Principles Governing the Operation of Synaptic Inhibition ...lobster.ls.huji.ac.il/idan/files/Albert_Neuron-2012.pdf · Principles Governing the Operation of Synaptic Inhibition in

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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

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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



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


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.


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



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


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


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


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


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


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


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


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.,


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


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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