We construct an electronic circuit for mimicking a single neuron's behavior in connection with the dynamics of the Hodgkin–Huxley mathematical model. Our results show that the electronic neuron, even though it contains binary-state circuitry components, displays a timing interplay between the ion channels, which is consistent with the corresponding timing encountered in the model equations. This is at the core of the mechanism determining not only the creation of action potentials but also the neuronal firing rate output. This work is suitable for educational purposes in physics, mathematical modeling, electronics, and neurophysiology and can be extended for implementation in networked neurons for more advanced studies of neuronal behaviors.

## I. INTRODUCTION

Neuroscience has benefited enormously from collaborative work in a variety of fields, in part because the building block of neurological systems, the neuron, happens to be a biological cell exhibiting physical, electrical, thermal, chemical, mechanical, and physiological properties. The neuron as a living cell performs a multitude of functions, all fundamentally dependent upon the capability of its permeable membrane to control in- and out-flows of ions. The resulting variable voltage across its membrane is the neuron's most valuable asset, with the action potential, shown in Fig. 1, as the bit of information neurons use to communicate. While performing experiments on the living neuron provides a wealth of information on how neurons work, the inclusion of physical and mathematical descriptions of the cell's behavior may help elucidate phenomena of difficult to understand from the experimental measurements alone. Moreover, developing equivalent physics-based electronic models with both experimental and computational outputs can be invaluable, not only from the scientific point of view but also from the inherently pedagogical opportunities the work offers.

In this manuscript, we revisit a silicon-based realization of the real biological neuron, along with results from computer simulations performed with mathematical equations developed by Alan L. Hodgkin and Andrew F. Huxley in the early 1950s.^{1} We describe the experimental electronic neuron^{2} in parallel with the computational work, demonstrating the implementation of the hardware counterparts of the biological neuron. Our study provides insights from the best of both experimental and computational approaches and can be used, with adequate adjustments, for teaching concepts in physics, electronics, mathematical modeling, and neurophysiology, in a wide range of levels.

## II. BACKGROUND

Mathematical modeling of neurons has been largely based on equations describing the physics of action potential generation across the cell membrane and the equivalent electronic circuitry. In the case of the Hodgkin and Huxley results, two defining factors made the realization of their model possible: (a) the selection of the squid giant axon, with a diameter about 0.5 mm, large enough for placing two electrodes inside the cell,^{3} and (b) the development of the voltage clamp technique allowing the transmembrane voltage to be held at a set level during the recording of experimental data. This technique, developed by the biophysicist Kenneth C. Cole,^{4,5} enabled Hodgkin and Huxley to obtain the membrane conductance changes, which happen at different membrane potentials, from the measurement of currents flowing across the membrane. They proposed that the electrical behavior of the membrane could be mimicked by the circuitry shown in Fig. 2, where the top graph depicts a representation of the membrane with three ion channels (leak, potassium, and sodium) and the corresponding circuitry displayed at the bottom of the same (Fig. 2).

In this representation, different concentrations of ions inside and outside of the cell correspond to the voltage across the plates of a capacitor with capacitance *C*, and flows of ions through the specific channels correspond to electric currents. We use the inverse resistance or conductance $ g = 1 / R$ in units of Siemens = 1/Ω. The flows controlled by these conductances, in parallel with the capacitor, correspond to the ionic currents. The ionic current is split into sodium ions (*I*_{Na}), potassium ions (*I*_{K}), and others including chloride ions all grouped together in the leak current ( $ I \u2113$). Hodgkin and Huxley conjectured that each of these three ionic currents would be determined by a driving force measurable as a voltage difference and a conductance. In this way, the sodium current, for example, would be given by the sodium channel conductance multiplied by the difference between the membrane potential and the sodium equilibrium potential, $ I Na= g Na(V\u2212 V Na)$, and likewise for the potassium $ I K= g K(V\u2212 V K)$ and leak $ I L = g L ( V \u2212 V L )$ currents. They demonstrated that the rising of the action potential was mostly associated with an increase in the conductance of sodium channels, while the dropping of the action potential was mostly associated with an increase in the conductance of potassium channels, consistent with the sodium ions flowing into the cell and the potassium ions flowing out of the cell.

An important aspect of Hodgkin and Huxley's work was that the changes they observed in the cell membrane permeability appeared to be dependent on the membrane potential, not on the membrane current. Based on their measurements, they therefore suggested that the electrical activity across the cell membrane could be represented by the electrical circuit shown in Fig. 2 (bottom), composed basically of a capacitor in parallel with three dynamic resistors (or conductors) along with their corresponding voltage sources.

## III. MODEL EQUATIONS

Action potentials are defined as a rapid (∼2 ms duration) upward spike on the membrane voltage from −70 mV to 40 mV, followed by a drop to −90 mV, and then a rise back to −70 mV (Fig. 1). As described in Sec. II, sodium and potassium ions are the most prominent contributors for the generation of action potentials. For this reason, Hodgkin and Huxley focused their differential analysis on these two types of ion channels and captured the effect of the remaining ion channels as a single time-independent conductance, the leak channel.

^{6,7}For example, the concentration of potassium ions inside the cell is higher than outside, leading to a negative equilibrium potential for potassium. In the case of sodium, the outside concentration of ions is higher than inside the cell, in this case, leading to a positive equilibrium potential for sodium. The variable resistors for these two paths in the circuit account for the fact that the amount of current that flows through those channels is time-dependent. Also, the values of the conductances depend on the membrane potential at a given moment in time. All these considerations are imbedded in the mathematical framework of the Hodgkin–Huxley neuronal model with four coupled ordinary differential equations,

^{1}as shown below

These differential equations are for the voltage difference *V*(*t*) between the potentials inside and outside of the cell in Eq. (1), the activation variable *m*(*t*) for sodium in Eq. (2), the inactivation variable *h*(*t*) for sodium in Eq. (3), and the activation variable *n*(*t*) for potassium in Eq. (4), with the parameter values of Eq. (1) shown in Appendix A, and the steady state values for the functions in Eqs. (2)–(4) shown in Appendix B. We solve the system of differential equations numerically using the standard Runge–Kutta fourth order method. These mathematical equations have been extended and applied to numerous studies^{8–12} and have stimulated an enormous amount of research. The model is an important landmark, being extremely influential in advancing the field of neuroscience.^{13,14}

## IV. MATERIALS AND METHODS

The electronic neuron in this work is based on a design developed by Sitt and Aliaga^{2} and is schematically depicted in Fig. 3, where the boxes for *R*_{Na} and *R*_{K} contain the electronic implementations of the sodium and potassium ion channels, respectively, detailed in Fig. 4 (top and bottom, respectively). The field-effect transistors (FETs) in the figure are adjusted to mimic the gates for the sodium channel activation (FET1), for the sodium channel inactivation (FET2), and the potassium channel activation (FET3). While no exclusions of the design by Sitt and Aliaga^{2} were made here, there were a few pieces of information, which were found to be missing in their presentation, including the values for the three reference voltages that feed into comparators before providing an input to the three corresponding FETs. Our construction of the circuitry therefore included some trial-and-error in figuring out viable values for these reference voltages, which we found to be $ V ref \u2212 Na 1\u22483.3$ V, $ V ref \u2212 Na 2\u22482.7$ V, and $ V ref \u2212 K\u22482.7$ V. These values for the reference voltages elicited the Hodgkin–Huxley-like dynamics we were looking for in the circuit.

Figure 5 shows the circuit used to provide the user-modified reference voltages. It includes a resistive divider to allow the use of positive feedback to alter the output of this circuit in order to introduce hysteresis at the input of a comparator (see the text below).

Another difficulty encountered in the setup was the substantial noise level^{15–17} generated by the output of FET3, as labeled in Fig. 4 (bottom). In fact, noise caused a significant fraction of the actual oscillations of the output and thus contributed highly to a poor membrane potential output. It was found that the comparator that fed into FET3 was reacting to the minute oscillations at its input. To amend this, we introduced hysteresis via part of a conventional positive feedback circuit outlined in an application note by National Semiconductor Corporation for the LM311 comparator that was being used. Doing so caused FET3s' output to be much more stable, resulting in a smoother trajectory for the time evolution of the membrane potential. If larger noise values are encountered, one can also use an ordinary op-amp as a comparator in the Schmitt trigger configuration,^{18} which allows for larger hysteresis (Fig. 6) .

## V. RESULTS AND DISCUSSION

Numerical solutions of the Hodgkin–Huxley model equations (1) through (4) yield time series for the corresponding four variables, with graphical representations shown, in the respective sequence, in Fig. 7. Similarly, the corresponding electronic neuron circuit produces a voltage output at the circuit site equivalent to a cell membrane, via the relative timings of changes between on- and off-states for the three FETs, as indicated in Fig. 4. The time evolution of the voltage across the electronic membrane shows the same overall features seen in the time evolution of the voltage across the membrane for the computational neuron. Equivalently, the transitions of the three FETs reproduce the relative timings of transitions for the activation/inactivation variables *n*, *m*, and *h* in the computational neuron. These timings are critical for the generation of action potentials since they control the dynamics of the ion channels for the in- and out-flow of ions across the membrane. The electronic outputs equivalent to the four traces shown in Fig. 7 are shown in Fig. 8.

In Fig. 9, we show a zoomed in graph of the superimposed computational variables *V*, *n*, *m*, and *h* (top) and the corresponding zoomed in graph of the superimposed electronic outputs (bottom). As the labels along with the distinct colors indicate in both top and bottom graphs of Fig. 9, the rise of the membrane voltage for the generation of an action potential (dotted blue line) starts with a rise in the sodium activation variable (green), a low value for the sodium inactivation variable, and also a downward trend for the potassium activation variable (red). At the point where the sodium activation variable (green) reaches a maximum, so does the action potential (dotted blue), and as the sodium activation trends downward, so does the action potential, concomitantly with the potassium activation reversing its trend to upward, followed by the same in the sodium inactivation. At the point of minimum for the action potential so is the case for the sodium activation, while the sodium inactivation has already changed its trend to upward and the potassium activation is already on a downward trend. The timing consistency we observe in our study between the numerical and the electronic outputs demonstrates the intricate dynamics taking place in the interplay between in- and out-flows of sodium and potassium ions across the neuronal membrane. These results are a central takeaway from the dual-approach of this study.

It is natural to establish a qualitative analogy between the Hodgkin–Huxley equations and the electronic model behaviors. Therefore, a comparison is made between the relative timings of the solutions to the Hodgkin–Huxley equations and their counterparts in the electronic neuron. One of these corresponding pairs is a direct voltage–voltage relationship between the membrane potential of the model neuron and that of the electronic neuron. The other three equations of the model, however, give variably sloped curves for the probabilities of activation or inactivation. The electronic neuron's analogs to these are FET outputs, which are binary by construction.

As much as the design of the electronic neuron is based on the dynamics contained in the Hodgkin–Huxley model equations, fundamental differences in scope and dimensionality do exist between the two approaches. The Hodgkin–Huxley output is the result of a numerical simulation biologically motivated and mathematically designed according to experimental measurements performed on real neurons. Furthermore, the model equations' variables *n*, *m*, and *h* are directly associated with probabilities considering the percentage of channels that are open in each case for the in- or out-flow of any specific type of ions, at any given time, in a continuous fashion. The electronic circuit representation, however, is a construct that operates on digital timescales and is not a system from which we can acquire anything but traditional physical quantities, like voltage and resistance, for example.

Furthermore, the electronic spike is generated by the charging of the membrane capacitor in response to a square-pulse stimulus, yielding the usual downward concavity of $ 1 \u2212 exp \u2009 ( \u2212 t / R C )$, where the experimental value used for the capacitor *C* was $ 0.047 \mu $ F and *R* is the parallel combination of the input capacitor (39 kΩ), the two resistors in the potassium channel (330 kΩ and 47 kΩ) and one of the resistors in the sodium channel (47 kΩ). The 330 kΩ resistor in the sodium channel has been bypassed by the opening of both FET1 and FET2 in that circuit (Fig. 4, top), which occurs very soon after the input pulse begins for the chosen reference potentials. This parallel combination gives an effective resistance of 20.2 kΩ, and a fit to the charging portion of the experimental pulse agrees with this value within 3%. The upward concavity observed in the Hodgkin-Huxley model spike results, basically, from the interplay between the smooth sigmoidal functions associated with the activation and inactivation functions of the ion channels. While it is feasible to reproduce such behaviors electronically, it would require a more elaborate experimental setting, beyond the scope of this work.

In Fig. 8, the three lower traces show the relative timings of the FET outputs. When compared to the timings of K activation, Na activation, and Na inactivation (three lower traces in Fig. 7), it can be seen that there are many similarities. For example, in both Figs. 7 and 8, the maxima of the Na activation trace (green) in both plots are positioned just prior to the minima of Na inactivation trace (orange) and also prior to the maxima of the K trace (blue).

An additional measure of the one-to-one connection between the model and the electronic neurons is displayed in Fig. 9, where the top graph shows the outputs for the model neuron and the bottom graph shows the outputs for the electronic neuron. The relative timings of the four traces in each of the two graphs do match, as they should.

## VI. CONCLUDING REMARKS

Modern-day electronics belong to a class of human achievement that is grounded in a robust ability to harness science at relatively small scales. This allows the design of electronic neurons where components play the role of fundamental features encountered in the biological neuron, and while not an exact replica of the biological cell, its electronic representation offers more accessibility, control, and flexibility than the real neuron in a petri dish. Our work on the electronic implementation of the Hodgkin–Huxley equations takes into account the dynamical output of the biological neuron, mimicked by the mathematical equations developed by Hodgkin and Huxley. Their experimental measurements on the giant axon of the squid gave them the physiological insight needed to infer the complicated underlying dynamics of opening and closing of ion channels, for the appropriate generation of action potentials. The implementation of the electronic neuron presented here, in association with the Hodgkin–Huxley model equations, embodies a setting for cross-disciplinary education in physics, electronics,^{19,20} mathematical modeling,^{21} and neurophysiology.^{22} Many mathematical models of neurons exist that contain differential equations or sometimes stochastic differential equations. It is well known that the former are well suited for integration by intelligently designed circuits. If direct integration is not feasible, then it should certainly be likely that one could build networks that exhibit nonlinear dynamics similar to that of brain cells. The rapidly growing field of neuromorphic hardware,^{23} for example, can capitalize on any new knowledge, leading to the reduction of the extensive computational time needed to execute complex numerical simulations based on more realistic neuronal models. Therefore, it is possible that much can be learned about neurons, the information-processing unit of the brain, via analogy to their silicon-based cousins.

### APPENDIX A: PARAMETER VALUES FOR Eq. (1)

### APPENDIX B: VOLTAGE-DEPENDENT RATES AND TIMINGS FOR Eqs. (2)–(4)

## REFERENCES

*Neuroscience, Exploring the Brain*

*Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition*

*Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting*

*The Art of Electronics*