Cellular function given parametric variation in the Hodgkin and Huxley model of excitability

Abstract

How is reliable physiological function maintained in cells despite considerable variability in the values of key parameters of multiple interacting processes that govern that function? Here, we use the classic Hodgkin-Huxley formulation of the squid giant axon action potential to propose a possible approach to this problem. Although the full Hodgkin-Huxley model is very sensitive to fluctuations that independently occur in its many parameters, the outcome is in fact determined by simple combinations of these parameters along two physiological dimensions: structural and kinetic (denoted S and K, respectively). Structural parameters describe the properties of the cell, including its capacitance and the densities of its ion channels. Kinetic parameters are those that describe the opening and closing of the voltage-dependent conductances. The impacts of parametric fluctuations on the dynamics of the system-seemingly complex in the high-dimensional representation of the Hodgkin-Huxley model-are tractable when examined within the S-K plane. We demonstrate that slow inactivation, a ubiquitous activity-dependent feature of ionic channels, is a powerful local homeostatic control mechanism that stabilizes excitability amid changes in structural and kinetic parameters.

Keywords: dimensionality reduction; excitability; homeostasis; inactivation.

Fig. 1.

A collage of slightly modified images extracted from the 1952 original report of Hodgkin and Huxley (1). (A) The table indicates ranges of cellular-level structural parameters. (B) The graphs depict protein-level kinetic parameters, expressed as six transition rate functions superposed with data points. The shaded areas added to the graphs suggest that a linear scaling of transition rate functions captures most of the underlying variance: <${\alpha }_n\left(v\right)$> = [0.85,1.15], <${\beta }_n\left(v\right)$> = [0.7,1.3], <${\alpha }_m\left(v\right)$> = [0.8,1.3], <${\beta }_m\left(v\right)$> = [0.7,1.35], <${\alpha }_h\left(v\right)$> = [0.85,1.15], <${\beta }_h\left(v\right)$> = [0.8,1.5]. B is modified with permission from ref. .

Fig. 2.

Minor change in parameters significantly affects membrane excitability. Realizations (10,000) of a full Hodgkin–Huxley model; each realization is uniquely defined by a vector of 10 parameters, expressed in terms of their scaling relative to the values chosen by Hodgkin and Huxley. Responses are classified to three excitability statuses (different colors): excitable (2,225), nonexcitable (4,884), and oscillatory (2,891). Subsets of the results (200 for each excitability status) are presented in polar plots (A–C): Given a list of 10 scaling parameters, the value of each parameter is depicted along its own (angular) axis, and the entire vector is depicted as a line that connects the 10 scaling parameters. The standard Hodgkin–Huxley model would be a line passing through 1 for all scaling parameters. Mean vectors are depicted by dashed lines. The histograms in D depict Euclidean distance between vectors of scaled parameters within each of the three excitability classes.

Fig. 3.

Theory inspired dimensionality reduction. (A) Idealized momentary current–voltage relations at different ratios of available sodium and potassium conductances [see chapter 11 in Jack et al. (27)]. In different phases of the action potential, different momentary current–voltage relations determine the dynamics. The black continuous line depicts the resulting current–voltage trajectory during an action potential. (A, Inset) A magnified version of the area at threshold (indicated in the main image by a circle), about which the system is linearized. (B) Histograms of the three excitability statuses, constructed from the data of Fig. 2 (10,000 Hodgkin–Huxley realizations), for each of the scaling parameters. Note that all parameters are freely fluctuating, simultaneously, over $\pm$25%.

Fig. 4.

Hodgkin–Huxley model in a 2D S–K plane. (A) Subsets of 100 realizations from each of the three excitability statuses of Fig. 2 are plotted together. (B) Realizations (30,000) of a full Hodgkin–Huxley model, covering parametric variations over the entire range indicated by Hodgkin and Huxley (Fig. 1), classified (different colors) to three excitability statuses: excitable (4,660), not excitable (12,271), and oscillatory (13,069). Linear regression through the excitable status cloud (blue) is depicted by a line, the equation of which is S = 4.4K – 1.6. (B, Inset) Same plot with excitable points omitted.

Fig. 5.

Response features are sensitive to the values of $S$ and $K$. (A) Five different instantiations for each of three S;K pairs; stimulation amplitude 14 $\mathrm{\mu }$A. (B) The integral of voltage response emitted during a simulated trace is sensitive to the position within the S–K plain. The integral, calculated by summing the voltage values of all data points along the trace, relative to −65 mV, is presented in arbitrary units: B, Left, gradual change of $K$ at $S=0.5$; B, Right, gradual change of $S$ at $K=0.5$. Point color depicts excitability class.

Fig. 6.

Machine learning classification of excitability status. Classification of excitability statuses (data of Fig. 2) using a linear kernel SVM; 80% training set. Each surface (oscillatory, excitable, and nonexcitable; color coded) represents the probability (z axis) of a given combination of $K$ and $S$ to give rise to its corresponding excitability status. Thus, for instance, the probability (Prob.) of a point $K=0.42$ and $S=0.7$ to yield an oscillatory (depicted green) excitability status is practically 1. The colored points at the top of the image are the actual data points, similar to the form of presentation used in Fig. 4B.

Fig. 7.

Slow inactivation of sodium channels stabilizes excitability in S–K plane. (Left) A schematic representation of sodium channel states, with many slow inactivation states. (Right) Demonstration of maintenance of excitability in S–K plane given parametric variation, controlled by activity-dependent transitions of sodium channels between available and not-available sets of states. The simulation describes a 200,000-step random walk process, beginning at 0.5;0.5 ($S;K$). The gray trajectory (squiggles-like) depicts a random walk where $K_{n+1}=K_n\pm 0.01$ and $S_{n+1}=S_n\pm 0.01$. The blue line depicts a walk where $K_{n+1}=K_n\pm 0.01$, and $S_{n+1}=S_n\pm 0.01-\left(2.6-4.4K\right)S+S\left(1-S\right)$. The slope corresponds to the fitted function of Fig. 4B.