Individual Neurons - Computational Explorations in Cognitive Neuroscience - page 63

probabilistic assumptions about the weight values), but
the particular assumptions of our approach end up being
more applicable to what the biological neuron seems to
be doing.

pothesis that the neuron is detecting and inhibition with
the null hypothesis (as we assumed above); 2) the frac-
tion of channels open gives the corresponding likeli-
hood v alue ( g e ￿ P (djh) for excitation and g i ￿

for inhibition), which is essentially what we as-
sumed already by computing the likelihood as a func-
tion of the sending activations times the weights; 3) the
baseline conductance levels, g e and g i ,representthe
prior probability values, P (h) and P (h) , respectively.
Note that the ratio form of the equation ensures that any
uniform linear scaling of the parameters cancels out.
Thus, even though the actual values of the relevant bi-
ological parameters are not on a 0 ￿ 1 scale and have
no other apparent relationship to probabilities, we can
still interpret them as computing a rationally motivated
detection computation.
The full equation for V m with the leak current (equa-
tion 2.9) can be interpreted as reflecting the case where
there are two different (and independent) null hypothe-
ses, represented by inhibition and leak. As we will
see in more detail in chapter 3, inhibition dynamically
changes as a function of the activation of other units in
the network, whereas leak is a constant that sets a basic
minimum standard against which the detection hypoth-
esis is compared. Thus, each of these can be seen as
supporting a different kind of null hypothesis. Although
neither of these values is computed in the same way as
the null likelihood in our example (equation 2.35), this
just means that a different set of assumptions, which can
be explicitly enumerated, are being used.
Taken together, this analysis provides a satisfying
computational-level interpretation of the biological ac-
tivation mechanism, and assures us that the neuron is
integrating its information in a way that makes good sta-
tistical sense.

2.7.3 Similarity of V m and P (hjd)
Finally, we can compare the equation for the equi-
librium membrane potential (equation 2.9 from sec-
tion 2.4.6) with the hypothesis testing function we just
developed (equation 2.31). For ease of reference the
equilibrium membrane potential equation is:

(2.37)

The general idea is that excitatory input plays the role
of something like the likelihood or support for the hy-
pothesis, and the inhibitory input and leak current both
play the role of something like support for null hypothe-
ses. Because we have considered only one null hypothe-
sis in the preceding analysis (though it is easy to extend
it to two), we will just ignore the leak current for the
time being, so that the inhibitory input will play the role
of the null hypothesis.
To compare the biological equation with our hypoth-
esis testing equation, we need to use appropriate val-
ues of the reversal potentials, so that the resulting mem-
brane potential lives on the same 0 ￿ 1 range that prob-
abilities do. Thus, we will assume that excitatory input
drives the potential toward 1 (i.e., E e = 1 ), and that
the inhibitory (and leak) currents drive the potential to-
ward 0 (i.e., E i = E l = 0 ). This makes sense con-
sidering that complete support (i.e., only excitation) for
the hypothesis should result in a probability of 1, and
complete absence of support (i.e., no excitation, all in-
hibition/leak) should result in a probability of 0. If we
substitute these values into the biological equation, we
get the following relationship:

2.8 The Importance of Keeping It Simple

So far, we have articulated a very simple view of the
neuron as a detector, and shown how this is consistent
with its biological properties, and with a mathematical
description of neural function based on hypothesis test-
ing. Thus, this detector model, together with the impor-
tance of graded processing and learning, provide a basis
for thinking that the neuron performs a relatively simple

P (djh)P (h)+P (djh)P (h)
(2.38)

The equations are identical under the following as-
sumptions: 1) excitation can be equated with the hy-