It wants the code, of course, so that it can *decode* signals and see
what the rest of the world is like. The task confronting a homunculus residing
in the brain, having access only to the incoming trains of spikes, would be to
guess --- that is, infer --- what the most likely state of the world is,
*given* that the spike train is thus-and-so. (Shades of Quine's
"sensory irritations.") There are no homunculi, but this is, in effect, the
problem one part of the brain must solve to make use of information coming from
another. If the code is well-designed, this decoding operation will be simple
and reliable, and ideally linear. It turns out, after applying Bayes's rule, that lots of neurons --- all the kinds the
authors have been able to examine in the necessary detail, or anyhow all those
they care to talk about --- can be nearly linearly decoded. More precisely:
pass the spike-train through a certain well-prescribed linear filter, the
"first Wiener kernel," and you get a good estimate of the mean of the stimuli
that will produce that spike-train. Adding quadratic terms to the filter
doesn't improve the fit much, so the first Wiener kernel *is* the neural
code for that neuron, or very nearly so, since it tells us how to decipher the
spikes.

Now that we can read the code, that we know *what* the spikes tell
us, we'd like to know *how much* they are telling us, how much
information a spike-train, or even a single spike, conveys. So long as we're
rummaging around in Norbert Wiener's toolkit for doohickeys like the first
kernel, we might as well check if there's anything there which would help with
this problem: and of course there is, namely information theory, priority for
which goes, however, to Shannon. (For that matter, Kolmogorov has strict
priority on the optimal linear estimator for time-series. Uncle Norbert was
unusually given to being a co-discoverer.) Given that a train averages so many
spikes per second, it's a fairly straightforward exercise to compute the upper
bound on how much information the train can carry, per second and per spike.
These limits were first calculated by MacKay and McCulloch in 1954, and can't
really be subverted --- the fewer constraints there are on possible patterns of
spiking, the more distinguishable spike-trains, and so each train carries (on
average) more information; by assuming only that the trains need to be
consistent with the average rate, and so ignoring the refractory period and
everything else, we're considering the least constrained ensemble of
spike-trains, and so getting the physical upper limit on the rate at which the
trains can convey information. The natural next question is to ask how close
real neurons and their spikes get to this limit. Here there is a difficulty,
because there is, surprisingly, no good way in general of measuring the rate of
information transfer in most actual, physical channels. Our authors have,
however, an ingenious way of underestimating it, using the measured statistics
of the spikes and the first Wiener kernel. (Since this is an underestimate of
the true rate of information transfer, and the MacKay-McCulloch limit is
certainly an overestimate of what's physically possible, we know the true
efficiency of the neural code is always greater than our estimate.) They then
proceed to calculate the baud rate, bits per spike, and efficiency of various
sensory and intermediate neurons; eye of newt is not *exactly* on the
list, but retinal ganglion cells in the salamander (species unspecified) are.
(A recent article adds two more types of neurons to the
collection.) The baud rate varies from cell type to cell type from just under
50 bits/second to over 300 bits/second; the information per spike, however, is
rather more constant, 2-3 bits/spike. The lowest estimated coding efficiency
is 20% (for those salamander cells); most are closer to 50%. For the one case
where naturalistic stimuli were provided, rather than just things which are
easy to cobble up in the lab, the efficiency was about 90%. In other words:
even when they're not doing the job they're designed for, many neurons come
within a factor of two of the fundamental physical limits on their performance;
they probably do much better on the stimuli for which they've adapted.

From here we got to some considerations on the reliability of neural
computation. There's a classical theory, going back to von Neumann, of how to
make reliable computers from unreliable parts, and it's been tacitly assumed,
for the most part, that it applies to the nervous system. But those coding
efficiencies should make us leery about thinking that neurons are especially
flaky, and it seems that in some cases individual neurons are as reliable as
the entire organism. The most impressive phenomena where this is not the case
is hyper-acuity, where the organism is able to respond to differences which are
smaller than what individual sensory cells should be able to detect, e.g.,
"echo-locating bats can apparent resolve jitter in the arrival time of their
echoes with a precision of 10 *nano*seconds" (p. 221, their italics).
The trick here is indeed to use lots of neurons, but to discriminate between
differences in patterns of activation from different stimuli, rather than
averaging all of them. (As the authors point out, this is the same trick as
the one that lets us see more than three colors!)

I'd have liked more at this point about how the neural code gets used in network computations, or, conversely, how to use the tricks we've learnt for breaking the code to get a handle on what the networks are computing and how, but this the authors largely reserve for future research (they have great hopes for the hippocampus and the motor cortex), along with a better understanding of the statistics of natural stimuli. They close with a "homage to the single spike," as a trustworthy and reliable carrier of a substantial amount of information, sometimes even responsible by itself for conscious sensations. "The individual spike, so often averaged in with its neighbors, deserves more respect."

This is obviously damn good science, and would make Spikes
worth reading though its authors wrote like literary critics. Thankfully, they
don't, and the book is a positive pleasure to read, clear, unpedantic, aware of
the history on which it builds, and possessed of a restrained but noticeable
and welcome sense of humor (e.g. citing George Carlin's "Seven Words You Can
Never Say on Television"). Between the response functions, the fondness for
working in the frequency domain rather than the time domain, and the references
to Horowitz and Hill, physicists, electrical engineers, probabilists, etc. will
feel quite at home; for the comfort of the less mathematically inclined among
the biologists, there are ninety pages of "mathematical asides," together
forming a decent course in information theory and time-series analysis
accessible to anyone who remembers how to integrate, and there is not a single
*model.* This is a really fun book which lays new paths across a very
important field, and is accessible to almost anyone with an education in the
natural sciences; in a word, a treasure.

xviii + 395 pp., diagrams, references, index

Neuroscience / Probability and Statistics

Currently in print as a hardback, ISBN 0-262-18174-6, US$95 [Buy from Powell's], and as a paperback, ISBN 0-262-68108-0, US$38 [Buy from Powell's]

11-12 May 1998 (minor revisions 13 Oct. 1999)