Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 51

2.6.2 The Activation Output You should observe a reasonably close fit between

the final values of act_eq for SPIKE with that of
NOISY_XX1 . However, with smaller g_bar_e val-
ues (e.g., .38), the NOISY_XX1 is somewhat below the
spiking act_eq . Achieving a much closer fit between
spiking and rate coded activations such as that displayed
in figure 2.15 requires different parameter values that
are not otherwise appropriate for this exploration. This
is due to the aliasing effects of discrete-time updat-
ing (i.e., coarse-grained digitization effects, like when
trying to display a photograph on a low-resolution dis-
play), which the spiking model is very sensitive to.

In this section, we explore the way in which the unit
computes its activation output. The main objective is
to understand the relationship between the spiking and
rate-code activation functions.

We will use the same

project as the previous section.

Press Defaults to start out with default parame-

ters.
From the previous section, we know that changing
the level of excitatory input will affect the membrane
potential, and the resulting rate coded activation value.
Now let's explore this relationship in the spiking activa-
tion function.

Change g_bar_e back to its default value of .4.,
and make sure act_fun is set to SPIKE .
An important aspect of spiking in real neurons is that
the timing and intervals between spikes can be quite
random, although the overall rate of firing remains pre-
dictable. This is obviously not evident with the single
constant input used so far, which results in regular fir-
ing. However, if we introduce noise by adding small
randomly generated values to the membrane potential,
then we can see some of this kind of effect, although
it is still not as dramatic as it would be with multiple
spiking inputs coming into the cell. Note that this ad-
ditional noise plays a similar role as the convolution of
noise with the XX1 function in the noisy XX1 function,
but in the case of the noisy XX1 we have a deterministic
function that incorporates the averaged effects of noise,
while here we are actually adding in the random values
themselves, making the behavior stochastic.

Set act_fun to SPIKE , and press Apply and then
Run .
Instead of the steady values during the input pre-
sentation period, you now see the oscillations caused
by the spiking mechanism (as we saw previously in
figure 2.12). Thus, as soon as the membrane poten-
tial crosses the threshold, the activation spikes, and the
membrane potential is reset (to a sub-resting potential of
0, reflecting the overshoot of the spiking mechanism).
Then, the potential climbs back up, and the process re-
peats itself.
The spacing between the spikes is inversely propor-
tional to the firing rate, but it can be hard to eyeball this
from the graph. Let's look at act_eq , the rate-code
equivalent spike-rate value as a function of the spike
train (see equation 2.18).

Click on the act_eq graph line (plotted in blue).

Change the variance of the noise generator
( noise_var in the control panel) from 0 to .005, and
press Apply and then Run .
It can be difficult to tell from a single run whether the
spike timing is random — the unit still fires with some
regularity.

Next, observe the effects of changing g_bar_e from
.4, first to .38 and then to .42.

Question 2.6 Describe and explain the effects on the
spike rate of decreasing g_bar_e to .38, and of in-
creasing it to .42.

Do many Run s on top of each other in the graph log.
Now you should see that the spike timing was actu-
ally so random that there is essentially a uniform dis-
tribution of spike times across these different runs (i.e.,
a spike could occur at any time step), but the rate code
equivalent activation ( act_eq ) nevertheless remained
relatively constant (i.e., it had only a few different val-
ues at the end of a run). This happens because the
precise time at which a spike fires depends greatly on

The empirically-measured rate-code equivalent for
the spiking activation function ( act_eq ) compares
fairly closely with the rate-code value computed di-
rectly as a function of the membrane potential (act for
NOISY_XX1 ), as we saw in figure 2.15.

To explore this relationship in the simulation, you
can switch between SPIKE and NOISY_XX1 for different
values of g_bar_e .