[Illustration: Figure 7—System diagram for electronic simulation of the
Hodgkin-Huxley model]
Given a suitable means of generating the conductance functions,
G_{Na}(v,t) and G_{K}(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 G_{Na} in the
electronic model. For inhibition, it is added to G_{K}. 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 G_{Na} on membrane potential, while
antifacilitation is due to inactivation of G_{Na}. 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.