 |
 Previous Article | Next Article 
The Journal of Neuroscience, March 1, 1998, 18(5):1893-1903
Interaction between Duration of Activity and Time Course of
Recovery from Slow Inactivation in Mammalian Brain Na+
Channels
Amir
Toib,
Vladimir
Lyakhov, and
Shimon
Marom
The Bernard Katz Minerva Center for Cell Biophysics, Department of
Physiology, Faculty of Medicine, Technion, and The Rappaport Institute
for Research in the Medical Sciences, Haifa 31096, Israel
 |
ABSTRACT |
NaII and NaIIA channels are the most abundant voltage-gated
channels in neonatal and adult cortex, respectively. The relationships between activity and availability for activation of these channels were
examined using the Xenopus expression system. The main
point of this work is that the time constant ( ) of recovery
from the unavailable (inactivated) pool is related to the duration
(t) of previous activation by a power law:
(t) = p · tD, with a scaling power
D congruent to 0.8 and 0.5 for NaII and NaIIA,
respectively, and p as a constant kinetic setpoint.
These relationships extend from tens of milliseconds to several minutes and are intrinsic to the channel protein. Coexpression of  1
auxiliary subunit, together with the  subunit of the NaIIA channel,
modulates the constant kinetic setpoint but not the scaling power of
the latter. The power law scaling between activity and availability is
not a universal property of ion channels; unlike that of voltage-gated sodium channels, the rate of recovery from slow inactivation of the
ShakerB channel is virtually insensitive to the duration of previous
stimuli. It is suggested that the power law scaling described here can
act as a molecular memory mechanism that preserves traces of previous
activity, over a wide range of time scales, in the form of modulated
reaction rates. This mechanism should be considered when theorizing
about the dynamics of threshold and firing patterns of neurons.
Key words:
sodium channel; ion channel; inactivation; excitability; power law; scaling
| |
INTRODUCTION |
Modulation of threshold potential by
changes in the availability of conductances is a powerful memory
mechanism that depends solely on the intrinsic properties of a neuron,
independent of changes in synaptic weight. In the past 30 years, there
have been many reports of slow changes in the availability of
voltage-gated sodium channels by means of a slow inactivation process
(e.g., Adelman and Palti, 1969 ; Chandler and Meves, 1970; Fox, 1976; Schauf et al., 1976; Brismar, 1977; Rudy, 1978; Almers et al., 1983;
Simoncini and Stühmer, 1987; Stühmer et al., 1987; Ruben et
al., 1992; Cummins and Sigworth, 1996; Featherstone et al., 1996;
Fleidervish et al., 1996; Hayward et al., 1997). Slow inactivation of
voltage-gated sodium channels was found in all membranes in which
appropriate experiments have been conducted (Ruben et al., 1992) and
extends over a wide range of time scales (tens of milliseconds to
minutes). The wide range of reported time scales of slow inactivation, together with the fact that the availability of voltage-gated sodium
channels is a key determinant of threshold potential, prompted us to
examine the relationships between activity and availability for
activation of these channels. More specifically, we explored the
possibility that a scaling relationship [Bassingthwaighte et al.
(1994), their references] exists between the reaction rates involved
in slow inactivation and the duration of previous depolarizations. A
scaling relationship, in this context, means that there are no
characteristic reaction rates for inactivation; instead, the effective
rate constants reflect the time allowed for inactivation to occur.
Voltage-gated sodium channels are usually regarded as either (1) not
important for long-term modulations extending beyond the envelope of an
action potential or (2) conferring a uniquely defined time scale that
can be directly derived from the process of slow inactivation. However,
if the rates of slow inactivation of voltage-gated sodium channels are
linked to the history of activity via a scaling relationship, the role
of these channels in activity-dependent processes must be reexamined,
because they can participate in forms of cellular memory over all
available time scales.
Types II and IIA voltage-gated sodium channels (Noda et al., 1986; Auld
et al., 1988) are the most abundant voltage-gated sodium channels in
the mammalian brain (Catterall, 1992). This report shows that the time
constant () for recovery of these channels from slow inactivation is
intrinsically related to the duration (t) of previous
history of depolarizations by a power law: (t) = p · tD, with a
constant kinetic setpoint (p) and a positive scaling power (D) over a wide range of time scales.
| |
MATERIALS AND METHODS |
Expression system
Currents were measured from the membranes of Xenopus
oocytes injected with cRNA coding for the brain channel types NaII
(Noda et al., 1986) or NaIIA (Auld et al., 1988) or for the rapidly inactivating ShakerB potassium channel (Tempel et al., 1987). cRNA for
the rat brain 1 subunit [Isom et al. (1994), their references] was
kindly given by I. Lotan (Wallner et al., 1993). Methods of DNA and
cRNA preparation are standard. Oocytes were harvested from mature
Xenopus females that were anesthetized by bathing in
ice-cold water. After the surgery, frogs were allowed to recover for
several hours in a small pool of water, and when fully active, they
were returned to the main tank. Oocytes were dissociated in 2 mg/ml
collagenase 1A (Sigma, St. Louis, MO). Stage V-VI defolliculated cells
were injected with 0.4-40 nl of mRNA ( 1 mg/ml). In coexpression experiments, the 1 cRNA was coinjected with the subunit at a
molar ratio of  1:1. Injected cells were incubated at 18°C for up
to 7 d in frog Ringer's solution.
Electrophysiological measurements
The resulting conductances were studied with patch-clamp (Hamill
et al., 1981) and two-electrode voltage-clamp (TEVC) techniques, using
Axopatch 200A and GeneClamp 500 amplifiers (Axon Instruments, Foster
City, CA), respectively. Pipets for patch-clamp experiments were made
from glass (#7052; Garner Glass Co., Claremont, CA) that was polished
to 3-5 M . Solutions for patch-clamp recordings were (in
mM): 96 NaCl, 2 KCl, 1 CaCl2, 1 MgCl2, and 10 HEPES, pH 7.5, and 95 KCl, 5 NaCl, 1 EGTA, and 10 HEPES, pH 7.5, for the pipet and bath solutions,
respectively. Intracellular aluminum silicate glass electrodes for TEVC
experiments were filled with 2 M KCl, resulting in
resistances ranging from 0.5 to 2 M. The bath solution for TEVC
experiments was (in mM): 95 NaCl, 3 KCl, 2 CaCl2, and 10 HEPES, pH 7.5. Experiments were
handled by a Quadra 800 (Apple Computers, Inc.) with PULSE software
(HEKA Electronic) and an ITC-16 computer interface (Instrutech
Corporation). Data were low-pass filtered at 2-5 kHz and sampled at
10-25 kHz. Long trains of pulses were delivered by a homemade pulse
generator that was triggered from the PULSE software. P/n leak
subtraction routines of the PULSE software were used. Some of the
experiments involved very long voltage pulses that caused nonspecific
conductances to appear. These conductances relaxed completely within 2 sec at a hyperpolarized membrane potential. Continuous perfusion during very long conditioning pulses (>3 sec) speeds up this relaxation.
Pulse protocols and analysis of slow recovery in voltage-gated
sodium channels
(1) The time constants for recovery from rapid inactivation
(Hodgkin and Huxley, 1952) were characterized at two voltages,  90 and
120 mV, by a standard double-pulse protocol.
(2) The kinetics of entry into the slow inactivation state was
examined. In these experiments, a conditioning pulse from 90 to 10
mV was applied to the membrane for various durations (1, 3, 10, 30, and
100 sec). Each conditioning pulse was followed by a recovery interval
at 120 mV for either 6 or 100 msec for NaII and NaIIA, respectively.
This interval, at 120 mV, allowed for a practically complete recovery
of the channels from rapid inactivation in the case of the NaII subunit [NaII()] and the brain 1 subunit of the Na channel
coexpressed with the NaIIA subunit [NaIIA( + 1)] and for
90% recovery in the case of the NaIIA subunit [NaIIA()]. A
test pulse to 10 mV was applied immediately after the recovery
interval. The fraction of channels in the slow inactivated state was
calculated from the ratio between the measured peak current response to
the test pulse and the peak current response evoked by a similar
procedure after a 15 msec conditioning pulse.
(3) Two types of pulsing protocols aimed at exploring the kinetics of
recovery from slow inactivation were performed.
Recovery from very long conditioning pulses. In these
experiments, a conditioning pulse from 90 to 10 mV was applied to the membrane for various durations (3, 10, 30, 100, and 300 sec). Each
conditioning pulse was followed by a 1 sec recovery interval at 90
mV. This interval allowed for a practically complete recovery of the
channels from rapid inactivation and was followed by a series of short
test pulses aimed at monitoring the process of recovery from slow
inactivation. The test pulses from 90 to 10 mV were delivered at a
frequency of 0.33, 0.5, or 1 Hz; no effect of the frequency of the test
pulses on the kinetics of recovery from slow inactivation was observed
within this range (which is an order of magnitude lower than the
slowest observed rate of recovery from rapid inactivation).
Recovery from shorter conditioning pulses. In these
experiments, the duration of the conditioning pulse (10, 30, 100, 300, 1000, and 3000 msec) allowed a standard double-pulse protocol to be
used to follow the time course of recovery. Each pair of pulses was
composed of a conditioning pulse from 90 to 10 mV followed by a
logarithmically varying recovery period at 90 mV and a test pulse to
10 mV. Note the overlap (at a conditioning duration of 3 sec) between
the two types of pulsing protocols.
The peak current amplitude of each nth test pulse
(In) was observed on-line, and the pulsing
was stopped by the experimenter when In
approached a maximum value (Imax). The
fraction of inactivation F at the time of the nth
test pulse was normalized to: F = 1 (In I1)/(Imax I1), where I1 is
the peak amplitude of the current that was evoked by the first test
pulse. The time constant was extracted by fitting a sum of
exponential functions to F. Fitting was done using
KaleidaGraph (Synergy Software).
| |
RESULTS |
Time-scale separation between rapid (Hodgkin-Huxley-like) and slow
inactivation processes
Voltage-gated sodium channels inactivate rapidly in response to a
short, fully activating depolarizing voltage pulse. This phenomenon,
which has been described and analyzed in the literature since the
publication of the Hodgkin and Huxley study (1952), occurs at the
milliseconds time scale and is demonstrated in Figure 1A and summarized in
Table 1. The subject of the present
report is a slower inactivation process, which involves reaction rates at the seconds-to-minute time scale and is demonstrated in Figure 1B. In these experiments, the membrane is pulsed to a
fully activating membrane potential for a duration t,
followed by a hyperpolarizing segment that is long enough to allow
recovery from the rapid Hodgkin-Huxley inactivation process. The
latter segment is then followed by a short depolarizing test pulse. The
fraction of inactivated channels, as a function of t, is
measured from the peak amplitude of the current that was evoked by the
test pulse. As indicated by the decay of availability in the
experiments of Figure 1C, within the examined range of up to
100-sec-long depolarization pulses, the channels are shifted into an
unavailable pool following uniquely defined reaction rates at the
seconds time scale (double exponential fits are depicted as
continuous lines).
View larger version (26K):
[in this window]
[in a new window]
|
Figure 1.
Inactivation of voltage-gated sodium channels.
A, Time scales of rapid inactivation rates, for both
NaII (top) and NaIIA (bottom) channels,
are demonstrated using a double-pulse protocol. The membrane is pulsed
to 10 mV from a holding potential of 90 mV (pulse duration is 10 msec). The membrane potential of the interval between the first and the
second pulses is 90 mV, and its duration is logarithmically increased
(note the time axes). The time constants of recovery from rapid
inactivation at 90 and 120 mV are summarized in Table 1.
B, Representative families of current
traces showing the level of slow inactivation are presented.
Oocytes are pulsed from a holding potential of 90 to 10 mV for a
duration t. After this conditioning depolarization, the
oocyte membrane potential is stepped to 120 mV, allowing for recovery
from the rapid inactivation process (6 or 100 msec for NaII and NaIIA,
respectively; see Materials and Methods, Table 1). The availability of
the channels for activation is then tested by an application of a short
depolarizing test pulse to 10 mV. The resulting test pulse
current traces are shown for both channel subtypes
(top, NaII; bottom, NaIIA); note the different time-scale bars (top and
bottom). The durations (t values) of the
conditioning pulses are depicted accordingly. C,
Kinetics of entry into the slow inactivation state is shown for NaII
(open triangles; n = 6; ±SD) and
NaIIA (open diamonds; n = 5; ±SD), together with double exponential functions depicted by
continuous lines (NaII, 1 = 0.6;
fraction = 0.33; 2 = 16.3; fraction = 0.67;
R > 0.99; NaIIA, 1 = 0.6;
fraction = 0.31; 2 = 21.9; fraction = 0.69;
R > 0.99). The time course of slow inactivation of
NaIIA channels, in response to physiologically realistic stimulation patterns, is depicted by broken lines. In these
experiments, during the conditioning phase, the membrane is held at
60 mV and pulsed to 0 mV for 2 msec at various frequencies
(filled diamonds, 16 Hz; filled
triangles, 45 Hz; filled inverted triangles, 125 Hz; filled circles, 250 Hz). Note that although the main
effect of a pulse train is completed within the first 30 sec (i.e., 480 pulses at 16 Hz to 7500 pulses at 250 Hz), accumulation of inactivation continues throughout the examined range (i.e., 1600 pulses at 16 Hz to
25000 pulses at 250 Hz).
|
|
Because the membrane potential of a neuron under physiological
conditions is not expected to be depolarized for seconds and because
this study is aimed at making a statement about the physiological significance of slow inactivation, NaIIA channels were exposed to
stimulation protocols that are similar in amplitude, duration, and
frequencies to firing patterns in neocortical neurons. The results
(Fig. 1C, dashed lines) show that when the
membrane is held at 60 mV and pulsed to 0 mV for 2 msec at
frequencies ranging from 16 to 250 Hz, the reduction of the
availability of the channels for activation is significant. It is
important to note that the pulse train data are fundamentally different
from the continuous data; the pulse train protocol, especially at high
frequencies, induces accumulation of channels in both rapid as well as
slow inactivation states and therefore cannot directly point at the physiological significance of slow inactivation alone. The
low-frequency data of Figure 1C is more informative in that
context. In the experiments that are described in the rest of this
report, the pulses to induce slow inactivation (defined here as
conditioning pulses) are continuous and optimized for kinetic
characterization rather than for physiological compatibility.
The time course of recovery from slow inactivation is scaled
according to the duration of the conditioning pulse
The time-scale separation, between the milliseconds range of rapid
activation-inactivation (Fig. 1A, Table 1) and the
many-seconds range of slow inactivation (Fig.
1B,C), allows one to reduce the kinetic scheme of the system to a simplified two-states scheme, A I, where A stands for the set of
all available-for-ion-conduction states, and I stands for
all unavailable (inactivated) states. In this simplified picture, the
classical Hodgkin-Huxley rapid inactivation (h gate)
belongs to the available set of states. The rate of recovery
I A is studied by applying a series of short depolarizing test pulses after t long conditioning
depolarizations. In these experiments, all the intervals between
depolarizations are at the seconds time scale; this is essential to
diminish the interference of fast gating processes, within the internal
structure of A, with the process of recovery from slow
inactivation. The results are shown in Figure
2. The time course of recovery from slow
inactivation, induced by 3-, 10-, 30-, 100-, and 300-sec-long conditioning depolarizations, is extracted from the change in the peak
of the current amplitude evoked by short depolarizing test pulses as
explained in Materials and Methods. The recovery can be adequately
described by a double exponential function (R > 0.99)
(Fig. 2A). The relationship between the two
components of recovery from inactivation will be explored in a separate
section of the Results, in which experimental protocols that are
optimized for that aim are analyzed. Regardless of this relationship
between the components, it is clear from Figure 2A
that the time course of recovery from inactivation is sensitive to the
conditioning-pulse duration. In particular, the value of the slower
time constant () for IA relaxation
increases systematically as a function of the duration (t)
of the conditioning pulse. A collection of many values of versus
t is shown on a log-log plot in Figure 2B, indicating that the longer the conditioning
depolarization is, the slower the time constant for recovery from
inactivation becomes. The behavior of the time constant as a function
of conditioning-pulse duration is the key observation of the present
report.
View larger version (25K):
[in this window]
[in a new window]
|
Figure 2.
Scaling relationship between the duration of
activation and the recovery rate from slow inactivation in
voltage-gated sodium channels. A, Time course of
recovery from slow inactivation induced by 3-, 10-, 30-, 100-, and
300-sec-long (t) conditioning depolarizations (top, NaII; bottom, NaIIA). The recovery
interval, at 90 mV, from the end of the conditioning pulse to the
beginning of the series of test pulses lasted 1 sec. The fraction of
inactivation was normalized and plotted as described in Materials and
Methods. Solid lines are double exponential fits to the
data. Insets, Recovery from a 100-sec-long conditioning
depolarization from 90 to 10 mV. Recovery from slow inactivation is
seen as a gradual increase in peak current responses to test pulses of
15 msec duration from 90 to 10 mV, delivered at a frequency of 0.5 Hz (top) or 0.33 Hz (bottom).
B, The value of the slower time constant for recovery from slow inactivation () shown increasing systematically as a
function of the duration t of the conditioning
depolarization in NaII (n > 6; ±SD) and NaIIA
(n = 5; ±SD) channel subtypes. Time constants were
extracted from the time course of recovery as shown in
A, with a test pulse frequency of 0.5 Hz. Solid
lines are power law functions of the form
(t) = p · tD
that were fit to the data (NaII, p = 0.3;
D = 0.84; R = 0.99; NaIIA,
p = 2.6; D = 0.46;
R > 0.99). Inset, The value of the
shorter time constant as a function of conditioning depolarization in NaII (open triangles; n > 5; ±SD) and
NaIIA (filled diamonds; n > 3;
±SD) channel subtypes. The relative amplitudes of this recovery
component (in the absence of the 1 subunit) are within the range of
0.29-0.53 and 0.09-0.33 for NaII and NaIIA, respectively, and do not
change systematically as the conditioning-pulse duration is increased.
C, Time course of recovery from slow inactivation of
NaII channels recorded in detached-macropatch configuration (i.e., the
channels are not in contact with intracellular factors). Inactivation
was induced by 3-, 10-, 30-, 100-, and 300-sec-long conditioning
depolarizations. The recovery interval, at 90 mV, from the end of the
conditioning pulse to the beginning of the series of test pulses lasted
1 sec. Solid lines are double exponential fits to the
data. Inset, Recovery from a 30 sec conditioning
depolarization from 90 to 10 mV. Recovery from slow inactivation is
seen as a gradual increase in peak current responses to test pulses of 10 msec duration from 90 to 10 mV, delivered at a frequency of 1 Hz. D, The value of the slower time constant ()
increasing systematically as a function of t. Mean
values (filled circles) are fitted with a power
law of the form: (t) = p · tD (p = 0.7; D = 0.75; R = 0.97).
|
|
The mean values of the slower time constants () for recovery from
slow inactivation are related to t by a power law (Fig. 2B, solid lines) of the form:
|
(1)
|
suggesting the existence of a scaling relationship between the
recovery rate from the unavailable set of states and the duration of
previous activity.
Scaling of recovery time course is intrinsic to the subunit of
the channel protein
Do the above-described relationships between activity and
availability arise from an intrinsic (physicochemical) channel
machinery? The power law scaling could stem from the Xenopus
oocyte intracellular milieu via a biochemical path (e.g., enzymatic
modulation that is activated by long depolarizing pulses) that might be
entirely irrelevant to the neurophysiological case. Taking advantage of the fact that NaII channels are efficiently expressed in oocytes, allowing one to record macroscopic Na currents in a detached patch of
membrane, we show in the experiments of Figure 2, C and
D, that the power law relationships between and
t are not produced by changes in the intracellular
environment of the oocyte during the conditioning phase; the scaling
phenomenon is clearly seen in the detached-patch mode, in which the
channel protein is not in contact with the cellular environment. There
are some differences between the results of TEVC and patch-clamp
recordings, especially at the faster time scales (compare Fig.
2C with A, top). These differences
might be related to the compromised accuracy of the TEVC technique in
describing the fast sodium channel kinetics [Ruben et al. (1997),
their references]. However, it is important to note that these
differences do not interfere with the key observation of the present
report (compare Fig. 2B with D).
Coexpression of the subunit together with the 1 subunit
introduces a kinetic offset in the relationship between the duration of
activity and the time course of recovery
The Na channel isoforms studied here have a much slower
inactivation kinetics, compared with their kinetics in mammalian cells, when expressed in Xenopus oocytes in the absence of a subunit [Isom et al. (1994), their references]. This becomes a major
concern if one is trying to generalize results on Na channel
inactivation from Xenopus oocyte experiments to neurons. To
overcome this problem, we conducted a set of experiments in which the
rat brain 1 subunit of the Na channel was coexpressed with the NaIIA
subunit. The modulatory effect of the 1 subunit on the fast
kinetics of NaIIA is demonstrated in Figure
3A. Slow inactivation of
NaIIA( + 1) channels is demonstrated in Figure 3B.
Current responses to a short depolarizing test pulse decrease in
amplitude as the duration of a preceding t-sec-long
depolarization conditioning pulse is increased. Figure 3C
summarizes the kinetics of entry of NaIIA( + 1) channels into the
slow inactivation state. The onset of slow inactivation of NaIIA
channels, in the presence of the 1 subunit, consists of two
components. These two components have comparable amplitudes but
distinctly separated time scales. An attempt to describe the entry
kinetics by a single exponent yields a less agreeable approximation
(depicted in Fig. 3C). The data of Figure 3 suggest that in
contrast to the fast inactivation kinetics, relaxation of the NaIIA
into slow inactivation is not markedly altered by 1 (compare Figs.
3C with 1C). As shown in Figure
4, the dependence of the slower rate of
recovery of the NaIIA( + 1) channel from slow inactivation on the
duration of the conditioning pulse is also quite similar to that of the
NaIIA channel. Interestingly, the kinetic setpoint
(p) is slightly smaller, 1.88 compared with 2.6 for
NaIIA( + 1) and NaIIA(), respectively. This small difference
is statistically significant (P = 0.028, paired two-sample
t test for means) and shows that NaIIA() is slower to
recover from inactivation, relative to NaIIA( + 1), over the
entire range of conditioning durations tested here. Coexpression of the
subunit of the NaIIA channel together with the 1 subunit uncovered another interesting feature that is apparently absent when
the subunit of the NaIIA channel is expressed alone; a close
examination of the curves of Figure 4A suggests that
as the conditioning duration increases, the time constant of the faster recovery component seems to increase, whereas its
relative amplitude decreases. This trend is demonstrated in the
inset of Figure 4B that shows the faster
time constant and its amplitude as a function of conditioning duration
for the case of Figure 4A. Note that in the absence
of the 1 subunit, there is no systematic relationship between the
faster time constant for recovery from slow inactivation and the
conditioning duration (Fig. 2B, inset). The interplay between the time constants and relative amplitudes of the
different components of recovery from inactivation will be more fully
addressed in other experiments (see Fig. 7).
View larger version (10K):
[in this window]
[in a new window]
|
Figure 3.
Effect of the 1 subunit on the entry of NaIIA
to slow inactivation. A, Typical responses of NaIIA()
and NaIIA( + 1) channels to a depolarization pulse from 90 to
10 mV. These traces were recorded from different
oocytes (of the same batch) 3 d after injection with cRNA of the
subunit alone (dashed line) or together with cRNA of
the 1 subunit (continuous line) in a 1:1 molar ratio. The presence of 1 subunit results in a faster activation and
inactivation kinetics. No systematic effect on current amplitude was
observed. B, A representative family of current
responses to test pulses from 120 to 10 mV, delivered 100 msec
after the end of a t long conditioning depolarization to
10 mV, showing the level of slow inactivation of NaIIA( + 1)
channels. C, Kinetics of entry into the slow
inactivation state for NaIIA( + 1) channels (open
symbols). For comparison, two types of exponential functions are fitted to the mean values (filled symbols), a
single exponent ( = 4.8 sec; R = 0.92) and a double
exponent (1 = 2.12; fraction = 0.55;
2 = 24.3; fraction = 0.45; R > 0.99).
|
|
View larger version (17K):
[in this window]
[in a new window]
|
Figure 4.
Scaling relationship between the duration of
activation and recovery rate from slow inactivation in NaIIA channels
that are coexpressed with the 1 subunit. A, Time
course of recovery from slow inactivation in NaIIA( + 1) channels
induced by 3-, 10-, 30-, 100-, and 300-sec-long (t)
conditioning depolarizations. The recovery interval, at 90 mV, from
the end of the conditioning pulse to the beginning of the series of
test pulses lasted 1 sec. The fraction of inactivation was normalized
and plotted as described in Materials and Methods. Solid
lines are double exponential fits to the data.
Inset, Recovery from a 100 sec conditioning
depolarization. Recovery from slow inactivation is seen as a gradual
increase in peak current responses to test pulses from 90 to 10 mV,
delivered at a frequency of 0.5 Hz. B, The value of the
slower time constant () increasing systematically as a function of
t. Time constants were extracted from the time course of
recovery as shown in A, with a test pulse frequency
of 0.5 Hz. Filled circles are the mean values at
each t. The solid line is a power law
function of the form (t) = p · tD that was fit to the mean values
(p = 1.88; D = 0.45;
R > 0.99). Inset, The relative
amplitude (filled diamonds, right
y-axis) and time constant (open circles,
left y-axis) of the faster component of recovery from
slow inactivation, for the cell shown in A, as a
function of conditioning-pulse duration.
|
|
The scaling power is not affected by the pulse protocol used for
recovery measurement
In this study, the method for observing the time course of
recovery is based on a series of brief depolarizing pulses at low frequencies (see Materials and Methods). This method is commonly used
when the recovery time scales are very long (e.g., Cummins and
Sigworth, 1996). Nevertheless, it is important to ascertain that
recovery data are not distorted by the series of test pulses. A
comparison was made to recovery data in which conditioning pulses were
followed by a varying recovery interval and a single test pulse. A
single set of 3-300-sec-long conditioning pulses, each of which was
followed by a single recovery interval and a single test pulse, lasts
1.5-2 hr. Figure 5 shows an exemplar
(5A) and a summary (5B) of results from three
oocytes that were injected with NaIIA( + 1) channels. Recordings
from these oocytes were stable throughout the entire pulse protocol. As
shown in Figure 5, the scaling power between conditioning duration and
recovery time constant is similar to the scaling power extracted from
series of recovery pulses.
View larger version (14K):
[in this window]
[in a new window]
|
Figure 5.
Scaling relationship between the conditioning
duration and recovery time course in long double-pulse protocols with
NaIIA( + 1) channels. A, Exemplar recovery time
courses that were extracted from double-pulse protocols as follows: A
short test pulse was applied 1 sec before each conditioning pulse. This
short test pulse (p#1 in the
inset) served as a control level for the calculation of
inactivation fraction (see Materials and Methods) and as a measure for
full recovery between pairs of pulses. Each T-sec-long conditioning pulse was followed by a recovery interval at 90 mV and a
single test pulse to 10 mV (p#2 in the
inset). This procedure was repeated for various recovery
intervals and conditioning durations. Insets, Control
and recovery pulse pairs for 300 sec conditioning duration.
Calibration: 4 msec, 4 µA. B, Relationship between
conditioning duration and recovery time constant fitted to a power law
function of the form: (t) = p · tD (p = 2.9;
D = 0.52; R = 0.96). Means and
SDs of recovery time constants are from three oocytes.
|
|
The scaling relationship between conditioning duration and
recovery time course persists over a range of voltages
The scaling of recovery time course was shown above at only one
voltage (90 mV). It is interesting to see whether the membrane potential, during the recovery phase, affects the scaling relationship between the duration of the conditioning pulse and the time course of
recovery from inactivation. A comparison between recoveries at 120,
90, and 60 mV is presented in Figure
6 for the case of NaIIA( + 1)
channels. Evidently, the scaling relationship is not unique to a
particular voltage; it persists over a wide range of voltages. As shown
in Figure 6, changing the recovery voltage from 90 to 120 mV does
not influence the dependence of the recovery time course on the
duration of the conditioning depolarization. In contrast, when the
membrane is held at 60 mV compared with the other two voltages during
the recovery phase, the recovery time course becomes slower as the
conditioning duration increases. This behavior at 60 mV might be
related to the inability to separate (at this relatively depolarized
level) between the different components of inactivation (see
below).
View larger version (12K):
[in this window]
[in a new window]
|
Figure 6.
Scaling relationship between the conditioning
duration and recovery time course that persists over a range of
recovery voltages. A, Exemplar recovery time courses at
three recovery voltages (60, 90, and 120 mV) are shown. The
membrane is stepped to a 10 mV conditioning voltage for a duration
t (10, 100, or 300 sec). Recovery was measured by
applying a series of test pulses to 10 mV and was normalized and
plotted as described in Materials and Methods. Note that at all three
recovery voltages, the time course of recovery becomes slower as the
duration of the conditioning pulse is increased. At 60 mV compared
with the other two voltages, the recovery time course seems to be more
sensitive to the duration of the conditioning pulse. B,
This becomes evident in this figure, where the value of the slower time
constant () is plotted as a function of the conditioning duration
t. The slower time constants were extracted from the
time course of recovery as shown in A. Filled
circles are the mean values at 120 mV; open
circles are the mean values at 60 mV. The scaling powers are
0.6 (R > 0.97) and 0.4 (R > 0.99) for 60 and 120 mV, respectively.
|
|
The relationships between different components of recovery from
inactivation in voltage-gated sodium channels: rapid, intermediate, and
slow kinetics
The experiments described above show that recovery from
inactivation in voltage-gated sodium channels is composed of at least three components. They are defined here as follows: a rapid
component (Fig. 1A, Table 1), an
intermediate component (which is observed as the first
component of recovery in, e.g., Figs. 2A,
4A), and a slow component (which was at
the focus of the present work). The time course of the slow component
was shown above to be scaled according to the duration of the
conditioning depolarization (Figs. 2B,D, 4B). Up to
this point, an effort was directed toward separating the slow component
from the other, faster components to allow for an accurate analysis.
Under natural conditions, however, these components overlap and operate
in concert to determine the time course of recovery from inactivation.
The results of Figure 7 reveal part of
the complexity of the relationships between the three recovery
components in NaIIA( + 1) channels. Figure 7 shows the relative
amplitudes and time constants of the rapid, intermediate, and slow
recoveries as a function of conditioning duration (t). This
figure was generated by exposing oocytes to the complete battery of
pulse protocols of recovery from inactivation (i.e., conditioning
durations from 10 msec to 300 sec). The data were extracted using the
two types of pulse protocols that are detailed in Materials and
Methods; filled circles depict data from double-pulse
experiments, whereas open circles depict data from
pulse-series experiments. At the milliseconds time scale, the rapid
(Hodgkin-Huxley-like) inactivation is dominant (Fig. 7, top
left). The time constant of this rapid component is
virtually insensitive to the duration of the conditioning pulse
(top right). As the conditioning-pulse duration
increases to the several-seconds time scale, the intermediate component
of recovery becomes dominant (Fig. 7, middle left).
The time constant of this component is sensitive to the
conditioning-pulse duration (middle right). At the
tens-of-seconds time scale, the slow, time-dependent component takes
over (Fig. 7, bottom). Note the hierarchic interplay between the relative amplitudes as the conditioning-pulse duration increases. A
similar interplay between the relative amplitudes as a function of
conditioning duration was recently reported by Hayward et al. (1997)
for muscle voltage-gated sodium channels in a mammalian expression
system. It is conceivable that if conditioning pulses longer than 300 sec were applied, additional components would be uncovered. (Note that
the difference between the measured amplitudes of the intermediate
component in double-pulse and pulse-series methods at the 3 sec
conditioning duration is probably attributable to (1) the inclusion of
the rapid component in the intermediate component while fitting
pulse-series data and (2) the compromised accuracy of fitting our
pulse-series data to a component that is shorter than 1 sec).
View larger version (21K):
[in this window]
[in a new window]
|
Figure 7.
The relationships between different components of
recovery from inactivation. The relative amplitudes and time constants
of the rapid (top), intermediate
(middle), and slow (bottom) recovery components are shown. Filled circles depict results of a
double-pulse procedure; open circles depict results of a
pulse-series procedure (see Pulse protocols and analysis of slow
recovery in voltage-gated sodium channels in Materials and Methods).
During long conditioning pulses (>3 sec), the oocytes were kept under
continuous perfusion at a rate of ~2-4 ml/min, a critical factor
when a kinetic analysis of the intermediate component of recovery is
sought (see Electrophysiological measurements in Materials and
Methods). Note the dynamics of the relative amplitudes as the
conditioning duration t is increased. Also note that
unlike that of the intermediate and slow components, the time constant
of the rapid component is insensitive to the duration of the
conditioning pulse.
|
|
The scaling relationship exhibited by voltage-gated sodium channels
is not a universal property of ion channels: a comparison with the
ShakerB channel
Although this study is about the mammalian brain voltage-gated
sodium channels, it is instructive to compare the results to the
behavior of other channel types. Such a comparison with the classic
ShakerB channel is described below, showing that the scaling relationship exhibited by voltage-gated sodium channels is not a
universal property of ion channels. Like voltage-gated sodium channels,
ShakerB potassium channels inactivate in a slow ("C-type") in
addition to a longer (milliseconds, "N-type") process of
inactivation (Hoshi et al., 1991). The inset to Figure
8A shows a series of superposed current responses to identical depolarizing voltage pulses
to +40 mV, separated by 1.5 sec intervals at 90 mV. Note that N-type
inactivation recovers completely from one pulse to the next. (The
symbols of Fig. 8A depict different
interpulse intervals.) As shown in Figure 8, B and
C, and unlike voltage-gated sodium channels, ShakerB
channels demonstrate a uniquely defined time scale for recovery from
slow (C-type) inactivation. This time scale (5 sec) is practically
independent of the duration of previous depolarization.
View larger version (11K):
[in this window]
[in a new window]
|
Figure 8.
ShakerB channels recover from slow inactivation in
a uniquely defined time scale. A, Evaluation of the
duration of interpulse interval (at 90 mV) that is required for
avoidance of accumulation of inactivation because of 50-msec-long
depolarizing test pulses to +40 mV is shown. These pulse parameters
(amplitude and duration) allow for a completion of the N-type
inactivation process. Slight accumulation of inactivation (saturating
at 5%) appears when pulses are delivered once every 750 msec
(open circles). Intervals of 1.5 sec (open
diamonds) and 3 sec (open squares) allow for complete recovery between test pulses. Inset, Fourteen
superposed current responses to identical 50-msec-long depolarizing
voltage pulses from 90 to +40 mV separated by 1.5 sec intervals at
90 mV are shown. B, ShakerB channels demonstrate a
uniquely defined time scale for recovery from slow (C-type)
inactivation. Slow inactivation was induced by conditioning
depolarizations to 0 mV from a holding potential of 90 mV. The
recovery interval, at 90 mV, from the end of the conditioning pulse
to the beginning of the series of test pulses lasted 1 sec. The
fraction of inactivation was normalized and plotted as described in
Materials and Methods, taking into account the positive sign of the
currents. This time scale (5 sec) is independent of the duration
t of the conditioning depolarization (open
circles, t = 3 sec; open
squares, t = 10 sec; open
diamonds, t = 30 sec; X,
t = 100 sec; and plus signs, t = 300 sec). Inset, Recovery from a
100 sec conditioning depolarization to 0 mV is shown. Recovery from
slow inactivation is seen as a gradual increase in peak current
responses to 20-msec-long test pulses from 90 to 0 mV, which were
delivered once per second. C, Comparison of the
relationship between the duration of depolarization and the rate of
recovery from slow inactivation of NaII (filled triangles, mean values from Fig. 2B) and
ShakerB (open circles, n > 3; ±SD)
is shown. Solid lines are power law functions of the form (t) = p · tD that were fit to the mean values (NaII,
p = 0.3; D = 0.84;
R = 0.99; ShakerB, p = 3.7;
D = 0.09; R > 0.99).
|
|
| |
DISCUSSION |
Physical interpretation
This report shows that voltage-gated sodium channels recover from
the unavailable inactivation pool at an effective rate that is related
by a power law to the time spent in that pool. Power law scalings are
characteristic of fractals. By itself, the fact that some aspects of
the gating of the voltage-gated sodium channels can be described by a
fractal is not a unique observation. Fractal behavior was described in
the gating of several ion channels in the past, (Liebovitch and
Sullivan, 1987) and there is an ongoing debate concerning the molecular
mechanisms (and the corresponding structural correlates) that underlie
this phenomenon [Bassingthwaighte et al. (1994), their
references). On the one hand, some models were proposed in which the
scaling relationship in ion channel gating is attributed to a
non-Markovian process. On the other hand, considering the statistical
and experimental uncertainties that are associated with the extraction
of time constants from long-lasting electrophysiological experiments,
one should always keep in mind the possibility that power law scaling
can arise from a (surprisingly small) sum of exponents (Sauve and
Szabo, 1985). Unfortunately, there is no statistical way to compare the goodness of fit of these two theoretical models (Markovian and fractal)
to the experimental data. As explained by Bassingthwaighte et al.
(1994), the problem is that both of these models can have an arbitrary
number of adjustable parameters; thus each model can be made to fit the
experimental data to unlimited accuracy. In fact, this property of
having an arbitrary number of adjustable parameters can be used to
relate the two models to each other (Liebovitch, 1989). For example,
one can think of the scaling relationship as resulting from a complex
Markovian arrangement (with multiple time-independent rates) that is
collapsed to (and described by) a single effective rate constant. Such
an arrangement can have a branching form, a chain form, or any other
complex form. Although it is not necessary to assume a particular
kinetic scheme, the principle of collapsing a complex Markovian
arrangement to a scaled effective rate can be exemplified by the
following mechanism:
|
|
After membrane depolarization, the I states absorb
channels so that as the duration of the depolarization conditioning
pulse becomes longer, the distribution of the channels shifts further and further to the right. After hyperpolarization (recovery phase), A is an absorbing state, and the effective recovery rate
becomes dependent on the distribution of channels between the different I states. A similar one-dimensional diffusion model was
theoretically explored by Millhauser et al. (1988) and was shown to
result in power law distributions of the form demonstrated in the
present study.
The results of the present report suggest that inactivation is a
complex process that involves a mixture of time-independent and
time-dependent rates. Moreover, Figure 7 shows that the relative amplitudes of the rates are time-dependent, an observation that was recently confirmed by Hayward et al. (1997) in a
mammalian expression system. The fact that there are at least three
distinct components of recovery from inactivation suggests that
inactivation is composed of at least three distinct "families" of
conformational states. One of these components (the rapid component) is
time-independent, suggesting a simple internal structure (e.g.,
ball-and-chain-like mechanism) that is mechanistically unrelated to
slow inactivation. This conclusion conforms with more direct approaches
to the problem of interaction between fast and slow inactivation in
voltage-gated sodium channels (e.g., Valenzuela and Bennett, 1994;
Featherstone et al., 1996). The other two components (the intermediate
and the slow components) correspond to complex Markovian or fractal (non-Markovian) arrangements. The observation that the relative amplitudes of the three components change with time in an hierarchical manner (Fig. 7, left column; Hayward et al., 1997)
simply reflects the fact that the three families of inactivation states
are connected to each other in some manner.
Neurophysiological implications
Regardless of the mechanistic interpretation above, from a
physiological point of view the significant observation is that the
macroscopic availability of voltage-gated sodium channels is related in
a scale-independent manner to the duration of previous activity. The
effective time constant for moving between the two functionally defined
states, IA, is proportional to
tD, where t is the time
spent in the unavailable state of origin, and D is a scaling
power. Because D is reported here to be positive, the longer
the channel molecule resides at the state of origin (i.e., the
inactivated pool), the harder it becomes to leave that state. This is a
basis for scale-independent memory, as opposed to the scale-dependent
memory phenomenon that is familiar from standard descriptions of
inactivation (e.g., fast inactivation and a refractory period that
takes place within a uniquely defined milliseconds range). This scaling
mechanism preserves traces of previous activity, over a wide range of
time scales, in the form of modulated reaction rates rather than
quantities of molecules that are available for activation [Marom
(1998), his references]. In the case of the availability of
voltage-gated sodium channels, one might expect such a scaling
relationship to have implications on the firing patterns of neurons,
because voltage-gated sodium channels can participate in a wide range
of activity-dependent modulations over a wide range of time scales.
The importance of the channel behavior described here to neuron firing
can be appreciated from Figure 9 (a
schematic version of Fig. 8C). Note that the relationships
between the rates of recovery from inactivation of sodium and potassium
channels are such that during ongoing activity that lasts up to the
point of intersection (depicted as t') each activation is
followed by a net increase in excitability (recovery of the
"restoring" potassium conductance from inactivation is slower than
is recovery of the "exciting" sodium conductance). Beyond this
point, each activation results in suppression of activity (recovery of
the exciting sodium conductance from inactivation is slower than is
recovery of the restoring potassium conductance). Simplified as it is,
the above picture suggests that the kinetic setpoints and the slopes of the (t) functions are important variables
that are expected to determine in a nontrivial manner the dynamics of
the neuronal response.
View larger version (16K):
[in this window]
[in a new window]
|
Figure 9.
The nature of interaction between slow
inactivation of voltage-gated sodium
[Na(V)] and potassium
[K(V)] channels: an
inference. A scheme of behavior for voltage-gated sodium and potassium
channels. The position of the point of
intersection (dashed line) and the relative
slopes of the functions are variables that reflect the composition of ion channels that is unique to a neuron at a given point
of time. It is suggested that these variables are dynamic (two-headed arrows) and influenced by the
developmental stage and by activity.
|
|
In summary, the results of this study suggest that neurons have a
memory capacity that is embedded in the machinery of excitability and
that is not delimited by particular time scales. Whether or not neurons
use this capability is a subject for further research, but until this
question is answered experimentally, there is no justification to
assume locality in time when theorizing about the dynamics of threshold
and firing patterns of single neurons and neurons in networks.
| |
FOOTNOTES |
Received Nov. 3, 1997; revised Dec. 11, 1997; accepted Dec. 15, 1997.
This work was partially supported by a grant from the Israel Science
Foundation. We thank E. Braun, D. Dagan, Y. Palti, and I. Perlman for
their help throughout the preparation of this manuscript.
Correspondence should be addressed to Dr. Shimon Marom, Department of
Physiology, Faculty of Medicine, Technion, Efron Street, Bat-Galim,
P.O. Box 9697, Haifa 31096, Israel.
| |
REFERENCES |
-
Adelman WJ,
Palti Y
(1969)
The effects of external potassium and long duration voltage conditioning on the amplitude of sodium currents in the giant axon of the squid, Loligo peali.
J Gen Physiol
54:589-606[Abstract/Free Full Text].
-
Almers W,
Stanfield PR,
Stühmer W
(1983)
Slow changes in currents through sodium channels in frog muscle membrane.
J Physiol (Lond)
339:253-271[Abstract/Free Full Text].
-
Auld VJ,
Goldin AL,
Krafte DS,
Marshall J,
Dunn JM,
Catterall WA,
Lester HA,
Davidson N,
Dunn RJ
(1988)
A rat brain Na+ channel alpha subunit with novel gating properties.
Neuron
1:449-461[Web of Science][Medline].
-
Bassingthwaighte JB,
Liebovitch LS,
West BJ
(1994)
In: Fractal physiology. New York: Oxford UP.
-
Brismar T
(1977)
Slow mechanism for sodium permeability inactivation in myelinated nerve fibre of Xenopus laevis.
J Physiol (Lond)
270:283-297[Abstract/Free Full Text].
-
Catterall WA
(1992)
Cellular and molecular biology of voltage-gated sodium channels.
Physiol Rev
72:S15-S48.
-
Chandler WK,
Meves H
(1970)
Slow changes in membrane permeability and long lasting action potentials in axons perfused with fluoride solutions.
J Physiol (Lond)
211:707-728[Abstract/Free Full Text].
-
Cummins TR,
Sigworth FJ
(1996)
Impaired slow inactivation in mutant sodium channels.
Biophys J
71:227-236[Web of Science][Medline].
-
Featherstone DE,
Richmond JE,
Ruben PC
(1996)
Interaction between fast and slow inactivation in Skm1 sodium channels.
Biophys J
71:3098-3109[Web of Science][Medline].
-
Fleidervish IA,
Friedman A,
Gutnick MJ
(1996)
Slow inactivation of Na+ current and slow cumulative spike adaptation in mouse and guinea-pig neocortical neurons in slices.
J Physiol (Lond)
493:83-97[Abstract/Free Full Text].
-
Fox JM
(1976)
Ultra-slow inactivation of the ionic currents through the membrane of myelinated nerve.
Biochim Biophys Acta
426:232-244[Medline].
-
Hamill OP,
Marty A,
Neher E,
Sakmann B,
Sigworth FJ
(1981)
Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches.
Pflügers Arch
391:85-100[Web of Science][Medline].
-
Hayward LJ,
Brown Jr RH,
Cannon SC
(1997)
Slow inactivation differs among mutant Na channels associated with myotonia and periodic paralysis.
Biophys J
72:1204-1219[Web of Science][Medline].
-
Hodgkin AL,
Huxley AF
(1952)
A quantitative description of membrane current and its application to conduction and excitation in nerve.
J Physiol (Lond)
117:500-544.
-
Hoshi T,
Zagotta WN,
Aldrich RW
(1991)
Two types of inactivation in Shaker K+ channels: effect of alterations in the carboxy-terminal region.
Neuron
7:547-556[Web of Science][Medline].
-
Isom LL,
De Jongh KS,
Catterall WA
(1994)
Auxiliary subunits of voltage-gated ion channels.
Neuron
12:1183-1194[Web of Science][Medline].
-
Liebovitch LS
(1989)
Testing fractal and Markov models of ion channel kinetics.
Biophys J
55:373-377[Web of Science][Medline].
-
Liebovitch LS,
Sullivan JM
(1987)
Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.
Biophys J
52:979-988[Web of Science][Medline].
-
Marom S
(1998)
Slow changes in the availability of voltage-gated ion channels: effects on the dynamics of excitable membranes [topical review].
J Membr Biol
161:107-116.
-
Millhauser GL,
Salpeter EE,
Oswald RE
(1988)
Diffusion models of ion-channel gating and the origin of power-law distributions from single-channel recording.
Proc Natl Acad Sci USA
85:1503-1507[Abstract/Free Full Text].
-
Noda M,
Ikeda T,
Suzuki H,
Takeshima H,
Takahashi T,
Kuno M,
Numa S
(1986)
Expression of functional sodium channels from cloned cDNA.
Nature
322:826-828[Medline].
-
Ruben PC,
Starkus JG,
Rayner MD
(1992)
Steady-state availability of sodium channels: interactions between activation and slow inactivation.
Biophys J
61:941-955[Web of Science][Medline].
-
Ruben PC,
Fleig A,
Featherstone D,
Starkus JP,
Rayner MD
(1997)
Effects of clamp rise-time on rat brain IIA sodium channels in Xenopus oocytes.
J Neurosci Methods
73:113-122[Web of Science][Medline].
-
Rudy B
(1978)
Slow inactivation of the sodium conductance in squid giant axons. Pronase resistance.
J Physiol (Lond)
283:1-21[Abstract/Free Full Text].
-
Sauve R,
Szabo G
(1985)
Interpretation of 1/F fluctuations in ion conducting membranes.
J Theor Biol
113:501-516[Web of Science][Medline].
-
Schauf CL,
Peneck TL,
Davis FA
(1976)
Slow inactivation in Myxicola axons.
Biophys J
16:771-778[Web of Science][Medline].
-
Simoncini L,
Stühmer W
(1987)
Slow sodium channel inactivation in rat fast-twitch muscle.
J Physiol (Lond)
383:327-337[Abstract/Free Full Text].
-
Stühmer W,
Methfessel C,
Sakmann B,
Noda M,
Numa S
(1987)
Patch clamp characterization of sodium channels expressed from rat brain cDNA.
Eur Biophys J
14:131-138[Web of Science][Medline].
-
Tempel BL,
Papazian DM,
Schwarz TL,
Jan YN,
Jan LY
(1987)
Sequence of a probable potassium channel component encoded at Shaker locus of Drosophila.
Science
237:770-775[Abstract/Free Full Text].
-
Valenzuela C,
Bennett Jr PB
(1994)
Gating of cardiac Na+ channels in excised membrane patches after modification by -chymotrypsin.
Biophys J
67:161-171[Web of Science][Medline].
-
Wallner M,
Weigle I,
Meera P,
Lotan I
(1993)
Modulation of the skeletal muscle sodium channel alpha subunit by the beta1 subunit.
FEBS Lett
336:535-539[Web of Science][Medline].
Copyright © 1998 Society for Neuroscience 0270-6474/98/1851893-11$05.00/0
This article has been cited by other articles:
|
|
|
|
 
Y. Dudai
Predicting not to predict too much: how the cellular machinery of memory anticipates the uncertain future
Phil Trans R Soc B,
May 12, 2009;
364(1521):
1255 - 1262.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
S. L. Hooper, E. Buchman, A. L. Weaver, J. B. Thuma, and K. H. Hobbs
Slow Conductances Could Underlie Intrinsic Phase-Maintaining Properties of Isolated Lobster (Panulirus interruptus) Pyloric Neurons
J. Neurosci.,
February 11, 2009;
29(6):
1834 - 1845.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. M Beggs
The criticality hypothesis: how local cortical networks might optimize information processing
Phil Trans R Soc A,
February 13, 2008;
366(1864):
329 - 343.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
A. C. Errington, T. Stohr, C. Heers, and G. Lees
The Investigational Anticonvulsant Lacosamide Selectively Enhances Slow Inactivation of Voltage-Gated Sodium Channels
Mol. Pharmacol.,
January 1, 2008;
73(1):
157 - 169.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
A. Kohn
Visual Adaptation: Physiology, Mechanisms, and Functional Benefits
J Neurophysiol,
May 1, 2007;
97(5):
3155 - 3164.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
P. J. Drew and L. F. Abbott
Models and Properties of Power-Law Adaptation in Neural Systems
J Neurophysiol,
August 1, 2006;
96(2):
826 - 833.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
M. Uebachs, C. Schaub, E. Perez-Reyes, and H. Beck
T-type Ca2+ channels encode prior neuronal activity as modulated recovery rates
J. Physiol.,
March 15, 2006;
571(3):
519 - 536.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
S. W. Jones
Are rate constants constant?
J. Physiol.,
March 15, 2006;
571(3):
502 - 502.
[Full Text]
[PDF]
|
|
|
|
|
|
|
N. Osorio, G. Alcaraz, F. Padilla, F. Couraud, P. Delmas, and M. Crest
Differential targeting and functional specialization of sodium channels in cultured cerebellar granule cells
J. Physiol.,
December 15, 2005;
569(3):
801 - 816.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
W. Ulbricht
Sodium Channel Inactivation: Molecular Determinants and Modulation
Physiol Rev,
October 1, 2005;
85(4):
1271 - 1301.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
G. N. Filatov, M. J. Pinter, and M. M. Rich
Resting Potential-dependent Regulation of the Voltage Sensitivity of Sodium Channel Gating in Rat Skeletal Muscle In Vivo
J. Gen. Physiol.,
July 25, 2005;
126(2):
161 - 172.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
G. Gilboa, R. Chen, and N. Brenner
History-Dependent Multiple-Time-Scale Dynamics in a Single-Neuron Model
J. Neurosci.,
July 13, 2005;
25(28):
6479 - 6489.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
N. Ulanovsky, L. Las, D. Farkas, and I. Nelken
Multiple Time Scales of Adaptation in Auditory Cortex Neurons
J. Neurosci.,
November 17, 2004;
24(46):
10440 - 10453.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
Y. Hayashida and A. T. Ishida
Dopamine Receptor Activation Can Reduce Voltage-Gated Na+ Current by Modulating Both Entry Into and Recovery From Inactivation
J Neurophysiol,
November 1, 2004;
92(5):
3134 - 3141.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. Hering, A. Feltz, and R. C. Lambert
Slow inactivation of the CaV3.1 isotype of T-type calcium channels
J. Physiol.,
March 1, 2004;
555(2):
331 - 344.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. M. Beggs and D. Plenz
Neuronal Avalanches in Neocortical Circuits
J. Neurosci.,
December 3, 2003;
23(35):
11167 - 11177.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
P. Darbon, A. Tscherter, C. Yvon, and J. Streit
Role of the Electrogenic Na/K Pump in Disinhibition-Induced Bursting in Cultured Spinal Networks
J Neurophysiol,
November 1, 2003;
90(5):
3119 - 3129.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
K. J. Kim and F. Rieke
Slow Na+ Inactivation and Variance Adaptation in Salamander Retinal Ganglion Cells
J. Neurosci.,
February 15, 2003;
23(4):
1506 - 1516.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
R. K Ellerkmann, V. Riazanski, C. E Elger, B. W Urban, and H. Beck
Slow recovery from inactivation regulates the availability of voltage-dependent Na+ channels in hippocampal granule cells, hilar neurons and basket cells
J. Physiol.,
April 15, 2001;
532(2):
385 - 397.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
S. Sokolov, R. G Weiss, E. N Timin, and S. Hering
Modulation of slow inactivation in class A Ca2+ channels by {beta}-subunits
J. Physiol.,
September 15, 2000;
527(3):
445 - 454.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
H. B. Nuss, E. Marbán;, C. W. Balke, L. Goldman, R. Aggarwal, S. R. Shorofsky;, J. dos Santos Cruz, L. F. Santana, C. A. Frederick, L. L. Isom, et al.
Whether "Slip-Mode Conductance" Occurs
Science,
April 30, 1999;
284(5415):
711a - 711.
[Full Text]
|
|
|
|
|
|
|
J. P O'Reilly, S.-Y. Wang, R. G Kallen, and G. K. Wang
Comparison of slow inactivation in human heart and rat skeletal muscle Na+ channel chimaeras
J. Physiol.,
February 15, 1999;
515(1):
61 - 73.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
K. Morgan, E. B. Stevens, B. Shah, P. J. Cox, A. K. Dixon, K. Lee, R. D. Pinnock, J. Hughes, P. J. Richardson, K. Mizuguchi, et al.
beta 3: An additional auxiliary subunit of the voltage-sensitive sodium channel that modulates channel gating with distinct kinetics
PNAS,
February 29, 2000;
97(5):
2308 - 2313.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
Top
Abstract
Introduction
