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Neuron
Article
Principles Governing the Operationof Synaptic Inhibition in DendritesAlbert Gidon1 and Idan Segev1,2,3,*1Department of Neurobiology, Alexander Silberman Institute of Life Sciences2Interdisciplinary Center for Neural Computation3The Edmond and Lily Safra Center for Brain SciencesThe Hebrew University of Jerusalem, Jerusalem 91904, Israel
*Correspondence: [email protected]
http://dx.doi.org/10.1016/j.neuron.2012.05.015
SUMMARY
Synaptic inhibition plays a key role in shaping thedynamics of neuronal networks and selecting cellassemblies. Typically, an inhibitory axon contactsa particular dendritic subdomain of its target neuron,where it often makes 10–20 synapses, sometimes onvery distal branches. The functional implications ofsuch a connectivity pattern are not well understood.Our experimentally based theoretical study high-lights several new and counterintuitive principlesfor dendritic inhibition. We show that distal ‘‘off-path’’ rather than proximal ‘‘on-path’’ inhibitioneffectively dampens proximal excitable dendritic‘‘hotspots,’’ thus powerfully controlling the neuron’soutput. Additionally, with multiple synaptic contacts,inhibition operates globally, spreading centripetallyhundreds of micrometers from the inhibitory syn-apses. Consequently, inhibition in regions lackinginhibitory synapses may exceed that at the synapticsites themselves. These results offer new insightsinto the synergetic effect of dendritic inhibition incontrolling dendritic excitability and plasticity andin dynamically molding functional dendritic subdo-mains and their output.
INTRODUCTION
Neurons are unique input-output devices. While their output is
generated at the soma and/or axon region, it is first and foremost
shaped by local processes in the dendritic tree (Koch and Segev,
2000; Hausser and Mel, 2003; Polsky et al., 2004; London and
Hausser, 2005; Spruston, 2008; Branco and Hausser, 2010).
The latter is covered with an abundance of inhibitory and excit-
atory synaptic inputs and with a mixture of voltage-dependent
membrane conductances that may trigger plasticity-inducing
signals, such as dendritic N-methyl-D-aspartate (NMDA) and
Ca2+ spikes (Hausser and Mel, 2003; Lynch, 2004; London and
Hausser, 2005; Magee and Johnston, 2005; Magee, 2007;
Sjostrom et al., 2008; Sejnowski, 2009). In the hippocampus
(Klausberger and Somogyi, 2008), the neocortex (Douglas and
330 Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc.
Martin, 2009; Helmstaedter et al., 2009), and other brain regions
(Tepper et al., 2004), individual inhibitory axons from distinct
input sources target specific dendritic subdomains, sometimes
very distal dendritic regions, where each axon may form 10–20
synapses (Thomson and Deuchars, 1997; Markram et al.,
2004; Kapfer et al., 2007; Silberberg and Markram, 2007). For
example, the axons of calretinin- and somatostatin-expressing
neurons contact the distal dendritic domain of the postsynaptic
target cell, parvalbumin-expressing basket cells target the soma
and proximal dendrites, and the axon of chandelier cells targets
very specifically the axons’ initial segment (Kisvarday and Eysel,
1993; DeFelipe, 1997; Defelipe et al., 1999; Markram et al., 2004;
Pouille and Scanziani, 2004). This domain-specific division of
labor between different inhibitory neuronal subclasses is ex-
pected to play a key role in selecting particular cell assemblies
(Runyan et al., 2010) and in shaping (e.g., synchronizing) their
activity (Cardin et al., 2009; Vierling-Claassen et al., 2010) and
in controlling local dendritic nonlinear and plastic process (Llinas
et al., 1968; Miles et al., 1996; Larkum et al., 1999; Komaki
et al., 2007; Sjostrom et al., 2008; Isaacson and Scanziani, 2011).
Our theoretical understanding of dendritic inhibition is
grounded on several, by now classical, analytical studies (Rall,
1964; Rinzel and Rall, 1974; Jack et al., 1975; Koch et al.,
1983, 1990; Hao et al., 2009). These studies mostly explore the
case of single inhibitory synapses impinging on passive dendritic
trees and focus on the impact of such inhibitory synapses on
the soma and/or axon’s initial segment. For example, how
‘‘visible’’ is the dendritic synaptic conductance change when
measured at the soma? What is the optimal locus of inhibition
that maximally prevents the excitatory current from reaching
the soma? These studies provided several important insights
that still dominate our present view on dendritic inhibition;
some of these predictions were later verified experimentally. In
particular, (1) inhibitory conductance change is highly local
(Liu, 2004; Mel and Schiller, 2004; Williams, 2004), (2) inhibitory
conductance change is alwaysmaximal at the inhibitory synaptic
contact itself (Jack et al., 1975), and (3) inhibition is maximally
effective in dampening the excitatory current reaching the
somawhen inhibition is located ‘‘on the path’’ between the excit-
atory synapse and the soma, rather than when it is located more
distally to the excitation (‘‘off-path’’ inhibition; Koch et al., 1983;
Hao et al., 2009).
Here we suggest that the spatial pattern of dendritic innerva-
tion by inhibitory axons—the domain-specific, targeting distal
Neuron
Principles Governing Dendritic Inhibition
branches and the multiple synapses per inhibitory axons—is
optimized to control local and global dendritic excitability and
plasticity processes in the dendritic tree, rather than to directly
affect excitatory current flow to the soma and/or axon region.
Toward this end, we defined a new measure for the impact of
dendritic inhibition—the shunt level (SL)—and solved Rall’s
cable equation (Rall, 1959) for SL for both single and multiple
inhibitory synapses. Using SL, we could systematically charac-
terize functional (as opposed to anatomical) inhibitory dendritic
subdomains and showed that an effective control of local
dendritic excitability requires a counterintuitive pattern of inhibi-
tory innervation over the dendrites. We verified our theoretical
predictions in detailed, experimentally based numerical models
of three-dimensional (3D) reconstructed excitable dendritic
trees receiving inhibitory synapses.
Our study enabled us (1) to propose a functional role for very
distal dendritic inhibition; (2) to demonstrate the regional effect
of multiple, rather than single, inhibitory synapses in terms of
the spread of their collective shunting effect in the dendritic
tree; and (3) to suggest an explanation as to why, in both cortex
and hippocampus, the total number of inhibitory dendritic
synapses per pyramidal cell is smaller (about 20%) than that of
excitatory synapses. This study thus provides a new perspective
on the biophysical design principles that govern the operation of
inhibition in dendrites.
RESULTS
The Shunt Level: A Tractable Measure for the Impactof Dendritic InhibitionWhen an inhibitory synapse is activated at a dendritic location,
i, a local conductance perturbation gi (a shunt) is induced in
the dendritic membrane. Depending on the reversal potential
of that synapse, either an inhibitory postsynaptic potential
(IPSP) is also generated or no potential change is observed
(a ‘‘shunting’’ or ‘‘silent’’ inhibition; Koch and Poggio, 1985).
Although the membrane shunt due to the activation of the
inhibitory synapses at i is highly local, its effect spreads to (i.e.,
is visible at) other dendritic locations (Rall, 1967; Koch et al.,
1990; Williams, 2004). Indeed, this spatial spread is reflected
by a change in input resistance, DRd, at location d. We define
the shunt level at location d, SLd, as,
SLd =DRd
Rd
(1)
where Rd is the input resistance in location d prior to the activa-
tion of gi. SLd is thus the relative drop in Rd at location d due to
the activation of single (ormultiple) steady conductance changes
at arbitrary dendritic locations (see Figures S8 and S9 and
related text available online for generalization to the transient
case). The value of SLd ranges from 0 (no shunt) to 1 (infinite
shunt) and depends on the particular dendritic distribution of
gis. For example, SLd = 0.2 implies that the inhibitory synapse
reduced the input resistance at location d by 20%, which is
also the relative drop in the steady voltage at d due to the inhibi-
tion after the injection of steady current at location d. Thus, in
order to characterize the effect of the inhibitory shunt in the
most general way, it is natural to ask how much increase in
excitatory current is required in order to exactly counter effect
the shunting inhibition. This is exactly what SL implies. Note
that the SL measure is applicable also for assessing the change
in input resistance due to excitatory synapses that, like inhibition,
exert a local membrane conductance change.
The spatial spread of SL can be solved using cable theory for
arbitrary passive dendritic trees receiving multiple inhibitory
synapses (see Experimental Procedures and Supplemental
Information). This solution provides several new and counterintu-
itive results regarding the overall impact of multiple inhibitory
dendritic synapses in dendrites and explains several experi-
mental and modeling results that were not fully understood prior
to the present study.
Dendritic ‘‘Hotspots’’ and Strategic Placementof InhibitionWe started with a geometrically simple case, whereby a single
inhibitory synapse impinges on a dendritic cylinder that is sealed
ended at one side and is coupled to an isopotential excitable
soma at the other (Figure 1A). The dendritic cylinder is comprised
of a hotspot (Magee et al., 1995; Schiller et al., 1997, 2000;
Larkum et al., 1999; Antic et al., 2010), which is modeled by a
cluster of 20 NMDA synapses, each randomly activated at
20 Hz (red circle and red synapse in Figure 1A). We then
searched for the strategic placement of the inhibitory synapse
that would effectively dampen this local dendritic hotspot.
Using numerical simulations for the nonlinear cable model
that includes the spiking soma and NMDA synapses depicted
in Figure 1A, we found that when the inhibitory conductance
change, gi, was placed distally (‘‘off-path’’) to the hotspot, the
rate of the soma action potentials (black trace in Figure 1B)
was reduced more effectively than when the same inhibitory
synapse was placed proximally (‘‘on-path’’) at the same distance
from the hotspot (orange trace in Figure 1B). Indeed, such
asymmetry in the impact of proximal versus distal inhibition for
dampening local dendritic hotspot was previously observed
in vitro (Miles et al., 1996; Jadi et al., 2012; Lovett-Barron
et al., 2012; see also Liu, 2004) and in simulations (Archie and
Mel, 2000; Rhodes, 2006), but the basis for this counterintuitive
result has remained unclear.
In order to provide an explanation for this result, we analytically
computed the value for SL at the hotspot (h) and thus assessed
the impact of inhibition at this location (Figures 1C–1E). In the
corresponding passive case, SLh at the hotspot that is due to
the inhibitory conductance change gi at location i can be ex-
pressed as the product of SL amplitude at location i (SLi) and
the attenuation of SL from i to h (SLi,h), i.e.,
SLh =SLi 3SLi;h: (2)
It can be shown (see Equations 4, 5, and 6 in Experimental
Procedures) that
SLi;h =Ah;i 3Ai;h; (3)
where Ah,i is the steady voltage attenuation from h to i (i.e., Vi / Vh
for steady current injected at h) and vice versa for Ai,h. Biophysi-
cally, Equation 3 can be explained as follows: depolarization
originating at h attenuates to i (Ah,i), where it changes the
Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc. 331
A
B
Soma
“Hotspot”
50 mV
300 ms
LS
ta
”to
pst
oh“
Distance of gi from “hotspot” (X)
On-p
ath
Off-path
0.3
0.2
0.1
-0.6 -0.4 -0.2 0 0.2 0.4
On-path Off-path
C
E
Proximal gi
Distal gi
Ei = V
rest
Rd (M
Ω)
SL a
t in
hib
. syn
.
0
0
0.1
0.2-0.2
0.2
0.4-0.4
0.3
-0.6
0.4
0.5
0
200
400
600
800
SL
Rd
Distance from “hotspot” (X)
Distance of gi from “hotspot” (X)
D “Hotspot”
00
0.2
0.2-0.2
0.4
0.4-0.4
0.6
-0.6
0.8
1
SL a
tte
nu
atio
n
to h
ots
pot
Distal gi
Proximal gi
Figure 1. Off-Path Inhibition Is More Effective than the Correspond-
ing On-Path Inhibition in Dampening a Local Dendritic Hotspot
(A) A model of a cylindrical cable (sealed end at L = 1) coupled to an iso-
potential excitable soma. Twenty NMDA synapses are clustered at the hotspot
located at X = 0.6; each synapse is randomly activated at 20 Hz. A single
inhibitory synapse (gi = 1 nS) is placed either distally or proximally at the same
electrotonic distance (X = 0.4) from the hotspot.
(B) Inhibitionof the somaticNa+ spikes ismore effectivewhen inhibition isplaced
distally to the hotspot (black synapse and corresponding black somatic spikes,
compared to orange synapse and corresponding orange somatic spikes).
Neuron
Principles Governing Dendritic Inhibition
332 Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc.
driving force for the inhibitory synapse. Consequently, the inhib-
itory synapse induces an outward current at i, resulting in
a reduction in local depolarization at i that propagates back to
site h (Ai,h). Consequently, the local conductance change at
the inhibitory synapse is also visible at other locations.
The asymmetry of the impact of distal versus proximal inhibi-
tion (Figures 1D and 1E) on location h (the hotspot) results from
the difference in the model’s boundary conditions, namely,
sealed-end boundary at the distal end and an isopotential
soma at the proximal end. This difference implies that the input
resistance and SLi (in cases of a fixed gi) also increase monoton-
ically with distance from the soma (Figure 1C and Equation 6 in
Experimental Procedures). Thus, the distal SLi (e.g., black circle
at X = +0.4, Figure 1C) is larger than that at the corresponding
proximal site (SLi at X = –0.4, orange circle). Additionally, the
overall voltage attenuation from the inhibitory synapses to the
hotspot and back to the synapses, and thus SLi,h (Equation 3),
is shallower for the distal synapses than for the proximal
synapses, because the latter is more affected by the somatic
current sink (Figure 1D, compare black arrowed dashed line to
the orange dashed line). The product of these two effects—the
initially larger SLi at the distal synapse and the shallower attenu-
ation of SLi from the distal synapse to the hotspot—implies that
SL at the hotspot (SLh) is larger for this synapse (Figure 1E). The
later conclusion also holds for transient inhibitory synaptic
conductance (Figures S8 and S9).
The above analysis considered the impact of the inhibitory
conductance change per se, namely, the case of a ‘‘silent inhibi-
tion,’’ whereby the reversal potential of the inhibitory synapse, Ei,
equals the resting potential, Vrest. Do the results depicted in
Figure 1 still hold when Ei is more negative than Vrest (hyperpola-
rizing inhibition)? Figure 2 shows that the advantage of the ‘‘off-
path’’ inhibition over the corresponding ‘‘on-path’’ inhibition in
dampening the hotspot is actually enhanced for hyperpolarizing
inhibition (compare Figure 1B to Figure 2B). Due to the asymme-
try in the boundary conditions, the distal synapse induces a larger
hyperpolarization at the hotspot compared to the proximal
synapse. Both the larger hyperpolarization and the larger SL at
the hotspot generated by the distal synapse are combined to
enhance its inhibitory impact on the hotspot (and thus on the
soma firing) as compared to the proximal synapse (Figure 2C
and see more detailed analysis in Figures S5–S7). These results
are also valid for different loci with respect to the hotspot of the
inhibitory synapses along the dendritic cable model (Figure S5).
Note that the results in Figures 1 and 2 hold for any dendritic
region producing inward current (e.g., via an a-amino-3-
hydroxy-5-methyl-4-isoxazole propionic acid [AMPA] synapse).
(C) SL (black line) and the input resistance (dashed line) as a function of
distance from the soma for the model shown in (A).
(D) Attenuation of SL from any dendritic site of the inhibitory synapse to the
hotspot (black line). The attenuation of SL from the synapse to the hotspot
(solid black line) is equal to the overall voltage attenuation from the hotspot to
the synapse and vice versa (black and orange dashed lines, for the corre-
sponding black and orange synapses depicted in A, respectively, Equation 3).
(E) The actual value of SL at the hotspot in the modeled cell depicted in (A) as
a function of the distance of the inhibitory synapse from the hotspot. Off-path
inhibitionattenuates lesssteeply compared to the respectiveon-path inhibition.
50 mV
300 ms
B
C“Hotspot”
Distance from “hotspot” (X)
0 0.2-0.2 0.4-0.4-0.6
V (
mV
)
-80
-79
-78
-77
-76
A
Soma
On-path Off-path
Proximal gi
Distal gi
Ei = V
rest - 10 mV
“Hotspot”
Figure 2. The Advantage of the ‘‘Off-Path’’ versus the ‘‘On-Path’’
Inhibition in Dampening a Dendritic Hotspot Is Boosted WhenInhibition Is Associated with Hyperpolarizing Reversal Potential
(A) Model as in Figure 1A but the inhibitory reversal potential, Ei, is 10mV more
negative than Vrest.
(B) The distal (off-path) inhibition is even more effective in dampening the
somatic spike firing (black trace) as compared to the corresponding on-path
inhibition (orange trace). Compare to Figure 1B.
(C) Voltage distribution for activation of either the distal (black) or the proximal
(orange) inhibitory synapse. Although the soma (located at X = –0.6) is more
hyperpolarized due to the proximal orange synapse, the hotspot is more hy-
perpolarized due to the black distal synapse.
SOMA
P
S
P
S
SOMA
SL
V
gi
0
0.2
0.4
0.6
0.8
1
0 0.5 1.0
Distance from soma (X)
gi
Figure 3. Steep Attenuation of SL in Distal Dendrites
Top: gi is located at a single terminal end of an idealized symmetrically
branched dendritic tree consisting of six identical stem dendrites of which the
structure of only one is fully shown (Rall and Rinzel, 1973). Bottom: attenuation
of SL (continuous line) and of steady voltage, V, (dotted line) from the distal
input dendritic terminal. Note the steep attenuation of SL toward the distal
dendritic terminals (blue arrow) compared to the attenuation of V (black arrow).
Neuron
Principles Governing Dendritic Inhibition
But the advantage of distal versus proximal inhibition at that
region is amplified in the voltage-dependent (nonlinear) case
(e.g., NMDA currents as in Figures 1B and 2B or active Ca+2 or
Na+ inward currents) because inhibition at the hotspot increases
the threshold for the activation of regenerative inward currents
(Jadi et al., 2012). We also note that the advantage of the ‘‘off-
path’’ inhibition over the corresponding ‘‘on-path’’ inhibition in
dampening a local dendritic hotspot is augmented in distal thin
dendrites because, in such branches, the asymmetry in (distal
versus proximal) boundary conditions is even larger than the
cylindrical case modeled in Figures 1 and 2 (Rall and Rinzel,
1973).
SL Spreads Poorly into the Thin Distal DendritesFigure 3 depictsSL in the case of an idealized branched dendritic
tree (Rall and Rinzel, 1973) receiving a single conductance
perturbation in a distal dendritic terminal. For comparison, the
steady voltage (V, dotted line) attenuation is also shown. V atten-
uation is steep from the distal (input) branch toward the branch
point (P) but is shallow in the direction of the sibling branch S
(Figure 3, black arrow) because of the sealed-end boundary
condition in this branch (Rall and Rinzel, 1973; Golding et al.,
2005). Similarly to V, SL attenuates steeply toward the soma;
however, in contrast to V, SL attenuates steeply toward terminal
S (blue line). This follows directly from Equation 3, as SL attenu-
ation from P to S depends on the (steep) voltage attenuation
from S to P (AS,P). Consequently, the impact of conductance
perturbation diminishes rapidly with distance in such thin
dendritic branches. Hence, excitatory currents in distal dendrites
are electrically ‘‘protected’’ from the inhibitory shunt, unless the
inhibitory synapses directly target these branches.
Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc. 333
SL
)%(
am
os
ta
tc
ap
mi.
bih
nI
B
A C
D
Distance from soma (μm) Distance from soma (μm)
0.1
0.2
0.3
00
200-200 400 600 800
“Somato-centric”
10
0 μ
min
hib
i. im
pa
ct
at
so
ma
50%
0%
0
10
20
30
40
50
-200 0 200 400 600 800
“Dendro-centric”
10
0 μ
m
SL
0.28
0
20 ms0.5
mV
1 m
V
E F
0.05
0.1
0.15
0.4 0.6 0.80.2
SL
Distance from junction (X)
2
4
816
0
0.2
0.25
0 1
gi
N = 2
N = 4
N = 8
N = 16
Figure 4. Global Spread and Enhanced Centripetal Accumulation of
Inhibitory Shunt in Trees with Multiple Inhibitory Synapses
(A) SL (color coded) in a model of a reconstructed CA1 pyramidal neuron
receiving a total of 15 inhibitory synapses (white dots; gi = 0.5 nS each, at
steady state) targeting distinct dendritic subdomains (basal, oblique, and
apical dendrites). Yellow traces represent EPSPs generated and recorded at
the sites denoted by yellow arrows before (continuous line) and after (dashed
line) the activation of the inhibitory synapses. The excitatory synapse on the
oblique branch is colocalized with one of the inhibitory synapses; the EPSP
generated by this synapse is less inhibited (SL = 0.2) than the excitatory
synapse located on the apical trunk (SL = 0.25), far from any one of the
inhibitory synapses. The EPSP was simulated using AMPA-like conductance
change (see Experimental Procedures).
(B) SL as a function of distance from the soma for the model shown in (A).
Inhibitory synapses are marked by black dots and excitatory synapses by
yellow dots.
(C and D) The impact of inhibition is depicted in color code (C) and as
a function of the distance from the soma (D). Inhibition is measured as
the percentage drop of the somatic voltage for steady current injected
Neuron
Principles Governing Dendritic Inhibition
334 Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc.
Elevated Centripetal Inhibition in DendritesIn the realistic case, the dendritic tree receives multiple inhibi-
tory synapses; even a single inhibitory axon typically contacts
the postsynaptic dendritic tree at multiple loci, often making
more than ten synapses in the postsynaptic dendritic tree
(Markram et al., 2004). We examined the implications of
multiple inhibitory synapses for SL in dendrites, using the model
of a reconstructed CA1 neuron (Golding et al., 2005) depicted
in Figure 4. This modeled neuron received inhibition at three
distinct dendritic subdomains: the basal, the apical, and the
oblique dendrites. In CA1, these morphological domains are
indeed innervated by inhibitory synapses arising from different
classes of inhibitory interneurons (for example, the axon of
bistratified cells target the basal and the oblique dendrites,
while the apical dendrite is targeted by the oriens lacunosum-
moleculare cells; Klausberger and Somogyi, 2008). We
assumed that each domain receives a cluster of five inhibitory
contacts (white dots).
The color-coded SL value induced by the activation of these
15 inhibitory synapses is shown in Figures 4A and 4B, superim-
posed on the modeled cell. As expected from the previous
section, SL spreads poorly (it attenuates steeply) in the direc-
tion of the dendritic terminals (Figure 4A, blue dendrites)
but, surprisingly, it spreads effectively (Figure 4A, red region)
hundreds of micrometers centripetally to the contact sites
themselves. Even more surprising was that SL became larger
in regions lacking inhibitory synapses compared to SL at the
synaptic sites themselves (Figure 4B). This is in contrast to
the prevailing view that the maximal effect of inhibition is
always at the synaptic site itself (Jack et al., 1975). This was
further demonstrated by simulation, whereby an excitatory
synapse in the proximal apical tree, far away from any inhibi-
tory synapse, was more inhibited than an excitatory synapse
contacting the oblique branches (compare the lower to the
upper excitatory postsynaptic potential [EPSP]; see Figure 4A;
continuous yellow line, before inhibition; dashed line, after
inhibition).
Note that the elevated centripetal increase in SL (red central
dendritic regions in Figure 4A) existed under a wide range of
conditions (Figure S3). Interestingly, we can show analytically
that such elevation in centripetal inhibition required at least
three inhibitory synapses encircling a dendritic region consisting
of multiple branches (Figure S2C).
For comparison, we also computed the impact of dendritic
inhibition as observed at the soma (the classical ‘‘somatocen-
tric’’ viewpoint). In Figures 4C and 4D, the same CA1 cell as in
Figures 4A and 4B was modeled, but here we computed the
percentage drop of somatic voltage from any dendritic location
at any given dendritic site for the neuron model and synaptic distribution as
in (A).
(E) Symmetrical starburst-like dendritic models consisting of multiple
(n = 2, 4, 8, 16) identical branches (L = 1) stemming from a common junction
(X = 0).
(F) SL for the corresponding models depicted in (E). Each branch receives
a single gi (1 nS) at X = 0.4 (dashed line). Note that for n = 8 (red) and n = 16
(blue), SL at the junction (lacking synapses) is larger than SL at the synaptic
sites.
SL
20
0S
L
0
0.08
DA
B
0.2 mV
100 ms
20 Hz
MC
PC
Distance from soma (μm)
C
SL
(%)
0
2
4
6
8
10
-400 0 400 800 1200
Figure 5. Inhibitory Shunt in Layer 5 Pyramidal Cell Dendrites Arising from a Single Martinotti Cell Inhibition
(A) Experimental scheme for simultaneous recordings from a synaptically connected layer 5 MC-to-PC pair. The PC morphology of this pair was reconstructed
and the locations of the 14 putative MC-to-PC synaptic contacts were identified (white dots in D; see Experimental Procedures).
(B) Somatic IPSPs recorded experimentally (blue trace) in the reconstructed PC shown in (D), after a train of action potentials (orange trace) in the presynaptic
MC (Silberberg and Markram, 2007). The black trace resulted from model fitting using the reconstructed PC model and the locations of the putative synapses.
(C) Computed SL arising from the activation of the 14 MC-to-PC synapses (0.15 nS each). Black dots mark the loci of the inhibitory synapses.
(D) Color-coded SL computed as in (C), superimposed on the modeled PC.
Neuron
Principles Governing Dendritic Inhibition
due to the 15 inhibitory synapses. When measured at the soma,
the largest impact of inhibition was obtained for depolarization
originating at distal dendrites, particularly for distal branches
receiving inhibitory synapses (red branches in Figure 4C). Note
that SL was very small in these distal branches (blue branches
in Figure 4A).
To analytically explain the counterintuitive results depicted in
Figures 4A and 4B, we constructed a symmetrical starburst-
like dendritic model consisting of multiple identical branches
stemming from a common junction (X = 0). Each of these
branches received an identical gi at a fixed distance (X = 0.4)
from the junction (Figure 4E). From Rall’s cable theory (Rall,
1959), it is straightforward to show that in such a structure, SL
at the junction remains constant, independent of the number of
stem branches (Figure 4F, all curves converge at X = 0).
However, increasing the number of branches (each with an
additional inhibitory synapse) had two consequences. First,
the local input resistance at each synapse was reduced and
therefore SLi at these sites was also reduced (Figure 4F, arrow;
Equation 6 in Experimental Procedures). Second, since the
input resistance at the junction was reduced with the increase
of the number of branches, the attenuation of SL from the junc-
tion to all the synaptic sites increased (Equation 3). Namely, the
synapses had progressively smaller shunting impact on each
other with increasing the number of branches. Together, these
results imply that when the number of branches is large enough,
SL at the junction (lacking synapses) may become larger than
SL at each of the synaptic sites. (The analytical solution for this
case is presented in Figure S3 and related text.)
Implications of Multiple Inhibitory Synapses forMartinotti-to-Layer 5 Pyramidal Cell ConnectionTo examine whether the above theoretical insights were appli-
cable to a real dendritic tree receiving specific inhibition at
known sites in a particular dendritic subdomain, we computed
SL in dendrites of a layer 5 pyramidal cell (PC) from the rat
somatosensory cortex, when inhibition was induced by the
single axon of a Martinotti cell (MC; Silberberg and Markram,
2007) with known loci of putative inhibitory synapses. MCs are
abundant in the rat neocortex, where they make up about 16%
of the population of cortical inhibitory cells (Markram et al.,
2004). These cells form short-term depressing g-aminobutyric
acid type A receptor (GABAAR) synapses on specific dendritic
domains of PCs (Kapfer et al., 2007; Silberberg and Markram,
2007; Berger et al., 2009). In layer 5, each MC axon makes an
average of 12 synaptic contacts on the PC apical dendrite
(Silberberg and Markram, 2007).
Based on experimental results by Silberberg and Markram
(2007) obtained from synaptically connected MC-to-PC pairs,
we constructed a detailed compartmental model of the post-
synaptic L5 PC in order to estimate the magnitude, time course,
and short-term dynamics of gi for the MC synaptic contacts
(see Experimental Procedures). Figure 5B shows the close
agreement between the model (black line) and the experimen-
tally recorded IPSPs (blue line) after the activation of a train of
spikes in the MC. Using this experimentally based estimate
of gi for each of the 14 inhibitory synapses (white dots in Fig-
ure 5D), we computed SL in the modeled PC (Figures 5C and
5D). In agreement with our theoretical predictions, SL was
Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc. 335
B
C
D
E
F
20 ms
SL
20
0 μ
m
0
0.12
Idend
amplitude (nA)
Vd
en
d (
mV
)
control
inhib.
0 0.4 0.8 1.2 1.6-80
-50
-20
10
Idend
gi
Idend
gi
Idend
1 nA
20 mV
0.5 nS
- Ca spike
- Na spike
+Ca spike
- Na spike
A
MCs
PC2 PC1
Figure 6. Converging Martinotti Cells’ Inhibition onto L5 PC
Dendrites Decouples the Dendritic Ca2+ Spike from the Soma Spikes
(A) Scheme for the disynaptic PC-MC-PC ‘‘loop,’’ whereby the activation of
a single PC (PC1) activates four MCs whose axons in turn converge on a single
PC (PC2; Berger et al., 2010).
(B) SL in PC model (Hay et al., 2011) resulting from the MC-to-PC ‘‘loop’’
depicted in (A), assuming a total of 48 MC synaptic contacts (white dots; see
Experimental Procedures). Inhibitory synapses are spatially distributed as
found experimentally (Wang et al., 2002). The ‘‘hot zone’’ in the PC apical
dendrite with voltage-dependent Ca2+ channels is marked by the dotted line.
(C) Dendritic Ca2+ spike (top red trace; red electrode in B) generated by an
excitatory-like postsynaptic current injection, Idend (1 nA; bottom red trace).
The dendritic Ca2+ spike triggered two somatic Na+ spikes (black traces; gray
electrode in B).
(D) Simultaneous injection of Idend and activation of the 48 MC synapses (gi in
gray) abolished both the Ca2+ and the Na+ spikes.
(E) As in (D) but Idend = 1.2 nA; Ca2+ spike has recovered but it does not trigger
somatic Na+ spikes.
(F) Dendritic potential (recorded 20 ms after Idend onset) as a function of Idendwith (black line) and without (dotted line) inhibition from MCs. Light gray area
indicates Idend intensities where Ca2+ spike was inhibited (as in D); dark gray
area indicates Idend intensities where Ca2+ spike was recovered but apical
dendrites were electrically decoupled from the soma (as in E).
Neuron
Principles Governing Dendritic Inhibition
relatively large in the main proximal shaft of the PC’s apical
dendrite and effectively spread into the dendritic region that is
surrounded by the MC synapses (red region), while it diminished
336 Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc.
in the distal apical tuft as well as in the oblique and basal
dendrites (blue).
In Figure 6, we explored the functional implications of the
spatial distribution of SL in PC dendrites receiving four MC
axons (48 inhibitory synapses, white dots in Figure 6B; Berger
et al., 2010), thus mimicking the MC-to-PC disynaptic ‘‘loop’’
(Silberberg and Markram, 2007; Berger et al., 2010). The
modeled layer 5 PC (Hay et al., 2011) faithfully replicated the
generation of dendritic Ca2+ spikes at a ‘‘hot zone’’ containing
a high density of Ca2+ channels (dashed line near the main apical
branch). Note that the model includes the increase in the Ihconductance with the distance from soma as was found exper-
imentally (Kole et al., 2006). Applying synaptic-like transient
excitatory current (Idend in Figure 6C) near the Ca2+ hot zone
resulted in the generation of a local Ca2+ spike in the PC model
(red trace in Figure 6C), followed by a burst of two somatic Na+
spikes (black traces in Figure 6C; Larkum et al., 1999). When
all 48 inhibitory synapses were activated, both the Ca2+ spike
and the resultant Na+ spikes were blocked (Figure 6D), in agree-
ment with recent experimental results (Murayama et al., 2009).
When the stimulus intensity, Idend, was increased, the local
Ca2+ spike was recovered but did not generate somatic Na+
spikes (Figure 6E). Thus, although the inhibitory synapses from
MCs did not contact the main apical shaft, MC inhibition effec-
tively electrically decoupled the dendritic Ca2+ spike from the
soma as well as decoupled the backpropagation of the Na+
spike from the soma to the dendrites (data not shown). There-
fore, MC inhibition may operate in PC dendrites directly on
the Ca2+ spike mechanism and/or on the electrical interaction
between the apical dendrite and the soma (Figure 6F). The
location of MC synapses on the oblique dendrites, as well as
on the distal apical branches (Figure 5D), and the large SL value
in these branches suggest that they may serve additional func-
tions, such as dampening local NMDA spikes in these branches.
We thus demonstrated that our theoretical predictions for
the spread of inhibitory conductance when multiple synapses
impinge on the tree hold for the realistic case of the MC-to-PC
connection. In particular, SL is elevated in central dendritic
regions lacking inhibition, namely the proximal apical trunk,
and this elevated inhibition is expected to decouple the two
spike initiation zones in L5 pyramidal cells: the soma and/or
axon region and the region in the vicinity of the main branch
point in the apical tuft.
DISCUSSION
The shunt level, SL, introduced in this study is a simple, intuitive,
and analytically tractable measure for assessing the impact
of inhibitory conductance change on dendritic cables. Solving
the cable equation for SL in arbitrary passive dendritic trees
receiving multiple inhibitory contacts has provided several
surprising results. In particular, we found that with multiple
synapses, SL spreads very effectively toward dendritic regions
encircled by these synapses and that it may become larger in
these regions than at the synaptic loci themselves. These
findings yielded several new insights regarding the functional
implications of the unique connectivity pattern of dendritic
inhibition. Importantly, although these insights are based on
Neuron
Principles Governing Dendritic Inhibition
the analytical solution for the steady-state case and for passive
dendrites (Figures 1, 2, and S1–S3), they nevertheless explain
simulated results obtained for corresponding nonlinear and
transient cases. In particular, we analyzed in detail the case of
an MC-to-PC inhibitory connection in layer 5 of the neocortex
(Figures 5 and 6), whereby the MC’s inhibitory synapses con-
tact the distal apical dendrites of the PC. Near the main apical
branch of the PC, a powerful Ca2+ spike could be evoked; this
spike interacts reciprocally with the soma to generate a burst
of Na+ spikes at the soma (BAC firing; Larkum et al., 1999).
Although the MC’s synapses are more distal than the Ca2+ spike
initiation region, we showed that they do effectively dampen the
Ca2+ spike (see Figure S12) and also electrically decouple the
apical dendrite from the soma, as expected from our analysis
of the corresponding passive case.
Anatomical versus Functional Inhibitory DendriticSubdomainsThe effective spread of SL into the dendritic region surrounded
by multiple inhibitory synapses (Figures 4 and 5) leads to
a spatially extended shunted dendritic domain beyond the
anatomical domain demarcated by these synapses. This spatial
spread of inhibitory shunt implies that in order to dampen excit-
atory and/or excitable dendritic currents, it is not necessary to
match each excitatory synapse with a corresponding adjacent
inhibitory synapse. Rather, by surrounding a dendritic region
with a few inhibitory contacts, it is possible to effectively dampen
the excitatory and/or excitable current that would be generated
in this region (Figures 5 and 6) and thereby effectively control
the neuron’s output. This may explain why in the neocortex
and the hippocampus, only�20% of the synapses are inhibitory
(DeFelipe and Farinas, 1992;Megıas et al., 2001; Merchan-Perez
et al., 2009).
Due to the extended centripetal spread of the inhibitory shunt,
different functional dendritic domains may interact with each
other and be formed dynamically by recruiting and/or omitting
various combinations of inhibitory synapses at strategic loci.
For example, when each of the group of five inhibitory synapses
in Figure 4A is individually active, then the functional dendritic
subdomain corresponding to each inhibitory subgroup is
spatially restricted. However, when all three inhibitory groups
of synapses are active together, as in Figure 4A, then the
functional dendritic domain that is shunted by the 15 inhibitory
synapses expands dramatically, effectively controlling the
excitatory and/or excitable charge (output) from a large portion
of the postsynaptic dendritic tree.
Why Does Inhibition Target Distal Dendrites?One surprising analytic result of this study is that distal off-path
inhibition is more effective than the corresponding on-path
inhibition for dampening a midway dendritic nonlinear hotspot
(Figures 1, 2, and S7, and see also experimental validation
in Miles et al., 1996; Lovett-Barron et al., 2012 and simulated
results in Archie and Mel, 2000; Rhodes, 2006). This result,
together with the result showing that SL spreads poorly to thin
distal branches (Figures 3, 4, 5, and 6), implies that in order to
control nonlinear process in distal dendritic branches, inhibitory
synapses should directly target the distal end of these branches.
We note that this result relies, in part, on the increase of the
input resistance (Rd) in distal branches (Rall and Rinzel, 1973;
Rinzel and Rall, 1974). However, in some cell types, the specific
membrane resistivity, Rm, along the main stem dendrite
decreases with distance from the soma (Magee, 1998; Stuart
and Spruston, 1998; Ledergerber and Larkum, 2010) and this
could lead to a decrease, rather than an increase, in Rd with
distance from the soma (Magee, 1998; but see Ledergerber
and Larkum, 2010). However, in a reconstructed model of
a layer 5 pyramidal cell (used in Figure 6), it is possible to show
in simulations that due to the thin diameter of distal dendritic
branches and the effect of the adjacent sealed-end boundary
conditions, even with the observed decrease in Rmwith distance
from the soma, Rd in thin distal branches still increases toward
the distal tips and, thus, the advantage of the off-path versus
on-path conditions still holds.
On-Path and Off-Path Inhibition: Somatic versusDendritic ViewpointsThe ‘‘on-path theorem’’ (Koch, 1998) states that the maximal
effect of inhibition in reducing the excitatory potential recorded
at the soma is achieved when inhibition is on the path between
the excitatory synapse and the soma (Rall, 1964; Jack et al.,
1975; Koch et al., 1983). At first glance, our findings (Figures 1
and 2) seem to contradict this classical result. However, we
searched for the strategic placement of inhibition so that it
most effectively dampens the inward current generated at the
locus of the excitatory synapses (or the ‘‘hotspot’’) itself, rather
than reducing the current reaching soma. Indeed, the powerful
impact of the off-path inhibition on the somatic firing as demon-
strated in Figures 1 and 2 is a secondary outcome of the
significant reduction of the inward current in the hotspot by
the distal inhibitory synapse: the more excitable the hotspot,
the more advantageous the distal inhibition compared to the
corresponding proximal inhibition.
In recent experiments, Hao et al. (2009) coactivated dendritic
inhibition, gi, and excitation, ge, while recording at the soma of
a CA1 pyramidal cell (somatocentric view). They derived an
arithmetic rule for the summation of the somatic EPSP and
IPSP, confirming the predictions of the on-path theorem also
for the case of multiple inhibitory and excitatory synapses.
Examining the effect of dendritic inhibition on dendritic spikes
invoked by ge, they found that the arithmetic rule does not hold
when gi and ge were coactivated on the same branch. This is
expected because, in this case, gi directly inhibits the dendritic
spike (large local SL). This case demonstrates that for dendrites
with active nonlinear currents (Murayama and Larkum, 2009;
Murayama et al., 2009; Kim et al., 2012; Palmer et al., 2012),
a dendrocentric view is required in order to characterize the
impact of dendritic inhibition. This is particularly true due to the
global and centripetal spread of inhibition in dendrites with
multiple inhibitory synapses.
Controlling dendritic nonlinear regenerative current such as
dendritic Ca2+ spike (Larkum et al., 1999), NMDA spike (Schiller
et al., 2000), and Na+ spikes (Kim et al., 2012) by inhibition
could be implemented either by increasing the threshold for
spike initiation (I/V curve is shifted to the right in Figure 6F) or
by suppressing an already fully triggered spike (reducedmaxima
Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc. 337
Table 1. Symbols
X, (Xi) Electrotonic distance (in units of the space constant, l)
from origin (to location i); (dimensionless).
L Electrotonic length (in units of l) of a dendritic branch;
(dimensionless).
V Steady membrane potentials, as a deviation from the
resting potential; (volt).
Ri Input resistance at location i; (U).
DRi Change in Ri due to synaptic conductance perturbation; (U).
gi Steady synaptic conductance perturbation at location i; (S).
SL Shunt level; (0 % SL % 1; dimensionless).
SLi Shunt level DRi / Ri due to activation of single or multiple
conductance perturbations; (0 % SL % 1; dimensionless).
Ri,j Transfer resistance between location i and location j; (U).
SLi,j Attenuation of SL (SLj/ SLi) for a single conductance
perturbation at location i; (0 % SLi,j % 1; dimensionless).
Ai,j Voltage attenuation, Vj/Vi, for current perturbation at
location i; (0 % Ai,j % 1; dimensionless).
r Dendritic-to-somatic conductance ratio; (Gdendrite/Gsoma;
dimensionless).
RN Input resistance at X = 0 for a semi-infinite cable; (U).
B Cable boundary condition; (Gdendrite/GN; dimensionless).
Neuron
Principles Governing Dendritic Inhibition
in Figure 6F; see Lovett-Barron et al., 2012). Dendritic off-path
inhibition is particularly potent because it effectively increases
the current threshold for spike initiation at the hotspot and,
therefore, it may effectively abolish the initiation of the dendritic
spike.
When the dendritic spike is fully triggered, then the on-path
inhibition is the preferred strategy for shunting the axial current
that flows from the hotspot to the soma, thus effectively reducing
the soma depolarization (‘‘somatocentric’’ view). This case is
essentially identical to the case studied theoretically by Rall
(1967), Jack et al. (1975), and Koch et al. (1983) and also in
experiments (Hao et al., 2009). However, regardless of whether
the spike at the hotspot is fully or only partially triggered, at the
hotspot itself (‘‘dendrocentric’’ view), the off-path inhibition is
always more effective in dampening the regenerative current
than the corresponding on-path inhibition (see Figure S11).
We note that branch-specific off-path distal inhibition is
also expected to powerfully affect the plasticity of excitatory
synapses in these branches, as this process depends on the
influx of (active) Ca2+ current either via NMDA-dependent
receptors or via voltage-dependent Ca2+ channels (Malenka,
1991; Malenka and Nicoll, 1993; MacDonald et al., 2006).
Robustness of the ResultsOur theoretical results are based on several simplifying
assumptions: we used an idealized starburst symmetrical model
to study the centripetal spread of SL in a steady state and in
most cases neglected the hyperpolarizing effect observed for
some inhibitory synapses. Since in vivo and in vitro studies
have demonstrated that inhibition often imposes a substantial
conductance change that is much larger than the conductance
change generated by excitatory synapses (Dreifuss et al.,
338 Neuron 75, 330–341, July 26, 2012 ª2012 Elsevier Inc.
1969; Borg-Graham et al., 1998; Marino et al., 2005; Monier
et al., 2008), analyzing SL on its own is partially justified.
However, when extending the analytic study to include more
complicated cases such as hyperpolarizing synapses (Figure 2
and see Supplemental Information), transient inhibitory con-
ductance change (Figures S8 and S9), the coactivation of
excitatory and inhibitory synapses (Figure S10), as well as using
numerical simulations for the nonlinear case, we showed that
the basic intuitions gained from the simplified models also hold
for many realistic cases.
In particular, because the centripetal spread of SL is already
expected for a starburst-like dendritic structure with three
inhibitory synapses and three branches (Figure S2 and related
text), the effective centripetal spread of SL is expected in any
dendritic structure with multiple inhibitory synapses encircling
a given dendritic region. This explains why we found a strong
centripetal spread of SL in a 3D reconstructed layer 5 PC
receiving MC inhibition (Figures 5C and 5D), in a layer 2/3
pyramidal cell receiving basket cell inhibition (Figure S4), in
hippocampal CA1 pyramidal neurons receiving inhibitory
synapses from multiple inhibitory sources (Figures 4A and 4B),
and in models of Purkinje cells and cortical spiny stellate
cells receiving multiple inhibitory synapses (data not shown).
Because individual inhibitory axons often form multiple (10–20)
synaptic contacts on the target dendritic tree, for most cases,
even single inhibitory axons are expected to form functional
dendritic subdomains with a strong centripetal inhibitory shunt-
ing effect.
In summary, this work advocates a ‘‘dendrocentric’’ viewpoint
for understanding how the neuron’s output is first and foremost
shaped in the dendrites, whereby excitatory and inhibitory
dendritic synapses interact with nonlinear membrane currents
before an output is generated at the axon. Our experimentally
inspired analytic study exposes several surprising principles
that govern this local dendritic foreplay.
EXPERIMENTAL PROCEDURES
SL in Dendritic Cables
The drop in the input resistance,DRd, at dendritic location d after the activation
of a single steady conductance perturbation, gi, at location i is given by Koch
et al. (1990):
DRd =Rd � R�d =
giR2i;d
1+giRi
; (4)
where Rd and R�d are, respectively, the input resistance prior to and after the
activation of gi (see definitions in Table 1).
The transfer resistance from i to d, Ri,d, is (Koch et al., 1983)
Ri;d =Rd;i =RiAi;d =RdAd;i : (5)
Combining Equations 4 and 5, we get that, due to the activation of the
conductance perturbation at location i, the relative drop in the input resistance,
SLd = DRd / Rd, is
SLd =
�giRi
1+giRi
�Ai;d 3Ad;i : (6)
The bracket denotes the amplitude of SL at the input location (d = i), which
depends on the product giRi. In contrast, the attenuation of SL from the input
location i to location d (SLi,d) is independent of gi (for a single gi) and is the
product of Ai,d 3 Ad,i. Consequently, SLi,d = SLd,i.
Neuron
Principles Governing Dendritic Inhibition
Using Equation 6 SLh at the ‘‘hotsopt’’ in Figures 1C and 1E is
SLh =
2664 giRN
tanhðL� XiÞ+ B1 + tanhXi
1+B1 + tanhXi
+giRN
37753SLi;h; (7)
where B1 = tanh(L) / r, r is the dendrite-to-soma conductance ratio, and
Xi is the distance of gi from the soma. The voltage attenuation (also for Figure
2C) in cylindrical dendrites is provided by Rall’s cable equations (Rall, 1959);
thus, SLi,h depends on whether gi is placed between the hotspot (h) and the
soma (‘‘on-path’’) or distally to the hotspot (‘‘off-path’’),
SLi;h =1� tanh L tanh X
1� tanh L tanh Xi
3B1tanh Xi + 1
B1tanh X + 1for 0RXiRX ðon-pathÞ;
SLi;h =1� tanh L tanhXi
1� tanh L tanhX3
B1tanh X + 1
B1tanh Xi + 1for XRXiRL ðoff-pathÞ:
SL in Arbitrary Dendritic Trees with Multiple Inhibitory Synapses
When multiple synapses (multiple gis) impinge on the dendritic tree, SL at any
location d is the result of sublinear interaction among the effects of individual
gis on the SL in this location. To analytically solve this case, it is useful to
consider the dendritic tree that has conductance perturbations (gi) at multiple
locations as a ‘‘new tree,’’ whereby each location with gi is a new node (a shunt
to ground). One can then iteratively compute (Rall, 1959) R�d at each location
d for this new tree and subtract the corresponding Rd in the original tree to
solve for DRd and SLd. This computation is simplified in the symmetrical
starburst-like model with identical stem branches depicted in Figures 4E
and 4F. To solve SL for this model, an equivalent two-cylinder structure is
constructed (Rall, 1967) with two conductance perturbations (see Figure S2
and related text).
In Figure 3, V and SL attenuations in the ideal branching dendrite were
computed using Equation 6 as in Rall and Rinzel (1973). For dendrites con-
sisting of 3D reconstructed morphology (Figures 4, 5, and 6), SL was
computed using ‘‘impedance’’ class in the NEURON simulation environment
(Hines and Carnevale, 1997).
In all the models used in this study, the axial resistance was Ra = 100 Ucm
and the specificmembrane capacitancewasCm = 1 mF/cm2. In Figures 4A–4D,
we used the reconstructed morphology of a CA1 pyramidal neuron (Golding
et al., 2005; Ascoli et al., 2007) with Rm = 15,000 U 3 cm2. In Figures 1 and
2, the model consisted of a sealed-end passive cylindrical cable (L = 1; Rm =
20,000 U 3 cm2) and diameter of 1 mm, coupled at X = 0 to an isopotential
soma such that r = 0.1. Inhibitory conductance change, gi, was 1 nS. In
addition to the passive membrane resistance, the somatic conductances in
Figures 1A and 1B and 2A and 2B included Na+ and K+ channels (model and
parameters, as previously described in Traub et al., 1991, with activation
and inactivation functions shifted by +15mV). In Figures 1A and 1B and 2A
and 2B, NMDA synapses were modeled (with gmax = 0.5 nS) as previously
described (Sarid et al., 2007). In Figure 4A, the excitatory synapse was
modeled by voltage-independent conductance with peak value of 0.5 nS
and rise and decay time constants of 0.2 ms and 10 ms, respectively.
Individual dendritic branches and inhibitory synapses in Figures 4E and 4F
were similar to the modeled dendrite in Figures 1 and 2 (without the soma)
with a branch diameter of 2 mm. In Figures 5C, 5D, and 6B, SL was computed
in a passive model after setting all voltage-dependent membrane con-
ductances to their value at the resting potential. The steady gi value used for
simulating the Martinotti inhibition and for computing SL in Figures 5C, 5D,
and 6B was the average conductance (0.15 nS) computed over the time
interval between the first and eighth IPSP shown in Figure 5B.
Conductance Magnitude and Waveform for MC-to-PC Inhibitory
Connection
In vitro recordings from a pair of connected layer 5 MCs to thick-tufted layer
5 PCs in rat somatosensory were kindly provided by Gilad Silberberg and
have been described previously (Silberberg and Markram, 2007). In short,
a train of eight action potentials was initiated in the presynaptic MC and the
resulting inhibitory postsynaptic potentials, IPSPs, were recorded at the
corresponding PC. This pair was reconstructed in 3D and the locations of
the putative MC synaptic contacts on the PC dendrite were identified. In the
PC model, Ih conductance was distributed in the dendrite; it was shown to
have a critical role in shaping the MC IPSPs in the PC (Kole et al., 2006; Silber-
berg and Markram, 2007). Leak conductance was adjusted such that the
measured membrane time constant was �17 ms (Le Be et al., 2007). The
MC-to-PC GABAergic synaptic conductance change was modeled as a sum
of two exponents (NEURON Exp2Syn) and with short-term depressing
dynamics (Markram et al., 1998). GABAA reversal potential was uniformly set
to –5mV relative to the resting potential.
A genetic algorithm (Druckmann et al., 2007) was used to fit the model’s
somatic IPSP (with the Martinotti inhibitory synapses at their putative loca-
tions) to the experimental trace. The parameters of the MC-to-PC synaptic
model and the short-term synaptic dynamics (Markram et al., 1998) were the
following: the time constant of recovery from depression (D); the time constant
of recovery from facilitation (F); the utilization of synaptic resources as used
analogously to Pr (e.g., release probability, U); the absolute strength (ASE) of
the synaptic connection (defined as the response when U equals 1); and the
rise (tR) and decay (tD) time constants of the synaptic conductance. Themodel
fit depicted in Figure 5B and used in Figures 6D–6Fwas obtained forASE,U,D,
F, tR, and tD using the respective values of 2.5 nS, 0.2, 574 ms, 1.5 ms, 2 ms,
and 23 ms. In Figures 6C–6E, the EPSC-like current injection, Idend, was
described as a sum of two exponents, amp 3 (�exp(�t/t1) + exp(�t/t2)) /
factor, where t1 = 4ms and t2 = 10ms, and amp is the amplitude of the injected
current after normalization by factor.
SUPPLEMENTAL INFORMATION
Supplemental Information includes twelve figures and can be found with this
article online at http://dx.doi.org/10.1016/j.neuron.2012.05.015.
ACKNOWLEDGMENTS
We thank Y. Yarom and D. Hansel for discussions of this work, M. Hausser, M.
London, A. Roth, B. Torben-Nielsen, H. Markram, S. Hill, F. Schurmann, and H.
Sompolinsky for their comments on earlier versions of this manuscript. This
work was supported by the EPFL fund for the Blue Brain Project, by the Gatsby
Charitable Foundation, and by the Hebrew University Netherlands Association
(HUNA).
Accepted: May 8, 2012
Published: July 25, 2012
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