Correlated Neuronal Activity: High- and Low-Level Views - Computational Neuroscience: A Comprehensive Approach - page 348

phases are averaged out, making the correlation flat everywhere except in the central
region. As in Figure 12.1, compensatory adjustments were made so that average
firing rates remained approximately the same; in this case the external excitatory
drive was slightly decreased as the connection strengths increased.
Figures 12.2d and 12.2e show that even such a simplified network may have quite
complex dynamics. Parameters in Figure 12.2d were identical to those of Figure
12.2b, except for two manipulations. First, for recurrent excitatory synapses only,
the synaptic time constant was increased from 3 to 10 ms; and second, to compen-
sate for this, the synaptic conductances were multiplied by 3/10. This generated
approximately the same firing rates and also preserved the average recurrent conduc-
tance level. However, as a consequence of these changes the correlations between
postsynaptic spikes almost disappeared. Thus, the tendency to fire in phase is much
larger when the characteristic timescales for excitatory and inhibitory synaptic events
are the same. This is reminiscent of resonance.
Figure 12.2e illustrates another interesting phenomenon. In this case the timescales
of both excitatory and inhibitory recurrent synapses were set to 10 ms, while the char-
acteristic time of all external input signals stayed at 3 ms. The mean conductance
levels, averaged over time, were the same as in Figure 12.2c, so the connections were
relatively strong. Now the peak in the cross-correlation histogram (Figure 12.2e) is
much wider than expected, with a timescale on the order of hundreds of milliseconds.
Such long-term variations are also apparent in the spike raster and in the firing rate
trace. This is quite surprising: firing fluctuations in this network occur with a char-
acteristic time that is at least an order of magnitude longer than any intrinsic cellular
or synaptic timescale. Discussion of the underlying mechanism is beyond the scope
of this chapter, but in essence it appears that the network makes transitions between
two pseudo steady-state firing levels, and that the times between transitions depend
not only on cellular parameters but also on how separated the two firing levels are.
In any case, a key point to highlight is that, in all examples we have presented, the
cross-correlation histograms show a common feature: a central peak with a shape
that resembles a double-exponential.

That is, the correlation function can be de-

scribed as

C max exp − |

|
t corr

t

C ( t )= , (12.2)

where t is the time lag and t corr , which determines its width, is the correlation time.
This function is related to many types of random processes [87, 42]. Indeed, below
we will use it to characterize the total input that drives a typical cortical neuron.
So far, we have looked at a variety of correlation patterns that a network may
display. Next, we take the point of view of a single downstream neuron that is driven
by this network. From this perspective we return to an important question posed
in the introduction: how does the response of a postsynaptic neuron depend on the
full set of correlated spike trains that typically impinge on it? Equation 12.2 will be
used as a rough characterization of those correlations. The answer will be presented
in two parts. First we will discuss some of the main factors determining whether
input correlations have an impact on the postsynaptic response, and roughly to what