Electrophorus electricus as a model system for the study of membrane excitability

The stunning sensations produced by electric fish, particularly the electric eel, Electrophorus electricus, have fascinated scientists for centuries. Within the last 50 years, however, electric cells of Electrophorus have provided a unique model system that is both specialized and appropriate for the study of excitable cell membrane electrophysiology and biochemistry. Electric tissue generates whole animal electrical discharges by means of membrane potentials that are remarkably similar to those of mammalian neurons, myocytes and secretory cells. Electrocytes express ion channels, ATPases and signal transduction proteins common to these other excitable cells. Action potentials of electrocytes represent the specialized end function of electric tissue whereas other excitable cells use membrane potential changes to trigger sophisticated cellular processes, such as myofilament cross-bridging for contraction, or exocytosis for secretion. Because electric tissue lacks these functions and the proteins associated with them, it provides a highly specialized membrane model system. This review examines the basic mechanisms involved in the generation of the electrical discharge of the electric eel and the membrane proteins involved. The valuable contributions that electric tissue continues to make toward the understanding of excitable cell physiology and biochemistry are summarized, particularly those studies using electrocytes as a model system for the study of the regulation of membrane excitability by second messengers and signal transduction pathways.     

Determinants of time-dependent membrane conductance. The nonrole of classical ion-membrane molecule interactions

We have examined the steady-state and time-dependent electrical properties of a model membrane system. The model assumes that the directed velocity and energy of ions moving through the membrane are determined by the applied electric field, ionic diffusion forces, and central elastic collisions between ions and membrane molecules. A simple analysis of the steady-state electrical properties of the model yields results identical with ones obtained previously using a more complex analysis procedure. The time-dependent conductance changes of the model in response to a step change in electric field strength when there is solution symmetry display three qualitative patterns dependent on the nature of the ion-membrane molecule interaction. One of the patterns of conductance change is quite similar to that observed in the sodium conductance system of a number of excitable tissues: an initial conductance rise to a maximum (activation) followed by a decay to a final steady-state value (inactivation). However, the correspondence between the time-dependent model behavior and known experimental behavior of excitable systems is only qualitative. We conclude that the classical ion-membrane molecule interactions we consider are not involved in determining time-dependent conductance processes in the excitable systems for which comparison is possible.     

Axon voltage-clamp simulations. I. Methods and tests.

This is the first in a series of four papers in which we present the numerical simulation of the application of the voltage clamp technique to excitable cells. In this paper we describe the application of the Crank-Nicolson (1947) method for the solution of the parabolic partial differential equations that describe a cylindrical cell in which the ionic conductances are functions of voltage and time (Hodgkin and Huxley, 1952). This method is compared with other methods in terms of accuracy and speed of solution for a propagated action potential. In addition, differential equations representing a simple voltage-clamp electronic circuit are presented. Using the voltage clamp circuit equations, we simulate the voltage clamp of a single isopotential membrane patch and show how the parameters of the circuit affect the transient response of the patch to a step change in the control potential.The stimulation methods presented in this series of papers allow the evaluation of voltage clamp control of an excitable cell or a syncytium of excitable cells. To the extent that membrane parameters and geometrical factors can be determined, the methods presented here provide solutions for the voltage profile as a function of time.     

Biological effects of oscillating electric fields: Role of voltage-sensitive ion channels

An alternating component of potential across the membrane of an excitable cell may change the membrane conductance by interacting with the voltagesensing charged groups of the protein macromolecules that form voltage-sensitive ion channels. Because the probability that a voltage sensor is in a given state is a highly nonlinear function of the applied electric field, the average occupancy of a particular state will change in an oscillating electric field of sufficient magnitude. This “rectification” at the level of the voltage sensors could result in conformational changes (gating) that would modify channel conductance. A simplified two-state model is examined where the relaxation time of the voltage sensor is assumed to be considerably faster than the fastest changes of ionic conductance. Significant changes in the occupancy of voltage sensor states in response to an applied oscillating electric field are predicted by the model.     

Intrinsic coherence resonance in excitable membrane patches

The influence of intrinsic channel noise on the spiking activity of excitable membrane patches is studied by use of a stochastic generalization of the Hodgkin-Huxley model. Internal noise stemming from the stochastic dynamics of individual ion channels does affect the electric properties of the cell-membrane patches. There exists an optimal size of the membrane patch for which the internal noise alone can cause a nearly regular spontaneous generation of action potentials. We consider the influence of intrinsic channel noise in presence of a constant and an oscillatory current driving for both, the mean interspike interval and the phenomenon of coherence resonance for neuronal spiking. Given small membrane patches, implying that channel noise dominates the excitable dynamics, we find the phenomenon of intrinsic coherence resonance. In this case, the relatively regular spiking behavior becomes essentially independent of an applied stimulus. We observed, however, the occurrence of a skipping of supra-threshold input events due to channel noise for intermediate patch sizes. This effect consequently reduces the overall coherence of the spiking.     

A theory of self-organization of an excitable membrane formed on the protoplasmic droplet isolated from Nitella

The process of self-organization of an excitable membrane of protoplasmic droplet of Nitella is studied theoretically by taking the interaction of the local electric current caused by the spatially non-uniform distribution of active domains into account. The theoretical model employed is that the surface membrane forms a mosaic structure composed of lipids and protein molecules, and that each element of the mosaic structure (domain) on the membrane has two distinct conformations corresponding to excited and resting states. The molecules constituting the membrane are derived from the inside of the protoplasm by diffusion. The excitability of the surface membrane appears suddenly after a morphogenetic structure of the membrane is formed with time on the surface of droplet. Time courses of the variation in membrane potential and in membrane resistance are calculated, and the results are compared with experimental data obtained with the protoplasmic droplet of Nitella.     

Modeling electroporation in a single cell. II. Effects Of ionic concentrations.

This study expands a previously developed model of a single cell electroporated by an external electric field by explicitly accounting for the ionic composition of the electroporation current. The previous model with non-specific electroporation current predicts that both the transmembrane potential V(m) and the pore density N are symmetric about the equator, with the same values at either end of the cell. The new, ion-specific case predicts that V(m) is symmetric and almost identical to the profile from the non-specific case, but N has a profound asymmetry with the pore density at the hyperpolarized end of the cell twice the value at the depolarized end. These modeling results agree with the experimentally observed preferential uptake of marker molecules at the hyperpolarized end of the cell as reported in the literature. This study also investigates the changes in intracellular ionic concentrations induced around an electroporated single cell. For all ion species, the concentrations near the membrane vary significantly, which may explain the electrical disturbances observed experimentally after large electric shocks are delivered to excitable cells and tissues.     

Application of the Hodgkin-Huxley equations to an electric field model for interaction between excitable cells

We previously described a model for the electrical transfer of excitation from one cell to the next which utilized the electric potential generated in the junctional cleft between the cells. Low-resistance connections between the cells were not used in the model, and it was assumed that the junctional membranes were excitable. This model was analyzed for the static case without capacitances and for the dynamic case in which capacitances were part of the circuit elements. For simplicity, the Na⁺ resistance (R_Na), after a threshold potential was exceeded, was allowed to decrease exponentially (to 1% of its initial value) within 0·25–1·0 ms, and possible changes in the K⁺ resistance were ignored. In this paper, we have incorporated the Hodgkin-Huxley equations into the operation of the lumped membrane units for the electrical equivalent circuit of the cell membrane. The parameters varied are the membrane capacitances, resistances, maximum Na⁺ conductance ($\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$), and the radial cleft resistance (R_jc). We demonstrated that our model worked very well, i.e. the successful transfer of action potentials was achieved, with the membrane units following Hodgkin-Huxley dynamics for changes in g_Na and g_K. The calculations indicate that transmission is facilitated when the junctional units have a higher $\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$ and a lower capacitance and when R_jc is elevated. Lowering the resistance of the junctional membrane units several fold, relative to the surface membrane units, also facilitated transmission; however, the absolute resistance of the junctional membrane was still well above the maximum value that would allow sufficient local-circuit current to flow to effect transmission. Thus, the electric field model provides an alternative means of cell-to-cell propagation between myocardial cells which is electrical in nature but does not require the presence of low-resistance connections between cells.     

Gene circuit designs for noisy excitable dynamics

Certain cellular processes take the form of activity pulses that can be interpreted in terms of noise-driven excitable dynamics. Here we present an overview of different gene circuit architectures that exhibit excitable pulses of protein expression, when subject to molecular noise. Different types of excitable dynamics can occur depending on the bifurcation structure leading to the specific excitable phase-space topology. The bifurcation structure is not, however, linked to a particular circuit architecture. Thus a given gene circuit design can sustain different classes of excitable dynamics depending on the system parameters.     

Alternative Models to Hodgkin–Huxley Equations

The Hodgkin and Huxley equations have served as the benchmark model in electrophysiology since 1950s. But it suffers from four major drawbacks. Firstly, it is only phenomenological not mechanistic. Secondly, it fails to exhibit the all-or-nothing firing mechanism for action potential generation. Thirdly, it lacks a theory for ion channel opening and closing activation across the cell membrane. Fourthly, it does not count for the phenomenon of voltage-gating which is vitally important for action potential generation. In this paper, a mathematical model for excitable membranes is constructed by introducing circuit characteristics for ion pump exchange, ion channel activation, and voltage-gating. It is demonstrated that the model is capable of re-establishing the Nernst resting potentials, explicitly exhibiting the all-or-nothing firing mechanism, and most important of all, filling the long-lasting theoretical gap by a unified theory on ion channel activation and voltage-gating. It is also demonstrated that the new model has one half fewer parameters but fits significantly better to experiment than the HH model does. The new model can be considered as an alternative template for neurons and excitable membranes when one looks for simpler models for mathematical studies and for forming large networks with fewer parameters.     

Control of wave propagation in a biological excitable medium by an external electric field

We present an experimental evidence of effects of external electric fields (EFs) on the velocity of pulse waves propagating in a biological excitable medium. The excitable medium used is formed by a layer of starving cells of Dictyostelium discoideum through which the waves of increased concentration of cAMP propagate by reaction-diffusion mechanism. External dc EFs of low intensities (up to 5 V/cm) are shown to speed up the propagation of cAMP waves towards the positive electrode and slow it down towards the negative electrode. Electric fields were also found to support an emergence of new centers, emitting cAMP waves, in front of cAMP waves propagating towards the negative electrode.     

Electrical and magnetic fields of the elementary bioelectric generator in the bounded anisotropic myocardium

Using the equations of electrodynamics of stationary currents, relationships were derived for calculating the characteristics of electric and magnetic fields of an elementary (dipole) bioelectric generator in a heterogeneous medium consisting of two regions namely, an anisotropic conducting region corresponding to the excitable myocardium tissue and an isotropic conducting or dielectric region corresponding to the space outside the myocardium where the measurement is made. The shape of distributions of the electric potential and magnetic induction at the myocardium surface was determined, and the effect of anisotropy on these distributions was estimate. Formulas for the identification of the local excited zone within the myocardium from electric and magnetic measurements outside the excitable tissue or on its surface were obtained.     

The Excitable Membrane: A Physiochemical Model

The model of the excitable membrane assumes common channels for Na⁺ and K⁺; the two ion species interact within the pores through their electrostatic forces. The electric field varies across the membrane and with time, as a result of ionic redistribution. Ionic flow is primarily controlled by energy barriers at the two interfaces and by Ca⁺⁺ adsorption at the external interface. When the membrane is polarized, the high electric field at the external interface acting on the membrane fixed charge keeps the effective channel diameter small, so that only dihydrated ions can cross the interface. The higher energy required to partially dehydrate Na⁺ accounts for its lower permeability when polarized. Depolarized, the channel entrance can expand, permitting quadrihydrated ions to pass; the large initial Na⁺ flow is the result of the large concentration ratio across the interface. The effect at the internal interface is symmetric; Na⁺ crosses with greater difficulty when the membrane is depolarized. Na⁺ inactivation occurs when the ion distribution within the membrane has assumed its new steady-state value. Calculations based on parameters consistent with physicochemical data agree generally with a wide range of experiments. The model does not obey the two fundamental Hodgkin-Huxley (HH) postulates (independence principle, ion flow proportional to thermodynamic potential). In several instances the model predicts experimental results which are not predicted by the HH equations.     

Modeling the electrical behavior of anatomically complex neurons using a network analysis program: Excitable membrane

We present methods for using the generalpurpose network analysis program, SPICE, to construct computer models of excitable membrane displaying Hodgkin-Huxley-like kinetics. The four non-linear partial differential equations of Hodgkin and Huxley (H-H; 1952) are implemented using electrical circuit elements. The H-H rate constants, and , are approximated by polynomial functions rather than exponential functions, since the former are handled more efficiently by SPICE. The process of developing code to implement the H-H sodium conductance is described in detail. The Appendix contains a complete listing of the code required to simulate an H-H action potential. The behavior of models so constructed is validated by comparison with the space-clamped and propagating action potentials of Hodgkin and Huxley. SPICE models of multiply branched axons were tested and found to behave as predicted by previous numerical solutions for propagation in inhomogeneous axons. New results are presented for two cases. First, a detailed, anatomically based model is constructed of group Ia input to an -motoneuron with an excitable soma, a myelinated axon and passive dendrites. Second, we simulate interactions among clusters of mixed excitable and passive dendritic spines on an idealized neuron. The methods presented in this paper and its companion (Segev et al. 1985) should permit neurobiologists to construct and explore models which simulate much more closely the real morphological and physiological characteristics of nerve cells.     

Kinetics of Excitable Membranes : Voltage amplification in a diffusion regime

An understanding of the properties of excitable membranes requires the calculation of ion flow through the membrane, including the effects of nonuniformity in the transverse membrane properties (mobilities, fixed charge, electric field). Permeability is apparently controlled at the external interface. Two factors may be involved here: the statistical blocking of pores by divalent cations, and activation energy. Only the former is included in the present treatment. When the total transmembrane voltage is varied, a redistribution in ionic concentration occurs. This can cause a change in boundary (zeta) potential, large in comparison with the applied voltage change—"voltage amplification." The result is a steep change in membrane conductance. The calculated flow curves are compared with experimental results. The Appendix gives an outline of the numerical method used for solving the boundary value problem with several diffusible ions, across a nonuniform regime.     

Quantitative description of end plate current and its reconstruction

An attempt was made to quantify the postsynaptic current based on the experimental data of the voltage clamp method. The conductance change in postsynaptic membrane was derived from the postsynaptic conductance of voltage clamped postsynaptic membrane, then it's temporal characteristics and dependence on the clamped voltage have been quantified. The temporal characteristics was found to be explained by the introduction of two schematic operators, active and inactive. This idea was applied to a simple electrical circuit model of the postsynaptic cells. Besides the change in postsynaptic potentials of normal synapse in excitable state was culculated as its application.     

Evidence against Hydrogen–Calcium Competition Model for Activation of Electrically Excitable Membranes

TO explain the voltage-dependent sodium permeability of excitable membranes, Stephens¹ proposed a model in which sodium-selective channels are normally blocked by calcium ions bound to negatively charged sites located near the outer end of the channels. The calcium ions can be displaced competitively by hydrogen ions, opening the channels to sodium. According to this model, depolarization of an excitable membrane causes an outward flow of hydrogen ions across the membrane. The consequent transient increase in hydrogen ion concentration at the outer surface of the membrane displaces calcium and opens the sodium channels. This model is particularly interesting because it is sufficiently specific to allow direct tests. Stephens shows that it is in general agreement with a variety of experimental data. To test the model further, we have determined the effect of variation in the internal and external concentration of hydrogen ions on sodium currents.     

Conformational flexibility of membrane proteins in electric fields. I. Ultraviolet absorbance and light scattering of bacteriorhodopsin in purple membranes

Bacteriorhodopsin of halobacterial purple membranes exhibits conformational flexibility in high electric field pulses (1-30 x 10(5) V m(-1), 1-100 micros). High-field electric dichroism data of purple membrane suspensions indicate two kinetically different structural transitions within the protein; involving a rapid (approximately 1 micros) concerted change in the orientation of both retinal and tyrosine and/or tryptophan side chains concomitant with alterations in the local protein environment of these chromophores. as well as slower changes (approximately 100 micros) of the microenvironment of aromatic amino acid residues concomitant with pK changes in at least two types of proton-binding sites. Light scattering data are consistent with the maintenance of the random distribution of the membrane discs within the short duration of the applied electric fields. The kinetics of the electro-optic signals and the steep dependence of the relaxation amplitudes on the electric field strength suggest a saturable induced-dipole mechanism and a rather large reaction dipole moment of 1.1 x 10(-25) C m ( = 3.3 x 10(4) debye) per cooperative unit at E = 1.3 x 10(5) V m(-1), which is indicative of appreciable cooperativity in the probably unidirectional transversal displacement of ionic groups on the surfaces of and within the bacteriorhodopsin proteins of the membrane lattice. The electro-optic data of bacteriorhodopsin are suggestive of a possibly general, induced-dipole mechanism for electric field-dependent structural changes in membrane transport proteins such as the gating proteins in excitable membranes or the ATP synthetases.