1Division of Neuroscience, Center for Theoretical Neuroscience, Baylor College of Medicine, Houston, Texas 77030; and 2Sloan Center for Theoretical Neurobiology, The Salk Institute, La Jolla, California 92037
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ABSTRACT |
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Wiest, Michael C., David M. Eagleman, Richard D. King, and P. Read Montague. Dendritic Spikes and Their Influence on Extracellular Calcium Signaling. J. Neurophysiol. 83: 1329-1337, 2000. Extracellular calcium is critical for many neural functions, including neurotransmission, cell adhesion, and neural plasticity. Experiments have shown that normal neural activity is associated with changes in extracellular calcium, which has motivated recent computational work that employs such fluctuations in an information-bearing role. This possibility suggests that a new style of computing is taking place in the mammalian brain in addition to current `circuit' models that use only neurons and connections. Previous computational models of rapid external calcium changes used only rough approximations of calcium channel dynamics to compute the expected calcium decrements in the extracellular space. Using realistic calcium channel models, experimentally measured back-propagating action potentials, and a model of the extracellular space, we computed the fluctuations in external calcium that accrue during neural activity. In this realistic setting, we showed that rapid, significant changes in local external calcium can occur when dendrites are invaded by back-propagating spikes, even in the presence of an extracellular calcium buffer. We further showed how different geometric arrangements of calcium channels or dendrites prolong or amplify these fluctuations. Finally, we computed the influence of experimentally measured synaptic input on peridendritic calcium fluctuations. Remarkably, appropriately timed synaptic input can amplify significantly the decrement in external calcium. The model shows that the extracellular space and the calcium channels that access it provide a medium that naturally integrates coincident spike activity from different dendrites that intersect the same tissue volume.
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INTRODUCTION |
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Numerous important neuronal processes are
sensitive to external calcium levels, e.g., calcium-dependent cell
adhesion (Rose 1998; Tang et al. 1998
;
Uemura 1998
), synaptic plasticity (Bear and
Malenka 1994
; Denk et al. 1996
),
neurotransmission (Katz and Miledi 1970
; Mintz et
al. 1995
; Qian et al. 1997
), ionotropic receptor
function (Xiong et al. 1997
), and metabotropic receptor function (Brown et al. 1995
; Kubo et al.
1998
). Thus if the external calcium level changes significantly
during normal neural activity, it could have computational effects by
influencing neural function in neighboring regions (Egelman and
Montague 1998
, 1999
; Montague 1996
; Smith
1992
; Vassilev et al. 1997
). In fact, several
studies observed significant fluctuations in external calcium at
relatively large spatiotemporal scales after electrical or
neurotransmitter stimulation (Benninger et al. 1980
;
Heinemann et al. 1990
; Lucke et al. 1995
;
Nicholson et al. 1978
; Pumain and Heinemann
1985
; Stanton and Heinemann 1986
) (see Fig.
1). Given that many important computational processes function on millisecond time scales and submicron spatial scales, we were led to ask how electrical events at
these smaller scales affect the external calcium level.
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In the mammalian brain, action potentials can propagate into the
dendrites of cortical and hippocampal neurons (Stuart and Sakmann 1994). These spikes cause large influxes of calcium
from the extracellular space (Helmchen et al. 1996
;
Magee and Johnston 1997
; Svoboda et al.
1997
). Previous computational work examining the properties of
calcium fluctuations around dendrites relied on rough estimates of the
magnitude and time course of calcium consumption (Egelman and
Montague 1998
, 1999
). Recent experiments have
allowed us to quantify the dynamics of calcium associated with single
experimentally measured dendritic spikes in hippocampal and cortical
pyramidal neurons (Fisher et al. 1990
; Helmchen
et al. 1996
; Jaffe et al. 1994
). These data,
which were not incorporated into previous models, allow a more rigorous
numerical estimate of how normal dendritic activity at millisecond time
scales could influence external calcium.
Using these experimental data and a computer simulation of the extracellular space, we addressed the issue of whether the measured increase in intracellular calcium caused by dendritic action potentials would be paralleled by a significant decrease in extracellular calcium in a region surrounding the activated dendrite. In this study, we examined several influences on external calcium fluctuations during a dendritic spike: 1) channel clustering, 2) background activity, 3) extracellular calcium buffering, 4) the geometric arrangement of coactive dendrites, and 5) dendritic spike modulation by synaptic stimulation.
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METHODS |
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Strategy
We approached these issues by using a finite difference model of
the extracellular space. The model is shown schematically in Fig.
2A. Simulations were
constrained by experimental data. We used realistic models of calcium
channels derived from patch recordings in hippocampal pyramidal cell
dendrites (Fisher et al. 1990; Jaffe et al.
1994
), "drove" these models with the voltage waveforms of
experimentally measured dendritic spikes (Fig. 2B), and
measured the time course, amplitude, and spatial extent of the external
calcium fluctuations that resulted. The segment of dendrite possesses
models of L-like, N-like, and T-like calcium channels whose operation
under a voltage-clamp step is shown in Fig. 2C. Figure
2D shows the calcium currents that result when the same
dendritic segment is activated with the experimentally measured
dendritic spike shown in Fig. 2B. The density of the calcium
channels was chosen to match the calcium entry per spike to values
reported by Helmchen et al. (1996)
.
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Experimentally measured back-propagating spikes
In our model, the dendritic voltage during a back-propagating
spike is clamped to action potential waveforms experimentally measured
in a CA1 hippocampal pyramidal cell dendrite 100 µm from the cell
body (waveforms courtesy of Dax Hoffman). By using this waveform to drive the calcium channel kinetics, we assume that the
dendritic calcium currents do not contribute appreciably to the
dendritic spike. This assumption is justified by comparing measurements of dendritic action potentials in the presence and absence
of calcium channel blockers (Fig. 3; N. Poolos and D. Johnston, personal communication). The difference between
the two voltage traces is negligible. We also assumed that an entire segment of dendrite (typically ~4 µm in our simulations) is
activated synchronously by the back-propagating action potential. The
conduction velocity of these dendritic spikes (>300 µm/msec)
justifies this assumption (Stuart and Sakmann 1994).
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Diffusion in the extracellular space
To simulate extracellular calcium diffusion, we developed a
three-dimensional model of neural tissue that includes an explicit representation of the extracellular space (Egelman and Montague 1998, 1999
). The basic cubic building blocks,
called intracellular units (IUs, 0.826 µm on a side), represent
membrane compartments that can be combined to represent dendritic
segments (Fig. 2A). In this study, all boundary elements are
separated by clefts of 20 nm. The extracellular space is further
subdivided into small boxes called extracellular space (ECS) units
(0.118 µm on a side). Each ECS unit contains a single state variable
that represents the average concentration in that volume. At each time
step, an ECS unit updates its concentration as a function of its
adjacent ECS neighbors and any calcium consumed or extruded by adjacent IUs. Each IU in the model has set densities of the different calcium channel types, which can be spread evenly over the surface of an IU or
clumped arbitrarily to represent the clustering of calcium channels.
The total volume of neural tissue simulated was chosen to be large
enough to avoid edge effects, typically (7 × 0.826 µm)3, where 0.826 µm is the side length of an IU.
External calcium diffusion in our model was set to match measurements
of the long-range (hundreds of microns) effective diffusion constant in
nervous tissue. Because of the impressive variation of those values
with brain region, temperature, and experimental method, we show in
Fig. 4A the effect of varying
our diffusion parameter. We note that the low diffusion
constant (8 µm2/s) is actually the value
measured in rat hippocampus (see Nicholson and Margaret
1987), the particular structure we modeled. This hippocampal
value, while apparently atypical, leads to a much larger external
calcium decrement during a dendritic action potential.
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To determine an effective diffusion constant, it is conventional to
define the tortuosity of a diffusive medium, which relates the
effective diffusion constant in the medium to the free diffusion constant in water (or other standard solution):
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Driving force for calcium entry
A modified Goldman-Hodgkin-Katz (GHK) model gives the calcium current
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(1) |
Channel models
Although a number of fine distinctions exist among dendritic
calcium channels, the three canonical types we used were a transient channel (T-like), a noninactivating channel (L-like), and a moderately inactivating channel (N-like). The Hodgkin-Huxley calcium channel models are represented in a standard way as functions of activation (mi) and inactivation
(hi) variables that together define a
"channel model" Ai(V, t)
where i indicates T-like, N-like, or L-like calcium channels. The exact form of the channel models and their dependence on
experimental data were taken from Fisher et al. (1990)
and Jaffe et al. (1994)
:
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Matching calcium influx to experimental data
We adjust in the channel models to fix the
total calcium consumption through a unit membrane area to agree with
the concentration change observed by Helmchen et al.
(1996)
during a back-propagating spike; if the single-channel
maximum permeability is a constant, this is equivalent to adjusting the
channel densities. Helmchen et al. (1996)
found a
151 ± 19 nM change in free calcium concentration per dendritic
spike and concluded that the observed change represents between 0.5%
and 1.0% of the total calcium influx, the rest having been buffered.
Assuming a dendritic diameter of 1.5 µm, these measurements yield a
total calcium flux per spike of 6,800-13,600 atoms/µm2 (depending on whether the 151 nM
calcium rise represents 1% or 0.5% of the total influx,
respectively). We assume equal densities of N-, L-, and T-type
channels, with a maximum permeability ratio of 1:1.38:1.5, respectively
(Johnston and Wu 1995
). These assumptions yield a total
maximum permeability (
) between 21 and 42 µm/s for
a 1 µm2 patch of membrane.
Resting external calcium level
To determine the influence of the initial external calcium
concentration on the consumption profile, we ran the ECS model with the
range of calcium between 0.8 and 2.0 mM (Fig. 4B). Figure 4C shows that in this physiological range, our results scale
linearly with the initial calcium concentration. We used a resting
external calcium concentration of 1.2 mM for the remaining simulations, as suggested by Pumain (1998).
Numerical integration
Our ECS model may be summarized by the difference equation
governing the change in the external calcium level,
Cj, at each time step,
t, in each ECS unit j. We write this change as
a sum of four terms: Xj represents
diffusion between neighboring ECS volumes,
Yj is the calcium consumption into cells
through voltage-gated channels, Zj is the
replenishment into the ECS by membrane calcium pumps, and
Bj represents an extracellular calcium
buffer
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RESULTS |
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Dendritic spike activity lowers local external calcium
Figure 5 shows our fundamental result: the change in external calcium through time in response to the dendritic spike shown in Fig. 2B. In Fig. 5A, the channel models are distributed evenly over the surface of the dendrite. Figure 5B shows that clustering the channels into small patches amplifies the external calcium decrement immediately above a patch. The bottom panels show that the calcium decrement is localized to within ~1µm of the activated dendrite.
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SIGNIFICANCE OF THE CALCIUM FLUCTUATION.
The results in Fig. 5 raise a critical question: what magnitude of
change in external calcium levels should be deemed significant? To
qualify as a signal, a change in extracellular calcium concentration must at least exceed normal statistical fluctuations. If N
is the number of ions in the volume of interest, the expected size of
the noise is , assuming the calcium ions act as independent random walkers. Hence, the expected size of a
statistical fluctuation in the average external calcium concentration depends on the volume of the averaging region. The results in Fig. 5
show average external calcium concentrations in an extracellular volume
20 nm ×118 nm ×118 nm = 2.78 × 105
nm3 (roughly the cleft volume above a typical
synaptic active zone). Using the
measure in
a volume of this size, fluctuations >0.1 mM would be considered
significant. Alternatively, if the calcium "reader" were an entire
overlying presynaptic terminal, then a reasonable estimate of the
averaging volume would be the volume of the entire synaptic cleft,
which is about 50 times larger than our present averaging volume
(Smith and Augustine 1988
). In this case, fluctuations
as small as 0.02 mM would be resolvable.
EXTRACELLULAR CALCIUM BUFFERS.
It is conceivable that calcium buffers in the extracellular space could
change the fluctuation results presented above. To examine this issue,
we included a simple buffer in our model of diffusion in the
extracellular space (see METHODS). We modeled the
extracellular buffer as a first-order binding process with a
Kd = 2 mM, as measured for calcium-binding
cadherins (Maurer et al. 1996). Because values for
ON and OFF rate constants
are not available in the literature, we conservatively chose the on and
off rate constants to equilibrate faster than the calcium dynamics. The
ON and OFF rates used were
Kon = 5/(ms · mM) and Koff = 10/ms.
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BACKGROUND ACTIVITY. We also asked how background neural activity affects the detection of a particular calcium signal. From one point of view, this background activity does not constitute noise in the system because calcium fluctuations caused by "background" activity may be interpreted as signals in their own right. That is, if we consider the external calcium level as a nonspecific measure of local activity, then any activity that affects external calcium is a legitimate signal, as opposed to noise. On the other hand, if we require the system to register the occurrence of a change caused by a specific dendrite in a tissue volume firing at some background rate, the background activity may be interpreted as noise.
Figure 7 shows the calcium level in a tissue volume consisting of a dendrite firing regularly at 30 Hz surrounded by axon terminals firing independently at Poisson rates of 10 Hz. The calcium level is measured at the surface of the dendrite. Although most of the significant dips in the external calcium concentration are caused by the dendrite's activity, some of the background fluctuations are of comparable size to the dendritic "signal." Thus a calcium sensor just outside the dendrite may not be able to reliably distinguish the dendritic spike signal from other nearby activity. This suggests that if the brain uses extracellular calcium levels to distinguish a signal within a specific diffusion-defined domain (e.g., a signal from a particular dendritic segment), it must boost the signal-to-noise ratio at specific calcium-sensing sites. We summarize some likely mechanisms for boosting the calcium signal in the DISCUSSION. In the absence of such boosting, the calcium signal may be used in a nonspecific, integrative fashion within each diffusion-defined calcium domain. Such domains are expected to be small, given the short range of the calcium fluctuation as shown in Fig. 5.
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External calcium may encode synchronous activity from distant cell bodies
To examine the dependence of the external calcium fluctuation on
coincident spike activity of neighboring dendrites, we simulated two
representative dendritic arrangements: crossing dendrites (Fig.
8) and dendritic bundles (Fig.
9). Coincident activity in neighboring
dendrites was expected to amplify the external calcium decrement. The
degree of the amplification depended on the geometric arrangement of
the dendrites. In the case where perpendicular dendrites met at a
specific point in space, coincident action potentials in the dendrites
amplified the external calcium signal by a factor of two (Fig. 8).
Figure 9 shows that bundles of parallel, coactive dendrites (such as
those found in Schmolke 1987) can dramatically amplify
and prolong the calcium decrement near the center of the bundle at
point C (compare to Fig. 5A, solid line), and
outside but between a second bundle, point D.
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Subthreshold synaptic input modulates the external calcium signal
The amplitude of a back-propagating dendritic spike
decreases as it travels away from the cell body. It has been shown that excitatory synaptic input preceding a dendritic spike can boost the
spike, save it from attenuation as it travels, and dramatically increase the calcium influx into more distal dendrites (Magee and Johnston 1997). To examine the effect of spike attenuation and boosting in distal hippocampal dendrites, we activated the model
with distal (~250 µm) dendritic spikes (Fig.
10). The unboosted spike is shown in
Fig. 10A (Control spike). Activation of the model with this
spike led to a very small external calcium decrement (Fig.
10B) that, by our reasoning above, would not be considered a
significant external calcium fluctuation. However, as mentioned, distal
spikes can be boosted in amplitude by synaptic input or by a
depolarizing current in the dendrite that emulates synaptic input
(Magee and Johnston 1997
; Fig. 10A, boosted
spike). Using this boosted dendritic spike to drive the model increased
the external calcium decrement (Fig. 10B) to a level
comparable to the fluctuations induced by the dendritic spike measured
at 100 µm from the soma (Fig. 5). The analogous result caused by
experimentally measured synaptic stimulation is shown in Fig.
11, which plots the external calcium
level under spikes paired and not paired with distal synaptic
stimulation (voltage traces from Fig. 1B of Magee and
Johnston 1997
). We note that we did not model synaptic dynamics
per se; rather we used effective spike waveforms measured during actual
synaptic stimulation to drive our model of calcium diffusion in the
extracellular space.
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This observed boosting effect could be combined with the effect
of coincident activity on external calcium to create a mechanism for
rapidly indexing specific tissue volumes (Egelman et al.
1998). In the simplest case the mechanism would work as
follows. We noted in the previous paragraph that synaptic input
to distal dendrites preferentially calls back-propagating spikes into
the dendritic branches that received recent synaptic input. Supposing
that the intersecting branches in Fig. 8 have just been synaptically
excited, each will draw down on external calcium as shown in Fig. 11,
and we might expect that near the intersection of these two branches the calcium decrement would be amplified. This expectation is confirmed
by the simulation results shown in Fig. 8. Thus synchronous activation
of particular synapses could be "flagged" by a deep calcium
decrement in a specific tissue volume. This sequence of events provides
a mechanism by which to identify a specific pattern of synaptic input
impinging on a tissue volume.
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DISCUSSION |
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Using detailed experimental data from the mammalian brain, we have
shown that increases in intracellular calcium during back-propagating spikes are likely to be paralleled by concomitant decreases in external
calcium near the activated dendrites (Figs. 4-11). The size of such
fluctuations can be modulated; for example, we showed that synaptic
input could change a small, insignificant external calcium fluctuation
into a larger, significant fluctuation. Likewise, clustering of calcium
channels amplifies external calcium fluctuations by as much as a factor
of two and possibly more (Fig. 5). The most dramatic changes in the
external calcium fluctuation are induced by arrangements of neighboring
coactive dendrites (Figs. 7 and 8). During a dendritic spike,
neighboring, coactive dendrites can amplify the external fluctuation by
a factor of four or more. In the same way that the intracellular
calcium level can reflect the firing rate of a particular neuron
(Helmchen et al. 1996), the extracellular calcium level
could encode the amount of coincident spike activity in a population of
neurons whose dendrites overlap. Hence, the external calcium changes
permit a local comparison of the activities of neurons whose cell
bodies may be widely separated and whose activity may be devoted to
different computations. Although not investigated in this study, we
expect calcium spikes or bursts of action potentials to lead to larger
external calcium fluctuations than those caused by a single
back-propagating spike.
Although experiments to test directly the predictions of our model
require daunting spatiotemporal resolution, experiments with lesser
resolution have shown that electrical or neurotransmitter stimulation
leads to large, slowly decaying decrements in external calcium
(Benninger et al. 1980; Heinemann et al.
1990
; Lucke et al. 1995
; Nicholson et al.
1978
; Pumain and Heinemann 1985
; Stanton and Heinemann 1986
). These experiments corroborate the
hypothesis that neural activity causes significant changes in
extracellular calcium at very small spatial and temporal scales, which
suggests the need for more definitive experimental tests of rapid
external calcium fluctuations.
Physiological readers of external calcium fluctuations
There are also data to suggest that rapid external calcium signals
are functionally important in the moment-to-moment operation of the
mammalian nervous system. For example, they may serve to modify
synaptic transmission on fast time scales. Experiments in the
cerebellum and hippocampus show that changes in probability of release
are proportional to the square of external calcium levels (Mintz
et al. 1995; Qian et al. 1997
). In this way,
changing external calcium levels in small tissue volumes could be
expressed as relative changes in the probability of release at synaptic terminals within the volume.
Furthermore, metabotropic glutamate receptors in the mammalian brain
possess a remarkable structural similarity to the external calcium
sensors of the parathyroid gland, suggesting that this important class
of glutamate receptor may also act as an external calcium sensor in the
brain. Such a possibility has indeed been demonstrated and studied with
mutational analysis (Kubo et al. 1998). An ionotropic
receptor has also been identified that is activated by changes in
external calcium (Xiong et al. 1997
).
Another example of a potential reader of external calcium
fluctuations that has important neural consequences is the
calcium-dependent cell adhesion molecule (cadherin) (see
Egelman, D. M., 1998, doctoral dissertation:
Computational properties of extracellular calcium dynamics.
www.cnl.salk.edu/~eagleman/). These molecules bind to one another
in the presence of sufficient calcium and dissociate if calcium levels
are low. Mammalian neural tissue expresses several cadherins which are
known to be localized to synaptic complexes (Fannon and Colman
1996
; Uchida et al. 1996
; Yamagata et al.
1995
). The role that cadherins play in hippocampal plasticity
was tested by Tang et al. (1998)
who used peptide
antagonists to the binding domain of the cadherins. Application
of these peptides to resting hippocampal slices had no effect on
baseline synaptic transmission; however, if long-term potentiation
(LTP)-inducing stimuli were used, the magnitude of the elicited
LTP was dramatically reduced in the presence of the peptides. The
peptides had to be present during the induction of LTP; adding the
peptides after LTP induction had no effect on the LTP elicited by the
stimulation. The interpretation of Tang et al. (1998)
,
which is also our interpretation, is that the stimulation protocol
caused decrements in external calcium that allowed the cadherins to
dissociate to reveal the sites where the peptides could bind and
diminish LTP (induction or expression). The idea is that this
inhibitory action arises by preventing reassociation of the cell
adhesion molecules. This interpretation predicts that raising external
calcium levels might prevent the influence of the peptides on LTP. In
fact, raising external calcium from 2.0 to 5.0 mM prevented the
inhibitory effects of the peptides.
The activity-dependent fluctuations in the external calcium level predicted by our simulations, as well as experimental measurements of external calcium changes, support the possibility that these external calcium sensors are actually used to signal among cells. Consequently, we suggest that external calcium has a pervasive, fluidlike influence throughout the interstices of the mammalian nervous system, allowing the spaces between neurons and glia to play a fundamental role in the computations carried out in the brain.
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ACKNOWLEDGMENTS |
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We thank Drs. Dan Johnson, Jeff Magee, Dax Hoffman, and Nick Poolos for generously sharing action potential data, calcium channel data, and technical expertise. We also gratefully acknowledge the helpful conversations and technical advice of Drs. David Jaffe and Costa Colbert. Figure 3 was provided by N. Poolos and D. Johnston.
This work was supported by the Center for Theoretical Neuroscience at Baylor College of Medicine, National Institutes of Health (NIH) Grants MH-52797 and DA-11723 (P. R. Montague), the United Negro College Fund/Merck Foundation and NIH grant MH-19547 (R. D. King), and the Biomedical Computation and Visualization Laboratory at Baylor College of Medicine (NSF-BIR-9412521). M. C. Wiest is a Kane Foundation Fellow and is supported by National Library of Medicine Grant 1T15LM07093 through the Keck Center for Computational Biology.
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FOOTNOTES |
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Address for reprint requests: P. R. Montague, Center for Theoretical Neuroscience, Division of Neuroscience, Baylor College of Medicine, One Baylor Plaza, Houston, TX 77030.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 7 July 1999; accepted in final form 20 October 1999.
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NOTE ADDED IN PROOF |
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Two recent experimental studies provided a direct demonstration of
functional depletion of extracellular calcium in the clefts of
individual synapses (Borst and Sakmann 1999;
Stanley 2000
). Borst and Sakmann (1999)
also showed that synaptic calcium depletion leads to changes in
synaptic transmission.
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REFERENCES |
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