Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 56

in the right responsivity range for conveying useful in-
formation, and it makes each neuron's responsivity de-
pendent on other neurons, which has many important
consequences as one can imagine from the above ex-
plorations.
Finally, we can peek under the hood of the simu-
lator to see how events are presented to the network.
This is done using something called a process ,whichis
like a conductor that orchestrates the presentation of the
events in the environment to the network. We interact
with processes through process control panels (not to
be confused with the overall simulation control panels;
see section A.10.1 in appendix A for more details).

expressions and manipulations. The mathematical lan-
guage of probability and statistics is particularly appro-
priate for describing the behavior of individual neurons
as detectors. The relevant parts of this language are in-
troduced here. We will see that they provide an inter-
esting explanation for the basic form of the point neu-
ron activation function described previously, which is
the main objective of the formalization provided here.
Note that there are other more complicated ways of an-
alyzing things that provide a more precise definition of
things like the weight values and the net input, which
we are less concerned about here (e.g., Hinton & Se-

jnowski, 1983; McClelland, 1998).
The most relevant branch of statistics here is hypoth-
esis testing . The general idea is that you have a hy-
pothesis (or two) and some relevant data or evidence ,
and you want to determine how well the hypothesis is
supported by these data. This provides an alternate lan-
guage for the same basic operation that a detector per-
forms: the data is the input, and the processing per-
formed by the detector evaluates the hypothesis that the
thing (or things) that the detector detects are present
given the data (or not). We can identify two important
hypotheses for a detector: 1) the hypothesis that the de-
tected thing is really “out there,” which we will label

To see the process control panel for this simulation,
press View on the detect_ctrl overall control panel
and select PROCESS_CTRL .
The Epoch_0 EpochProcess control panel will ap-
pear. The Run and Step buttons on our overall con-
trol panel work by essentially pressing the correspond-
ing buttons on this process control panel. Try it. The
ReInit and NewInit buttons initialize the process
(to start back at digit 0 in this case) — the former reuses
the same random values as the previous run (i.e., start-
ing off with the same random seed), while the latter
generates new random values. You can also Stop the
process, and GoTo a specific event. Although the simu-
lation exercises will not typically require you to access
these process control panels directly, they are always
an option if you want to obtain greater control, and you
will have to rely on them when you make your own sim-
ulations.

; and 2) the nu ll hypothesis that this thing is not out
there, labeled h . What we really want to do is com-
pare the relative probabilities of these hypotheses being
true, and produce some output that reflects the extent
to which our det ec tion hypothesis ( h ) wins out over the
null hypothesis ( h ) given the current input.
The result after the detailed derivation that follows
is that the probability of h given the current input data

Go to the PDP++Root window. To continue on to
the next simulation, close this project first by selecting
.projects/Remove/Project_0 . Or, if you wish to
stop now, quit by selecting Object/Quit .

(which is written as P (hjd) ) is a simple ratio func-
tion of two other functions of the relationship between
th e h ypotheses and the data (written here as f(h; d) and

):

2.7 Hypothesis Testing Analysis of a Neural

Detector

(2.22)

[Note: This section contains more abstract, mathe-
matical ideas that are not absolutely essential for un-
derstanding subsequent material. Thus, it could be
skipped, but at a loss of some depth and perspective.]
One of the primary ways of expressing and using
a computational level description is via mathematical

Thus, the resulting probability is just a function of how
strong the support for the detection hy pothesis h is over
the support for the null hypothesis h . This ratio func-
tion may be familiar to some psychologists as the Luce
choice ratio used in mathematical psychology models