# Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability

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Contributed by Eve Marder, December 28, 2019 (sent for review September 23, 2019; reviewed by Alain Destexhe and Idan Segev)

## Significance

A long-standing issue in physiology is the robustness of cellular functions to variations in concentrations and kinetics of biomolecules. The robustness is believed to reflect functional relations between many underlying biological components. Here we study the relationship between sodium and potassium channel parameters and membrane excitability. We used the dynamic clamp, a computer-controlled system that essentially makes it possible to do simulations with biological cells and ionic channels. We established a real-time closed-loop interaction between a genetically controlled population of excitability-relevant ion channels and a low-dimensional mathematical description of excitability. The results provide insights into how robustness of excitability benefits from the variability of history-dependent time scales that ion channels display.

## Abstract

Excitability—a threshold-governed transient in transmembrane voltage—is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin–Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model’s sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in *Xenopus* oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin–Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.

The canonical Hodgkin–Huxley model of excitability (1) consists of four dynamical variables (membrane voltage and three protein state variables) and more than 10 parameters. Several of the parameters represent actual, measurable physical entities (membrane capacitance, ionic concentrations inside and outside the cell, and densities of ionic channel membrane proteins). Other parameters shape the six exponential functions relating membrane voltage to probabilities of transitions between protein states (2⇓–4). The considerable sensitivity of the model to parametric variations—especially protein densities and kinetics—is not on par with the robustness of many biological systems, as revealed in experiments showing a high degree of resilience to variation in values of measurable parameters (5⇓⇓–8). Since the Hodgkin–Huxley model is biophysically solid, it is generally assumed that its sensitivity reflects functional relationship between parameters (7). Hence, a low-dimensional representation of Hodgkin–Huxley parameter space within which the seemingly complicated and parameter-sensitive system becomes tractable would help to understand how a biological system could control its state and adapt to changes using a simple and physiologically relevant process.

Recently, a biophysically oriented parameterization of the Hodgkin–Huxley model was introduced (4), offering a framework for understanding control of excitability amid changes in protein densities and their kinetics. In the resulting phase diagram, the excitability status of a given Hodgkin–Huxley realization is determined by rational functions fully defined in terms of Hodgkin–Huxley parameters along two physiological dimensions: structural and kinetic. The structural dimension (denoted S) is a measure for the relative contribution of maximal exciting (i.e., sodium) conductance. The kinetic dimension (denoted K) is a measure of the relative contribution of restoring voltage-dependent rate functions that pull the membrane back to its hyperpolarized potential (i.e., closure of sodium channels and opening of potassium channels). Thus, a point in the S–K phase diagram represents many different possible sets of the model parameters, explicit instantiations of a Hodgkin–Huxley model that give rise to similar functional outcome. Examined in the S–K plane, the Hodgkin–Huxley model reveals order that is impossible to detect at higher-dimensional representations. The three different excitability statuses are clustered in phases: nonexcitable, excitable, and oscillating (depicted in Fig. 1, *Left*). Note that the S–K phase diagram is different from a phase space where each point depicts a unique membrane state of a given instantiation and where lines connecting such states depict phase portraits, a trajectory in time (e.g., refs. 9⇓⇓–12).

The present study aims to experimentally uncover the S–K phase diagram, with actual biological components rather than mathematical modeling. The challenge is this: By definition, a cell at a specific moment in time is an instantiation of one point in the S–K phase diagram, one set of parameters. It is therefore impossible to systematically cover the S–K plane of a given biological cell (neuron, cardiac myocyte, etc.) by manipulating the kinetic and structural features of its constituents, the ionic channel proteins. Collecting data from many individual cells of similar type would not help, because similar types have a tendency to be residents of same phase (e.g., cardiac myocytes are mostly oscillating, and cortical neurons are mostly excitable but not oscillatory). Here we face the challenge of systematically sampling the Hodgkin–Huxley S–K phase diagram by combining the established methodology of heterologous expression of channel proteins in *Xenopus* oocytes (13, 14) and hard real-time dynamic clamp (15⇓⇓–18). With this approach we experimentally reconstruct the first phase transition in the S–K diagram: the transition between the nonexcitable and excitable phases (depicted in Fig. 1, *Right*). Moreover, we show that directional “walk” within the S–K plane exposes hysteresis in the organization of the phase diagram, which we explain in terms of channel protein slow and activity-dependent gating that potentially enables control of excitability amid parametric variation (3, 4).

## Results and Discussion

The *Xenopus* oocyte protein expression system is frequently used in physiological studies of ionic channels; several fundamental studies of ion channel structure–function relations were made using this simple and experimentally elegant system (e.g., refs. 19⇓–21). For all practical purposes the oocyte is an ideal electrophysiological ghost: it is a large and spherical (i.e., isopotential) leaky capacitor, it does not express significant voltage-sensitive membrane conductances, and it readily expresses functional conductances following injection of ionic channel coding mRNAs. Wedding heterologous *Xenopus* oocyte expression with dynamic clamp makes it possible to split the system’s components between those that are biologically expressed in the oocyte membrane and those that are computationally expressed in the dynamic clamp algorithm. Fig. 2*A* demonstrates the efficacy of this approach in generating a biosynthetic excitable system: An oocyte is impaled by two sharp electrodes. Signals from the voltage measuring electrode are read by a real-time processor that calculates Hodgkin–Huxley sodium and potassium currents, feeding the sum of these currents back to the oocyte through the current injecting electrode. The oocyte contributes membrane capacitance and leak conductance, the linear components; the dynamic clamp algorithm contributes sodium and potassium voltage-dependent conductance, the nonlinear components. The nature of the system’s response (Fig. 2*A*, *Bottom*) depends on the Hodgkin–Huxley parameters implemented in the dynamic clamp algorithm.

Following Ori et al. (4), the structural dimension (S) is defined as

To experimentally construct an S–K phase diagram in the dynamically clamped *Xenopus* oocyte, a random list of *Methods*). The resulting phase diagrams of two experiments are presented in Fig. 2*B*. Similar to the theoretical phase diagram of Fig. 1, *Right*, the plot of Fig. 2*B* shows well-structured S–K planes with soft but relatively well-defined borders that separate nonexcitable from excitable phases (colors depict response amplitude clusters, classified to four bins indicated in horizontal color bars). In Fig. 2*B*, *Top*, the parameters of sodium and potassium conductance were taken from the Hodgkin–Huxley canonical model. In Fig. 2*B*, *Bottom*, the potassium conductance parameters are those of the Kv1.3 channel (22⇓–24).

To validate the reduction to an S–K space, we take the above experimental system a step further by relocating the voltage-dependent potassium conductance from the dynamic clamp algorithm into the biological domain (Fig. 3*A*). This is achieved by injection of mRNA that codes the voltage-dependent Kv1.3 potassium channel (25). Within a few days, the channels are extensively expressed in the oocyte membrane. As demonstrated in Fig. 3*A*, *Bottom*, upon activation of the dynamic clamp, excitability emerges with biological capacitance, leak, and potassium conductance, while the sodium conductance and its related kinetics are expressed computationally. (Our attempts to implement the inverse experimental condition, where sodium conductance is biologically expressed, did not succeed; the expression of sodium conductance was too weak to support full-blown excitability.)

Since there is no standard model for excitability with Kv1.3 conductance, we express the structural dimension in terms of actual maximal conductance normalized to membrane capacitance; thus, *B* and *Methods*). We take the voltage-dependent rate functions of the Kv1.3 conductance (22, 24) as reference to express the kinetic dimension; hence,

The phase diagrams of three different experiments are presented in Fig. 4, where S–K planes with well-defined borders that separate nonexcitable from excitable phases validate the reduction of the original Hodgkin–Huxley high-dimensional parameter space to the lower S–K phase diagram. Borders that separate nonexcitable from excitable phases in 12 different experiments are summarized in Fig. 4, *Bottom Right*.

The power of the parameterization may be further appreciated by observing multiple instantiations of the same S–K coordinate, demonstrating that the outcome is quite resilient to the actual set of parameters used to determine a given coordinate (Fig. 5). Even delicate response features (e.g., the poststimulus subthreshold depolarization in the rightmost traces of Fig. 5, *Bottom*), not accounted for in the construction of the theory (4), are nicely caught by the S–K coordinates.

Over the years since Hodgkin and Huxley presented their canonic model, physiologists have continued to explore the complex behavior of voltage-gated ion channels. In particular, not all transition rates are voltage-dependent, and their characteristic time scales span a wide range that extends from submilliseconds to minutes (26⇓⇓⇓–30). The results presented in Figs. 2, 4, and 5 are limited to the short, millisecond time scale; hence, slow activity-dependent effects could not be detected. However, slow channel protein gating and its impacts on response dynamics might be exposed by traveling in a directional manner within the S–K diagram; for instance, by moving up and down along a ramp within the diagram. The kinetics of Kv1.3 are particularly relevant in this context as they involve voltage and state-dependent transition rates and a mix of slow and fast reactions spanning a wide range of time scales (22, 24). Moreover, these kinetics of Kv1.3 were suggested to have significant impacts on excitability on longer time scales (31⇓⇓–34). Fig. 6, *Left*, shows membrane responses of a Kv1.3-expressing oocyte as a function of S–K coordinates in a directed walk within the diagram. A gradual (up and down ramp; total trajectory 450 s) change is implemented in *Left*, top left corner of the diagram), the oocyte responds to the stimulus with a self-propelled depolarization that becomes a fully blown action potential. At some point, as might be expected, the structural exciting force (S) is so high and the kinetic restoring force (K) is so low (0.95, 0.42, depicted by arrow in Fig. 6, *Right*) that the membrane cannot hyperpolarize back to resting potential and remains stuck in a depolarized, not excitable value. Upon return, clear hysteresis is revealed, reflecting recovery of the Kv1.3 from long-lasting inactivation. Fig. 6, *Right*, *Insets*, show two repetitions of the same ramp protocol in another oocyte, demonstrating reversibility of the hysteresis phenomenon. Such hysteresis is not seen with the Hodgkin–Huxley original model and is in line with reports pertaining to impacts of slow Kv1.3 inactivation on adaptive membrane excitability.

## Concluding Remarks

We live in a time marked by capacity to collect data at ever-increasing speed and resolution. As a result, it is tempting to use these data to construct numerical models of increased dimensionality, making them more and more biologically realistic. To avoid the fallacy of attributing importance to each and every measurable parameter, good practice combines methods that point to functional relations between parameters (35) and formulation of low-dimensional phase diagrams. However, reduction of dimensionality—a Via Regia to formal understanding—also comes at a price. In many cases it is a unidirectional path where measurables are abstracted and compressed to an extent that loses the explicit properties of the physiological data from the abstract representation. Consequently, once an abstract low-dimensional model is constructed, evaluation of impacts and subsequent incorporation of new biological features into the low-dimensional model become challenging, if at all possible.

Here we approached the problem by implementation of a methodology that has a long and successful history in membrane physiology: system identification using closed-loop control (i.e., voltage clamp, patch clamp, and dynamic clamp). We describe an experimental–theoretical hybrid, a framework enabling bidirectional real-time interaction between abstract low-dimensional representation and real biological entities. This is not a post hoc fitting procedure; rather, it is a live experiment where the impacts—of a given biological component—on the abstract low-dimension representation are identified by implementing a real-time closed loop design.

Specifically, combining dynamic clamp and heterologous expression of ionic channel proteins in *Xenopus* oocytes, we constructed an excitable system composed of a mix of biologically and computationally expressed components. This experimental configuration enabled systematic sampling of the Hodgkin–Huxley parameter space. The resulting phase diagram validates a theoretically proposed diagram (4).

A spectrum of Hodgkin–Huxley single-compartment representations exists, extending from concrete and computationally intensive Markov kinetic models of channel state transitions, to abstract models that are computationally efficient yet biophysically less realistic (10, 11, 36). The S–K phase diagram is situated in between, touching both ends. On one hand, its two dimensions are expressed in physiologically accessible parameters, and on the other hand, the two dimensions are intimately related to the abstract nonlinear oscillator inspired models with S linked to the cubic polynomial expression that provides fast positive feedback, and K is related to the recovery variable that introduces slow negative feedback. As such, the S–K phase diagram may serve as a common ground to relate various representations in this spectrum to each other.

The shape of the S–K phase diagram proposed in ref. 4 and experimentally constructed in this report suggests that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. Moreover, we show that the basic ingredients for such control—namely, memory and adaptation—are manifested in the phase diagram as a natural outcome of ion channel slow inactivation kinetics.

Many theoretical and experimental analyses show that the wide range of temporal scales involved in slow inactivation is sliced thinly to a degree effectively equivalent to a continuum of scales, indicative of the extensive network of configurations within which the channel protein may diffuse giving rise to slow activity-dependent gating and adaptive firing patterns (28, 32, 37⇓⇓⇓⇓⇓⇓–44). Indeed, slow activity-dependent gating was suggested as a means for maintenance and control of membrane excitability. Specifically, activity dependence of protein kinetics at relatively slow time scales, entailed by multiplicity of protein states, was pointed at as a general automatic and local means for stabilization of cellular function, independent of protein synthesis, and operates over a wide—minutes and beyond—range of time scales (3, 4, 45, 46). Thus, precisely because these ion channels do not have a single, fixed time constant encoded in their molecular structure but rather slide through multiple states, cells have a built-in mechanism to smoothly function over a larger range of firing patterns and voltages. A similar argument holds for the wide range of time scales contributed by the plethora of different Kv channels (25), which also can expand stable operating ranges. This partially mitigates the control problem that cells face: getting it right may not require the perfect match between channel numbers that might otherwise be necessary. Viewed from another angle, multiple states of channel inactivation and recovery from inactivation necessarily result in hysteresis, and the time scales of that hysteresis become a memory mechanism (32, 34, 38) so that cells can use it to keep track of their recent pattern of activity and inactivity. This again expands the time course over which patterns of activity can influence the way the cell responds to physiological inputs. Interestingly, we usually think of the fastest membrane events (action potentials) having little lasting effect on the cells in which they are seen; but, looking only at the fast voltage deflections such as action potentials hides the effects of the slower channel dynamics that influence future events.

It remains to be seen how far the approach described here may be used in system identification of excitable membranes more complicated than the minimal, two-conductance single-compartment Hodgkin–Huxley configuration. Certainly, cells that contain many different types of ion channels will show a range of time scales and history dependence (47). Developing intuition into how a given set of firing properties depends on conductance densities of many channels may require new kinds of principled dimensionality reduction to complement brute force numerical simulations.

## Methods

Clusters of *Xenopus* oocytes were kindly provided by N. Dascal’s laboratory (Tel Aviv University, Tel Aviv, Israel). Individual oocytes were separated from their clusters by standard mechanical and enzymatic treatment and kept at 18 to 20 °C overnight prior to mRNA injection. The mRNAs were prepared from Kv1.3 carrying vectors kindly provided by Alomone Labs (Jerusalem). Oocytes were allowed to express injected mRNA over 2 to 6 d before electrophysiological experiments. A two-electrode voltage clamp system (NPI TURBO TEC-03X) and a National Instruments board (NI 625x series) were used to control the experiments. A sufficiently short loop duration (40 μs) was achieved within Real Time Experiment Interface (RTXI; www.rtxi.org) environment implemented in CPP software. The sequence of an experiment was as follows: An oocyte situated in a perfusion chamber was impaled with two Agarose cushioned electrodes prepared as described elsewhere (14). The bath solution, under continuous perfusion, was composed of 96 mM NaCl, 2 mM KCl, 1 mM *B*). Leak conductance in Kv1.3-injected oocytes (

Three comments on difficulties associated with technical aspects of the experimental approach employed here are as follows: 1) Completion of a protocol such that S–K coordinates are properly characterized entails voltage clamp procedures for maximal conductance, leak, capacitance, and maintenance of relatively stable resting potential, while going through the S–K plane. Thus, stable electrophysiological settings are necessary for a typical experiment lasting ca. 30 min. In our hands, continuous superfusion of the bath medium and use of Agarose cushioned electrodes promoted such stability. 2) Another difficulty arises due to the hardware used. Sometimes, the real-time processor and/or the limits of the current injection device were not sufficient to catch up at high S and low K values, giving rise to ringing about the peak of the spike. We assume that advanced hardware can do better. 3) We were not able to implement an experimental condition where sodium conductance was biologically expressed, whereas potassium conductance is computationally expressed. In our hands, the expression of sodium conductance is too weak to support full-blown excitability. Higher expression would have introduced problems in both estimation of maximal conductance and dynamically clamping the fast sodium current, which might be solved by conducting the experiments at lower temperatures.

### Data Availability.

Dynamic clamp data files are archived in Mendeley Data (https://data.mendeley.com/datasets/72pv9sfxkw/1) (48), enabling construction of the phase diagrams described in the present manuscript.

## Acknowledgments

This work was partially supported by research grants from the Leir Foundation (E.M. and S.M.), the National Institutes of Health (E.M.), and the Israel Science Foundation (S.M.). The authors thank Leonid Oddesky, Tamar Galateano, and Yael Abuchatzera for technical support; Nathan Dascal’s group (Tel Aviv University) for generous supply of oocytes; Alomone Labs (Jerusalem) for channel carrying vectors; and Omri Barak, Erez Braun, Daniel Dagan, and Michele Giugliano for helpful comments.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: marom{at}technion.ac.il or marder{at}brandeis.edu.

Author contributions: E.M. and S.M. designed research; H.O., H.H., and S.M. performed research; S.M. analyzed data; and E.M. and S.M. wrote the paper.

Reviewers: A.D., Center National de la Recherche Scientifique; and I.S., Hebrew University.

The authors declare no competing interest.

Data deposition: Dynamic clamp data files are archived in Mendeley Data (https://data.mendeley.com/datasets/72pv9sfxkw/1).

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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