Amplifying and Linearizing Apical 
Synaptic Inputs 
to Cortical Pyramidal Cells. 
jvind Bernander, Christof Koch * 
Computation and Neural Systems Program, 
California Institute of Technology, 139-74 
Pasadena, Ca 91125, USA. 
Rodney J. Douglas 
Anatomical Neuropharmacology Unit, 
Dept. Pharmacology, 
Oxford, UK. 
Abstract 
Intradendritic electrophysiological recordings reveal a bewildering 
repertoire of complex electrical spikes and plateaus that are dif- 
ficult to reconcile with conventional notions of neuronal function. 
In this paper we argue that such dendritic events are just an exu- 
berant expression of a more important mechanism - a proportional 
current amplifier whose primary task is to offset electrotonic losses. 
Using the example of functionally important synaptic inputs to the 
superficial layers of an anatomically and electrophysiologically re- 
constructed layer 5 pyramidal neuron, we derive and simulate the 
properties of conductances that linearize and amplify distal synap- 
tic input current in a graded manner. The amplification depends 
on a potassium conductance in the apical tuft and calcium conduc- 
tances in the apical trunk. 
*To whom all correspondence should be addressed. 
519 
520 Bernander, Koch, and Douglas 
1 INTRODUCTION 
About half the pyramidal neurons in layer 5 of neocortex have long apical dendrites 
that arborize extensively in layers 1-3. There the dendrites receive synaptic input 
from the inter-areal feedback projections (Felleman and van Essen, 1991) that play 
an important role in many models of brain function (Rockland and Virga, 1989). 
At first sight this seems to be an unsatisfactory arrangement. In light of traditional 
passive models of dendritic function the distant inputs cannot have a significant 
effect on the output discharge of the pyramidal cell. The distal inputs are at least 
one to two space constants removed from the soma in layer 5 and so only a small 
fraction of the voltage signal will reach there. Nevertheless, experiments in cortical 
slices have shown that synapses located in even the most superficial cortical layers 
can provide excitation strong enough to elicit action potentials in the somata of 
layer 5 pyramidal cells (Cauller and Connors, 1992, 1994). These results suggest 
that the apical dendrites are active rather than passive, and able to amplify the 
signal en route to the soma. Indeed, electrophysiological recordings from cortical 
pyramidal cells provide ample evidence for a variety of voltage-dependent dendritic 
conductances that could perform such amplification (Spencer and Kandel, 1961; 
Regehr et al., 1993; Yuste and Tank, 1993; Pockberger, 1991; Amitai et al., 1993; 
Kim and Connors, 1993). 
Although the available experimental data on the various active conductances pro- 
vide direct support for amplification, they are not adequate to specify the mecha- 
nism by which it occurs. Consequently, notions of dendritic amplification have been 
informal, usually favoring voltage gain, and mechanisms that have a binary (high 
gain) quality. In this paper, we formalize what conductance properties are required 
for a current amplifier, and derive the required form of their voltage dependency by 
analysis. 
We propose that current amplification depends on two active conductances: a 
voltage-dependent K + conductance, gK, in the superficial part of the dendritic 
tree that linearizes synaptic input, and a voltage-dependent Ca 2+ conductance, 
gca, in layer 4 that amplifies the result of the linearization stage. Spencer and Kan- 
del (1961) hypothesized the presence of dendritic calcium channels that amplify 
distal inputs. More recently, a modeling study of a cerebellar Purkinje cell suggests 
that dendritic calcium counteracts attenuation of distal inputs so that the somatic 
response is independent of synaptic location (De Schutter and Bower, 1992). A 
gain-control mechanism involving both potassium and calcium has also been pro- 
posed in locust non-spiking interneurons (Laurent, 1993). In these cells, the two 
conductances counteract the nonlinearity of graded transmitter release, so that the 
output of the interneuron was independent of its membrane voltage. The principle 
that we used can be explained with the help of a highly simplified three compart- 
ment model (Fig. 1A). The leftmost node represents the soma and is clamped to 
-50 mV. The justification for this is that the time-averaged somatic voltage is re- 
markably constant and close to -50 mV for a wide range of spike rates. The middle 
node represents the apical trunk containing gca, and the rightmost node represents 
the apical tuft with a synaptic induced conductance change gsyn in parallel with 
gK. For simplicity we assume that the model is in steady-state, and has an infinite 
membrane resistance, 
Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 521 
Isoma 
g 
K syn 
Vsoma 
ECa 
EK 
1- Esyn 
B Passive response and targets C Activation curves 
Linearized 
100 200 
gsyn (nS) 
3O 
 2O 
10 
o 
-50 
o lO 
v (ray) 
Figure 1: Simplified model used to demonstrate the concepts of saturation, 
linearization, and amplification. (A) Circuit diagram. The somatic compart- 
ment was clamped to Voma = -50 rnV with Eca= 115 mV, EK = --95 rnV, 
Eu = 0 rnV, and g - 40 nS. The membrane capacitance was ignored, since 
only steady state properties were studied, and membrane leak was not included for 
simplicity. (B) Somatic current, Ioma, in response to synaptic input. The pas- 
sive response (thin dashed line) is sublinear and saturates for low values of g,u. 
The linearized response (thick solid line) is obtained by introducing an inactivating 
potassium conductance, GK ("gA" in c). A persistent persistent GK results in a 
somewhat sub-linear response (thick dashed line; "gM" in c). The addition of a cal- 
cium conductance amplifies the response (thin solid line). (C) Analytically derived 
activation curves. The inactivating potassium conductance ("IA") was derived, but 
the persistent version ("IM") proved to be more stable. 
522 Bernander, Koch, and Douglas 
2 RESULTS 
Fig. lB shows the computed rela{onship between the excitatory synaptic input 
conductance and the axial current, Io,a, flowing into the somatic (leftmost) com- 
partment. The synaptic input rapidly saturates; increasing g,yn beyond about 50 nS 
leads to little further increase in Iso,a. This saturation is due to the EPSP in the 
distal compartment reducing the effective synaptic driving potential. We propose 
that the first goal of dendritic amplification is to linearize this relationship, so that 
the soma is more sensitive to the exact amount of excitatory input impinging on 
the apical tuft, by introducing a potassium conductance that provides a hyper- 
polarizing current in proportion to the degree of membrane depolarization. The 
voltage-dependence of such a conductance can be derived by postulating a linear 
relationship between the synaptic current flowing into the somatic node and the 
synaptic input, i.e. Io,,a = constant 'gun. In conjunction with Ohm's law and 
current conservation, this relation leads to a simple fractional polynominal for the 
voltage dependency of gK (labeled "gA" in Fig. 1C). As the membrane potential 
depolarizes, gK activates and pulls it back towards EK. At large depolarizations gK 
inactivates, similar to the "A" potassium conductance, resulting overall in a linear 
relationship between input and output (Fig. lB). As the slope conductance of this 
particular form of gK can become negative, causing amplification of the synaptic 
input, we use a variant of 9K that is monotonized by leveling out the activation 
curve after it has reached its maximum, similar to the "M" current (Fig. 1C). Incor- 
porating this non-inactivating K + conductance into the distal compartment results 
in a slightly sublinear relationship between input and output (Fig. lB). 
With #K in place, amplification of Io,,a is achieved by introducing an inward 
current between the soma and the postsynaptic site. The voltage-dependency of the 
amplification conductance can be derived by postulating Io,,a = 9ain  constant. 
#sun. This leads to the non-inactivating #ca shown in Fig. 1C, in which the overall 
relationship between synaptic input and somatic output current (Fig. lB) reflects 
the amplification. 
We extend this concept of deriving the form of the required conductances to a 
detailed model of a morphologically reconstructed layer 5 pyramidal cell from cat 
visual cortex (Douglas et al., 1991, Fig. 2A;). We assume a passive dendritic tree, 
and include a complement of eight common voltage-dependent conductances in its 
soma. 500 non-NMDA synapses are distributed on the dendritic tuft throughout 
layers 1, 2 and 3, and we assume a proportionaltry between the presynaptic firing 
frequency fin and the time-averaged synaptic induced conductance change. When 
fin is increased, the detailed model exhibits the same saturation as seen in the 
simple model (Fig. 2B). Even if all 500 synapses are activated at fin = 500 Hz only 
0.65 nA of current is delivered to the soma. This saturation is caused when the 
synaptic input current flows into the high input resistances of the distal dendrites, 
thereby reducing the synaptic driving potential. Layer 1 and 2 input together can 
contribute a maximum of 0.25 nA to the soma. This is too little current to cause 
the cell to spike, in contrast with the experimental evidence (Cauller and Connors, 
1994), in which spike discharge was evoked reliably. Electrotonic losses make only 
a minor contribution to the small somatic signal. Even when the membrane leak 
current is eliminated by setting Rrn to infinity, Ioma only increases a mere 2% to 
0.66 nA. 
Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 523 
Layer 3 
Layer 4 
Layer 5 
B 
2 
Current delivered to soma 
0 o 
' Layer 5] ' 
ssive dendrlte 
Layers 1-3: 
Llnearlzed 
and amplified 
Layers 1-3: 
?assve dendrte 
lOO 
fln (Hz) 
2OO 
o 
C Activation Curves 
gK 
960 -40 -20 0 
V (mV) 
o 
Input-Output behavior 
Act 1ve, 
predicted / Active 
5O 150 20O 
fin (Hz) 
Figure 2: Amplification in the detailed model. (A) The morphology of this 
layer V pyramidal cell was reconstructed from a HRP-stained cell in area 17 of the 
adult cat (Douglas el al., 1991). The layers are marked in alternating black and 
grey. The boundaries between superficial layers are not exact, but rough estimates 
and were chosen at branch points; a few basal dendrites may reach into layer 6. 
Axon not shown. (B) Current delivered to the soma by stimulation of 500 AMPA 
synapses throughout either layer 5 or layers 1-3. (C) Derived activation curves for 
gs: and Sea. Sigmoidal fits of the form g(V) = 1/(1 + e(v'a'-v)/K), resulted in 
KK = 3.9 rnV, VaaIf,K -- --51 mV, Kca = 13.7 mV, Vaalf,Ca ---- --14 reV. (D) 
Output spike rate as a function of input activation rate of 500 AMPA synapses 
in layers 1-3, with and without the derived conductances. The dashed line shows 
the fou rate predicted by using the linear target Loma as a function of fi, in 
combination with the somatic f - I relationship. 
524 Bernander, Koch, and Douglas 
Vm 
' ' 
t (msec) 
Figure 3: Dendritic calcium spikes. All-or-nothing dendritic Ca 2+ calcium 
spikes can be generated by adding a voltage-independent but Ca2+-dependent K + 
conductance to the apical tree with gmax -- 11.4 nS. The trace shown is in response 
to sustained intradendritic current injection of 0.5 nA. For clamp currents of 0.3 nA 
or less, no calcium spikes are triggered and only single somatic spikes are obtained 
(not shown). These currents do not substantially affect the current amplifier effect. 
By analogy with the simple model of Fig. 1, we eliminate the saturating response by 
introducing a non-inactivating form of gz spread evenly throughout layers 1-3. The 
resulting linearized response is amplified by a Ca + conductance located at the base 
of the apical tuft, where the apical dendrite crosses from layer 4 to layer 3 (Fig. 2A). 
This is in agreement with recent calcium imaging experiments, which established 
that layer 5 neocortical pyramidal cells have a calcium hot spot in the apical tree 
about 500-600/rn away from the soma (Tank et al., 1988). Although the derivation 
of the voltage-dependency of these two conductances is more complicated than in 
the three compartment model, the principle of the derivation is similar (Bernander, 
1993, Fig. 2C;). We derive a Ca + conductance, for a synaptic current gain of 
two, resembling a non-inactivating, high-threshold calcium conductance. The curve 
relating synaptic input frequency and axial current flowing into the soma (Fig. 2B) 
shows both the linearized and amplified relationships. 
Once above threshold, the model cell has a linear current-discharge relation with a 
slope of about 50 spikes per second per nA, in good agreement with experimental 
observations in vitro (Mason and Larkman, 1990) and in vivo (Ahmed et al., 1993). 
Given a sustained synaptic input frequency, the somatic f-I relationship can be used 
to convert the synaptic current flowing into the soma Isoma into an equivalent output 
frequency (Abbott, 1991; Powers et al., 1992; Fig. 2D). This simple transformation 
accounts for all the relevant nonlinearities, including synaptic saturation, interaction 
and the threshold mechanism at the soma or elsewhere. We confirmed the validity 
of our transformation method by explicitly computing the expected relationship 
between fin and four, without constraining the somatic potential, and comparing 
the two. Qualitatively, both methods lead to very similar results (Fig. 2D): in the 
Amplifying and Linearizing Apical Synaptic Inputs to Cortical Pyramidal Cells 525 
presence of dendritic gc, superficial synaptic input can robustly drive the cell, in a 
proportional manner over a large input range. 
The amplification mechanism derived above is continuous in the input rate. It does 
not exhibit the slow calcium spikes described in the literature (Pockberger, 1991; 
Amitai et al., 1993; Kim and Connors, 1993). However, it is straightforward to add a 
calcium-dependent potassium conductance yielding such spikes. Incorporating such 
a conductance into the apical trunk leads to calcium spikes (Fig. 3) in response to 
an intradendritic current injection of 0.4 nA or more, while for weaker inputs no 
such events are seen. In response to synaptic input to the tuft of 120 Hz or more, 
these spikes are activated, resulting in a moderate depression (25% or less) of the 
average output rate, fou (not shown). 
In our view, the function of the dendritic conductances underlying this all-or-none 
voltage event is the gradual current amplification of superficial input, without am- 
plifying synaptic input to the basal dendrites (Bernander, 1993). Because gca 
depolarizes the membrane, further activating gca, the gain of the current amplifier 
is very sensitive to the density and shape of the dendritic gca. Thus, neuromod- 
ulators that act upon gc control the extent to which cortical feedback pathways, 
acting via superficial synaptic input, have access to the output of the cell. 
Acknowledgements 
This work was supported by the Office of Naval Research, the National Institute of 
Mental Health through the Center for Neuroscience, the Medical Research Council 
of the United Kingdom, and the International Human Frontier Science Program. 
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