Computer simulations of high-pass filtering in zebrafish larval muscle fibres
1 MRC Functional Genetics Unit, Department of Human Anatomy and Genetics,
University of Oxford, South Parks Road, Oxford, OX1 3QX, UK
2 University of Alberta, Department of Biological Sciences, Biological
Sciences Building, Edmonton, Alberta, Canada, T6G 2E9
* Author for correspondence (e-mail: declan.ali{at}ualberta.ca)
Accepted 15 June 2005
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Summary |
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Key words: ion channel, sodium, potassium, activation, inactivation, action potential
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Introduction |
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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 [IK(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.
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Materials and methods |
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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 k
cm2 and axial resistance of 35.4
cm. 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 cm2. 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.
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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 cm2,
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.
|
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Results |
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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.
|
Determination of the parameters that affect once-only or repetitive firing
Co-varying V50 of activation and inactivation and time constant of inactivation 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
V50,act values for the sodium current from 50 mV to
+5 mV in 5 mV steps, and for each value of V50,act we
varied the V50,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 V50,act and
V50,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.
|
|
Slope of activation and inactivation of the sodium current
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.
Altering potassium current parameters
Simultaneously varying the V50,act and
V50,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
V50,act and V50,inact were varied over
this same range.
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Discussion |
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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 510 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 35 mV and
membrane-charging time constants to the order of 1215 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 V50,act and V50,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 V50,act and
V50,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 IK(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 V50,act and V50,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.
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