Computer simulations of high-pass filtering in zebrafish larval muscle fibres

The zebrafish, Danio rerio Hamilton, is a particularly convenient model for experimental investigations into developing nervous and muscle systems. The advantages it offers include the invariant pattern of development of the somatic motor system (especially the trunk muscles that underlie swimming) and its ready accessibility to electrophysiological studies(Drapeau et al., 1999; Westerfield, 2000). Despite these advantages, the electrophysiological properties of the developing muscle in larvae of zebrafish have received surprisingly little attention(Buckingham and Ali, 2004; Buss and Drapeau, 2000). Zebrafish axial muscle is typical of teleost muscle in that it consists of both red and white fibre types(Greer-Walker and Pull, 1975; van Raamsdonk et al., 1979)with different behavioural roles (Buss and Drapeau, 2002).

A number of neurones in different organisms respond to a sustained depolarization with a short burst of spikes, sometimes with only one spike(Buss et al., 2003; Eliasof et al., 1987; Korn et al., 1990; Rothe et al., 1999; Torkkeli and French, 2002; Tsutsui et al., 2001). This represents a form of high-pass filtering, in which the input to the neurone is differentiated when in the increasing direction. The frequency of the phenomenon of once-only firing suggests that it is an important physiological adaptation. The ionic basis of this phenomenon, however, is often poorly understood. Furthermore, except in the case of sensory neurones showing rapid adaptation to sustained stimuli, the behavioural significance of high-pass filtering in many neurones is not clear. The white muscle fibres of the tail of developing Danio rerio support once-only, or single-spike, firing and are very convenient for patch-clamp studies(Buckingham and Ali, 2004; Buss and Drapeau, 2000; Drapeau et al., 1999), hence they offer a convenient model for the study of high-pass filtering in a well-defined behavioural context.

The phenomenon of once-only firing in fish muscle is thought to be an adaptation to the functional role of these muscles in swimming, which in larvae involves rhythmic contractions at rates up to 30 per second(Buss and Drapeau, 2000). Once-only firing presumably ensures that the muscle can be activated at high frequencies without evoking tetanic contractions, which may compromise regular, rhythmic contractions. Currently, little is known of the ionic basis of once-only firing, although it occurs in a number of different neurones in both vertebrates and invertebrates. The spider VS-3 slit-sense organ, for example, contains two neurones, of which one exhibits once-only firing while the other fires tonically in response to a sustained stimulation(Seyfarth and French, 1994). In computer simulations of these neurones, altering two parameters – the slope of the steady-state inactivation curve and the time constant of inactivation of the sodium current – was sufficient to switch between these two patterns of firing (Torkkeli and French, 2002). Similarly, large cells in the corpus glomerulosum of adult filefish, Stephanoplepis cirrhifer, fire only once in response to a prolonged depolarization(Tsutsui et al., 2001). This phenomenon has been modelled by a computer simulation(Tsutsui and Oka, 2002), but the ionic events causing it were not explored. Immature retinal ganglion neurons (RGN) exhibit single-spike firing during postnatal days P1 to P11,thought to be a consequence of insufficient calcium-activated potassium current [I_K(Ca)], while retinal amacrine cells fire single spikes because of insufficient Na⁺ conductance and a lack of removal of Na⁺ inactivation(Eliasof et al., 1987).

In a recent paper in which we described the steady-state and kinetic properties of a sodium and potassium current present in the inner muscles of the zebrafish (Buckingham and Ali,2004), we suggested that once-only firing may arise from the fact that the sodium current is rapidly inactivated and requires a rebound to more hyperpolarized voltages before sufficient channels are reactivated. Since at a resting potential of approximately –70 mV in the larvae(Buss and Drapeau, 2000) and approximately –80 mV in the adult(Westerfield et al., 1986) up to 70% of sodium current is inactivated(Buckingham and Ali, 2004), a second spike might only be possible after a hyperpolarization. We proposed that this hyperpolarization is provided by the rapidly inactivating potassium current. An alternative explanation (Adrian and Bryant, 1974; Bryant,1962; Bretag, 1987)is that a voltage-gated chloride current provides a shunt, preventing a second spike during a prolonged depolarization. Since it is extremely difficult to manipulate the steady-state and kinetic properties of sodium channels experimentally, we here report a computer model of zebrafish muscle incorporating the sodium and potassium channels, to determine whether the properties of these ion channels as described are sufficient to explain once-only firing. The rationale of our approach is first to model the voltage-clamp currents as previously described and then to systematically alter the measured parameters until once-only firing is lost, i.e. repetitive firing is obtained. This approach therefore effectively predicts the properties of these ion currents that determine once-only or repetitive firing.

The cell was modelled with a single segment of diameter of 12 μm, length of 70 μm (based on measurements of muscle fibres in situ; D.W.A. and S.O.B., personal observations), specific membrane resistance of 1 kΩcm^–2 and axial resistance of 35.4 Ωcm. The model exploited the built-in feature of NEURON in which the `d_lambda' value (the fraction of the length constant at 100 Hz) of 0.1 was used to set the number of compartments. Because this value produced a length constant considerably greater than the length of the segment, the simulations contained only one compartment. A non-specific passive conductance was inserted and set to 0.001 S cm^–2. The reversal potentials for sodium and potassium ions were set to +50 mV and –77 mV, respectively, and the reversal potential for the passive current was set to –70 mV, unless otherwise specified. To mimic a cell being recorded with a patch pipette, current was injected and current or voltage recorded at the same point, in the middle of the cell's length. Parameters were changed as described in the Results section. The values for the steady-state properties were taken from Buckingham and Ali(2004), and the values are shown in Table 1. The time constants for inactivation were taken from the original data over the range–35 mV to +60 mV, and the data for ranges more negative than –70 mV were taken as the rate of recovery from inactivation, assuming inactivation and recovery from inactivation to be the same simple transition passing in opposite directions. The rate of activation was estimated from the data in Buckingham and Ali (2004) and adjusted to produce current traces in response to voltage-clamp steps that resembled the actual recordings. These time constants were supplied to the programme as look-up tables (Tables 2, 3) for interpolation. To simplify the model, and where experimental data are lacking or incomplete, we made some assumptions about the time constants of activation and inactivation of the currents. We assumed that inactivation and recovery from inactivation can be represented by a simple set of transitions, the time constants for which in both directions have an identical voltage dependency. The time constants of activation for both sodium and potassium are difficult to measure with any accuracy and therefore have not been reported. For potassium, a simple voltage-independent value of 1 ms was assumed, whereas for sodium the values shown in Table 2 were estimated from fig. 4 of Buckingham and Ali(2004) and adjusted to produce currents that resembled those recorded in situ. Values for the time constant of inactivation for potassium were derived from Buckingham and Ali(2004) and are shown in Table 3. Pilot simulations revealed a sharp dependency of the time course of the sodium current upon the inactivation time constant, especially at membrane potentials at which sodium currents begin to appear (–40 to –30 mV). Since the rate of inactivation and the rate of recovery from inactivation at these potentials are unknown, we adjusted the values for this transition at these membrane potentials until currents with realistic time courses were achieved. The values that were obtained in this way (0.3 ms at –50 mV and 0.2 ms at–30 mV) are in reasonable agreement with the experimental data.

Reliable values for the V₅₀ and slope of activation and inactivation for the sodium and potassium currents are available(Buckingham and Ali, 2004), so we considered these to be fixed parameters, except where these parameters were under study. The maximum conductances of the sodium and potassium currents were adjusted until their peak currents matched those in real voltage-clamp experiments; maximum conductance values of 7 and 0.21 S cm^–2,respectively, were found to result in currents of the appropriate amplitudes(Fig. 1). No other parameters had to be adjusted to produce currents that resembled those observed in similar experiments in situ.

Currents in response to a series of depolarizations from –70 to +60 mV closely resembled those recorded in situ(Fig. 1), although the V_50,act was shifted by ∼10 mV in the depolarizing direction and the Boltzmann slope of activation also reduced. We attempted to adjust the time constants of activation to correct this but found that,although compensatory shifts in V₅₀ could be achieved by speeding activation and slowing inactivation, the I–Vcurves and the time courses of the currents differed from those recorded in vivo. We therefore conclude that the difference between the model and in situ values for V_50,act and the Boltzmann slope is due to inaccuracies in assuming that the peak inward current is proportional to the maximum conductance at any membrane potential. These inaccuracies arise because the similar values for the time constants of activation and inactivation introduce errors in estimating Hodgkin Huxley parameters (m and h) from peak currents. Imperfect space clamp in whole-cell patch-clamp recordings of muscles may be a further source of error. Improper voltage control will lead to an underestimation of current amplitudes and increases in the time constants of activation. To confirm that adequate voltage clamp conditions were met, we ensured that series resistances were less than 5 MΩ before compensation, that Na⁺ currents reached their peak within 500 μs and that the Na⁺ currents reversed within 5–10 mV of their theoretical reversal potential(Sontheimer et al., 1992; Buckingham and Ali, 2004). In addition, any cells showing uncontrolled regenerative inward currents were omitted from our study (Eliasof et al.,1987). As a result, we favour the interpretation that the muscle cells in our study were space clamped well enough to reduce this source of error to acceptable levels. However, since our simulations explore the effects of varying steady-state and kinetic properties as independent variables, we proceeded with the values described above.

We ran simulations of current-clamp experiments to determine the resultant firing properties of the model. Currents less than 2 nA did not evoke spikes,but currents of ≥3 nA evoked one, and only one, spike(Fig. 2A). More than one spike was never observed at currents up to 10 nA, at which level of depolarization the first spike began to appear distorted. As was reported for spikes recorded in situ (Buckingham and Ali,2004), the action potentials recorded in our simulations were graded, being of lower amplitude when evoked from more depolarized potentials(Fig. 2B).

In current-clamp experiments (Buckingham and Ali, 2004), zebrafish muscle can follow a series of depolarizing pulses with action potentials, provided the pulses are separated by a hyperpolarization. The same phenomenon is supported by our modelled cell(Fig. 3A). Indeed, the model also accurately mimicked the graded recovery of the action potential as the inter-pulse interval is increased (Fig. 3B), as well as the dependence of the rate of spike recovery upon the inter-stimulus holding potential (Fig. 3B). In our previous paper, we surmised that once-only firing is a result of a requirement for inter-stimulus hyperpolarization, whose effect is to remove inactivation of the sodium current. An analysis of the value of the inactivating state variable of the sodium current showed that hyperpolarization to around –90 mV between two successive pulses does indeed significantly increase the value compared with hyperpolarization to–70 mV (Fig. 3C). To determine whether this difference in the inactivation state particle is responsible for differences in the rate of spike recovery, we performed a dual pulse simulation similar to that in Fig. 3A and halted it 0.025 ms before the second stimulation, at which point we set the value of the inactivation particle to 0.1205, the value obtained at the same time when the simulation is run with an interpulse 0.7 nA hyperpolarization. The simulation was then allowed to continue. The resulting second spike showed partial recovery to full amplitude(Fig. 3D). This provides evidence that, in our simulation at least, the difference between spike recovery at these two potentials is attributable to the inactivation state of the sodium current.

To determine the parameters of the sodium current that determine whether a sustained depolarization will result in a single spike or a train of spikes,we systematically varied a number of parameters in the model of the sodium current. In the first set of simulations, we varied the V_50,act values for the sodium current from –50 mV to+5 mV in 5 mV steps, and for each value of V_50,act we varied the V_50,inact from –90 mV to +5 mV in 5 mV steps. The number of action potentials during a 50 ms depolarization from a resting potential of –70 mV was counted. The runs were then sorted according to whether they elicited no action potentials, only one action potential or more than one action potential. The data were then plotted as a two-dimensional matrix with V_50,act and V_50,inact as the independent variables(Fig. 4). Because the action potentials were graded, we chose to select the criterion for an action potential as any spike that crosses +20 mV. Increasing the depolarizing current shifts the diagonal border between once-only firing (red region, Fig. 4A) and no response (black region, Fig. 4A) but does not significantly shift the areas in which repetitive firing occurs (green region, Fig. 4A, reading from left to right). This more amply illustrates our previous finding that increasing the level of depolarization does not lead to repetitive firing. When the time constant of inactivation is doubled (slowing the rate of inactivation), the area of parameter space over which repetitive firing is obtained is increased(Fig. 4B). However, increasing the amplitude of the depolarizing currents still did not greatly increase the area of repetitive firing.

It seemed plausible that the current densities of both sodium and potassium currents might affect the firing pattern. We therefore mapped the firing pattern of the model against the conductance density for both currents(Fig. 5). Altering either conductance density did not induce repetitive firing except at very high sodium current densities (more than a factor of 10; Fig. 5A). It has been suggested that slowing the rate of inactivation of voltage-gated sodium currents can result in changes in firing properties(Cantrell and Catterall,2001). A marked change in firing pattern was observed in our model by slowing the rate of inactivation of the sodium channel by multiplying it by 2 (Fig. 5B) or 4(Fig. 5C).

A shift from once-only firing to repetitive firing in spider mechanosensory cells required a change in the slope of activation of a sodium current(Torkkeli and French, 2002). We simultaneously varied the slope of both activation and inactivation for the sodium current from ±3 to ±9.8 in 0.2 steps. Although two spikes could be elicited for some values, repetitive firing was never obtained over these ranges of values.

Simultaneously varying the V_50,act and V_50,inact of the potassium current between –40 mV and +10 mV and between –60 mV and +10 mV, respectively, did not produce repetitive firing. Increasing the time constant of activation of the potassium current from 1 to 2, 4, 8 or 16 ms also failed to produce either repetitive firing or dampened oscillations. Similarly, doubling the time constant of inactivation failed to produce repetitive firing as the V_50,act and V_50,inact were varied over this same range.

Here, we model the ion currents we reported in a previous recent publication to determine whether such a description of those currents is sufficient to explain the phenomenon of once-only firing observed in zebrafish larval skeletal muscle. The model demonstrates that without having to alter the values of V_50,act and V_50,inact or the slopes of activation or inactivation or the levels of current expression,these currents do indeed generate once-only firing, once the time constants for inactivation and activation are adjusted to produce currents that resemble those observed in voltage-clamp experiments. This argues that it is not necessary to invoke any extra currents, such as voltage-gated chloride currents, although it does not show that such currents do not contribute to the phenomenon in situ.

Our model differs from in vivo patch-clamp recordings in that it does not take into account possible errors arising from inadequate space clamp. Simulated current injections produced membrane potential deflections that did not differ along the length of the fibre. The same was seen when the model cell was divided into 50 compartments (data not shown). Thus, the model had good space clamp. Whole-cell patch-clamp recordings from muscle, however,suffer from imperfect space clamp, as well as errors arising from the series resistance of the recording pipette. In our previous study, we confirmed adequate space clamping of the muscle fibres by only using cells that did not show any uncontrolled regenerative inward currents and in which the Na⁺ currents occurred within several hundred microseconds from the onset of the voltage steps. Lastly, we ensured that Na⁺ currents reversed within 5–10 mV of the theoretical reversal potential. It is possible that a lack of adequate space clamping could negate our ability to record more than a single spike, but we were confident that we had adequate voltage control of our cells and that the single spike phenomenon we recorded is also seen in vivo. Buckingham and Ali(2004) estimate that series resistance in their recordings could amount to 3–5 mV and membrane-charging time constants to the order of 12–15 μs after 85%series resistance compensation using the amplifier's circuitry. These differences in space clamp are not taken into account by our model.

The values for the inactivation rate of the sodium current had to be adjusted to produce sodium currents under voltage clamp with properties resembling those of in situ currents. This is probably attributable to errors in estimating these values from patch-clamp recordings, arising from the fact that it is difficult to measure rates of activation from rapidly inactivating currents where activation is incomplete before significant inactivation occurs. Although this would be a potentially serious source of error in the model if in situ data were used naively, it is assumed that at least some of this error is compensated for by the adjustments to the values of time constant performed to produce realistic currents.

When V_50,act and V_50,inact are mapped (Fig. 4) against each other in terms of the firing properties that they produce, it can be seen that the values obtained in voltage-clamp experiments on zebrafish muscle in situ lie in a position away from where repetitive firing is obtained. This suggests that currents that can be described in this way are unlikely to mediate repetitive firing unless significant changes to their properties are made. Where the data are mapped as in Fig. 4, the minimum change in V_50,act and V_50,inact required to bring about repetitive firing is represented by the shortest line from the in situ values to the borderline of the area in which repetitive firing is obtained. The coordinates of the end of this line represent the steady-state values at which repetitive firing is obtained, and in the model this represents a convergence of the steady-state curves to bring about an increased overlap(Fig. 6). A number of studies have shown that some steady-state parameters and maximum current density of voltage-gated sodium channels can be modulated by intracellular enzymes such as tyrosine kinases and phosphatases (Alroy et al., 1999; Hilborn et al.,1998; Ratcliffe et al.,2000) or by second messengers such as G-protein subunits (Ma et al., 1994, 1997). However, the model reported here suggests that the inactivation rate of the sodium current must be at least quadrupled if the firing properties of the cell are to change from once-only firing to repetitive discharges. Alternatively, the model suggests that such a change of firing properties could be effected by a simultaneous rightward shift in inactivation and a leftward shift in activation. These steady-state changes differ from those of Torkkeli and French(2002), except that both mechanisms are due primarily to the amount of sodium channel inactivation. However, a number of other groups have found a variety of mechanisms for once-only firing. For instance, a large amount of sodium inactivation coupled with an insufficient sodium conductance is thought to be responsible for the single-spike firing in retinal amacrine cells of the tiger salamander(Eliasof et al., 1987). Neurons of the medial nucleus trapezoid body relay synaptic information and follow presynaptic action potentials at frequencies up to 600 Hz(Wu and Kelly, 1993). These neurons fire single spikes in response to long depolarizing steps and can be induced to fire multiple action potentials when potassium channels are blocked by dendrotoxin (Brew and Forsythe,1995). Single-spike firing may also be developmentally regulated when the influence of an I_K(Ca) is strong and results in once-only firing at early postnatal stages(Rothe et al., 1999). We conclude from our findings that the changes necessary to affect the firing properties of the cells are greater than are likely to be observed in real cells and that the sodium channels of zebrafish muscle are optimized to prevent repetitive discharge and ensure once-only firing even at high levels of excitation.

Our parameter mapping approach has shown that, in our computer model, the parameter that most affects whether once-only or repetitive firing is obtained is the time constant of inactivation of the sodium current. Although the most common use of the somatic tail muscle will be in swimming, it is likely that it is also used in postural behaviours, not all of which require rapid pulsatile contractions. Indeed, our model suggests that, with the steady-state and kinetic properties reported using voltage clamp, the muscles would not be able to maintain a sustained contraction, making tonic posture control difficult. It is tempting to speculate that a switch from phasic to tonic control of the muscle might be made possible by modulation of the sodium current, which effects an increase in the time constant of inactivation together, possibly, with an effective increase in the sodium current density or the V_50,act and V_50,inact. As argued above, there is evidence that such mechanisms exist, although it is unclear whether alterations in kinetics or steady-state properties through intracellular modulation of voltage-gated ion channels would be fast enough to allow such switching. Further, adaptations for once-only firing are also evident in two elements of the afferent pathway: motor neurones to these muscles respond to a sustained depolarization with a short burst consisting of only a few spikes (Buss et al.,2003), and the Mauthner neuron, which is presynaptic to the primary afferents, also exhibits once-only firing(Charpier et al., 1995; Hatta and Korn, 1998; Korn et al., 1990). It is of interest that three consecutive elements in this motor pathway are primed to ensure high-pass filtering.

We have shown in a previous paper that the maximum rate of full spiking in response to pulsatile stimulation is dependent upon the inter-stimulus membrane potential (Buckingham and Ali,2004), and we suggested that this might reflect the dependence of the recovery from inactivation of the sodium channel. This model confirms this, as artificially resetting the state of the sodium inactivation particle to a value obtained at resting potentials of –90 mV restores the amplitude of the following spike. As yet, we know of no way of imitating this experimentally.

Our findings show no role for the potassium current in ensuring once-only firing, as no perturbation of the potassium current succeeded in bringing about repetitive firing.