Network activity determines 
spatio-temporal integration in single cells 
jvind Bernander, Christof Koch * 
Computation and Neural Systems Program, 
California Institute of Technology, 
Pasadena, Ca 91125, USA. 
Rodney J. Douglas 
Anatomical Neuropharmacology Unit, 
Dept. Pharmacology, 
Oxford, UK. 
Abstract 
Single nerve cells with static properties have traditionally been viewed 
as the building blocks for networks that show emergent phenomena. In 
contrast to this approach, we study here how the overall network activity 
can control single cell parameters such as input resistance, as well as time 
and space constants, parameters that are crucial for excitability and spario- 
temporal integration. Using detailed computer simulations of neocortical 
pyramidal cells, we show that the spontaneous background firing of the 
network provides a means for setting these parameters. The mechanism 
for this control is through the large conductance change of the membrane 
that is induced by both non-NMDA and NMDA excitatory and inhibitory 
synapses activated by the spontaneous background activity. 
1 INTRODUCTION 
Biological neurons display a complexity rarely heeded in abstract network models. 
Dendritic trees allow for local interactions, attenuation, and delays. Voltage- and 
*To whom all correspondence should be addressed. 
43 
44 Bernander, Koch, and Douglas 
time-dependent conductances can give rise to adaptation, burst-firing, and other 
non-linear effects. The extent of temporal integration is determined by the time 
constant, and spatial integration by the "leakiness" of the membrane. It is unclear 
which cell properties are computationally significant and which are not relevant 
for information processing, even though they may be important for the proper 
functioning of the cell. However, it is crucial to understand the function of the 
component cells in order to make relevant abstractions when modeling biological 
systems. In this paper we study how the spontaneous background firing of the 
network as a whole can strongly influence some of the basic integration properties 
of single cells. 
1.1 Controlling parameters via background synaptic activity 
   dV  
The input resastance, Rn, s defined as 7/', where dV s the steady state voltage 
change in response to a small current step of amplitude dI. Jin will vary throughout 
the cell, and is typically much larger in a long, narrow dendrite than in the soma. 
However, the somatic input resistance is more relevant to the spiking behavior of 
the neuron, since spikes are initiated at or close to the soma, and hence 
(henceforth simply referred to as Ri,) will tell us something of the sensitivity of the 
cell to charge reaching the soma. 
The time constant, r,, for a passive membrane patch is R,. C,, the membrane 
resistance times the membrane capacitance. For membranes containing voltage. 
dependent non-linearities, exponentials are fitted to the step response and the 
largest time constant is taken to be the membrane time constant. A large time 
constant implies that any injected charge leaks away very slowly, and hence the cell 
has a longer "memory" of previous events. 
The parameters discussed above (Ri,, r,n) clearly have computational significance 
and it would be convenient to be able to chane them dynamically. Both depend 
directly on the membrane conductance Gm= /F', so any change in Gm will change 
the .parameters. Traditionally, however, Gm has been viewed as static, so these 
parameters have also been considered static. How can we change Grn dynamically? 
In traditional models, Gm has two components: active (time- and voltage. 
dependent) conductances and a passive "leak" conductance. Synapses are mod- 
eled as conductance changes, but if only a few are activated, the cable structure 
of the cell will hardly change at all. However, it is well known that neocortical 
neurons spike spontaneously, in the absence of sensory stimuli, at rates from 0 to 
10 Hz. Since neocortical neurons receive on the order of 5,000 to 15,000 excitatory 
synapses (Larkman, 1991), this spontaneous firing is likely to add up to a large total 
conductance (Holmes & Woody, 1989). This synaptic conductance becomes crucial 
if the non-synaptic conductance components are small. Recent evidence show in- 
deed that the non-synaptic conductances are relatively small (when the cell is not 
spiking) (Anderson et al., 1990). Our model uses a leak R, = 100,000 klcrn , 
instead of more conventional values in the range of 2,500-10,000 klcrn . These 
two facts, high R, and synaptic background activity, allow Ri, and r,n to change 
by more than ten-fold, as described below in this paper. 
Network activity determines spatio-temporal integration in single cells 45 
MODEL 
A typical layer V pyramidal cell (fig. 2) in striate cortex was filled with HRP dur- 
ing in vivo experiments in the anesthetized, adult cat (Douglas et al., 1991). The 
3-D coordinates and diameters of the dendritic tree were measured by a computer- 
assisted method and each branch was replaced by a single equivalent cylinder. This 
morphological data was fed into a modified version of NEURON, an efficient sin- 
gle cell simulator developed by Hines (1989). The dendrites were passive, while the 
soma contained seven active conductance. s, underlying spike generation, adaptation, 
and slow onset for weak stimuli. The model included two sodium conductances (a 
fast spiking current and a slower non-inactivating current), one calcium conduc- 
tance, and four potassium conductances (delayed rectifier, slow 'M' and 'A' type 
currents, and a calcium-dependent current). The active conductances were modeled 
using a Hodgkin-Huxley-like formalism. 
The model used a total of 5,000 synapses. The synaptic conductance change 
in time was modeled with an alpha function, g(t) = "'"ete-t/t,'"'. 4,000 
synapses were fast excitatory non-NMDA or AMPA-type (tpek = 1.5 rnsec, gpek = 
0.5 nS, Eo = 0 mV), 500 were medium-slow inhibitory GABAA (tpa = 
10 msec, gpak  1.0 nS, Eeo = -70 mV), and 500 were slow inhibitory GABAB 
(tpea = 40 msec, gpea = 0.1 nS, Ereo = -95 mV). The excitatory synapses were 
ls concentrated towards the soma, while the inhibitory ones were more so. For a 
more detailed dcription of the model, see Bernander et al. (1991). 
Rin, non-NMDA 
Rin, non-NMDA and NMDA 
 
o  i o ,i i  7 
Background frequency (Hz) Background frluency (Hz) 
Figure 1: Input resistance and time constant as a function of background 
frequency. In (a), the solid line corresponds to the "standard" model with passive 
dendrites, while the dashed line includes active NMDA synapses as described in the 
text. 
46 Bernander, Koch, and Douglas 
3 RESULTS 
3.1 Ri,and ', change with backgromd frequency 
Fig. 1 illustrates what happens to R/,and ', when the synaptic background activ- 
ities of all synaptic types are varied simultaneously. In the absence of any synaptic 
input, R/,= 110 Mfi and 'm = 80 rnsec. At 1 Hz background activity, on av- 
erage 5 synaptic events are impinging on the cell every rnsec, contributing a total 
of 24 nS to the somatic input conductance Gin Because of the reversal potential 
of the excitatory synapses (0 mV), the membrane potential throughout the cell is 
pulled towards more depolarizing potentials, activating additional active currents. 
Although these trends continue as f is increased, the largest change can be observed 
between 0 and 2 Hz. 
Figure 2: Spatial integration as a function of background frequency. 
Each dendrite has been "stretched" so that its apparent length corresponds to its 
electrotonic length. The synaptic background frequency was 0 Hz (left) and 2 Hz 
(right). The scale bar corresponds to 1 A (length constant). 
Activating synaptic input has two distinct effects: the conductance of the post- 
synaptic membrane increases and the membrane is alepolarized. The system can, 
at least in principle, independently control these two effects by differentially vary- 
ing the spontaneous firing frequencies of excitatory versus inhibitory inputs. Thus, 
increasing f selectively for the GABAB inhibition will further increase the mem- 
brane conductance but move the resting potential towards more hyperpolarizing 
Network activity determines spatio-temporal integration in single cells 47 
potentials. 
Note that the 0 Hz case corresponds to experiments made with in vitro slice prepa- 
rations or culture. In this case incoming fibers have been cut off and the spontaneous 
firing rate is very small. Careful studies have shown very large values for Ri, and 
'm under these circumstances (e.g. Spruston & Johnston, 1991). In vivo prepara- 
tions, on the other hand, leave the cortical circuitry intact and much smaller values 
of Ri, and rm are usually recorded. 
3.2 Spatial integration 
Varying synaptic background activity can have a significant impact on the electro- 
tonic structure of the cell (fig. 2). We plot the electrotonic distance of any particular 
point from the cell body, that is the sum of the electrotonic length's 
associated with each dendritic segment i, where ;j =  is the electrotonic 
length constant of compartment j, lj its anatomical length and the sum is taken 
over all compartments between the soma and compartment i. 
Increasing the synaptic background activity from f = 0 to f = 2 Hz has the effect 
of stretching the "distance" of any particular synapse to the soma by a factor of 
about 3, on average. Thus, while a distal synapse has an associated L value of 
about 2.6 at 2 Hz it shrinks to 1.2 if all network activity is shut off, while for a 
synapse at the tip of a basal dendrite, L shrinks from 0.7 to 0.2. In fact, the EPSP 
induced by a single excitatory synapse at that location goes from 39 to 151 /V, a 
decrease of about 4. Thus, when the overall network activity is low, synapses in the 
superficial layer of cortex could have a significant effect on somatic discharge, while 
having only a weak modulatory effect on the soma if the overall network activity is 
high. Note that basal dendrites, which receive a larger number of synapses, stretch 
more than apical dendrites. 
3.3 Temporal integration 
That the synaptic background activity can also modify the temporal integration 
behavior of the cell is demonstrated in fig. 3. At any particular background fre- 
quency f, we compute the minimal number of additional excitatory synapses (at 
greak -- 0.5 nS) necessary to barely generate one action potential. These synapses 
were chosen randomly from among all excitatory synapses throughout the cell. We 
compare the case in which all synapses are activated simultaneously (solid line) 
with the case in which the inputs arrive asynchronously, smeared out over 25 msec 
(dashed line). If f = 0, it requires 115 synapses firing simultaneously to generate 
a single action potential, while 145 are needed if the input is desynchronized. This 
small difference between inputs arriving synchronized and at random is due to the 
long integration period of the cell. 
If the background activity increases to f - I Hz, 113 synchronized synaptic 
inputs ..spread out all over the cell--are sufficient to fire the cell. If, however, 
the synaptic input is spread out over 25 msec, 202 synapses are now needed in 
order to trigger a response from the cell. This is mainly due to the much smaller 
value of rm relative to the period over which the synaptic input is spread out. Note 
48 Bernander, Koch, and Douglas 
that the difference in number of simultaneous synaptic inputs needed to fire the 
cell for f - 0 compared to f = I is small (i.e. 113 vs. 115), in spite of the more 
than five-fold decrease in somatic input resistance. The effect of the smaller size of 
the individual EPSP at higher values of f is compensated for by the fact that the 
resting potential of the cell has been shifted towards the firing threshold of the cell 
(about -49 rnV). 
E 0 
E 
.... Unsynchronized input 
Synchronized input 
Background equency (Hz) 
Figure 3: Phase detection. 
A variable number of excitatory synapses were fired superimposed onto a constant 
background frequency of I Hz. They fired either simultaneously (solid line) or 
spread out in time uniformly during a 25 msec interval (dashed line). The y axis 
shows the minimum number of synapses necessary to cause the cell to fire. 
3.4 NMDA synapses 
Fast excitatory synaptic input in cortex is mediated by both AMPA or non-NMDA 
as well as NMDA receptors (Miller et al., 1989). As opposed to the AMPA synapse, 
the NMDA conductance change depends not only on time but also on the post- 
synaptic voltage: 
G(V,t) = 1.05. 
1+ . [Mg+]. e-?V' 
(1) 
where '] = 40 msec, '2 = 0.335 rnsec, r = 0.33 mM -1, [Mg2+] : 1 rnM, 
7 = 0.06 mV -1. During spontaneous background activity many inputs impinge 
on the cell and we can time-average the equation above. We will then be left with 
a purely voltage-dependent conductance. 
We measured the somatic input resistance, Ri,, by injecting a small current pulse in 
the soma (fig. 4) in the standard model. All synapses fired at a 0.5 Hz background 
frequency. Next we added 4,000 NMDA synapses in addition to the 4, 000 non- 
Network activity determines spatio-temporal integration in single cells 49 
NMDA synapses, also at 0.5 Hz, and again injected a current pulse. The voltage 
response is now larger by about 65%, corresponding to a smaller input conductance, 
even though we are adding the positive NMDA conductance. This seeming paradox 
depends on two effects. First, the input conductance is, by definition, 'd/ _ G(V)+ 
-DA' (V - Er), where G(V) is the conductance specified in eq. (1). For the 
synapse this derivative is negative below about -35 reV. Second, due to the 
excitation the membrane voltage has drifted towards more depolarized values. This 
will cause a change in the activation of the other voltage-dependent currents. Even 
though the summed conductance of these active currents will be larger at the new 
voltage, the derivative  will be smaller at that point. In other words, activation 
of NMDA synapses gives a negative contribution to the input conductance, even 
though more conductances have opened up. 
Next we replaced 2,000 of the 4,000 non-NMDA synapses in the old model with 
2,000 NMDA synapses and recomputed the input resistance as a function of synap- 
tic background activity. The result is overlaid in figure la (dashed line). The curve 
shifts toward larger values of Jin for most values of f. This shift varies between 
50 % - 200 %. The cell is more excitable than before. 
-60 
-61 
-62 
-64' 
-65' 
-66 
0 
200 400 600 800 1000 
t (msec} 
Figure 4: Negative input conductance from NMDA activation. 
At times t = 250 msec and t = 750 msec a 0.05 nA current pulse was injected 
at the soma and the somatic voltage response was recorded. At t = 500 msec, 
one NMDA synapse was activated for each non-NMDA synapse, for a total of 8,000 
excitatory synaptic inputs. The background frequency was 0.5 Hz for all synapses. 
4 DISCUSSION 
We have seen that parameters such as R/., r., and L are not static, but can 
vary over about one order of magnitude under network control. The potential 
computational possibilities could be significant. 
50 Bernander, Koch, and Douglas 
For example, if a low-contrast stimulus is presented within the receptive field of 
the cell, the synaptic input rate will be small and the signal-to-noise ratio (SNR) 
low. In this case, to make the cell more sensitive to the inputs we might want to 
increase R/,. This would automatically be achieved as the total network activation 
is low. We can improve the SNR by integrating over a longer time period, i.e. by 
increasing rm. This would also be a consequence of the reduced network activity. 
The converse argument can be made for high-contrast stimuli, associated with high 
overall network activity and low R/, and rm values. 
Many cortical cells are tuned for various properties of the stimulus, such as orien- 
tation, direction, and binocular disparity. As the effective membrane conductance, 
Gin, changes, the tuning curves are expected to change. Depending on the exact 
circuitry and implementation of the tuning properties, this change in background 
frequency could take many forms. One example of phase-tuning was given above. 
In this case the temporal tuning increases with background frequency. 
Acknowledgements 
This work was supported by the Office of Naval Research, the National Science 
Foundation, the James McDonnell Foundation and the International Human Fron- 
tier Science Program Organization. Thanks to Tom Tromey for writing the graphic 
software and to Mike Hines for providing us with NEURON. 
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