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The Journal of Neuroscience, May 15, 2002, 22(10):4142-4152
The Spatial Dimensions of Electrically Coupled Networks of
Interneurons in the Neocortex
Yael
Amitai1, 2,
Jay R.
Gibson1,
Michael
Beierlein1,
Saundra L.
Patrick1,
Alice M.
Ho1,
Barry W.
Connors1, and
David
Golomb2
1 Department of Neuroscience, Brown University,
Providence, Rhode Island 02912, and 2 Department of
Physiology and Zlotowski Center for Neuroscience, Faculty of Health
Sciences, Ben-Gurion University, Beer-Sheva 84105, Israel
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ABSTRACT |
Inhibitory interneurons of the neocortex are electrically
coupled to cells of the same type through gap junctions. We studied the
spatial organization of two types of interneurons in the rat somatosensory cortex: fast-spiking (FS) parvalbumin-immunoreactive (PV+) cells, and low threshold-spiking (LTS)
somatostatin-immunoreactive (SS+) cells. Paired recordings in layer 4 demonstrated that both the probability of coupling and the coupling
coefficient drop steeply with intersomatic distance, reaching zero
beyond 200 µm. The dendritic arbors of FS and LTS cells were
reconstructed from electrophysiologically characterized,
biocytin-filled cells; the two cell types had only minor differences in
the number and span of their dendrites. However, there was a markedly
higher density of PV+ cells than SS+ cells. PV+ cells were densest in
layer 4, while SS+ cell density peaked in the subgranular layers. From these data we estimate that there is measurable electrical coupling (directly or indirectly via intermediary cells) between each
interneuron and 20-50 others. The large number of electrical synapses
implies that each interneuron participates in a large, continuous
syncytium. To evaluate the functional significance of these findings,
we examined several simple architectures of coupled networks
analytically. We present a mathematical method to estimate the average
summated coupling conductance that each cell receives from all of its
neighbors, and the average leak conductance of individual cells, and we
suggest that these have the same order of magnitude. These quantitative results have important implications for the effects of electrical coupling on the dynamic behavior of interneuron networks.
Key words:
FS cells; LTS cells; inhibitory interneurons; gap
junctions; dendritic fields; coupling coefficient; coupling
conductance; network architecture
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INTRODUCTION |
The circuitry of the neocortex has
traditionally been represented by maps of neurons that are
interconnected by axons and chemical synapses (Braitenberg,
1978 ; White, 1989). This is an impoverished
view, however, because there is now strong evidence that electrical
synapses are also a frequent and important feature of neocortical
circuits. Electrical synapses are most prevalent between inhibitory
interneurons (Galarreta and Hestrin, 1999; Gibson
et al., 1999). The circuits defined by electrical synapses can
be highly specific; among two common types of interneurons in the
neocortex, the large majority of electrical synapses interconnect cells
of the same type (Gibson et al., 1999). The importance
of electrical synapses to the function of the neocortex is still poorly
understood, but recent studies suggest a role in neuronal synchronization and rhythm generation (Benardo, 1997;
Beierlein et al., 2000; Galaretta and Hestrin,
2001; Deans et al., 2001).
Understanding what electrical synapses do in the neocortex, and
precisely how they do it, will require quantitative information about
the patterns of neural circuits defined by electrical connections. Important issues include the incidence of connectivity, the
organization and the size of coupled assemblies of neurons, and the
strength of the electrical coupling each cell has with other cells.
These kinds of data have been hard to come by. The anatomical basis of
electrical synapses is the gap junction (Bennett, 1977),
a structure visible only with electron microscopy. Very few studies have described gap junctions between neurons in the mammalian forebrain, and most of these have been between certain types of interneurons in either the neocortex (Sloper, 1972) or
hippocampus (Kosaka, 1983; Kosaka and Hama,
1985). Unlike chemical synapses in the cerebral cortex, gap
junctions have been seen most frequently at dendrodendritic and
dendrosomatic sites (Sloper and Powell, 1978;
Tamás et al., 2000; Szabadics et al.,
2001). Although this is very important information, it does not
reveal the scale of electrical coupling at the level of larger
interneuronal circuits.
We have used data derived from dual recordings of electrically coupled
neurons in the rat somatosensory cortex, anatomical reconstructions and
immunohistochemistry for specific markers of GABAergic neurons, and
theoretical analysis to study the spatial distribution of two coupled
populations of interneurons in the neocortex. Our goal has been to
provide quantitative answers to the following questions: Do coupled
neurons form small, restricted clusters or large, continuous networks?
On average, how many neurons are coupled to each individual neuron?
What are the effects of gap junctions on the biophysical properties of
the neurons and the network? Our data suggest that GABAergic neurons of
the neocortex form large electrically interconnected networks, where
each neuron contacts tens of other neurons. We also show that the input
conductance attributable to nonjunctional membrane and that
attributable to the sum of electrical synapses onto all other neurons
have similar magnitudes; this implies that approximately one-half of
the input conductance measured experimentally is contributed by gap junctions.
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MATERIALS AND METHODS |
Slice preparation and recording. Thalamocortical
slices 400 µm thick were obtained from Sprague-Dawley rats aged
postnatal day 14 (P14) to P21, as described previously (Gibson
et al., 1999). The slices were incubated for 1 hour and then
placed in a submersion chamber at 32°C for recording. The bathing
solution contained (in mM): 126 NaCl, 3 KCl, 1.25 NaH2PO4, 2 MgSO4, 26 NaHCO3, 10 dextrose, and 2 CaCl2,
saturated with 95% O2/5% CO2.
Micropipettes were filled with (in mM): 135 K-gluconate, 4 KCl, 2 NaCl, 10 HEPES, 0.2-4 EGTA, 4 ATP-Mg, 0.3 and GTP-Tris, 0.5-10
phosphocreatine-Tris, pH 7.25, 295 mOsm. In some experiments,
neurobiotin or biocytin (4 mg/ml) was added to the normal filling
solution. All recordings were made in current-clamp mode, under
infrared-differential interference contrast visualization. All neurons
were classified according to their firing pattern in response to an
injection of a square current pulse as either fast spiking (FS) cells
or low-threshold spiking (LTS) cells (details given in Gibson et
al., 1999). When depolarized, FS cells fired with high
frequencies of narrow action potentials, with little or no frequency
adaptation. LTS cells had a tendency to fire on the rebound when
depolarized from more negative membrane potentials, their spikes were
broader, and they exhibited clear frequency adaptation. To characterize
the electrical coupling between two cells, a step current was injected
into one cell and the voltage responses of both cells were measured
(see also Gibson et al., 1999). The coupling coefficient
(CC) was defined as the ratio between the steady-state voltage
deflection of the postjunctional cell and that of the prejunctional
cell. Cells are defined as "electrically coupled" if the measured
CC between them is >0.01, the smallest that can be reliably
distinguished above the membrane voltage noise.
Histological procedures. Slices that contained stained cells
were fixed in 4% paraformaldehyde in 0.1 M phosphate
buffer, transferred to 30% sucrose, resectioned to 80 µm, and
reacted with the avidin-biotin-peroxidase [avidin-biotin complex
(ABC)] procedure (Vector Laboratories, Burlingame, CA). For
immunohistochemistry, Sprague-Dawley rats aged P16-P18 were
anesthetized with 30 mg/kg pentobarbital, perfused intracardially with
5 ml of heparinized saline followed by 4% paraformaldehyde in 0.1 M phosphate buffer, pH 5.4, for 25 min. Brains were
removed, hemisected, and placed in fixative for an additional 2 hr
before changing to 0.1 M phosphate buffer. Subsequently,
tissue was cryoprotected in 30% sucrose/0.1 M phosphate
buffer, pH 7.4, overnight. The tissue was sliced at 60 µm along the
thalamocortical plane (Agmon and Connors, 1991), which
is approximately parallel to the barrel rows. Tissue was washed three
times in PBS (0.1 M phosphate/0.15 M NaCl at pH
7.4) before incubation in 0.5% H2O2 for 1 hr.
Slices were washed three times in PBS followed by three more washes in
Tris-buffered saline (TBS; 0.05 M Tris/0.15 NaCl, pH 7.4)
for 10 min each. The slices were incubated overnight at room
temperature with shaking in primary antiserum for either somatostatin
(SS; Peninsula, San Carlos, CA) or parvalbumin (PV; Sigma, St. Louis,
MO). Final concentrations of each antiserum were 1:1000 and 1:400,
respectively, including 10% normal goat serum, 2% bovine serum
albumin, 0.5% Triton X-100, and TBS (all purchased from Sigma). On day
2, the tissue was washed three times in TBS and incubated 3 hr at room
temperature in biotinylated anti-rabbit IgG (Vector Laboratories) using
a 1:200 final dilution including 10% normal goat serum, 2% bovine
serum albumin, 0.5% Triton X-100, and TBS. After several rinses, an
ABC Elite kit (Vector Laboratories) was used to visualize somatostatin
or parvalbumin.
Morphometric analysis. Stained cells were digitally
reconstructed at 40× magnification with a Neurolucida system
(MicroBrightField Inc., Colchester, VT), and the dendritic branching
patterns were evaluated using a standard Sholl analysis (Sholl,
1956). Sections reacted for PV or SS were viewed under the
light microscope at 10× magnification and mapped with Neurolucida
software in seven different, randomly selected sections taken from
three different animals. Stained cells were counted in at least
2-mm-wide vertical strips across all layers of the primary
somatosensory cortex. Background staining was sufficient to allow
determination of the borders of cortical laminas. In some
cases, adjacent sections were stained for cytochrome oxydase to reveal
layer 4 and the barrel structures.
Cell density in thin sections was calculated with the Neurolucida
software. For additional analysis, the coordinates of cells in mapped
sections were converted into ASCII files and analyzed using a routine
written in Matlab. In this routine, cell density was calculated by
dividing each section into 30 × 50 µm rectangular bins, and a
sliding average of the number of cells was performed in 4 × 4 such rectangles. To calculate the volume density of the cells, we made
the following measurements and assumptions: (1) The maximal depth of
focus (z-axis) was measured with Neurolucida and found to be
~12 µm. (2) The collapse of the tissue along the z-axis
was estimated to be ~60% (Benes and Lang, 2001); thus
12 µm represents 30 µm of unfixed tissue thickness. (3) Because the somata of many viewed cells in the thin plane of view are cut in the
middle, we added another 5 µm for each side of the section. Accordingly, the tissue thickness (z-axis) was additionally
corrected to a value of 40 µm for calculations of cell density by
volume. We call this corrected measure the "effective thickness" of
the tissue.
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RESULTS |
Morphology of dendritic trees of FS and LTS interneurons
Ultrastructural studies suggest that gap junctions form between
dendrites, or between dendrites and somata, of inhibitory neurons
(Tamás et al., 2000; Szabadics et al.,
2001). Thus, the potential for such a junction exists wherever
dendrosomatic membranes of two cells are in close proximity. We
analyzed the spatial extent of the dendritic trees of FS and LTS
interneurons. The dendrites of eight FS cells and seven LTS cells that
were well stained with biocytin were fully reconstructed (Fig.
1). The dendritic trees of both
types had variable profiles, and there was no clear correlation between
the physiological type and any common morphological classification of
dendritic pattern such as "bitufted" or "multipolar." Other studies of neocortical interneurons have also concluded that the somatodendritic morphology of these cells does not distinguish their
subtype (Kawaguchi and Kubota, 1997; Bayraktar et
al., 2000). Sholl analysis of the dendrites revealed some
quantitative differences between the two cell types (Fig.
2A,B).
The total proximal dendritic length (<200 µm from the soma) was
~17% larger for FS cells than for LTS cells (1690 ± 640 µm
and 1440 ± 890 µm, respectively), because FS cells had more
primary dendrites and proximal branching (Fig. 1). However, the
dendrites of FS cells rarely extended beyond 400 µm, whereas some LTS
cells possessed branches that extended >600 µm. These longer
dendrites were usually vertically oriented, ascended toward the pia,
and account for the long tail in the LTS Sholl histogram (Figs. 1,
2B) and the small deviation to the right in the
cumulative probability plot (Fig. 2C). For both cell types,
~80-90% of their total dendritic length occurred within 200 µm
from the soma (Fig. 2C). We conclude that the dendritic profiles of FS and LTS cells in layer 4 are similar in their general outline, and show only minor quantitative differences.
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Figure 2.
Sholl analysis of the inhibitory neurons from
Figure 1 reveals more primary branches for FS cells
(A) and a "tail" of longer branches in LTS cells
(B). Comparing the cumulative length of the two cell
types shows that for both, 80-90% of the dendrites occurred within
200 µm of the soma. FS cells, closed circles; LTS cells,
open squares.
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The spatial distribution of parvalbumin- and
somatostatin-immunoreactive cells
Previous studies from our laboratory showed that FS cells were
generally parvalbumin immunoreactive (PV+), while most LTS cells were
somatostatin immunoreactive (SS+) (Gibson et al., 1999). Studies from other laboratories have also concluded that PV+ and SS+
cells correspond to two such specific and nonoverlapping populations of
interneurons, distinguished by morphology and electrophysiology (Kubota et al., 1994; Gonchar and Burkhalter,
1997; Kawaguchi and Kubota, 1997). We used these
molecular markers to analyze the spatial organization of the two
classes of interneuron populations quantitatively. Sections of barrel
cortex were cut in either the thalamocortical plane angle as used for
our electrophysiology experiments (Agmon and Connors,
1991) or in the tangential plane parallel to the pia and then
processed for PV or SS immunoreactivity. Both immunostaining methods
stained only partial dendritic arbors. There were, however, striking
differences in the patterns of axonal immunostaining. Single SS+ axons
were seen coursing for hundreds of microns through single sections,
most often along the vertical dimension (Fig.
3A). There was an especially
dense plexus of SS+ axons in layer 1. These features are consistent
with the axonal arborization features of Martinotti cells, which have
vertically projecting axons that arborize in layer 1, and which are SS+
(Kawaguchi and Kubota, 1997). In contrast, PV+ axons
could rarely be traced for any significant length. Instead, PV sections
had numerous clear rings of stained boutons surrounding the somata of
unstained cells. These were especially prominent in layer 4 (Fig.
3B), and are consistent with the general conclusion that
many PV+ cells are basket cells (Hendry et al., 1989;
Akil and Lewis, 1992).
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Figure 3.
Immunohistochemistry for somatostatin
(A) and parvalbumin (B) in layer 4. Arrows in A point to a somatostatin-positive axon
crossing upward. Arrows in B point to punctate
parvalbumin-positive terminal staining around the somata of cells.
Scale bars: A, 100 µm; B, 20 µm.
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The average density of PV+ cells across the entire cortical thickness
was larger than that of SS+ cells (92 ± 9 vs 67 ± 12 cells/mm2, respectively). Because our estimated
effective thickness of the sections is 40 µm (see Materials and
Methods), we calculated that the average neuron density by volume ( )
was ~2300 ± 225 cells/mm3 for PV+ cells and
1675 ± 300 cells/mm3 for SS+ cells.
Both interneuron types were present in layers 2 through 6, but they had
very different distributions of density. The density of PV+ cells was
especially high in layer 4, whereas SS+ cells were more concentrated in
lower laminas (Fig. 4). Tangential
sections through layer 4 revealed a higher density of PV+ cells inside the barrel borders (data not shown; Sanchez et al.,
1992; McMullen et al., 1994). Autocorrelation of
the radius vector of each cell against all other cells in tangential
sections through several layers did not reveal any anisotropy for both
cell types (data not shown).
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Figure 4.
Average density of somatostatin-positive and
parvalbumin-positive cells in the different cortical laminas.
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The spatial distribution of the two interneuron types was also
highlighted by a bin-averaging process (Fig.
5). Each histological section was divided
into rectangular bins of 30 × 50 µm (the smaller side parallels
the pia), and the smoothing process averaged an area of 4 × 4 such bins (120 × 200 µm). This averaging area roughly matches
the dimensions of the dendritic fields of the neurons, which show a
clear vertical bias (Jin et al., 2001). Sholl analysis demonstrates that such an area encloses ~80% of the average
dendritic tree. Electrophysiological recordings also verified that
there was no electrical coupling between cells whose somata were >200 µm apart (see below). Figure 5 displays three representative sections for each immunoreactive cell type. PV+ cells formed a clear band of
high-density patches in layer 4 (Fig. 5A) and another weaker band in layer 6. The distribution of SS+ cells was more irregular, with
high-density patches in the lower laminas (Fig. 5).
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Figure 5.
The spatial distribution of parvalbumin-positive
cells (A) and somatostatin-positive cells
(B). Three-example sections of each cell type are
presented. The top panels depict the raw data. The
bottom panels illustrate by color-coding the average cell
density in bins of 30 × 50 µm, deducted from a smoothing
process of 4 × 4 such bins. Note that the maximal density
observed was 0.5 cells/bin (red), thus eight cells in a
rectangle of 120 × 200 µm. The corresponding cortical laminas
are marked to the left.
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Figure 5 also reveals the degree of potential overlap between dendritic
fields in these thin sections. One would expect such an overlap if
there are at least two cell bodies in a rectangle of 120 × 200 µm, which corresponds to light blue on the color scale (0.125 cells
in each single bin of 30 × 50 µm). It is apparent from Figure
5 that even in these thin sections, and for the relatively sparse SS+ cell network, there is almost a continuity of overlapping dendritic fields. In a three-dimensional network of cells, such a
continuity of dendritic fields is bound to be robust.
Probability of coupling and coupling coefficients
What determines the likelihood that two interneurons will be
coupled? The fact that the dendritic fields of two adjacent neurons overlap does not necessarily mean that their dendrites are in contact
or that there are gap junctions interconnecting them. To estimate more
directly the relationship between cell proximity and electrical
coupling, we performed electrophysiological experiments. Pairs of
inhibitory cells were recorded in layer 4 of the barrel cortex.
Electrical coupling was common between pairs of interneurons of the
same type and rare between pairs of interneurons of different types, as
described previously (Gibson et al., 1999; Deans
et al., 2001). Of 125 cell pairs consisting of the same types
of interneurons, 75 were electrically connected (FS-FS, 52 of 88 pairs, or 59%, were coupled; LTS-LTS, 23 of 37 pairs, or 62%, were
coupled). Data for the two types of cells were pooled, because statistical analysis did not reveal any differences.
A critical issue for this study is the distance-dependence of
electrical coupling. For each pair of recorded cells, the distance x between the centers of the somata and the angle relative
to the vertical orientation of the cortex were measured. If the two cells were electrically connected, we measured the coupling coefficient (CC). The probability
PE(x) that two neurons recorded
simultaneously were electrically coupled, and the histogram of
CC(x), were deduced from the data (Fig.
6A,B).
Examining the relationship between PE(x) and CC(x) and the
angle between the two coupled cells as defined above did not reveal
anisotropy (data not shown). Obviously, the product
PE(x) × CC(x), which is
proportional to the total amount of electrical conductance a cell
receives from its coupling to other cells at a distance x
from it, decays rapidly with x (Fig. 6C). No
coupled pairs were observed at distances of >200 µm. We can infer
from this, together with the average dendritic spread (Fig.
2C), that most directly coupled pairs of neurons have
substantial dendritic overlap.
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Figure 6.
Histograms of the probability
PE that two cells are coupled
(A), the coupling coefficient CC
(B) and PE × CC (C) as a function of the distance
between the cells. Data are based on recordings from pairs of both FS
and LTS cells.
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When electrical coupling is detected between two neurons, it may occur
through a direct gap junctional contact or indirectly via intermediary
cells that connect the two recorded cells. Without a structural
analysis of each cell pair, it is not possible to distinguish direct
from indirect coupling. We define M as the average number of
cells that are directly connected by gap junctions to each interneuron.
We define ME as the average number of neurons that are electrically coupled to each neuron either by direct contact
or indirectly via intermediary neurons. ME is
limited, in practice, to neurons coupled strongly enough (via any
route) to be measurable under our recording conditions. The smallest CC we were routinely able to measure with confidence was
~0.01, because of membrane potential noise (which typically had a SD of ~0.16 mV) and our limited averaging protocol (Gibson et
al., 1999). ME is an upper bound for
M (assuming that all the direct electrical connections
between cells have CCs that are larger than the confidence
limit). We can calculate ME in a volume of tissue using the following formula:
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(1)
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The integral in Equation 1 is approximated by sums over shells,
such that in each shell the values of PE are
those in Figure 6A. Each shell is 50 µm wide; the number
of shells (m) is 4, rj = j × 50 µm and r0 = 0:
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(2)
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ME is therefore proportional to the cell
density . For example, when a local for a given cell type is
2000 cells/mm3, the average
ME in this area will be 28. From our data, values vary between 1600 cells/mm3 for SS+ cells in
layer 4 to 3650 cells/mm3 for PV+ cells in layer 4. Thus ME varies between 22 for the lowest density
of SS+ cells and 51 for the highest density of PV+ cells.
We cannot determine the exact relationship between
ME and M using the information
currently available. We can deduce, however, that M is >1.
Before introducing a general argument, we present a simple example.
Consider a very large network in which each neuron is electrically
coupled at random to M other neurons. The closest, most
strongly coupled cell pairs are highly likely to be coupled directly
(cf. Tamás et al., 2000). Because CC
is, at most, ~0.2 for directly coupled pairs (Fig.
6B), then CC for a pair indirectly coupled
through one intermediary will be  0.04, and with two intermediaries
CC will be 0.008 (i.e., too small to measure readily). If
only first-order coupling is detectable, then
ME = M. If
ME is at most second-order coupling (i.e., only one intermediate cell) and if we neglect parallel routes of coupling, then ME = M2.
Thus, even for our lowest cell densities, ME is
~25 and M is between 5 and 25.
But are the assumptions behind this simple example plausible? There are
three main problems. First, a neuron can be indirectly coupled to other
neurons through more than one intermediate neuron, thus creating a
certain degree of overlap in the network, and yielding a smaller
ME. Second, if each third-order neuron is
coupled to more than one second-order neuron, it may have a coupling
coefficient above the detection threshold, and this may cause an
increase of ME. Third, our simple example does
not consider dimensionality, whereas we know that neurons are coupled
primarily to neighboring neurons. Assessing all of these factors
demands a more thorough analysis. Nevertheless, we can make several
biologically realistic assumptions: (1) neurons are connected at random
with the probability given by P(x); (2) P does
not depend on any other factor except x; and (3) the
statistics of the electrical connections are homogeneous. To obtain a
more general result, we need to use concepts borrowed from percolation
theory. This theory describes how something (e.g., ionic current) flows
through the random interconnections of lattices (e.g., networks of
neurons) (Stauffer and Aharony, 1992). Using the above
assumptions and percolation theory, one can show that if M < 1, each neuron is coupled (directly or indirectly) only to a
small number of other neurons, and ME is not
much larger than 1. Because our experimental results show that
ME is of the order of a few tens, M
should be >1. If M > 1, the same theory tells us that
almost all of the neurons belong to one large connected network, or
syncytium, in which all cells are coupled to each other through other
cells that belong to the network (Erdös and R nyi,
1960; Traub et al., 1999). This analysis cannot
exclude cases in which electrical coupling exists only between neurons within spatially restricted patches. However, assuming a uniform likelihood that dendrites of same-type interneurons will create electrical synapses when they are adjacent, the cellular distribution as revealed by immunohistochemistry does not support such inhomogeneity within the somatosensory area.
Models of electrically coupled networks
The density of electrical coupling within a cell network will have
important consequences for the estimated values of certain biophysical
properties. In particular, conductances attributable to electrical
coupling will add to membrane conductances, and significant errors can
be made if network effects are not taken into account. In the following
analysis, we will estimate the contribution of electrical coupling to
the input conductances of each cell as measured experimentally. Two
parameters in particular are likely to be affected:
gL, the intrinsic leak conductance of each cell
(i.e., the leak conductance of the non-gap junctional membrane), and
GE, the strength of each electrical connection between two neurons. To estimate these values, one has to assume a
model of the network architecture. We will first examine how these
biophysical parameters depend on the specific architecture of the
network, and then suggest a method to estimate them from data that can
be generated experimentally. All network architecture models
can be represented by the following steady-state equations (for
i from 1 to N):
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(3)
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where Vi is the voltage of the
ith neuron and, at the neuronal rest
state Vi = 0, gL is
the leak conductance, GE is the coupling conductance between any two cells, and Ii is the
current injected into the ith neuron. For
simplicity, we consider models without heterogeneity: all of the
neurons have the same gL and all the existing
coupling strengths are GE. Because electrical
synapses are usually symmetrical (i.e., the coupling conductance from
cell A to cell B is equal to the coupling conductance from cell B to
cell A) (Galarreta and Hestrin, 2001), the matrix
Jij is symmetric such that
Jij = Jji.
Jij is 1 if electrical coupling exists between
neurons i and j, and is zero otherwise. We
consider cases in which constant current is injected to the neuron with
an index i = 0 only. Summing Equation 3 over all of the
neurons, we obtain
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(4)
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Next we examine a few simple network architectures. We
start with a model that has only two cells and then show that the influence of the network should be considered by using more realistic architectures.
The two-cell model
Traditionally, the coupling conductance between two cells,
GE, has been calculated from the measured
coupling coefficient, CC, assuming a two-cell model
(Bennett, 1977). We consider here the simple case where
the two cells are identical. This architecture (Fig.
7A) includes two cells, each
with an input conductance gL (resistance
RL = 1/gL), which
are coupled through a resistor RE (conductance
GE = 1/RE). The
cells have indices 0 and 1. Step current I0 is
injected into cell 0. The coupling coefficient in the steady state is
CCi = Vi/V0 (for i  0). The conductance GE can be estimated
from the following relationship:
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(5)
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For GE  gL (and
therefore V1 V0),
Equations 4 and 5 become: gL = I0/V0,
GE = gL × CC1.
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Figure 7.
Architectures of network models. A, Two
cells coupled by a gap junction with a conductance
GE. B, A cell (0) coupled to
M other cells. C, One-dimensional architecture.
Each cell is coupled to M/2 cells on its left and
M/2 cells on its right. Each coupling connection
in B and C has a conductance
GE = gE/M. Cells in all architectures have
a leak conductance gL (not specified in
C).
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However, this model does not take into account the fact that each cell
may be coupled to many other cells. Obviously, the data we described
for networks of neocortical interneurons require more elaborate models.
One cell coupled to M other cells
In this architecture (Fig. 7B), cell number 0 is
electrically coupled to M other cells, which are not coupled
to each other. It is considerably more realistic than the two-cell
model, because we concluded above that M is much greater
than 1. The total coupling conductance on the 0th
cell is gE = M GE.
The steady-state voltages of cells in response to current injection
into cell 0 are given by the following equations:
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(6)
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(7)
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From Equation 7, one can calculate
gE/gL knowing
Vi and V0:
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(8)
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For large M, these equations become:
I0/V0 = gL + gE and
Vi = gEV0/(M gL).
As we show below (see Appendix B), the sum  i 0
CCi can be calculated from experimental data and
is useful for estimating network parameters. From Equation 8, this sum
can be computed by:
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(9)
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The sum is approximately equal to
gE/gL for large
M or small gE, and we show below that
under these conditions, one can estimate this ratio experimentally.
What happens if we try to estimate gL (the
nonjunctional membrane conductance) and gE (the
total junctional conductance) for one cell coupled to M
other cells but use the traditional approach that assumes the simple
two-cell model? The two-cell model makes the approximation
gL = I0/V0. If we take into
account a large M, Equation 6 tells us that
I0/V0 = gL + gE. Hence, our error is a factor of (gL + gE)/gL, and this error
biases the calculation of both gL and
gE upward. For example, if
gL = gE, the error
is a factor of 2. Even if the coupling conductance
GE (between two cells) is relatively small, the
value of the total coupling conductance gE = M GE can be of the same order as the leak
conductance of the cell, gL. Using the two-cell
model can therefore lead to unacceptably large errors.
Networks with one-dimensional architecture
The two architectures described above are too simplistic, but they
do demonstrate that network effects should be considered when one
estimates gL and gE. In
practice, of course, we know that not all the neurons are directly
coupled to the recorded neurons. Furthermore, as shown in Figure 6, the
probability that two cells are coupled depends strongly on the distance
between them. Therefore, we next consider a model with spatially
decaying connectivity.
This architecture is one-dimensional, a long chain of neurons, that
stretches from  N/2 to N/2, where N 
M. Each neuron has an index i and each is directly
coupled to M other neurons that are arrayed symmetrically to
either side. Thus, the cell at i = 0 is coupled to
cells j = M/2 ... 0 ... M/2 (Fig.
7C). Cells near the edges are connected to a number of
neurons that is smaller than M. The network is studied for
large N. The version of Equation 3 for this system is
|
(10)
|
The stimulus current Ii is
I0 for i = 0, and it is 0 otherwise. Using Fourier series, we obtain
![V<SUB><UP>i</UP></SUB>=<FR><NU>I<SUB>0</SUB></NU><DE>2&pgr;</DE></FR><LIM><OP>∫</OP><LL>0</LL><UL>2&pgr;</UL></LIM>d&thgr;<FR><NU><UP>cos</UP>(i&thgr;)</NU><DE>g<SUB><UP>L</UP></SUB>+<FR><NU>2g<SUB><UP>E</UP></SUB></NU><DE>M</DE></FR> <LIM><OP>∑</OP><LL><UP>j=1</UP></LL><UL><UP>M/2</UP></UL></LIM>[1−<UP>cos</UP>(j&thgr;)]</DE></FR>](/content/vol22/issue10/fulltext/4142/img011.gif)
|
(11)
|
We cannot evaluate the integral in Equation 11 exactly for
M > 2, and instead compute it numerically. An example
is shown in Figure 8, where we use the
parameters M = 28 (the value derived from a typical
cell density of 2000 cells/mm3); M is
close to ME, and gE = gL. When current I0 is
injected into the center cell, the voltage Vi
decreases gradually to cells i = 14 and 14 (i.e.,
cells M/2 and M/2) and then drops sharply at
cells i = 15 and 15, after which it decreases
gradually again. These sharp jumps of Vi occur
because neuron number 14 (or M/2) is directly coupled to the
injected neuron (i = 0), but neuron number 15 (or
M/2 + 1) is only indirectly coupled to it.
Interestingly, when gE and
gL have similar values, the values of
Vi for i that are just larger than
M/2 are not negligible at all. For example, for
gE = gL and
M = 28, V15/V14 = 0.32. Yet, for this specific example, indirect connections fall
below the experimental confidence level.
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|
Figure 8.
The coupling coefficient,
CCi, as a function of the cell number
i for a one-dimensional architecture with M = 28 and gE/gL = 1. The values were computed either by solving Equation 3 or by
numerical integration of Equation 11; the two results are equal. Note
the jump in CCi between cells 14 and 15 (i = M/2 and i = M/2 + 1). The
dashed line denotes the confidence level of CC = 0.01.
|
|
The sum i0 CCi, which is
useful for calculating gE, can be calculated
exactly in two limiting cases. In the limit of large M, we
show in Appendix A that
|
(12)
|
Namely, (gL/gE)
i0 CCi = 1 for M
  . Similarly, we can show that i0
CCi = gE/gL for
gE/gL 1.
What is the value of i0 CCi in
a parameter regime not close to these limits? The dependence on
connectivity, M, for three values of
gE/gL (0.5, 1, and 2) is
shown in Figure 9A. Figure 9A demonstrates that Equation 12 holds for large
M. Moreover, the dependence of i0
CCi on M is weak, unless M
is small (<10). Figure 9B demonstrates further that for a
specific M in any of the architectures examined, the value
of (gL/gE)
i0 CCi decreases gradually and
slowly from 1 as a function of
gE/gL. The solid line
represents the one-dimensional architecture and M = 28,
as examined in the example above. Even for
gE/gL = 2, the value
of (gL/gE)
i0 CCi is 0.95, demonstrating
that the value of the sum i0
CCi is close to
gE/gL even beyond the two
limiting cases described above.
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|
Figure 9.
Effects of network architectures on
(gL/gE)
i0 CCi. A, The
dependence of (gL/gE)
i0 CCi at steady state on
M for one-dimensional architecture and three values of
gE/gL: 0.5 (solid
line), 1 (dotted line), and 2 (dashed line).
B, The dependence of
(gL/gE)
i0 CCi at steady state on
gE/gL for three
architectures: one-dimensional architecture (solid line),
one cell coupled to M other cells (dotted line),
and two-dimensional architecture (dashed line). M = 28 for all the architectures. Calculations for the
one-dimensional architecture were carried out as in Figure 8. Equation 9 was used for calculating CC for the architecture with one
cell coupled to M other cells. In the two-dimensional
architecture, cells are located on a two-dimensional grid, at positions
x = (i , j), where i and
j are integers and is the grid unit
length. Cells at positions (i1,
j1) and (i2,
j2) are coupled if
 3. Calculations were performed by solving Equation 3.
|
|
We also consider a two-dimensional model, in which neurons are located
on a two-dimensional lattice and are electrically connected if the
distance between them is smaller than a certain value. The behavior of
this model is found to be similar to the behavior of the
one-dimensional model, as represented by the dashed line in
Figure 9B. This line is only slightly above the line for the one-dimensional model with the same M. For comparison, we
present also the dependence of
(gL/gE)
i0 CCi for M = 28 as a function of
gE/gL for the
architecture of one neuron coupled to M other neurons (Eq. 9). This line is only slightly below the line for the one-dimensional
model with the same M.
Together, the important result of this analysis is that the value of
(gL/gE)
i0 CCi depends only weakly on
the model architecture. For all the architectures we examined,
(gL/gE)
i0 CCi has the asymptotic
value of 1 for large M or small
gE/gL, and for
gE ~ gL, the
difference between one architecture and another is <5%. Therefore, we
propose to use the sum i0 CCi
for estimating gE/gL.
Estimating gL and
gE from measurements
Unfortunately, we cannot develop a method for estimating
GE(i, j) for a specific connection
between two specific neurons, i and j, in the
network. In many cases, however, the dynamic behavior of neuronal
networks can be described by knowing gE and
gL (Chow and Kopell, 2000). We
show above that the sum i0 CCi
can be used for estimating the ratio between gE
and gL. Furthermore, we can use the sum
i Vi to estimate
I0/gL (Eq. 4). It is
obviously impractical to measure the sums i
Vi and i0
CCi directly in experiments, because of the
limited number of neurons that can be recorded. Instead, we have
developed a method for estimating these values, and therefore for
estimating gE (the total conductance of a cell
from all of its electrical connections) and gL
(the intrinsic leak conductance of a cell), by averaging over many experiments, and we present it in detail in Appendix B.
For a given set of dual intracellular measurements, the estimations for
gE and gL depend only on
the cell density . As increases, gL
decreases and gE increases (Fig.
10). For a typical of 2000 cells/mm3, gE contributes
approximately one-half of the measurable input conductance
(gL = 11 nS and
gE = 10 nS). For the extreme case of close to 4000 cells/mm3,
gE/gL is close to 2. In
every reasonable scenario, the values of gE and
gL are of the same order of magnitude. The fact
that the values of gL and
gE are similar means that if the membrane conductance of the cell, gL, is estimated
naively as I0/V0, as it generally is in systems of uncoupled cells, the estimated value of
gL would be approximately twice the correct
value.
View larger version (14K):
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|
Figure 10.
The dependence of gL
(solid line) and gE (dashed
line) on the cell density , calculated using Equations B3 and
B4, for the data obtained experimentally.
|
|
What are the implications of these values on the values of
GE? Although we cannot measure
GE directly, we can estimate the order of
magnitude of its average value. Considering an average value of
= 2000 cells/mm3, we have estimated that
gE = 10 nS and M, the number of
neurons that are directly coupled to each neuron, is between 5 ( ) and 28 (ME). Thus, for this cell density,
GE varies between 0.36 nS and 2 nS.
| |
DISCUSSION |
We have investigated the spatial distribution of two coupled
groups of neocortical interneurons. From our morphological studies we
conclude that the somadendritic morphology of FS and LTS cells is
similar, with ~80% of the dendritic trees <200 µm from the soma.
However, FS and LTS cells differ considerably in their densities and
laminar distribution. Electrophysiological recordings from same-type
neuronal pairs demonstrated that the probability of electrical coupling
and the coupling coefficient declined with the distance between somata.
Electrical coupling never occurred when the distance between somata was
>200 µm. Our computations suggest that a single neuron is
electrically coupled to tens of other neurons, implying that each type
of interneuron forms a highly interconnected network over large
cortical areas.
Estimating parameters of electrical coupling
The strength of electrical coupling between two cells is easily
calculated when these cells are isolated (Bennett,
1977). For lack of better methods, the "two-cell model" has
usually been used to determine the electrical coupling strength between
cell pairs even in highly connected systems (Gibson et al.,
1999; Galarreta and Hestrin, 2001). However,
this method can provide a good approximation only if the effects on the
recorded cell pair of other, unrecorded, coupled cells are minimal,
namely when gE is small compared with gL. Here we show that for the networks of
neocortical interneurons, gE = MGE is not small in comparison with
gL; rather, the two have similar magnitudes. The
two-cell model cannot be used in these systems because it does not take
into account two important factors: (1) the increase of the effective
leak conductance
(I0/V0) because of the
current flow to other neurons, and (2) the fact that electrical coupling between two cells can be mediated through other neurons.
We present here an approach for estimating the average membrane
conductance, gL, and the average total
electrical conductance attributable to electrical junctions onto a
single neuron, gE, based on a large number of
measurements and on solving the voltage profiles in several models. We
estimate that the gL and
gE of inhibitory interneurons in the neocortex
are of the same order of magnitude (i.e., gap junctions contribute
approximately one-half of the input conductance measured during
electrophysiological experiments. The exact ratio between
gL and gE depends on the density of the cells within a coupled neuronal network.
A recent study of neocortical interneurons from connexin36
(Cx36) knock-out (KO) mice demonstrated that, indeed, the
mean input resistance of cells in the KO is ~30-40% higher than
that of wild-type cells (Deans et al., 2001). This
increase is not as high as our theoretical results predict, but it is
also very likely that the KO cells achieve some partial compensation of membrane conductance during development. Nevertheless, this finding supports our prediction that gap junctions provide a major contribution to the total measured conductance of these cells.
The architecture of electrically coupled networks of neurons has been
studied mostly in noncortical tissues, such as the retina. There, the
effect of gap junctions was examined using two-dimensional architectures in which each neuron is coupled only to its few nearest
neighbors (hence M = 4 for a rectangular grid)
(Naka and Rushton, 1967; Gold, 1979;
Poznanski and Umino, 1997). Analysis reveals that when
current is injected into a cell within such a nearest-neighbor
architecture, the size of voltage deflections in other cells decreases
almost exponentially with the distance from the injected cell. In
contrast, each cell in our system is coupled to tens of other cells, so
models with large M are more appropriate. In such models,
voltage deflections decay only weakly over short distances, in
agreement with our experimental data. Interestingly, a study of
electrical coupling between neurons in the inferior olive revealed
spatial coupling patterns very similar to ours (Devor and Yarom,
2002): neurons in the olive are highly likely to be coupled if
their intersomatic distance is <100 µM, the dependence
of PE and CC on distance is
comparable with what we find here, and the number of connected cells is
estimated to be between 10 and 40. It will be interesting to perform
similar investigations in other parts of the CNS to test the generality of these rules about the architecture of electrical networks.
Potential sources of bias
Our calculations have several potential sources of bias. First,
the cell density was calculated from random sections and corrected only
for shrinkage of the slice thickness. Our estimate is smaller than the
density of PV+ cells in another study using stereometry (~7000
cells/mm3; Ren et al., 1992).
However, the relative ratio of PV+ neurons in that study is ~54% of
all GABAergic cells, which is much higher than most other estimates in
rodent somatosensory cortex (Gonchar and Burkhalter,
1997, Kawaguchi and Kubota, 1997). It is also possible that the probability of coupling (Fig. 6A)
was underestimated because some dendritic arbors were severed by the
slicing procedures. The signal-to-noise ratio prevents the detection of
the least effective connections. All of these technical limitations are likely to bias our results toward weaker electrical coupling. Even so,
our modeling results are robust to our experimental results and to a
wide range of cell densities.
Another possible source of error arises from the relatively small
electrophysiological sample from which the probability of connectivity
was deduced. Nevertheless, the probability data agree well with recent
morphological data. Double-staining for Cx36 reporter
genes and specific markers of interneurons suggested that not
all PV+ and SS+ interneurons express Cx36, and the proportion of
double-labeled cells was similar to the proportion of
electrophysiologically coupled neighboring cells in the sample
(Dean et al., 2001).
Our modeling work does not consider heterogeneities in
gL, variations in the electrical coupling
strength GE, and sparse connectivity in which
neurons are not necessarily coupled to each other, even if they are
adjacent. These factors could bias our estimates of gE and gL. Qualitative
estimation of the bias will require more theoretical work on models
with relatively elaborate architectures. Preliminary modeling results
with sparse and spatially decaying connectivity (data not shown)
indicate that the effects of sparseness on the estimated values of
gE and gL are small.
Functional implications
Electrical coupling is likely to affect both the properties of
single cells and the properties of cellular networks as a whole. Gap
junctions increase the effective leak conductance of neurons and thus
decrease their passive time constants (Andreu et al., 2000). This may cause faster reaction times to stimulus-induced changes and may make firing times follow the membrane potential more
faithfully. The minimal current needed to initiate firing will increase
with electrical coupling as well, and the frequency-current dependence
of a neuron will be shifted to the right, (i.e., to larger current
regimes) (Holt and Koch, 1997).
At the network level, it has been suggested that a population of
inhibitory neurons in vivo synchronizes its spikes and
entrains populations of excitatory neurons via their inhibitory
chemical synapses (Buzsáki et al., 1983). Data
from brain slices do indeed confirm that this scenario can occur under
certain conditions (Whittington et al., 1995,
Jefferys et al., 1996; Draguhn et al., 1998). Theoretical work, however, has shown that
synchronization through inhibition has a fundamental limitation because
of the joint effects of sparse connectivity and heterogeneity in the cellular intrinsic properties (Golomb and Hansel, 2000;
Neltner et al., 2000; Golomb et al.,
2001). Synchrony is destroyed by the heterogeneity of the
network at weak inhibitory conductances and because of sparseness at
strong inhibitory conductances. There is, therefore, only a
restricted window of conductance strengths (if at all) in which network
firing synchrony can be achieved by inhibition. Synchronization through
electrical coupling is an alternative mechanism. Indeed, theoretical
studies have most commonly claimed that strong electrical coupling in
neuronal systems tends to increase spike synchronization (Traub
et al., 2001); if it is weak, other patterns of synchrony may
emerge (Chow and Kopell, 2000). In local networks,
electrical coupling is a much more robust mechanism for spike
synchronization than inhibitory connections (Golomb et al.,
2001). Our study shows that the electrical coupling
(gE) between groups of interneurons is strong,
and therefore electrical coupling may play an important role in
synchronizing the firing patterns of interneurons.
Studies of LTS cells provide evidence for the spatial dimensions of
coupled networks (Beierlein et al., 2000). When a
network of LTS neurons is activated selectively with agonists, IPSPs
become synchronized over distances of ~400 µm. In
connexin36 knock-out mice, long-range synchrony is absent
(Deans et al., 2001), implying that it is a
coupling-dependent, collective network effect. These data are
consistent with two of our conclusions: (1) the electrical coupling
conductance gE is relatively strong, and (2)
interneurons form large, extensive electrically coupled networks. The
fact that synchronization of IPSPs (and probably spikes) does not have a much larger correlation distance may be a result of extensive sparseness and heterogeneity of the network.
There may be other roles for large-scale, electrically connected
networks of interneurons. In many cases, FS-type inhibitory neurons of
the sensory neocortex have more broadly tuned receptive fields than
regular-spiking neurons (Swadlow and Weyand, 1987; Simons and Carvell, 1989; Swadlow, 1989;
Gibber et al., 2001); long-range electrical coupling
among interneurons might account for this. Furthermore, the spread of
activity through coupled networks of interneurons could create a
"surround inhibition" effect around focal areas of activation.
| |
FOOTNOTES |
Received Dec. 4, 2001; revised Feb. 1, 2002; accepted Feb. 15, 2002.
This research was supported by National Institutes of Health Grants
NS25983 and DA125000 (B.W.C.), United States Israel Binational Science
Foundation Grants 9700043 (Y.A., B.W.C.) and 9800015 (D.G.), and Israel
Science Foundation Grant 59/98 (Y.A.). We thank E. Bienenstock, D. Hansel, and C. Meunier for helpful discussions.
Correspondence should be addressed to Dr. Yael Amitai, Department of
Physiology, Faculty of Health Sciences, Box 653, Ben-Gurion University,
Beer-Sheva 84105, Israel. E-mail:
yaela{at}bgumail.bgu.ac.il.
| |
APPENDIX A:  i 0 CCi for
large M |
In this Appendix, we prove that for the one-dimensional
architecture,
|
(A1)
|
We first note that, from Equation 4 and the definition of
CCi,
|
(A2)
|
Therefore, Equation A1 is correct if we can prove that
|
(A3)
|
We start by computing the sum in Equation 11 and obtaining
|
(A4)
|
The first two terms in the square brackets are finite and go to
zero after division by M. The third term diverges at
 = 0. We need to show that its contribution to the integral is
as small as we wish, provided that M is large enough. To do
this, we first see that this term divided by M is finite
near 0:
![<FR><NU><UP>sin</UP>(M&thgr;/2)<UP>sin</UP>(&thgr;)</NU><DE>M[1−<UP>cos</UP>(&thgr;)]</DE></FR>=1+O(&thgr;<SUP>2</SUP>).](/content/vol22/issue10/fulltext/4142/img023.gif)
|
(A5)
|
This term reaches maximum at = 0. For small values
and large M, the term behaves as 2 sin(M/2)/(M). We choose a value 0
2 such that the contributions to the integral (Eq. A5) from the ranges [0, 0] and [2 0, 2] are small. For the
range [0, 2 0], the
term in Equation A5 is bounded:
![<FENCE><FR><NU><UP>sin</UP>(M&thgr;/2)<UP>sin</UP>(&thgr;)</NU><DE>M[1−<UP>cos</UP>(&thgr;)]</DE></FR></FENCE>≤<FR><NU>1</NU><DE>M</DE></FR><FENCE><FR><NU><UP>sin</UP>(&thgr;)</NU><DE>1−<UP>cos</UP>(&thgr;)</DE></FR></FENCE>≤<FR><NU>1</NU><DE>M</DE></FR><FENCE><FR><NU><UP>sin</UP>(&thgr;<SUB>0</SUB>)</NU><DE>1−<UP>cos</UP>(&thgr;<SUB>0</SUB>)</DE></FR></FENCE>.](/content/vol22/issue10/fulltext/4142/img024.gif)
|
(A6)
|
The second inequality is a result of the fact that the function
sin()/[1 cos()] is monotonically decreasing in the
interval (0, 2), and is antisymmetric around = . Hence,
the contribution of the third term in the square brackets of Equation A4 can be made arbitrarily small by choosing a large enough
M, and Equation A4 becomes Equation A3.
| |
APPENDIX B: Estimating gL and
gE |
In this appendix, we present a method for estimating the
sums i Vi and
i0 CCi by averaging over many
experiments. Assuming homogeneous networks, we define
PE(x) as the probability that two
cells at positions x1 and
x2 = x1 + x are
electrically coupled (in this appendix, x means a
three-dimensional coordinate). The sum i
Vi = V0 + i0 Vi, in response to a
specific current I0 is estimated to be
![[V(0)]<SUB><UP>pop</UP></SUB>+&rgr;∫dxP<SUB><UP>E</UP></SUB>(x)V(x),](/content/vol22/issue10/fulltext/4142/img025.gif)
|
(B1)
|
where [V(0)]pop is the average of the
membrane voltages over the entire neuronal population to which current
is injected and is the cell density. By including the probability
PE(x), we take into account in the
integral in Equation B1 only the voltage of neurons that show response
to the current injected to neuron number "0." Similarly, the sum
i0 CCi can be described
by:
|
(B2)
|
The integrals in Equations B1 and B2 are approximated by sums over
shells, such that in each shell the values of PE
and CC are those given in Figure 6. The parameters
gL and gE can be obtained from the following equations:
![I<SUB>0</SUB>/g<SUB><UP>L</UP></SUB>=<LIM><OP>∑</OP><LL><UP>i</UP></LL></LIM> V<SUB><UP>i</UP></SUB>=[V<SUB>0</SUB>]<SUB><UP>pop</UP></SUB>+<FR><NU>4&pgr;&rgr;</NU><DE>3</DE></FR> <LIM><OP>∑</OP><LL><UP>j=1</UP></LL><UL><UP>m</UP></UL></LIM>(r<SUP><UP>3</UP></SUP><SUB><UP>j</UP></SUB>−r<SUP><UP>3</UP></SUP><SUB><UP>j−1</UP></SUB>)P<SUB><UP>E</UP></SUB>(j)V(j),](/content/vol22/issue10/fulltext/4142/img027.gif)
|
(B3)
|
|
(B4)
|
| |
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TOP
ABSTRACT
INTRODUCTION
