VOLTAGE OSCILLATIONS IN THE BARNACLE GIANT MUSCLE FIBER Catherine Morris and Harold Lecar, Laboratory of Biophysics, National Institute of Neurological and Communicative Disorders and Stroke, National Institutes of Health, Bethesda, Maryland 20205 abstract Barnacle muscle fibers subjected to constant current stimulation produce a variety of types of oscillatory behavior when the internal medium contains the Ca+ + chelator EGTA. Oscillations are abolished if Ca++ is removed from the external medium, or if the K+ conductance is blocked. Available voltage-clamp data indicate that the cell's active conductance systems are exceptionally simple. Given the complexity of barnacle fiber voltage behavior, this seems paradoxical. This paper presents an analysis of the possible modes of behavior available to a system of two noninactivating conductance mechanisms, and indicates a good correspondence to the types of behavior exhibited by barnacle fiber. The differential equations of a simple equivalent circuit for the fiber are dealt with by means of some of the mathematical techniques of nonlinear mechanics. General features of the system are (a) a propensity to produce damped or sustained oscillations over a rather broad parameter range, and (b) considerable latitude in the shape of the oscillatory potentials. It is concluded that for cells subject to changeable parameters (either from cell to cell or with time during cellular activity), a system dominated by two noninactivating conductances can exhibit varied oscillatory and bistable behavior. INTRODUCTION Voltage-clamp studies of the barnacle muscle (Keynes et al., 1973; Hagiwara et al., 1969; Hagiwara et al., 1974; Murayama and Lakshminarayanaiah, 1977; Beirao and Lakshmina-rayanaiah, 1979) indicate that the fiber possesses a simple conductance system consisting of voltage dependent Ca++ and K+ channels, neither of which inactivates appreciably. Current-clamp studies, however, show complicated oscillatory voltage behavior (Hagiwara and Nakajima, 1966; Murayama and Lakshminarayanaiah, 1977). In this paper we ask whether a system of two noninactivating conductances can, in fact, account for such phenomena. A mathematical study shows that this simple system can predict much of the barnacle fiber behavior, although the simplest model fails to explain some areas of behavior. Of the two species of ion conductances in barnacle muscle (voltage- and time-dependent), neither shows fast inactivation, although accumulation of permeating ions produces time-dependent reduction of K+ currents (Keynes et al., 1973) and may also account for the slow but variable decay of Ca++ channel currents. Two noninactivating conductances would constitute an unusually simple conductance apparatus, compared with other crustaceans (cf. Mounier and Vassort, 1975; Hencek and Zachar, 1977). There is preliminary evidence for a Ca++-activated gK in the barnacle fiber (Murayama and Lakshminarayanaiah, 1977). In our analysis this process would not enter into consideration of the primary oscillation, but it might effect slow modulations to produce "bursting" phenomena similar to those modelled for Biophys. j. © Biophysical Society • 0006-3495/81/07/193/21 $1.00 Volume 35 July 1981 193-213 193 Aplysia neuron (Plant and Kim, 1976; Plant, 1976). Since our current clamp records were obtained from EGTA-perfused fibers, any Ca++-dependent slow processes would necessarily be depressed. We do, however, indicate how Ca+ + accumulation at the inner surface of the membrane may act as a very slow "inactivation" process. With /K and /Ca dominating the membrane, it would be reasonable to anticipate a limited repertoire of voltage behavior in barnacle fibers. Under physiological conditions this is indeed the case: the barnacle fiber responds to nerve stimulation like other crustaceans, with graded depolarizations. But EGTA-perfused fibers subjected to constant current stimulation exhibit a variety of complicated voltage responses. One is compelled to ask whether the two noninactivating conductances would be sufficient to produce voltage oscillations as variable in character as those illustrated in the literature (cf. Hagiwara and Nakajima, 1966; Murayama and Lakshiminarayanaiah, 1977). Other factors, such as failure of radial space clamp (caused by membrane invaginations), ion accumulation, slow inactivation, inhomogeneous distribution of channel types, and further species of conductances, could all lend additional complexity to the system and perhaps even cause additional oscillations. In this paper we consider the voltage oscillations induced under current clamp, and explore the possibility that they are produced by nothing more than two noninactivating conductances distributed homogeneously in a well space-clamped membrane. The observed behavior is compared with that predicted from a simple theoretical model. Although the model will be seen to give a reasonably good quantitative account of the current-clamp data, accurate fitting of the data for a limited set of conditions is not our main concern, since the extant voltage-clamp data are variable (cf. Keynes et al., 1973; Hagiwara et al., 1974). Rather, we study the range of behavior available to an excitable system of two nonlinear, noninactivating conductances, showing how the system predicts not only oscillations, but also pronounced qualitative variations in the character of the oscillations for different sets of parameters. Although this does not, of course, preclude involvement of other factors, it suggests that they have only a modulating influence, and are not fundamental to the oscillations. LIST OF ABBREVIATIONS OF MODEL PARAMETERS / = applied current (fiA/cm2) /L, /c, /K = leak, Ca++, and K+ currents, respectively (ftA/cm2) £l> ge*i Sk = maximum or instantaneous conductance values for leak, Ca++, and K+ pathways, respectively (mmho/cm2) ica = conductance constant for nonlinear /& (mmho/cm2) V = membrane potential (mV) Ye* = equilibrium potential corresponding to leak, Ca++, and K+ conductances, respectively (mV) M - fraction of open Ca++ channels N - fraction of open K+ channels MJV), Njy) = fraction of open Ca++ and K+ channels, at steady state ^uiV), Xjv(IO = rate constant for opening of Ca++ and K+ channels (s_1) ^m. = maximum rate constants for Ca++ and K+ channel opening (s_1) V\ = potential at which A/. = 0.5 (mV) V2 = reciprocal of slope of voltage dependence of Mm (mV) K3 = potential at which TV. = 0.5 (mV) V4 = reciprocal of slope of voltage dependence of 7V„ (mV) 194 Biophysical Journal Volume 35 1981 [Ca++]i, [Ca++]0 = internal and external Ca++ concentration (mM) K = accumulation-layer constant (cm"') F = Faraday constant C = membrane capacitance (/tF/cm2) METHODS Large specimens of the barnacle Balanus nubilus (Pacific Bio-Marine Laboratories Inc., Venice, Calif.) were used. The barnacle was sawed into lateral halves, and the depressor scutorum rostralis muscles were carefully exposed. Individual fibers were dissected, the incision starting from the tendon. The other end of the muscle was cut close to its attachment on the shell and ligatured. Isolated fibers were either used immediately or kept for up to 30 min in standard artifical seawater (ASW; see below) before use. Experiments were carried out at room temperature of ~22°C. The muscle chamber, internal and external electrodes, internal perfusion system and current clamping system were essentially similar to those described by Keynes et al. (1973). The composition of the normal internal solution was: K acetate (180 mM), glucose (600 mM), Tris OH (12.5 mM), and EGTA (2.5 mM). For some experiments the internal solution consisted of Cs acetate (180 mM), TEA chloride (60 mM), glucose (485 mM), Tris OH (12.5 mM), and EGTA (2.5 mM). The external solutions were either normal ASW (385 mM NaCl, 10 mM KC1, 10 mM CaCl2, 50 mM MgCl2, and 10 mM TES), high-Ca++ ASW (325 mM NaCl, 10 mM KC1, 100 mM CaCl2, and 10 mM TES, or Ca++-free ASW as normal ASW except CaCl2 substituted by MgCl2). Most of the results are presented in conjunction with theoretical computations of voltage oscillations. The differential equation systems were integrated using the MLAB program developed at National Institutes of Health (NIH) (Knott, 1979). This program employs the Geary-Nordsieck algorithm for numerical integration, which is adequate for our problem. Much of the analysis makes use of methods of nonlinear mechanics, and various analytical results are given in the text in order to gain insight into the requirements for oscillation. The Model In accord with the voltage-clamp experiments of Keynes et al. (1973), the model is assigned two independent voltage-dependent conductances, gK and gCa, each having a sigmoid voltage dependence. The relaxation times with which these conductances approach new values after voltage changes are given as bell-shaped functions of voltage. We shall assume for simplicity that the relaxation kinetics are first order, since precise kinetics are not essential for the description of all excitation effects (cf. Lecar and Nossal, 1971; FitzHugh, 1961). In general we have used linear relations for the instantaneous current-voltage curves through open-channels. Although our own experiments on the instantaneous Ca++ current in barnacle muscle (unpublished data) show departures from linearity, under situations of high permeable-ion gradient, in all but one instance we stick for the sake of simplicity to the linear driving force approximation. The exception occurs when we look at the Ca++ conductance system in isolation from gK, for under these circumstances there is no countering force to keep the system in the nearly linear region. Fig. 1 shows the equivalent circuit hypothesized for a space-clamped patch of sarcolemma membrane. The equations describing the membrane behavior are: / = C V + gL{VL) + gc,M(V- KCa) + gK N(V - VK) M = \M(V) [MSV) - M] (1) N = \N(V) [N„(V) - TV]. The parameters and variables are defined in the list of abbreviations. The variables M and TV are Morris and Lecar Voltage Oscillations in the Barnacle Giant Muscle Fiber 195 1 Figure 1 Equivalent circuit for a patch of space-clamped barnacle sarcolemma. analogous to the Hodgkin-Huxley (1952) "w" and "«" parameters. M and N are the fraction of channels open at any given time and, by rather elementary statistical arguments, are given the forms (e.g., Lecar et al., 1975; Ehrenstein and Lecar, 1977): Af.(K) = 1/2 {1 + tanh [(V - K,)/K2]} XM(K)=X^cosh([F- K,]/2K2) N„(V) = 1/2 {1 + tanh [(V - V3)/V,]\ X^(K) = \;cosh([K- K,]/2K,). (2) Eqs. 1 and 2 represent a third-order nonlinear system of the Hodgkin-Huxley form, which we shall use to explain the excitation behavior of the barnacle muscle. Values for the parameters were drawn from the voltage-clamp literature or from our own voltage-clamp data. In many cases we shall focus on the way in which the character of the solutions changes as the parameters are made to vary. RESULTS Experimental Observations Voltage behavior of current-clamped fibers was observed under three different conditions: with Ca++-free external solution (to minimize gCi), with Cs/TEA-containing internal perfusion solution (to minimize gK), and with solutions that optimize both gK and gCi. K+ Conductance In Ca++-free solutions fibers produce no voltage oscillations. The membrane response to small stimuli is not discernably different from a passive RC circuit, but as the stimuli increase, a small active response is seen (Fig. 2a); instead of maintaining a voltage that would be dominated by the drop across the leakage resistance, the membrane partially repolarizes, leaving a small peak where the purely /?C-response is truncated. At the end of the stimulus the voltage (which has usually slightly depolarized again) returns almost exponentially to rest. The straightforward interpretation is that the larger stimuli activate a nonlinear K+ conductance, thereby producing an outward, hyperpolarizing current. The subsequent slow polarization of the K+ plateau is probably due to K+ accumulation during the prolonged stimulus (Keynes et al., 1973) and/or to a small residual Ca++ current, but not to gK inactivation. 196 Biophysical Journal Volume 35 1981 30 20 10 0 v inv) -10 -20 -30 V: 200 0 TIflE IflSEC) ALL-K« SYSTEJ1 Figure 2 Plateau potentials of the K+ system, (a) Voltage responses (in millivolts) of a fiber to increasing current stimulus (480, 960, and 1200 ^A/cm2). External saline: Ca++-free ASW. Internal saline: K+ perfusate. In this and subsequent records, the duration (in milliseconds) of the current stimulus is indicated by arrowheads, (b) Computed voltage response of model to increasing currents. These curves represent numerical solutions to Eq. 3 for / - 25,100, and 400. Other parameters were as follows: gc = 0, gti _ 8, gL - 3, KK - -70, VL--50, \N - 1/15, C - 20, V3--1.0, K4 - 14.5. Initial conditions: K(0)--50, yv(0) - N„ (- 50). The value for gL was set at 3 instead of 2 to simulate roughly the increased leak frequently encountered in Ca++-free solutions. Ca++ Conductance The behavior of gK-blocked fibers is somewhat more complex. Again, no oscillations are observed, and very small stimuli produce simple /?C-like responses. With increasing stimuli, however, a threshold is reached and bistability in the membrane voltage becomes evident. Beyond a threshold (--17 mV) the membrane voltage shoots rapidly to about +20 mV and remains in a depolarized state with a slow repolarizing droop. This is a "Ca++ plateau action potential" (Fig. 3a, top trace). When the stimulus pulse ends, the fiber does not return to rest, but simply undergoes a small voltage drop caused by the removal of the stimulating current. Between hundreds of milliseconds and several seconds later, as the plateau voltage continues to decline, the voltage does reach a threshold and falls precipitously to rest (Fig. 3c). The threshold for returning to rest occurs at a more depolarized voltage (~ 0 mV) than for activation. The plateau action potential is readily interpreted as indicating that the only operative conductance system of significance (when gK is blocked) is a nonlinear gCa. Threshold behavior would result from the negative resistance provided by the inward, depolarizing /Ca; bistability would follow from lack of voltage-sensitive inactivation in the negative resistance element. Oscillations: K+ and Ca++ Conductances Together The voltage behavior of the current-clamped fiber when either gK or gCi is minimized is relatively simple and predictable; essentially, stimulation results in one of two types of plateaus. One might anticipate that when both conductances are recruited simultaneously, a voltage plateau intermediate between those for all K+ and all Ca++ would result. This is not, however, the case; the voltage behavior, over a wide range of current stimulus, is oscillatory. Small stimuli produce small, essentially passive depolarizations, but once a threshold voltage (which is in the same range as the all-Ca++ threshold) is reached, oscillations appear. Morris and Lecar Voltage Oscillations in the Barnacle Giant Muscle Fiber 197 Figure 3 Plateau potentials of the all-Ca+* system, (a) Voltage responses (in millivolts) of a fiber to two values of current stimulus, 60 jjA/cm2 (lower trace) and 240 jtA/cm2 (upper trace). Note that for the smaller current the voltage reaches its plateau value after the stimulus is over. Time is expressed in milliseconds. External saline: high-Ca++ ASW (100 mM Ca++). Internal saline: Cs/TEA/EGTA perfusate, (b) Voltage response (mV) of the model (nonlinear driving force) to increasing currents. These curves represent^ numerical solutions to eq. 3 and 7, for / - 15, 25, and 50. Other parameters were as follows: gK - 0,gc = 40, gL- 2, [Ca++]; - 0, [Ca++]0 - 100, KL - -35,XW - 0.1, C - 20, VI - 10, V2 -15. Initial conditions K(0) = - 35, M(0) = M„ (- 35). Time is expressed in milliseconds, (c) Responses of a fiber to two subsequent current stimuli (360 iiA) 1 min apart (upper trace first). Duration of the stimulus is 100 ms, so that these records emphasize the poststimulus part of the response. These represent particularly long plateau action potentials, but are typical in shape. Voltage, mV; Time, s. (d) Poststimulus voltage response of the model with an accumulation term. This curve^ represents a solution to Eq. 8 for / - 0. Other parameters were as follows: gK - 0, gCt - 40, gL - 2, [Ca++]„- 100, VL - -35, VI - 10, V2 - 15, C- 20, K= -10"4. Initial conditions K(0) - +28, [Ca++]; (0) - 0.001. Voltage, mV; time, s. Oscillations of barnacle muscle fibers are decidedly not stereotyped (see also Hagiwara and Nakajima, 1966; Murayama and Lakshminarayanaiah, 1977); they are variable and sometimes complex. Depending on experimental parameters such as leakage or Ca++ conductance, the oscillations can either be damped or continuous. Usually, but not always, oscillations cease at the end of the stimulus and the fiber returns to rest. Fig. 4a shows a sequence of damped oscillations for various values of applied current. Although the amplitude and frequency of the oscillations are dependent on stimulus strength, increasing the stimulus does not produce a monotonic change in oscillation amplitude and frequency. Eventually, as / increases, the voltage excursions decrease and the rate of damping of the oscillations increases. Factors other than current strength also influence the character of the oscillations. The external calcium ion concentration is particularly influential. Fibers that show little or no 198 Biophysical Journal Volume 35 1981 100 150 TIflE tnSEC) j \ J \ J YJ bii m A ...... . A 100 150 TlflE IHSEC) 250 TIME (nSECI Figure 4 Voltage oscillations in fibers with Ca++ and K+ conductances. External saline: either ASW (ai, aii, aiii, bi, biii, cii), Ca++-free ASW (biv), or, in all others, high Ca++ ASW. Internal saline: K+ perfusate. Voltage, mV; time, ms. (a) Voltage responses of a fiber to varying current stimulus. Current from top to bottom: 180, 540, and 900 /iA. (b) Effect of varying [Ca++]0. Current stimulus, 240 iiA. i, ii, and hi, iv are from two different fibers. In i and ii, [Ca++]„ is increased from 10 mM (i) to 100 mM (ii). In in and iv it is decreased from 10 mM (i) to 0 mM (iv). The contrast between i and ni also illustrates how variable the oscillatory response can be from fiber to fiber. Trace iv illustrates the RC response of the fiber in 0 mM Ca++. (c) Variety of oscillatory characteristics. Current stimuli are 180, 600, 180, 360,960, and 600 nA from i through vi. Morris and Lecar Voltage Oscillations in the Barnacle Giant Muscle Fiber 199 oscillations in normal saline (10 mM Ca++) usually will oscillate when 100 mM Ca++ is substituted. Fibers which oscillate in normal saline change their mode of oscillation in 100 mM Ca++ saline (Fig. 4b). The "piggyback" electrode assembly that perfuses saline through the cell interior produces variable degrees of damage to the invaginating membrane, which leads to individual variation in the nonspecific leak. During the voltage-clamp experiments, leak conductance of the barnacle fiber tends to increase as the preparation ages (personal observation). Current-clamped preparations, as they age, eventually either cease oscillating in response to stimuli or produce more and more damped oscillations. The variability of gL from preparation to preparation and with time in a given preparation may be a major factor in the lack of stereotyped behavior in barnacle fibers. It is worth pointing out in greater detail the oscillatory behavior types exhibited by the fiber, which encompass a broad range. Some fibers produce almost sinusoidal oscillations, and these are damped to varying degrees (Fig. 4cii and 4civ). Other fibers produce trains of spikes, with the details of their spike shape and their periodicity varying considerably (Fig. 4ci and 4civ). Quite frequently, fibers display a bistable type of oscillation: at the end of the stimulus the voltage falls to a slightly lower mean level, but continues to oscillate (usually for not > 100-200 ms) and then falls to rest (Fig. 4a/). Occasionally this bistability appears chaotic, with the fiber producing sporadic bursts of activity (Fig. 4cvi). How the individual conductances can produce the type of plateau behavior shown by the barnacle fiber seems fairly clear. It is a bit more difficult to decide intuitively whether these two nonlinear conductances, neither of which inactivates appreciably, could interact to create the array of behavior that oscillating fibers display. The mathematical analysis which follows is consistent with the simplest interpretation of the single-conductance experiments, and shows that a system of two noninactivating nonlinear conductances oscillates under a wide variety of conditions, and that the oscillations can take many forms, dependent on the values of conductance parameters and applied current. Unlike oscillations observed under current clamp, current oscillations observed under voltage clamp cannot occur in a system having only voltage-dependent conductances. Keynes et al. (1973) showed that voltage-clamp oscillations diminish or disappear when radial space clamp is improved (by "inflating" the fiber to open up the membrane clefts). In our analysis we assume perfect space clamp. We have done additional computations using two patches of membrane, each obeying Eq. 1, but separated by a series resistance. For current-clamp conditions we found that the two-patch model does not give oscillations qualitatively different from the space-clamped model, although the detailed quantitative solutions change. For voltage-clamp conditions, on the other hand, there is an optimum range of cleft resistance for which the cleft membrane shows current oscillations even though the surface membrane is under potential control. The oscillations seen under voltage clamp may therefore be induced as approximate current-clamp oscillations of the cleft membrane when departure from radial space clamp is taken into account. The two-patch computation simulated the voltage-clamp oscillations and also their diminution (by varying the cleft-resistance parameters). Thus, it seems reasonable to conjecture that, because of the complicated membrane geometry, the 200 Biophysical Journal Volume 35 1981 system responsible for voltage oscillations under current clamp can also account for current oscillations under voltage clamp under conditions that are, by definition, imperfect. ANALYSIS To explore the contribution of the components of the overall conductance system, we snail examine in turn the behavior of each of the two nonlinear conductances operating in isolation from the other. The behavior of the two nonlinear conductances acting in concert will be examined afterward. This should demonstrate how the interaction of the two nonlinear conductances produces novel behavior. Responses of System with a Single Voltage-dependent Conductance Consider a system in which either the Ca++ or the K+ system is working in isolation. Figs. 2b and 3c show that the all-Ca++ system has bistable responses and a characteristic threshold, whereas the all-K+ system exhibits a graded response with a transient peak followed by a decay to a voltage plateau. Let us generalize and write the equation of a single-conductance system as: I-CV+gL{V-VL)+gp(V-Vd A-X(K) - M] Here m can be either M or TV, and the subscript i stands either for Ca or K. The main qualitative difference between the two cases is whether V{ is less than or greater than VL. For these systems, Kq, > VL and VK < VL. Other parameters describing the voltage-dependent conductances are not very different for the two systems. Let us study the behavior of Eqs. 3 in the V, /i-phase plane. We can write the equations for the nullclines (that is, the V = 0 and (i = 0 isoclines) explicitly as (K- 0 nullcline) Kfci) - (/ + Sl^l + ^M^)/(gL + gi(i), (4a) (m = 0 nullcline) m( V) = m~( V). (4b) Eq. 4 shows that the V = 0 nullcline is a bilinear function of n, which varies between the values V(0) - (/ + gLvL) I gL, and W) - V + gLVL + g;V{) / (gL + g-X (5) Fig. 5 shows nullclines for typical all-K+ and all-Ca++ conditions, illustrating the qualitative difference between the two cases. When V, < VL (all-K+) there can be only one singular point, whereas when Vx > VL (all-Ca++) there can be as many as three singular points. The fi = 0 nullcline is approximately the same in both cases, with slight shifts along the voltage axis and slight differences in steepness. Morris and Lecar Voltage Oscillations in the Barnacle Giant Muscle Fiber 201 PHASE PLANES TWO VALUES CURRENT -50 -iO -30 -20 -10 0 10 20 30 10 v < nv) C I-*! 0.8 0.6 0.1 0.2 INTRACELLULAR Cf»» , III C ■ ACCUnULATION / a ^^0^ 0.2 -50 -10 -30 -20 -10 0 10 20 30 10 v invi -50 -25 V -I-H 0 25 50 Figure 5 (a) V,N phase plane for all-K+ system. Dotted curve shows a typical phase trajectory when applied current is switched from 0 to 400. V - 0 nullclines are shown for various values of applied current, /. The intersection of the N - 0 and V - 0 nullcline for each value of current is the sole singular point for the all-K+ system for that set of parameters and is always a stable node. Parameters used are gL - 3, gc - 0, g* - 8, VL - 50, VK--70, V3--1.0, VA - 14.5, \N - 1/15, C - 20. Voltage; V is expressed in millivolts. (b)V, M phase plane for all-Ca+ + system. Parameters are those given for Fig. 3b except for values of /. Dotted curve is the phase trajectory when / is switched from 0 to 100. The dashed curve is the M - 0 isocline. The two solid curves are the V = 0 isoclines for / - 0 and / - 100. The / - 0 nullclines intersect at three points. A, B, and C. The points A and Care stable nodes, whereas B is a saddle point. The pattern of the three singular points leads to bistable behavior as explained in the text. When / is switched to 100 there is only one singular point, C; this stable node determines the plateau voltage. The line S-S represents the threshold separatrix, sketched in approximately, (c) The effects of intracellular Ca+ + accumulation shown on the V, M phase plane. The dotted line is the M - 0 nullcline and tlje series of solid lines are V - 0 nullclines for increasing degrees of Ca accumulation (from right to left, gc, is decreased arbitrarily as follows: 40, 30, 25, 20, 10. Concurrently [Ca++]i is increased as follows: 0.001,0.1, 0.5, 1.0, 10. Other parameters are as for Fig. 3b, but with / - 0. Ca++ accumulation causes the V - 0 nullcline to shift to the left, ultimately causing the stable point C to disappear, the system reverting to monostable behavior about the resting point A. The computer response of the all-Ca++ system, with a linear instantaneous Ca++ I-V relation, gives an unrealistically high voltage plateau. The problem is that, although the linear I-V relation implies some outward current, the experiments were done with close to zero internal Ca++. Thus, a better quantitative fit to the plateau is obtained by modifying the driving force term in Eq. 3. It is more accurate to write: /c - -gc.MR(V, [Ca++L, [Ca++]0), (6) where, in general, the conductance gCa would be a function of the permeating ion concentration, but can be obtained by empirical fit to the instantaneous Ca /-Kcurve; and the function 202 Biophysical Journal Volume 35 1981 R is derived from some electrodiffusion or barrier model of permeation through an ionic channel. Although our own preliminary data indicate that a two-barrier model provides a better fit to the voltage-clamp data, for the present we shall make do with an electrodiffusion expression, since the exact form is not crucial for calculating the plateau, as long as the expression makes some sense for the case of zero internal Ca++. With the electrodiffusion form, the driving force function of Eq. 6 becomes: RIV rCa-1 rCa-1 ) - V{l ~ «Ca"],/[Ca"].)e»p (K/12.5)} tf(K,LCa j„lCa J0) - [i _ Cxp (K/12.5)] (?) The solutions shown in Fig. 3d are ones using this nonlinear expression for /Ca, and in subsequent discussions of the all-Ca++ system the nonlinear form is assumed. Later, when we study the K+ and Ca++ systems together, we revert to the linear approximation, because the voltage never leaves the linear region. Let us return to a consideration of the phase planes of the individual conductance systems. By definition, the intersection points of two nullclines are equilibrium positions of V and fi, that is, singular points of Eqs. 3. The patterns of these singular points and the stability of the system in the neighborhood of each singular point can qualitatively explain the observed behavior. For the all-K+ system, each nullcline pair shown in Fig. 5a intersects only once, at a singular point which is stable (V = 0 nullclines for several values of applied current are illustrated; the n = 0 nullcline does not change as / is varied). For the larger currents (/ = 100, / = 400), the trajectories reach a voltage peak as they cross the V = 0 nullcline. This represents the early peak seen in Fig. 2b; note that the peak occurs before breaches its steady state and that the voltage then declines. Such behavior, sometimes referred to as a graded action potential, has been observed and discussed for synthetic systems having a single voltage-dependent conductance (Mueller and Rudin, 1967; Muller and Finkelstein, 1972). Consider now the all-Ca++ system (Fig. 5b). There are three singular points in the phase plane for the value / = 0. Stability analysis shows that the points A and C are stable and that B is a saddle point. What happens when we go from / = 0 to, for example, / = 100? Because of the change in /, the system is described by a new V = 0 nullcline, to the right of the original one, but the M = 0 nullcline is not altered. The system now has only one singular point (C), which is approached as the potential is depolarized under the influence of / (the trajectory for the depolarization is shown as a dotted line). This singular point is stable, representing a plateau voltage of the sort shown in Fig. 3a". What happens to the system when we return from / = 100 to / = 0? The initial conditions are now C rather than A. The phase plane reverts to the original one with three singular points, including the saddle point, B. Saddle points always give rise to limiting curves called "separatrices," which cannot be crossed by phase trajectories. Thus, when the current is returned abruptly to zero, the system cannot cross the separatrix and return to the stable point, A. The only singular point available to the system is C, so that when the stimulus is turned off, the phase trajectory simply runs between C and C, which leaves the system in a depolarized state. Thus, the observed persistence of the voltage plateau when the stimulus is turned off is a manifestation of bistability. Were an impulse of current used to raise instantaneously the voltage past the region where the separatrix Morris and Lecar Voltage Oscillations in the Barnacle Giant Muscle Fiber 203 intersects the K-axis, bistability would be demonstrable in the / = 0 phase plane alone. Except for the use of a nonlinear term for the driving force, the behavior of the all-Ca++ system is exactly that of the theoretical K,w-reduced Hodgkin-Huxley equations devised by FitzHugh (1961) to explain threshold behavior in a qualitative way. FitzHugh's K,m-reduced system describes the discontinuous threshold behavior of a hypothetical system with a single activating conductance and no inactivation. The all-Ca++ barnacle fiber is a realization of this mathematical possibility. Perturbations Caused by Ca+ + Accumulation The all-Ca++ version of Eq. 3 predicts stable plateau action potentials, but the real fiber does eventually repolarize. The mechanism directly responsible for repolarization has not been isolated. As it could be simple Ca++ accumulation or some form of slow inactivation, we now breifly describe in terms of the phase plane analysis how a slow process like Ca++ accumulation brings about repolarization. Accumulation of Ca+ + intracellular^ will simultaneously cause gCa to decrease and [Ca++]j to increase. To get a qualitative idea how this will affect the response, we need only look at the V = 0 nullclines as we alter these two parameters (changes in these parameters do not change the M = 0 nullcline since they do not appear in Eq. 4b). Note (Fig. 5c) that the upper stable point and the unstable point converge and then disappear as Ca++ accumulation proceeds (i.e., as ga decreases and [Ca++]j increases; the decrements and increments in these parameters are arbitrarily chosen). When the unstable point drops out, along with its separatrix, only one stable point remains and the system moves toward it, repolarizing as it does. A more direct way to demonstrate the termination of the plateau is to include an equation for /Ca-dependent intracellular Ca++ accumulation. We substitute the steady-state value of A/„ for the term M(t), since ion accumulation has very much slower kinetics than gating. The resultant differential equations are: dV - C-1 {/ - ^{V- VL) + gCiM„(V)VR(V, [Ca+ + ](, [Ca+ + ]0)h and d[Cda,++]i - K\(CF)~l gcMSV)VR(V, [Ca++]i( [Ca++]0)}. (8) Although the second equation is oversimplified (it does not take into account various sinks for intracellular Ca++), it describes the major feature of accumulation. The value for the constant K was chosen to represent a Ca++-accumulating compartment of 1 fim on the inner side of the membrane. Fig. 3d is the computed solution of Eqs. 8, which uses initial conditions that simulate the end of a stimulus pulse. After a gradual decline in voltage, there is a precipitous fall to rest. The actual mechanism for repolarization could be more complex than Ca+ + accumulation, since Hagiwara and Nakajima (1966) have shown that Ca++ modulates the height of the Ca++ spike in a manner not consistent with simple changes in the driving force. As in some other Ca++-excitable membranes (Eckert and Brehm, 1979), the Ca++ conductance in barnacle muscle may be inactivated by elevated [Ca++]j. Such a mechanism, however, would also be modelled, qualitatively, by concomitantly increasing [Ca++]j and decreasing gCi. 204 Biophysical Journal Volume 35 1981 Limit-cycle oscillations The most interesting feature of the barnacle muscle behavior under current clamp is the voltage oscillation, which occurs only when the Ca++ and K+ system are both operative. As different explanations have been adduced for the oscillations, we wish to explore how well the oscillations are explained by the two-conductance system itself with no additional mechanisms. It can be shown (Fig. 6) that the V, M, N system of Eq. 1 predicts current-induced oscillations resembling the observed behavior. Numerical simulations, however, such as those of Fig. 6, require the simultaneous adjustment of several parameters, and it is not easy to learn the exact requirements for oscillation from numerical study of the full third-order system. To study the oscillating state in some generality, we make use of the different relaxation times of the Ca++ and K+ conductances. We study a reduced set of equations, in which the Ca++ system is assumed to be so much faster than the K+ system that gCi is instantaneously in steady state at all times [i.e., M = Mjy)]. For this approximation Eq. 1 becomes 1 ~ C^7 + SdV ~ K) + gcM~{V) (V- KCa) + gkN(V - KJ, at and ™ = \N(V)(N„(V) - N). (9) We shall call this the V,N reduced system, following the terminology of FitzHugh (1969). Eq. 9 describes a second-order system, whose properties can be visualized on the F,7V-phase plane. It might be asked why we expect Eq. 9 to be a reasonable approximation to the full system. The justification lies in Tikhonov's theorem, utilized by Plant and Kim (1976; see their appendix). Eqs. 1 are of the proper form for the Kand TV differential equations to satisfy an 20 10 o -10 V (MV) -20 -30 -10 "5°0 40 80 120 160 200 TIr\E (MSEC) Figure 6 Examples of oscillations computed from the full V, M, N third-order system (Eq. 1). The parameters used for these computations are: gL - 2, VL--50, - 100, KK - — 70, \u - 1.0, \N - 0.1, V\ - 0, V2 - 15, K3 - 10, V4 - 10, C - 20. The resting potential was taken to be -50 mV. For the broken line ( + ), g-'-**ly->lv--v°i. (■■> gaiV, - VK) In the usual fashion (Minorsky, 1962), Eqs. 9, when thought of as a general second order system of the form V=MKN), N=f2(V,N), (12) will have stability properties determined by the character of the eigenvalues of the pair of equations linearized about the singular point, S. The eigenvalues (p), in turn, are the solutions of the characteristic equation: y ' 3" dNhF \dVdN dVdNJs V ' In order to have a stable limit cycle, both roots must be positive if they are real or have a positive real part if they are complex. Since the first term in brackets in Eq. 13 is equal to the sum of the roots, and the second term in brackets is the product of the roots, the necessary condition for both roots positive is: 6W dVdNjs K ' 206 Biophysical Journal Volume 35 1981 Figure 7 Figure 8 Figure 7 Nullclines in V, N that give a stable limit cycle. The nullclines are labeled; the spiral traces the trajectory of the system when / is changed from 0 to 300. The point at the intersection is an unstable node. Vertical and horizontal bars on the nullclines indicate the direction in which the trajectory must cross. Computations are based on the parameters used in Fig. 9, but with / - 300. The sufficient condition for oscillation is given by Eq. IS. Essentially, the condition states that the intersection must occur somewhere in the negative-resistance part of the Ca++ current-voltage curve. This is the ascending limb of the bump in the V — 0 nullcline, as shown in the figure. An analogous condition for a somewhat different reduced system is given by Plant and Kim (1976). Voltage, V, mV. Figure 8 Plot of real and imaginary parts of eigenvalue (p, in Eq. 13) of the linearized V, A'-reduced equations as current is varied. The curve shown is the root with plus sign, the negative root leading to a mirror reflection (not shown) across the real axis. When the eigenvalues are real, they are represented by points on the real line. The shaded loop marks the range of current values for which the real part is positive, so that there is a stable limit cycle. The imaginary part of the eigenvalue is approximately equal to the oscillation frequency, so that the value on the imaginary axis shows the pattern of frequency variation with applied current. Parameters used in these computations are those of Fig. 9, but with / varied as indicated by the numbers adjacent to the curve. The first inequality guarantees at least one positive root and the second guarantees that both roots have the same sign. From Eq. 9 we can substitute the explicit expressions for the partial derivatives of Eq. 14, to obtain the following inequalities: ga ldM\ \ dVh (Vc, ~ K) >gL + SkWs + gcMK) + CXN(KS), and Sea (^)s (VCi - Vs)