[From Bill Powers (2008.12.20.0725 MST)]
Martin Taylor 2008.12.19.23.5 –
“Frequency” of
impulses is verbally different, but conceptually indistinguishable from
the inverse of average inter-impulse interval. You seem to use it here to
assert that “frequency” matters, not inter-impulse interval.
When you said that Atherton et al. plotted frequency and I
demurred, it was because they didn’t plot the inverse of average
inter-impulse intervals, they plotted the inverse of individual
inter-impulse intervals.
And I would say that this gives a maximally noisy frequency plot. But if
you find physical effects that are related to those rapidly varying
inter-impulse intervals in a simple and relatively linear way, I would
certainly not say you should use a frequency measure instead. However,
you’re using averaged measures from the start when you speak of the 10%
and 90% levels – those can be measured only over many impulses.
It makes no difference to me whether you write your equations using p or
q if you have defined the relationship between p and q. If p = 1/q, and I
prefer q, I can always go through the equations and substitute q for p.
It’s the same equation either way – except, as noted, for the ease of
solving it (and, come to think of it, except for singularities).
I don’t want to be a crotchety old reactionary who rejects something just
because he doesn’t know a lot about it. But my only means of judging
information theory is to compare it with what I know, which is the
electronic circuitry associated with what you call information channels
and what I call signal pathways, and with my small acquaintance with
neurology. So far I haven’t seen any result from information theory that
couldn’t be replicated using suitable passive or active filters – and in
fact, that seems to be the only way to put information theory into
practice. This has given me the impression that information theory is an
unnecessarily long way around the barn. If the door is just to your left,
why go all the way around to your right to get to it?
I have never had a problem with
using frequency of nerve impulses as a working hypothesis about the
important parameter for neural signals. I do have an issue with asserting
that it is, and especially with asserting that it is the only parameter,
thereby denying that other features, of the signal in one nerve fibre and
of its relation with signals in other nerve fibres, might
matter.
Take a look at this:
[
http://en.wikipedia.org/wiki/Postsynaptic_potential
](http://en.wikipedia.org/wiki/Postsynaptic_potential)In it, we find the following:
···
===================================================================
Algebraic summation
Postsynaptic potentials are subject to
summation, either spatially or temporally.
Spatial
summation: If a cell is receiving input at two synapses that are
near each other, their postsynaptic potentials add together. If the cell
is receiving two excitatory postsynaptic potentials, they combine so that
the membrane potential is depolarized by the sum of the two changes. If
there are two inhibitory potentials, they also sum, and the membrane is
hyperpolarized by that amount. If the cell is receiving both inhibitory
and excitatory postsynaptic potentials, they can cancel out, or one can
be stronger than the other, and the membrane potential will change by the
difference between them.
Temporal
summation: When a cell receives inputs that are close together in
time, they are also added together, even if from the same synapse. Thus,
if a neuron receives an excitatory postsynaptic potential, and then the
presynaptic neuron fires again, creating another EPSP, then the membrane
of the postsynaptic cell is depolarized by the total of the EPSPs.
==========================================================================
The relationship of which you speak does not exist between nerve fibers,
but inside dendrites and cell bodies where excitatory or inhibitory
post-synaptic potentials generated by neurotransmitters can interact.
Here, indeed, the interactions are always happening impulse by impulse,
on a very rapid time scale where microseconds and nanoseconds
matter. There can be, in “electrical” neurons with very small
cell-wall capacitance, summation effects so that two impulses can cause
an immediate output impulse if the effects coincide within a millisecond
or so. But in most cells, the summation effects are smaller and spread
over a much longer time, so we have to talk about average ion
concentrations, with a single impulse causing only a very small change in
the excitatory post-synaptic potential, or EPSP. There are neurons with
such a large cell-membrane capacitance that after a steady input signal
suddenly disappears, the output frequency of impulses decreases
exponentially over five or ten seconds or longer – the so-called
“afterdischarge” in the innocent language of pre-electronic
neurology.
I’m sure you could represent this effect as a declining probability of
firing, but to deal with it at that level of abstraction is to lose the
connection with the lower-level aspects of the physical model of reality.
And since temporal averaging is happening inside the dendrites and cell
body, the discrete nature of the incoming impulses is lost before the
mechanisms of generating an output impulse come into play – any logical
calculations refer to imaginary processes, not what is actually happening
inside the cell to the best of our knowledge. There are few types of
neurons in which there is any one-to-one correspondence or
synchronization of incoming and outgoing impulses.
These observations are part of my resistance to the use of information
theory in models like ours. There may be analyses in which
information theory leads to conclusions that could not be reached any
other way, but I doubt that they apply at the engineering levels of
modeling.
So those hypotheses are what I
think I prefer. It is possible for evidence to choose between your
preference and mine by showing that neither relative impulse timing nor
individual inter-impulse intervals affect anything downstream (or that
they do), but I know of no such data at the moment, not being a
neurophysiologist. In the absence of data we are both entitled to sustain
our own preferences.
I think we can agree that the timing of impulses arriving at synapses has
very strong and immediate effects on downstream processes, and that it
makes no difference whether we measure the occurrences of those impulses
in terms of the interval or the reciprocal of the interval. How we
measure them makes no difference in the processes. When we look at those
processes, as far as they have been established in the physical model, we
see that there is temporal averaging involved in them and that there
definitely are processes going on between arrivals of impulses. This
physical model works on a microsecond-by-microsecond scale where neither
interval nor frequency of impulses exists. When an impulse arrives, the
EPSP or IPSP suddenly changes, and when another impulse arrives it
changes again. Between impulses the membrane potential declines at a rate
determined by metabolism, diffusion, and recombination as well as
leakiness across the membrane capacitance. It makes no difference when
the impulses arrive; we can compute what will happen in any case. An EPSP
rises suddenly and decays slowly until the next impulse arrives and it
rises again. The resulting degree of depolarization determines how long
after an impulse occurs the next impulse will occur. There is nothing
left for information theory to explain at this level.
Best,
Bill P.