Correlated Neuronal Activity: High- and Low-Level Views - Computational Neuroscience: A Comprehensive Approach - page 358

time may have a very strong impact on the response of a postsynaptic neuron [69].
This is an interesting observation because little is known about the dynamic role of
this parameter.

12.6.3 Quantitative relationships between input and output

The solution to the non-leaky model of Equation 12.6 consists in the moments of T ,

, T 2 and so forth. For each of these moments there are three sets of analytic
expressions, because details of the solutions depend on the relative values of m and
s.

T

Here we only discuss the expressions for the average interspike interval T ,

which is the inverse of the mean firing rate, but T 2 and therefore the CV ISI can also
be obtained in closed form [69].
When m > s,

V q − V reset

m

T = . (12.7)

In this case there is a strong positive drift toward threshold. Even when Z is equal
to 1 the total input is positive; in other words, the voltage gets closer to threshold

in every time step, whether the fluctuating component is positive or negative. The
mean firing rate behaves as if the input were constant and there were no fluctuations.
This can be seen in Figure 12.6, which plots the mean firing rate and the CV ISI of
the model neuron as a function of s for various combinations of the other two input
parameters. The values of m are indicated in each column, and the three curves
in each plot correspond to t corr equal to 1, 3 and 10 ms, with higher correlation
values always producing stronger responses and higher variability. Continuous lines
and dots correspond to analytic solutions and simulation results, respectively. Notice
how, when m=0.02, the firing rate stays constant for s below 0.02, although the
variability increases most sharply precisely within this range.
When m=0,

−

V q V reset

2t corr s 2

2 ( V q − V reset )

s

−

T = + . (12.8)

Clearly, the average interspike interval decreases with both t corr and s.Inth s
case there is no drift, no net displacement; the voltage advances toward thresh-
old when Z =+1 and retreats toward the barrier when Z

1. Under these condi-
tions the neuron is driven exclusively by fluctuations. The middle column of Figure
12.6corresponds to this regime. As can be seen, the variability of the neuron also
increases monotonically with s and t corr .
Finally, when m

= −

≤ s,

V q − V reset

m

2

T = + t corr ( c − 1 ) ( exp ( − a V q ) − exp ( − a V reset )) , (12.9)

wherewehavedefined

s
m

c ≡

1
mt corr (

a ≡

) .

(12.10)

c 2 − 1