John R. Clay

Relationship between membrane excitability and single channel open-close kinetics

We have developed a novel technique for simulating the influence of the effects of single channel kinetics on the voltage changes associated with membrane excitability. The technique uses probability distribution functions for the durations of channel open- and closed-state lifetimes, which can be calculated for any model of the ion conductance process. To illustrate the technique, we have used the Hodgkin and Huxley model of nerve membrane ion conductances to simulate channel kinetics during predetermined voltage changes, such as a voltage jump and an action potential. We have also simulated the influence of channels on voltage changes in a free running, non-voltage-clamped patch of membrane 1 micron2 or less in area. The latter results provide a direct illustration of the relationship between fluctuations of membrane excitability and fluctuations in channel open- and closed-state lifetimes.

A quantitative description of the E-4031-sensitive repolarization current in rabbit ventricular myocytes.

We have measured the E-4031-sensitive repolarization current (IKr) in single ventricular myocytes isolated from rabbit hearts. The primary goal of this analysis was a description of the IKr kinetic and ion transfer properties. Surprisingly, the maximum time constant of this component was 0.8 s at 33-34 degrees C, which is significantly greater than the value of 0.18 s previously reported under similar conditions in the original measurements of IKr from guinea pig ventricular myocytes. The primary, novel feature of our analysis concerns the relationship of the bell-shaped curve that describes the voltage dependence of the kinetics and the sigmoidal curve that describes the activation of IKr. The midpoint of the latter occurred at approximately +10 mV on the voltage axis, as compared to -30 mV for the point on the voltage axis at which the maximum time constant occurred. Moreover, the voltage dependence of the kinetics was much broader than the steepness of the activation curve would predict. Taken together, these results comprise a gating current paradox that is not resolved by the incorporation of a fast inactivated state in the analysis. The fully activated current-voltage relation for IKr exhibited strong inward-going rectification, so much so that the current was essentially nil at +30 mV, even though the channel opens rapidly in this voltage range. This result is consistent with the lack of effect of E-4031 on the early part of the plateau phase of the action potential. Surprisingly, the reversal potential Of /Kr was ~15 mV positive to the potassium ion equilibrium potential,which indicates that this channel carries inward current during the latter part of the repolarization phase of the action potential.

A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

The Hodgkin and Huxley (HH) model predicts sustained repetitive firing of nerve action potentials for a suprathreshold depolarizing current pulse for as long as the pulse is applied (type 2 excitability). Squid giant axons, the preparation for which the model was intended, fire only once at the beginning of the pulse (type 3 behaviour). This discrepancy between the theory and experiments can be removed by modifying a single parameter in the HH equations for the K+ current as determined from the analysis in this paper. K+ currents in general have been described by IK=gK(V−EK), where gK is the membranes K+ current conductance and EK is the K+ Nernst potential. However, IK has a nonlinear dependence on (V−EK) well described by the Goldman–Hodgkin–Katz equation that determines the voltage dependence of gK. This experimental finding is the basis for the modification in the HH equations describing type 3 behaviour. Our analysis may have broad significance given the use of IK=gK(V−EK) to describe K+ currents in a wide variety of biological preparations.

Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory

Injection of a current pulse of brief duration into an aggregate of spontaneously beating chick embryonic heart cells resets the phase of the activity by either advancing or delaying the time of occurrence of the spontaneous beat subsequent to current injection. This effect depends upon the polarity, amplitude, and duration of the current pulse, as well as on the time of injection of the pulse. The transition from prolongation to shortening of the interbeat interval appears experimentally to be discontinuous for some stimulus conditions. These observations are analyzed by numerical investigation of a model of the ionic currents that underlie spontaneous activity in these preparations. The model consists of: Ix, which underlies the repolarization phase of the action potential, IK2, a time-dependent potassium ion pacemaker current, Ibg, a background or time-independent current, and INa, an inward sodium ion current that underlies the upstroke of the action potential. The steady state amplitude of the sum of these currents is an N-shaped function of potential. Slight shifts in the position of this current-voltage relation along the current axis can produce either one, two, or three intersections with the voltage axis. The number of these equilibrium points and the voltage dependence of INa contribute to apparent discontinuities of phase resetting. A current-voltage relation with three equilibrium points has a saddle point in the pacemaker voltage range. Certain combinations of current-pulse parameters and timing of injection can shift the state point near this saddle point and lead to an interbeat interval that is unbounded . Activation of INa is steeply voltage dependent. This results in apparently discontinuous phase resetting behavior for sufficiently large pulse amplitudes regardless of the number of equilibrium points. However, phase resetting is fundamentally a continuous function of the time of pulse injection for these conditions. These results demonstrate the ionic basis of phase resetting and provide a framework for topological analysis of this phenomenon in chick embryonic heart cell aggregates.

Diffusion models for firing of a neuron with varying threshold

Abstract A stochastic model for the firing of a neuron with refractory properties is treated analytically. Refractory behavior is modeled by a threshold function θ(t) which is infinite immediately after the neuron fires, as well as during the absolute refractory period, and then decreases monotonically to the quiescent threshold level, θ∞, during the relative refractory period. Using Walds identity, input-output relations are derived analytically for the exponential threshold which has a time constant equal to the membrane time constant. A method for computing these relations for a general threshold is presented and is explicitly used for the general exponential threshold and the Hagiwara threshold, θ(t) = θ∞ eα/t, where a is a constant.

Determining K+ channel activation curves from K+ channel currents

Abstract Potassium ion channels are generally believed to have current-voltage (IV) relations which are linearly related to driving force (V–EK), where V is membrane potential and EK is the potassium ion equilibrium potential. Consequently, activation curves for K+ channels have often been measured by normalizing voltage-clamp families of macroscopic K+ currents with (V–EK), where V is the potential of each successive step in the voltage clamp sequence. However, the IV relation for many types of K+ channels actually has a non-linear dependence upon driving force which is well described by the Goldman-Hodgkin-Katz relation. When the GHK dependence on (V–EK) is used in the normalization procedure, a very different voltage dependence of the activation curve is obtained which may more accurately reflect this feature of channel gating. Novel insights into the voltage dependence of the rapidly inactivating IA channels Kv1.4 and Kv4.2 have been obtained when this procedure was applied to recently published results.

Effects of Divalent Cations on the E-4031-Sensitive Repolarization Current, I Kr , in Rabbit Ventricular Myocytes

The effects of divalent cations on the E-4031-sensitive repolarization current (I(Kr)) were studied in single ventricular myocytes isolated from rabbit hearts. One group of divalent cations (Cd2+, Ni2+, Co2+, and Mn2+) produced a rightward shift of the I(Kr) activation curve along the voltage axis, increased the maximum I(Kr) amplitude (i.e., relieved the apparent inward rectification of the channel), and accelerated I(Kr) tail current kinetics. Another group (Ca2+, Mg2+ and Sr2+) had relatively little effect on I(Kr). The only divalent cation that blocked I(Kr) was Zn2+ (0.1-1 mM). Under steady-state conditions, Ba2+ caused a substantial block of I(K1) as previously reported. However, block by Ba2+ was time dependent, which precluded a study of Ba2+ effects on I(Kr). We conclude that the various effects of the divalent cations can be attributed to interactions with distinct sites associated with the rectification and/or inactivation mechanism of the channel.

Membrane Current and Membrane Potential from Single-Channel Kinetics

The problem we wish to address in this chapter is how an action potential is constructed from the time- and voltage-dependent kinetics of single-channel currents. In order to solve this problem, we must know how channels behave in a free-running, unclamped membrane. When channels open or close, membrane voltage changes. Since opening and closing are themselves voltage dependent, channel kinetics are self perturbing. This chapter presents a model of channel kinetics in an unclamped membrane and how channel fluctuations are related to membrane excitability.

Comparison of the pacemaker properties of chick embryonic atrial and ventricular heart cells.

SummaryWe have investigated the pacemaker properties of aggregates of cells dissociated from the atria and ventricles of 10 to 14-day-old chick embryonic hearts using a two-microelectrode current and voltage-clamp technique. These preparations usually beat spontaneously and rhythmically in tissue culture medium containing 1.3mm potassium with a beat rate typically in the range of 15–60 beats per minute. The beat rate results show considerable variability, which precludes any statistically significant comparison between the spontaneous activity of atrial and ventricular cell preparations at 10–14 days of development. However, the shapes of pacemaker voltage changes do exhibit differences characteristic of cell type. Spontaneous atrial preparations rapidly depolarize from maximum diastolic potential (∼−90 mV) to a plateau range of pacemaker potentials (−80 to −75 mV). The membrane subsequently depolarizes more gradually until threshold (∼−65 mV) is reached. In contrast, spontaneously beating ventricular cell preparations slowly hyperpolarize after maximum diastolic potential to the −100 to −95 mV range before gradually depolarizing toward threshold. Voltage-clamp analysis reveals a virtual lack of any time-dependent pacemaker current in atrial preparations. These preparations are characterized by an approximately linear background current (Ibg) having a slope resistance of ∼100 KΩ cm2. Ventricular preparations have a potassium ion pacemaker current with slow kinetics (IK2), and a second time-dependent component (Ix) which is activated at potentials positive to −65 mV. The background current of these preparations displays inward rectification. Computer simulations of pacemaking reveal that the initial rapid phase of pacemaker depolarization in atrial cells is determined by the membrane time constant, which is the product of membrane capacitance and the slope resistance ofIbg. The hyperpolarization after maximum diastolic potential of ventricular cells is caused byIK2. The final slow phase of depolarization in both cell types is caused in part by the steady-state amplitude of the fast inward sodium current (INa). This component has negative slope conductance which effectively increases the slope resistance in the vicinity of threshold compared to TTX-treated preparations. This mechanism is sufficient to produce interbeat intervals several seconds in duration, even in the absence of time-dependent pacemaker current, provided that the background current is at the appropriate level.

Optimal stimulus shapes for neuronal excitation

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.