varying the external ion concentrations, Hodgkin and Huxley were able to resolve the transmembrane current into two “active” components, the potassium ion current and the sodium ion current. They found that while the membranepermeabilities to chloride and most other inorganic ions were relatively constant, the permeabilities to both potassium and sodium were strongly dependent on membrane potential. In response to a suddenly applied (step) depolarization, the sodium permeability rises rapidly to a peak and then declines exponentially to a steady value. The potassium permeability, on the other hand, rises with considerable delay to a value which is maintained as long as the membrane remains depolarized. The magnitudes of both the potassium and the sodium permeabilities increase monotonically with increasing depolarization. A small imposed depolarization will result in an immediately increased sodium permeability. The resulting increased influx of sodium ions results in further depolarization; and the process becomes regenerative, producing the all-or-none action potential. At the peak of the action potential, the sodium conductance begins to decline, while the delayed potassium conductance is increasing. Recovery is brought about by an efflux of potassium ions, and both ionic permeabilities fall rapidly as the membrane is repolarized. The potassium permeability, however, falls less rapidly than that of sodium. This is basically the explanation of the all-or-none spike according to the Modern Ionic Hypothesis.

Figure 5—Hodgkin-Huxley representation of small area of axon membrane

Figure 6—Typical responses of sodium conductance and potassium conductance to imposed step depolarization

By defining the net driving force on any given ion species as the difference between the membrane potential and the equilibrium potential for that ion and describing permeability changes in terms of equivalent electrical conductance changes, Hodgkin and Huxley reduced the ionicmodel to the electrical equivalent inFigure 5. The important dynamic variables in this equivalent network are the sodium conductance(G{Na})and the potassium conductance(G{K}). The change in the sodium conductance in response to a step depolarization is shown inFigure 6B. This change can be characterized by seven voltage dependent parameters:

1. Delay time—generally much less than 1 msec2. Rise time—1 msec or less3. Magnitude of peak conductance—increases monotonically with increasing depolarization4. Inactivation time constant—decreases monotonically with increasing depolarization.5. Time constant of recovery from inactivation—incomplete data6. Magnitude of steady-state conductance—increases monotonically with increasing depolarization7. Fall time on sudden repolarization—less than 1 msec.

1. Delay time—generally much less than 1 msec

2. Rise time—1 msec or less

3. Magnitude of peak conductance—increases monotonically with increasing depolarization

4. Inactivation time constant—decreases monotonically with increasing depolarization.

5. Time constant of recovery from inactivation—incomplete data

6. Magnitude of steady-state conductance—increases monotonically with increasing depolarization

7. Fall time on sudden repolarization—less than 1 msec.

Figure 6Bshows the potassium conductance change in response to an imposed step depolarization. Four parameters are sufficient to characterize this response:

1. Delay time—decreases monotonically with increasing depolarization2. Rise time—decreases monotonically with increasing depolarization3. Magnitude of steady-state conductance—increases monotonically with increasing depolarization4. Fall time on sudden repolarization—8 msec or more, decreases slightly with increasing depolarization.

1. Delay time—decreases monotonically with increasing depolarization

2. Rise time—decreases monotonically with increasing depolarization

3. Magnitude of steady-state conductance—increases monotonically with increasing depolarization

4. Fall time on sudden repolarization—8 msec or more, decreases slightly with increasing depolarization.

In addition to the aforementioned parameters, the transient portion of the sodium conductance appears to exhibit an accommodation to slowly varying membrane potentials. The time constants of accommodation appear to be those of inactivation or recovery from inactivation—depending on the direction of change in the membrane potential(18). The remaining elements in the Hodgkin-Huxley model are constant and are listed below:

Figure 7—System diagram for electronic simulation of the Hodgkin-Huxley model

Given a suitable means of generating the conductance functions,GNa(v,t)andGK(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 avoltage 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 inFigure 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 isgenerally 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 toGNain the electronic model. For inhibition, it is added toGK. 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 ofGNaon membrane potential, while antifacilitation is due to inactivation ofGNa. The occurrence of one form of response or the other is determined by the relative importance of the two mechanisms(18). Grundfest(20)has mentionedboth 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 neuronswould 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 inReference 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 increasedcapacitance—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 Mooreet al.(23)inAplysiaand 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 behaviorand also all of the subthreshold somatic and dendritic behavior outlined onpage 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

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