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The Journal of Neuroscience, February 15, 2003, 23(4):1506
Slow Na+ Inactivation and Variance Adaptation in
Salamander Retinal Ganglion Cells
Kerry J.
Kim and
Fred
Rieke
Department of Physiology and Biophysics, University of Washington,
Seattle, Washington 98195
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ABSTRACT |
The retina adapts to the temporal contrast of the light inputs. One
component of contrast adaptation is intrinsic to retinal ganglion
cells: temporal contrast affects the variance of the synaptic inputs to
ganglion cells, which alters the gain of spike generation. Here we show
that slow Na+ inactivation is sufficient to produce
the observed variance adaptation. Slow inactivation caused the
Na+ current available for spike generation to depend
on the past history of activity, both action potentials and
subthreshold voltage variations. Recovery from slow inactivation
required several hundred milliseconds. Increased current variance
caused the threshold for spike generation to increase, presumably
because of the decrease in available Na+
current. Simulations indicated that slow Na+
inactivation could account for the observed decrease in excitability. This suggests a simple picture of how ganglion cells contribute to
contrast adaptation: (1) increasing contrast causes an increase in
input current variance that raises the spike rate, and (2) the
increased spike rate reduces the available Na+
current through slow inactivation, which feeds back to reduce excitability. Cells throughout the nervous system face similar problems
of accommodating a large range of input signals; furthermore, the
Na+ currents of many cells exhibit slow
inactivation. Thus, adaptation mediated by feedback modulation of the
Na+ current through slow inactivation could serve as
a general mechanism to control excitability in spiking neurons.
Key words:
contrast adaptation; slow Na+
inactivation; modulation of Na+ current; retinal
ganglion cell; models for spike generation; adaptation; spike-frequency
adaptation; retinal signal processing
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Introduction |
Sensory signals are highly variable,
and these variations occur on a wide range of time scales. To encode
these signals efficiently, sensory systems adapt, trading sensitivity
to slowly changing aspects of the input for increased sensitivity to
rapid changes. Adaptation permits sensory neurons to use their limited
dynamic range effectively. Visual neurons adapt to the mean light
intensity (for review, see Walraven et al., 1990 ) and to the amplitude
of the variations about the mean (i.e., the spatial and temporal contrast) (for review, see Shapley, 1997; Meister and Berry, 1999). Although the properties of mean and contrast adaptation have been studied in detail, we have an incomplete understanding of the mechanisms responsible. Here we investigate the contribution of Na+ currents in retinal ganglion cells to
temporal contrast adaptation.
Changes in temporal contrast cause adaptation in cells throughout early
visual pathways, including those in retina (Shapley and Victor, 1978;
Sakai et al., 1995; Smirnakis et al., 1997; Chander and Chichilnisky,
2001) and cortex (Albrecht et al., 1984; Ohzawa et al., 1985;
Sanchez-Vives et al., 2000a). Contrast adaptation in the retina
includes contributions from the retinal circuitry (Sakai et al., 1995;
Rieke, 2001) and intrinsic properties of ganglion cells (Kim and Rieke,
2001b). Retinal cells have a high gain for light increments and
decrements about a steady mean, causing saturation at the onset of a
high contrast stimulus (Burkhardt et al., 1998). Contrast adaptation
permits recovery from such saturation by reducing gain in the
maintained presence of time-varying lights.
Changes in temporal contrast alter the variance and in some cases the
mean of synaptic inputs to visual neurons. The increase in variance
reduces the gain of spike generation in retinal ganglion cells (Kim and
Rieke, 2001b). Thus, a component of contrast adaptation is produced by
a variance-induced change in the input-output relationship of a
ganglion cell. Ca2+-activated
K+ currents shape the input-output
relationship of many cells through effects such as spike-frequency
adaptation (for review, see Sah and Davies, 2000). However, neither
K+ nor Ca2+
currents contribute substantially to variance adaptation in ganglion cells; instead, variance adaptation is generated by properties of the
Na+ currents (Kim and Rieke, 2001b).
We investigated the mechanisms permitting ganglion cells to adapt to
the input current variance. These experiments led to three main
conclusions: (1) slow inactivation reduced the
Na+ current available for spike generation
after both subthreshold depolarizations and spikes; recovery from slow
inactivation required several hundred milliseconds; (2) slow
inactivation caused the fraction of the
Na+ current available for spike generation
to depend on the variance of the input currents to the cell; and (3)
simulations showed that action potentials rather than subthreshold
voltages dominated the extent of slow Na+
inactivation. These simulations reproduced the experimentally observed
variance adaptation. Thus, the mechanism for variance adaptation was
simple: increased current variance caused a higher firing rate, which
in turn reduced the available Na+ current
and increased spike threshold.
Some of this work has been published previously in abstract form (Kim
and Rieke, 2001a).
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Materials and Methods |
Experimental procedures
Dissection. All experiments used retinas from larval
tiger salamanders (Ambystoma tigrinum) from Charles
Sullivan (Nashville, TN). Animal procedures followed protocols approved
by the Administrative Panel on Laboratory Animal Care at the University
of Washington (Seattle, WA). Procedures for preparing and dissociating
the retina to obtain isolated cells have been described previously (Kim
and Rieke, 2001b). Cells were continuously superfused during recording with a HEPES Ringer's solution containing (in
mM): 136 NaCl, 2 KCl, 1.5 CaCl2, 10 glucose, 2 NaHCO3, 1.6 MgCl2, and 3 HEPES; pH was adjusted to 7.4 with NaOH, and osmolarity was 270-275
mOsm. Experiments were performed at 20-22°C.
Ganglion and amacrine cells were distinguished from other cell types by
their ability to generate Na+ spikes. Data
were collected only from cells that were able to generate repetitive
action potentials with a width of < 2 msec and a minimum
amplitude of  60 mV above the resting voltage. In two experiments we
labeled the ganglion cells by retrograde transport of rhodamine dextran
through the optic nerve (Lukasiewicz and Werblin, 1988). After
dissociation, >95% (178 of 185) of the cells with morphology similar
to that of the recorded cells were ganglion cells. We did not attempt
to distinguish between ganglion cell types.
Patch recording procedure. Voltage and current responses of
ganglion cells were measured using perforated-patch or whole-cell recordings. Similar results were obtained in each case. Patch pipettes
were filled with a solution containing (in mM):
115 Cs-aspartate, 20 CsCl, 10 HEPES, 1 N-methylglucamine
(NMG) EGTA, and 0.2 CaCl2; pH was adjusted to 7.2 with NMG-OH, and osmolarity was 260-265 mOsm. For whole-cell
recordings, 1 mM ATP and 0.2 mM GTP were added to the internal solution. For
perforated-patch recordings, the pipette tip was filled with an
amphotericin-free solution, and the pipettes were back-filled with
internal solution with an additional 1 mg/ml solubilized amphotericin-B
(Sigma, St. Louis, MO). Filled pipettes had resistances of
3-5 M , and the series resistance during recording was 10-20 M.
Ganglion cells had resistances between 1 and 4 G and capacitances
between 10 and 20 pF.
Na+ current inactivation and recovery were
measured in voltage-clamp recordings. To isolate
Na+ currents,
K+ and Ca2+
currents were suppressed by replacing K+
with Cs+ in both internal and external
solutions and adding 0.1 mM
Cd2+ to the external solution. Block of
the Ca2+ current was confirmed in some
cells by adding 100 nM TTX to the external solution; this
eliminated all inward current in response to depolarizing voltage
steps. To improve the voltage clamp, the Na+ current was reduced by replacing
up to 100 mM of external Na+
with NMG+. Voltages have been corrected
for junction potentials, which were <10 mV.
Adaptation of the spike response of a cell to changes in the
variance of the injected current was measured in current-clamp recordings. Random noise with a Gaussian distribution (bandwidth, 0-50
Hz) was injected into the cell, and the variance of the distribution was changed periodically without changing the bandwidth. The mean holding current was between 0 and 10 pA, resulting in a firing rate of
2-6 Hz. Under these conditions, the mean voltage was between  60 and
55 mV.
Data analysis
We quantified adaptation of spike generation to the variance of
the injected current using a static nonlinearity model (Brenner et al.,
2000; Chichilnisky, 2001; Kim and Rieke, 2001b). This model separates a
time-independent (or static) nonlinearity in the response of the cell
from true adaptive changes in sensitivity. The model describes the
transformation of injected current into the probability of spiking of
the cell. Adaptation caused the current-to-spikes transformation to
change with the variance of the injected current. The model does
not attempt to account for the dynamics of adaptation, but
instead describes the steady-state effect of adaptation on the
current-to-spikes transformation.
The static nonlinearity model predicts the spike probability as a
function of time by passing the injected current through a linear
filter and applying a static nonlinearity to the filter output. The
linear filter estimates the time dependence of the relationship between
injected current and spike probability. The static nonlinearity
captures the abrupt increase in firing probability associated with the
membrane voltage crossing threshold for spike generation. The linear
filter was estimated by calculating the average current waveform
preceding a spike. The static nonlinearity was estimated by comparing
the output of this filter (the filter convolved with the injected
current) with the measured spiking probability.
Variance adaptation in experiments and simulations (see below) was
analyzed using this model. The linear filter and static nonlinearity
were calculated from several minutes of measured or simulated responses
to random injected currents. This procedure was repeated for currents
of several variances. The first 2 sec of data after a change in
variance were discarded to allow variance adaptation to reach a steady
state. As reported previously (Kim and Rieke, 2001b), adaptation to
changes in the variance of the injected current could be captured by a
change in the linear filter. Thus, the response of the cell was
described as a variance-dependent linear filter followed by a
variance-independent static nonlinearity. Changes in the amplitude of
the linear filter were used to quantify the extent of adaptation.
Computer simulation of ganglion cells
As described in Results, Na+
currents in ganglion cells underwent both fast (Hodgkin and Huxley,
1952) and slow (Crill, 1996; Fleidervish et al., 1996; Mickus et al.,
1999) inactivation. We used a computer simulation to test whether the
measured properties of the Na+ current
contributed to variance adaptation. The simulation provided a
relatively simple description of the cell while capturing activation and inactivation of the Na+ current in
detail. Other conductances, particularly
K+ and Ca2+
conductances, are known to shape the spike response of a cell (Hodgkin
and Huxley, 1952; Fohlmeister and Miller, 1997). However, adaptation of ganglion cells to changes in current variance was not
altered by K+-,
Ca2+-, or
Ca2+-activated currents (Kim and Rieke,
2001b); to minimize the number of parameters in the model, these
currents were not simulated separately. We begin by describing the
simulated Na+ current and then describe
how this current was incorporated in a spiking model for the ganglion cell.
Na+ current simulation. The
Na+ current was simulated using the
Hodgkin-Huxley formalism with the addition of two slow inactivation variables: s1 and
s2. The
s1 variable had both slow onset and recovery. Because of its slow kinetics, changes in
s1 were dominated by subthreshold
voltages. We refer to this form of inactivation as spike independent.
The s2 variable describes slow
inactivation that was entered after a spike and recovered slowly. We
refer to this form of inactivation as spike dependent.
In the Hodgkin-Huxley formalism, the activation variable m
and inactivation variable h are independent. We assume the
same is true for s1 and
s2. Although the assumption of
independent inactivation gates is not likely to hold (Hille, 2001),
this simulation captured the measured properties of the
Na+ current and thus allowed us to explore
the role of slow inactivation in variance adaptation. Furthermore, the
relative simplicity afforded by this assumption permitted us to
eliminate all free parameters in the simulation.
The m, h, s1,
and s2 variables describe the fraction
of gating particles in the Na+ channel
that are in the permissive state. The Na+
current is given by the probability that all gating particles are in
the permissive state (i.e., that the channel is open and not
inactive):
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(1)
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where V is the membrane voltage,
ENa is the
Na+ equilibrium potential, and
GNa is the maximal
Na+ conductance.
The activation and inactivation variables were described by first-order
kinetics (Fig. 1) (e.g., the
spike-independent slow inactivation particle
s1 obeyed):
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(2)
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where  s1 and
 s1 are rate constants describing the entry
into and recovery from slow inactivation. With first-order kinetics, the s1 variable exponentially
approaches its steady-state value at a given voltage. The rate
constants, s1 and s1,
were calculated from measurements of the steady-state value,
s , and the exponential time
constant  s1:
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(3)
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and
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(4)
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Spike-dependent slow inactivation, described by
s2, was entered only after a spike.
Thus at voltages below spike threshold, the recovery rate constant
s2 was simply the inverse of the recovery time
constant:
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(5)
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The rate constants for the h and m
variables were from Hodgkin and Huxley (1952). The activation variable,
m, was described by the rate constants:
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(6)
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and
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(7)
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where V is in millivolts and
m and m are in
msec 1. The fast-inactivation variable
h was described by:
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(8)
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and
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(9)
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The spike-independent slow inactivation variable
s1 was described by:
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(10)
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and
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(11)
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The spike-dependent slow inactivation variable
s2 was described by:
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(12)
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Onset of spike-dependent slow inactivation was simulated by
multiplying s2 by a factor of 0.77 after each Na+ spike (see Results).
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Figure 1.
Schematic of ganglion cell simulation. The
simulation describes the transformation between the input current and
membrane voltage of a cell. The cell was simulated as a single
compartment with leak, capacitive, cellular noise, and
Na+ currents (see Eq. 13). If an
Na+ spike was generated, the simulation was switched
to voltage clamp and forced to repolarize along the voltage trajectory
of an experimentally measured action potential. The
Na+ current parameters were updated continuously
during this voltage trajectory. After the voltage returned to rest, the
simulation was switched back into current clamp, and the voltage was
controlled again by the currents.
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Note that the rate constants controlling the slow inactivation
variables s1 and
s2 are several orders of magnitude
smaller than those for the fast inactivation variable h.
Model for spike generation. The
Na+ current described above formed the
basis of the simulation describing how input currents were converted to
membrane voltage (Fig. 1). The input to the simulation was the injected
current Istim, and the output was the
membrane voltage V, including action potentials. In addition to the Na+ current, the simulation
included leak and capacitive currents and a Gaussian
current noise Inoise (bandwidth, 0-50
Hz) to simulate cellular noise. These currents were summed to
obtain an equation describing the dynamics of the membrane voltage:
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(13)
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Here Cm is the membrane
capacitance, Gleak is the leak
conductance, and Eleak is the reversal
potential for the leak conductance. The
Na+ current was calculated from Equation 1. The ganglion cell was simulated as a single isopotential
compartment, because ganglion cells are believed to be electrotonically
compact (Taylor et al., 1996), and our isolation procedure removed all
but one or two short (< 40 µm) processes.
Equation 13 determines how the voltage in the simulated cell depends on
injected current and is able to generate the rising phase of an action
potential. It cannot, however, produce action potentials with a
reasonable shape, because it lacks the voltage-dependent K+ conductance that rapidly repolarizes
the cell. Thus, to reproduce the voltage changes produced by action
potentials, we forced the voltage in the simulation to follow the
trajectory of an experimentally measured action potential when the
membrane voltage indicated that an action potential was being generated
(i.e., whenever a trigger voltage  was reached). During the action
potential, the activation and inactivation variables of the
Na+ current were updated continuously in
response to the voltage. This allowed spikes to alter the state of the
Na+ current without simulating all
conductances shaping the action potential. After return to rest, the
simulation was switched back to current clamp, and the currents were
again free to perturb the membrane voltage.
Computer simulations were performed in Igor Pro
(Wavemetrics, Lake Oswego, OR) using a temporal
integration scheme described by MacGregor (1987). This integration
scheme allows for relatively large step sizes without instability in
the integration. Step size for integration was  0.1 msec; smaller step
sizes gave no noticeable increase in accuracy. The measured action
potential used during the voltage-clamp periods was interpolated to the temporal resolution of the integration step size. Values of the parameters in the simulation are given in Table
1. was chosen to be 15 mV so that
only Na+ spikes could cause the switch to
voltage clamp. The simulation had no free parameters: the rate
constants describing the m and h gates were taken
from Hodgkin and Huxley (1952), and all other parameters are the
average value from experimental cells (n 4) (Table
1). In Results we discuss how alterations in the simulation parameters
affected adaptation.
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Results |
Spike generation in retinal ganglion cells adapts to the variance
of the current injected into the cell (Kim and Rieke, 2001b). The
experiments and simulations described below indicate that slow
inactivation of the Na+ current can
account for most of this adaptation. First, we show that increasing the
injected current variance lowered the gain and increased the threshold
of spike generation. Second, we show that increasing the current
variance reduced the magnitude of the Na+
current through slow Na+ inactivation.
Third, we simulate the measured properties of the Na+ current and show that slow
Na+ inactivation can account for much of
the change in gain associated with variance adaptation. Together these
results provide a simple mechanistic description for variance
adaptation: increased variance decreases excitability by decreasing the
available Na+ current.
Effects of variance on spike generation and
Na+ current
Variance changes gain and threshold of spike generation
Variance adaptation was measured by injecting Gaussian current
fluctuations of two variances into a current-clamped ganglion cell and
recording the spike responses. We used the static nonlinearity model
(see Materials and Methods) to determine how the current variance
affected the gain and kinetics of spike generation. The model
characterizes spike generation at a single current variance by passing
the injected current through a linear filter (Fig. 2A) and applying a
static nonlinearity (Fig. 2B). The linear filter estimates the time dependence of the relationship between injected current and spike probability. The static nonlinearity captures the
thresholding and rectification of spike generation. Linear filters and
static nonlinearities were compared for several variances.
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Figure 2.
Effects of variance on current-to-spikes
transformation. The static nonlinearity model was used to characterize
the transformation between injected current and the firing rate.
Gaussian current fluctuations (bandwidth, 0-50 Hz) with variances of
16 and 144 pA2 were injected into a ganglion cell.
The model parameters were calculated from 100 sec of recording.
A, Linear filters for the two injected current
variances. B, Static nonlinearities for the two current
variances. The nonlinearity describes the relationship between the
output of the filters (A) and the measured firing
rate. C, Histograms of local voltage maxima for each
current variance. Maxima were defined as data points with amplitudes
larger than those of surrounding points, including both action
potential peaks (> 0 mV) and subthreshold maxima (less than 20 mV).
Bin size was 0.76 and 1.19 mV for low and high variance. The
inset plots the cumulative distribution of subthreshold
maxima. The change in threshold was estimated from the voltages at
which the cumulative distributions reached 0.99.
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The linear filter and static nonlinearity at each variance are unique
up to a single scale factor (Brenner et al., 2000; Chichilnisky, 2001;
Kim and Rieke, 2001b). Thus, scaling the y-axis of the
filter in Figure 2A and the x-axis of the
static nonlinearity in Figure 2B by the same factor
does not change the prediction of the model, because the effect of
changing the filter amplitude is offset by the change in the static
nonlinearity. Although adaptation could affect the shape of both the
filter and static nonlinearity, we found that the static nonlinearities
could be made to overlap with an appropriate choice of scale factor
(Fig. 2B). This allowed spike generation to be
described as a variance-dependent linear filter followed by a
variance-independent static nonlinearity. As a consequence, changes in
excitability (e.g., because of changes in threshold for spike
generation) were captured by a change in the filter amplitude.
Changes in variance altered both the amplitude and the time-to-peak of
the linear filter, as shown in Figure 2A. The
decrease in time-to-peak of the linear filter with increasing variance appeared to be a general property of refractoriness (see Fig. 8). The
reduction in the filter amplitude meant that the change in spike
probability for a given current change was smaller at high variance
than low variance. Similar changes in the linear filter were seen in
all 10 cells tested; on average, a ninefold change in variance
decreased the filter amplitude by 35 ± 4% (mean ± SEM).
The decrease in filter amplitude could be produced by an increased
threshold for spike generation. With an increased threshold, a larger
current would be required to generate a spike, thereby lowering the
sensitivity of spike generation to the injected current. To test for
such an effect, we measured spike threshold during periods of high and
low current variance.
The change in threshold with variance was estimated from the largest
depolarizations that failed to lead to a spike. Figure 2C
shows a histogram of the local voltage maxima, identified as 1 msec
time windows in which the measured voltage exceeded that in the
adjacent 1 msec windows (adjacent sampling points). Subthreshold voltage fluctuations form the large, broad peak in the histogram centered near 50 mV. Action potentials create the small peak centered near +5 mV. The histogram is 0 in the voltage range from 30
to 10 mV, because the voltage of the cell reaches these values only during an action potential, and action potentials have local maxima only at their peak.
Subthreshold voltage maxima extended to larger depolarizations when the
current variance was increased. We quantified this shift using the
cumulative distributions of subthreshold maxima (Fig. 2C,
inset). In principle, threshold could be determined from the
largest subthreshold maxima, the point at which the cumulative distribution reaches 1.0. This definition is impractical with finite
data because it relies on a single measurement, the single largest
subthreshold point observed. Instead, we estimated relative thresholds
for the two variances from the voltages that exceeded 99% of the local
maxima. This provided a reliable estimate of shifts in the largest
subthreshold voltages. In this cell, threshold was 1.8 mV higher during
high variance than during low variance. The mean increase in threshold
was 1.6 ± 0.3 mV (mean ± SEM; nine cells) for a ninefold
increase in injected current variance. Ignoring local maxima for 100 msec after each action potential had little effect on the apparent
threshold change, indicating that it was not caused by the absolute
refractory period. The subthreshold voltage changes produced by
moderate contrast light inputs are ~ 10 mV in amplitude. Hence a
change in threshold of 1-2 mV represents a substantial change in sensitivity.
Increased variance reduces the available
Na+ current
We found previously that the Na+
current was necessary for variance adaptation in ganglion cells,
whereas K+ or
Ca2+ currents were not (Kim and Rieke,
2001b). The central role of the Na+
current in adaptation and its impact on the threshold for spike generation suggested that adaptation might be caused by changes in the
amount of available (i.e., not inactive)
Na+ current and a resulting increase in
threshold. Consistent with this idea, we found that the
Na+ current was influenced by the
magnitude of previous voltage fluctuations.
Ganglion cells were voltage-clamped at 60 mV and subjected to voltage
fluctuations of two variances under conditions that isolated
Na+ currents (see Materials and Methods).
After the voltage fluctuations, the Na+
current was allowed to recover from fast inactivation by holding at
60 mV for 20 msec. The Na+ current was
then measured by stepping the voltage to 0 mV (Fig. 3A). Figure 3B
shows that the Na+ current depended on the
voltage variance. In this cell, the smaller voltage fluctuations
reduced the peak Na+ current by 4% and
the larger fluctuations by 17%. A similar dependence of the
Na+ current on previous voltage variance
was seen in each of 17 cells studied.
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Figure 3.
Na+ current is sensitive to
preceding voltage fluctuations. A, Voltage-clamp
stimulus. The Na+ current was measured in response
to a brief test pulse to 0 mV after 2 sec of Gaussian voltage
fluctuations (bandwidth, 0-50 Hz). A 20 msec recovery period at a
holding voltage of 60 mV was imposed between the noise and test
pulse. B, Average Na+ currents after
noise with variances of 0, 25, and 100 mV2 are
shown. Responses from four pulses after uncorrelated noise stimuli were
averaged together to prevent the specific immediate history of the
noise from affecting the Na+ current.
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We studied the recovery kinetics of the
Na+ current by varying the time between an
inactivating stimulus and the test pulse used to measure the peak
Na+ current. The
Na+ current recovered slowly after
inactivation by a voltage step or voltage noise. Figure
4B shows
Na+ currents measured at different times
after inactivation caused by voltage noise (Fig. 4A).
In all 12 ganglion cells studied, the Na+
current continued to recover from inactivation for times well beyond
the 10 msec expected from Hodgkin-Huxley kinetics. The time course of
recovery of the Na+ current is shown in
Figure 4C, which plots the ratio of the
Na+ currents with and without inactivation
against the recovery time. Full recovery required 1-4 sec.
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Figure 4.
Time course of recovery from noise and voltage
steps. A, Voltage-clamp stimulus used to measure slow
inactivation produced by noise. The Na+ current was
measured by stepping the voltage to 0 mV at various times
(trec) after a period of voltage
noise (100 mV2 variance; bandwidth, 0-50 Hz).
B, Na+ currents in response to the
test pulse. From smallest to largest, traces correspond to
trec of 10, 100, and 900 msec.
C, Time course of recovery of peak
Na+ current after inactivation by noise. The
Na+ current at each trec
was normalized to that for a trec of 900 msec. This ratio is plotted against the recovery time.
D, Voltage-clamp stimulus used to measure slow
inactivation produced by voltage steps. The Na+
current was measured at various times after a 500 msec depolarization
from 80 to 20 mV. E, Na+ currents
produced by the test pulse in D. From smallest to
largest, traces correspond to trec of 20, 110, and 1100 msec. F, Time course of recovery of peak
Na+ current after inactivation by voltage step.
Na+ currents were normalized to that for a
trec of 3 sec.
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Recovery of the Na+ current from
inactivation was similar when inactivation was produced by voltage
steps rather than a fluctuating voltage (Fig. 4D).
Figure 4E shows the Na+
currents measured at different times after inactivation caused by a
voltage step. Figure 4F shows the time course of
recovery. Again the Na+ current recovered
much more slowly than expected from Hodgkin-Huxley kinetics. Thus,
both noise and step stimuli caused Na+
channels to enter an inactive state from which they recovered slowly.
Properties of slow Na+ inactivation
Recovery of the Na+ current from fast
inactivation after an action potential typically requires a few
milliseconds (Hodgkin and Huxley, 1952; Hille, 2001). As shown above,
the ganglion cell Na+ current exhibited
slow inactivation with a recovery time constant several orders of
magnitude longer. There are two distinct functional components of this
slow inactivation. The first decreased the Na+ current after depolarizations above
spike threshold; recovery from this spike-dependent inactivation had a
time constant of a few hundred milliseconds. The second component of
slow inactivation had a slower onset and recovery; this
spike-independent component was controlled by subthreshold voltage fluctuations.
Spike-dependent slow Na+ inactivation
The Na+ current recovered slowly
after brief voltage-clamp pulses exceeding spike threshold (Fig.
5A). We refer to the large Na+ currents in response to these voltage
pulses as Na+ spikes. The
Na+ current declined exponentially in
amplitude in response to a series of pulses (Fig. 5B). This
form of slow inactivation reflected the all-or-none properties of spike
generation; no slow inactivation occurred when the pulses were too
small to produce an Na+ spike, and
increasing the length or amplitude of the depolarizing pulse did not
produce additional inactivation. We therefore refer to this component
of slow inactivation as spike dependent.
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Figure 5.
Spike-dependent slow inactivation.
A, Decline in Na+ current in response
to a series of voltage pulses. The cell was repeatedly stepped from
50 to 0 mV for 5 msec to generate Na+ spikes. The
interval between pulses was 20 msec. B, Peak
Na+ current amplitude as a function of pulse number.
Peak Na+ currents were normalized by that produced
by the first pulse. Normalized Na+ currents are
shown for interpulse intervals of 20 and 125 msec. Solid
lines are single exponential fits to the data.
C, Steady-state peak Na+ current as a
function of interpulse interval measured from fits in B.
Solid lines are single exponential fits to the data used
to estimate the recovery rate constant. D, Fractional
reduction in peak Na+ current produced by each pulse
calculated from the fits in B. E, Rate
constant s2 for recovery from inactivation. The rate
constant was calculated from the inverse of the time constant of the
fits in C. Different symbols plot measurements from four
cells. The solid line is an exponential fit to
s2 for all four cells.
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Spike-dependent slow inactivation was characterized by a first-order
kinetic model. Some fraction of the
s2-gating variable is permissive,
while the remainder, 1 s2, is
slowly inactivated and nonpermissive. A fraction of the non-inactive
Na+ current enters the slow-inactive state
after a spike; recovery from slow inactivation occurs with a
voltage-dependent rate constant, s2. We varied
the duration and holding voltage between pulses to determine the
recovery rate constant, s2, and the change in s2 after a pulse.
The time course of recovery from spike-dependent slow
Na+ inactivation was measured from the
steady-state Na+ current for pulse series
like that in Figure 5A. The steady-state amplitude was
reached when recovery from inactivation between pulses balanced
inactivation produced by each pulse. Lengthening the interval between
pulses increased the steady-state amplitude of the
Na+ current. Figure 5C plots
the steady-state amplitude against the interpulse interval for two
holding voltages. The recovery rate constant,
s2, was estimated from the time constant of
the exponential fits in Figure 5C. This rate constant is
plotted against the recovery voltage in Figure 5E. All four
cells studied showed a decrease in the rate of recovery from
spike-dependent slow Na+ inactivation with
increasing voltage.
The amount of inactivation produced by each pulse was determined by
correcting the reduction in Na+ current
between successive pulses for recovery during the intervening interval.
Figure 5D shows the fractional reduction in peak
Na+ current for different interpulse
intervals. Fitting these data with exponentials and extrapolating to an
interpulse interval of 0 (i.e., no recovery) estimated the fractional
reduction in available Na+ current
immediately after a pulse, s . The
average fractional reduction in Na+
current produced by each pulse was s = 0.23 ± 0.05 (mean ± SEM; four cells). Thus, after each
pulse the Na+ current was reduced to 77%
of its initial value.
Spike-independent slow Na+ inactivation
The spike-dependent slow Na+
inactivation described above did not fully explain the behavior of the
Na+ current. Subthreshold changes in
voltage also caused slow inactivation; this spike-independent component
of inactivation had both slow onset and recovery.
Spike-independent slow inactivation was described as a first-order
process analogous to Hodgkin-Huxley fast inactivation. In this case, a
single voltage-dependent time constant describes the approach to steady
state after a change in voltage. The onset and recovery rate constants
s1 and s1 were
determined from the time constant and steady-state values of slow
inactivation (see Eq. 2, Materials and Methods). To reduce errors in
measurement, we used voltage-clamp stimuli that produced large changes
in steady-state inactivation. For depolarized voltages at which
steady-state inactivation was large, we measured the rate of onset of
inactivation following a step from a hyperpolarized voltage (Fig.
6A). For hyperpolarized voltages at which steady-state inactivation was small, we measured the
time constant for recovery from a step producing strong inactivation (Fig. 6D). Combining these measurements
provided estimates of the voltage-dependent rate constants
s1 and s1.
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Figure 6.
Spike-independent slow inactivation.
A, Inactivation at depolarized voltages.
Top, Test pulses were delivered to measure the
Na+ current before and after an inactivating step of
variable duration and voltage. A 20 msec recovery period at 80 mV
between the inactivating pulse and the second test pulse allowed
recovery from fast inactivation. Bottom, The ratio of
the Na+ currents produced by the two test pulses is
plotted against tinact for two voltages.
Measured points were fit with single exponentials (smooth
curves) with time constants of 1160 msec for a
Vinact of 55 mV and 620 msec for a
Vinact of 35 mV. B, C,
Na+ currents in response to test pulses at a
Vinact of 55 mV (B)
and a Vinact of 35 mV
(C). From smallest to largest, solid
lines correspond to trec of 1600, 250, and 40 msec. The dotted line is
Na+ current produced by
V1. D, Inactivation at
hyperpolarized voltages. Top, Two test pulses were
delivered to measure the Na+ current before and
after inactivation produced by a 500 msec step to 20 mV followed by a
variable duration and voltage recovery period. Bottom,
The ratio of the peak Na+ currents produced by the
two test pulses is plotted against trec at
two voltages. The measured points were fit by a product of two
exponentials (smooth curves) with time constants of 130 and 770 msec for a Vrec of 95 mV and time
constants of 100 and 890 msec for a Vrec of
75 mV. E, F, Na+ currents in
response to test pulses at a Vrec of 95 mV
(E) and at a Vrec of
75 mV (F). From smallest to largest,
solid lines correspond to
trec of 20, 170, and 1380 msec. The
dotted line is the Na+ current
produced by V1.
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Figure 6A shows the procedure used to measure the
time constant and steady-state value of inactivation at depolarized
voltages (greater than or equal to 60 mV). A test pulse was delivered to measure the Na+ current in the absence
of slow inactivation. This was followed by 1 sec at 80 mV to allow
recovery from spike-dependent slow inactivation produced by the test
pulse. An inactivating pulse of variable duration was then delivered at
one of several voltages, followed by a second test pulse to measure the
available Na+ current. Figure 6,
B and C, shows the
Na+ currents for several inactivation
periods at 55 and 35 mV. Dashed lines show the
Na+ currents before inactivation. The step
to 55 mV, well below spike threshold, produced substantial
inactivation, provided the inactivation period was >100 msec.
The effect of slow inactivation on the available
Na+ current was measured from the ratio of
the Na+ currents in response to the first
and second test pulses. Figure 6A, bottom,
plots this ratio against the inactivation period for inactivation
voltages of 55 and 35 mV. The ratio measured at 35 mV does not
extrapolate back to 1 for an inactivation duration of 0 msec, because
the voltage step exceeded spike threshold and thus produced a large
Na+ current and spike-dependent slow
inactivation; this reduced the available
Na+ current ~20% (see above). The time
constant with which spike-independent slow inactivation approached a
steady-state value was estimated from single exponential fits to the
Na+ current ratio at each inactivation
voltage (Fig. 6A, smooth curves). The
steady-state value of spike-independent slow inactivation was estimated
by extrapolating the fit to infinite time and subtracting the
contribution of spike-dependent slow inactivation.
Figure 6D shows the procedure used to measure the
time constant and steady-state value of inactivation at hyperpolarized
(less than or equal to 60 mV) voltages. After a brief voltage step to
measure the initial Na+ current, a 500 msec period was imposed to allow recovery from spike-dependent slow
inactivation. The voltage was then increased to 20 mV for 500 msec,
causing both spike-dependent and spike-independent slow inactivation.
The cell was allowed to recover at one of several voltages for a
variable time. Finally, the Na+ current
was remeasured. Figure 6, E and F, shows
Na+ currents for several recovery times.
The ratio of the Na+ currents before and
after inactivation is plotted as a function of the recovery time (Fig.
6D, bottom). A double exponential was fit
to the Na+ current ratio at each
inactivation voltage. The faster time constant (100-400 msec) was
similar to that measured for recovery from spike-dependent slow
inactivation. The slower time constant, 500-1500 msec, described
recovery from spike-independent slow inactivation. The extrapolated
value of the fit at infinite recovery time gave the steady-state value.
Experiments such as that in Figure 6 characterized the first-order
kinetic model of spike-independent slow inactivation. The onset and
recovery rate constants, s1 and
s1, were calculated from the time constant and
steady-state value of the spike-independent inactivation using
Equations 3 and 4. Values for s1 and
s1 are plotted against voltage in Figure
7A,B. These measurements were fit (smooth curves) using Equations 10 and 11 describing the
s1 variable to estimate the voltage
dependence of each rate constant. Fits were forced to go to zero at
large positive voltages for s1 and large
negative voltages for s1.
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Figure 7.
Rate constants describing spike-independent slow
inactivation. A, The recovery rate constant,
s1, plotted versus voltage. The solid
line is an exponential fit calculated from Equation 10.
B, The entry rate constant, s1,
plotted versus voltage. The solid line is a saturating
exponential fit calculated from Equation 1. s1 and
s1 were determined from the steady-state value and the
slow time constant from the fits in Figure 6 using Equations 3 and 4.
Different symbols correspond to measurements from
different cells.
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Recovery from spike-independent slow inactivation was approximately
three times slower than recovery from spike-dependent slow
inactivation. For example, at 50 mV, s1  0.6 1 sec1, whereas
s2 2 4 sec1. This substantial difference in
recovery kinetics suggests that the Na+
current may have two distinct mechanistic components of slow inactivation (see Discussion).
Slow Na+ inactivation can account for
variance adaptation
The experiments described above indicate that slow inactivation
causes the amount of available Na+ current
to depend on the past history of voltage changes. The simulations
described below show that slow Na+
inactivation can account for the ability of ganglion cells to adapt to
the variance of their input current. Spike-dependent slow inactivation
made the dominant contribution to variance adaptation in the
simulation. Thus increasing variance caused a higher spike rate in the
simulation, which in turn reduced the available
Na+ current and decreased excitability.
Motivation for simulation
We found previously that Na+ currents
but not K+ or
Ca2+ currents were required for variance
adaptation (Kim and Rieke, 2001b). Thus to simplify the simulation, we
incorporated the Na+ current properties
described above without explicit simulation of
K+ and Ca2+
currents (see Materials and Methods). By itself, such a simulation will
not generate action potentials, because K+
and other currents are required to shape the action potential and
repolarize the cell after an Na+ spike.
Instead, spikes were generated by forcing the voltage of the simulated
ganglion cell through the trajectory of an experimentally measured
action potential once the Na+ current had
generated the rising phase. Once the voltage returned to rest, the
simulated cellular currents were free to influence the voltage. The
criterion level at which the simulation switched modes was 15 mV,
well beyond the experimental threshold for action potential generation.
Table 1 gives the parameters of the simulation, and Equations 6-12
give the rate constants describing the Na+
current. As a test of the Na+ current
description, we compared the measured and predicted reduction in
Na+ current produced by the voltage
fluctuations of Figure 3A. The simulation predicted a
reduction in Na+ current by a factor of
0.22, within the range observed experimentally. The simulation showed
no change in Na+ current without slow
Na+ inactivation. From this we conclude
that the simulation captured the key features of slow
Na+ inactivation during voltage fluctuations.
Slow inactivation is necessary for variance adaptation
in simulation
The contribution of slow inactivation to adaptation in the
simulation was determined using the static nonlinearity model (Fig. 2)
(see Materials and Methods). Experimentally, increasing the current
variance decreased the amplitude and time-to-peak of the linear filter
(Fig. 2A). Slow inactivation of the
Na+ current could explain the changes in
filter amplitude.
Figure 8 compares variance adaptation in
real and simulated ganglion cells. Figure 8A shows
the linear filters and static nonlinearities for a real ganglion cell.
Figure 8, B and C, shows linear filters and
static nonlinearities for the simulated ganglion cell with (Fig.
8C) and without (Fig. 8B) slow
Na+ inactivation. In all cases the static
nonlinearities overlapped (insets), and hence the effects of
variance on the simulated responses were restricted to changes in the
linear filter.
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Figure 8.
Comparison of variance adaptation in real and
simulated ganglion cells. Several minutes of Gaussian current
fluctuations (bandwidth, 0-50 Hz) of 16 and 144 pA2
variance were injected into experimental and simulated cells.
Adaptation in the computer simulated spike train was analyzed using the
static nonlinear model and treated identically to that of an
experimental cell. A, Adaptation in an experimental
cell. Left, Linear filters. Right,
Filters with the time axes normalized to the time-to-peak of each
filter. B, Simulation lacking slow
Na+ inactivation. C, Simulation with
slow Na+ inactivation. Insets show
that the static nonlinearities overlapped in all cases. Simulation
parameters are given in Table 1 and Equations 6-12.
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A change in the time-to-peak of the filter occurred both in the
simulations based on models of the Na+
current and in simple threshold-crossing models with a refractory period (data not shown). This change was absent when the refractory period was removed from the threshold-crossing model. Similar changes
in time-to-peak of the filter have been seen in other threshold-crossing models (Pillow and Simoncelli, 2003). Thus, the
change in time-to-peak was attributable to a time-dependent nonlinearity (refractoriness) rather than a variance-dependent change
in the behavior of the cell. Refractoriness in the simulation, resulting from the membrane time constant and the repolarization after
the action potential, reproduced the change in time-to-peak seen
experimentally (Fig. 8B,C). We did not study this
effect further because it was an inherent property of any cell with a refractory period.
The change in amplitude of the filter reflected a change in
excitability mediated by slow Na+
inactivation. To separate changes in amplitude from those in kinetics,
the time axes of the linear filters were normalized by the
time-to-peak. Figure 8A-C shows these rescaled
filters (right). Increasing the variance decreased the
filter amplitude in the real cell (Fig. 8A) and in
the simulation with slow Na+ inactivation
(Fig. 8C) but did not produce a change in amplitude without
slow Na+ inactivation (Fig.
8B). With slow Na+
inactivation, the filter amplitude was reduced by 20-25% for a
ninefold change in variance, close to that seen experimentally.
Slow inactivation contributed to adaptation in the simulation by
reducing the available Na+ current when
the current variance increased. This effect was dominated by
spike-dependent slow inactivation. Figure
9B shows the simulated voltage
responses to injected currents of low and high variance.
Spike-independent and spike-dependent slow inactivation are shown in
Figure 9C,D. All panels in Figure 9 show behavior of the simulated cell several seconds after a change in variance. After
a variance change, inactivation approached its steady-state value with
time constants of 0.15 and 1 sec for spike-dependent and
spike-independent slow inactivation. Both components of slow inactivation increased with increasing variance.
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Figure 9.
Increase in slow Na+
inactivation with variance. Gaussian current fluctuations (0-50 Hz) of
16 and 144 pA2 were injected into a simulated
ganglion cell. A, The 16 pA2 variance
current fluctuations. The 144 pA2 current
fluctuations were identical to that shown except for a scaling factor.
B, Voltage output of a simulated ganglion cell.
C, Value of the spike-independent slow inactivation
variable s1. D, Value of the
spike-dependent slow inactivation variable
s2. Several seconds of current injection at
each variance were delivered before time 0, so variables had time to
reach steady state.
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Spike-dependent slow inactivation increased substantially with
variance, as shown in Figure 9D. The greater firing rate at high variance combined with the slow recovery from inactivation caused
s2 to depend strongly on variance.
However, spike-independent slow inactivation changed little (3-5%)
when the variance increased, as shown in Figure 9C. When
spike-dependent slow inactivation was removed from the simulation,
~90% of the variance-induced change in available
Na+ current disappeared, whereas removal
of spike-independent slow inactivation had little effect. Thus,
spike-dependent slow inactivation accounted for the majority of the
variance adaptation.
Variance adaptation is robust to changes in
simulation parameters
The simulations described above indicate that slow
Na+ inactivation can account for variance
adaptation. Next we investigated the robustness of adaptation to the
parameters of the simulation. As described above, adaptation was
measured by the variance-dependent change in amplitude of the linear
filters in the static nonlinearity model. We explored parameters that
led to spike rates near the physiological range of 2-6 Hz.
Not surprisingly, changes in the parameters controlling the extent of
slow-dependent Na+ inactivation affected
the extent of adaptation. Varying the rate constant of recovery from
spike-dependent slow inactivation, s2, or the
fractional decrease in s2 following a
spike, s, between the experimental
extremes changed the extent of adaptation by up to 40%. The
spike-dependent inactivation produced much larger changes than using
the extreme measured values of the rate constants and voltage
dependence of spike-independent slow inactivation, s1 and s1, which
changed the extent of adaptation by <10%. Adaptation was also
sensitive to the membrane capacitance and leak conductance, which
varied substantially between cells (Table 1). Variations in these
parameters likely represent differences between different ganglion cell
types as well as variability introduced by the isolation procedure. The
resulting cell-to-cell variability in membrane time constant and in
spike shape caused up to 50% changes in the extent of adaptation.
Changes in other parameters of the model produced only small effects on
the extent of adaptation. The rate constants for the h and
m variables of the Na+ current
are of particular interest. Using and from a recent ganglion
cell model (Fohlmeister and Miller, 1997) rather than the
Hodgkin-Huxley values had little effect (< 10%) on variance adaptation. Adaptation was also changed minimally by shifts of up to 10 mV in the voltage dependence of the Na+ current.
The above results show that the specific values of the parameters in
the simulation influence the extent of variance adaptation but not its
presence. Much of the cell-to-cell variability in adaptation can be
explained by the measured variability in parameters controlling
adaptation. The robustness of variance adaptation to changes in the
parameters of the simulation comes about because adaptation is
dominated by spike-dependent slow inactivation. After an action
potential, slow inactivation reduces the available Na+ current and decreases excitability for
several hundred milliseconds.
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Discussion |
The experiments described above lead to three conclusions about
how retinal ganglion cells adapt to the variance of their input
currents: (1) the Na+ currents of retinal
ganglion cells exhibit spike-dependent and spike-independent slow
inactivation; (2) slow inactivation reduces the available
Na+ current when the variance of the input
current of the cell increases, leading to a decrease in excitability;
and (3) the role of slow inactivation in variance adaptation holds for
a wide range of cellular and Na+ current
parameters. Below we discuss the role of slow
Na+ inactivation in contrast adaptation
and more generally in controlling neural excitability.
Contrast adaptation, variance adaptation, and slow
Na+ inactivation
Slow Na+ inactivation influences
neural excitability directly by controlling the available transient
Na+ current (Fleidervish et al., 1996;
Colbert et al., 1997; Jung et al., 1997) and indirectly through
modulation of persistent Na+ currents
(Schwindt and Crill, 1995; Crill, 1996; Taddese and Bean, 2002). Slow
inactivation is usually studied by measuring the influence of current
or voltage steps on the Na+ current.
Increased inactivation decreases the Na+
current and decreases excitability.
Retinal ganglion cells also show alterations in excitability caused by
slow inactivation. We studied slow Na+
inactivation induced by Gaussian current fluctuations because they
mimic physiological inputs to ganglion cells more closely than current
steps. The variances of fluctuating currents used here approximate
those of the synaptic inputs to a ganglion cell for high and low
contrast light inputs. Recovery from slow inactivation required several
hundred milliseconds. Thus, after a change in the variance of the input
currents, the available Na+ current will
reach a steady-state level relatively slowly. This gradual approach to
steady state will cause the excitability of the cell to reflect the
voltage changes occurring in the previous 0.2-0.5 sec.
Slow inactivation caused the available Na+
current to depend on the past history of subthreshold and
superthreshold voltage changes. Other neurons also exhibit
spike-dependent and spike-independent slow
Na+ inactivation (Colbert et al., 1997;
Mickus et al., 1999). If the Na+ channels
enter the slow inactive state only from the open state, one
inactivation mechanism could give rise to both spike-dependent and
spike-independent slow inactivation (Mickus et al., 1999). However, the
substantial difference in the recovery kinetics of slow inactivation in
ganglion cells suggests either distinct mechanisms or that recovery
from inactivation depends on the past history of voltage changes.
Slow Na+ inactivation can explain the
component of contrast adaptation intrinsic to spike generation in
ganglion cells (Kim and Rieke, 2001b): increases in temporal contrast
increase the variance of the input currents to a ganglion cell, causing
a decrease in available Na+ current
because of slow inactivation and hence decreasing excitability. The
decrease in excitability increases the dynamic range of the cell,
permitting a ganglion cell to encode a wider range of input currents.
Slow inactivation reduced the available
Na+ current during high variance by
30-40%. This slow inactivation contributes to a fast kinetic
component (time constant of < 1 sec) of temporal contrast
adaptation; other mechanisms operating in the retinal circuitry operate
~10 times more slowly (Smirnakis et al., 1997; Chander and
Chichilnisky, 2001; Kim and Rieke, 2001b; Rieke, 2001).
Slow Na+ inactivation and sustained
Na+ currents
The persistent current is a sustained
Na+ current that inactivates exclusively
through slow inactivation (Crill, 1996). This current typically
activates at voltages 10-15 mV below the transient Na+ current and is smaller by 2-3 orders
of magnitude. The persistent current can alter cell excitability by
providing an inward current that helps depolarize a cell to spike
threshold (Schwindt and Crill, 1995; Colbert et al., 1997; Koizumi et
al., 2001). Changes in the availability of the persistent current
through slow inactivation can modulate excitability.
Persistent currents have been found in retinal neurons (Koizumi et al.,
2001). However, the effects of slow inactivation (described in
Results) appear distinct from those produced by modulation of a
persistent current. Our simulation had a non-inactivating Na+ current in a narrow voltage range
because of the overlap of the activation and fast inactivation gating
variables (the window current) (French et al., 1990), and modulation of
this current to more closely match the persistent current did not
affect the extent of adaptation.
Slow inactivation as a general modulator of excitability
A variety of cellular and network mechanisms allows cells
throughout the nervous system to accommodate a wide range of input signals. These mechanisms operate on time scales ranging from tens of
milliseconds to minutes. They also are controlled by a variety of
statistical properties of the input signals, including the mean and
variations about the mean. In several cases these adaptation mechanisms
have been linked to functional properties of adaptation (Sanchez-Vives
et al., 2000b; Kawai, 2002; Smith et al., 2002).
In motion-sensitive neurons on the fly visual system, the time scale of
adaptation itself adapts (Fairhall et al., 2001). Thus the rate of
onset of adaptation after an increase in motion variance depends on the
duration of the previous period of low motion variance. This provides
an interesting challenge for understanding the underlying mechanisms,
because scaling of the adaptation time scale is not expected from a
simple mechanism characterized by a single time constant. However,
scaling of the recovery kinetics of slow
Na+ inactivation has been observed (Toib
et al., 1998) and could contribute to a rescaling of the adaptation
time course. It will be interesting to determine whether this
observation applies more generally.
Two observations suggest that adaptation mediated by control of the
available Na+ current through slow
inactivation may be a general mechanism. First, slow
Na+ inactivation is a common, although not
universal, property of voltage-activated
Na+ currents. In pyramidal cells, for
example, Na+ currents in the dendrites
show substantially greater slow inactivation than those in the soma
(Colbert et al., 1997 |
TOP
ABSTRACT
Introduction
