[Martin Taylor 2004.05.08.2121]
[From Bill Powers (2004.05.08.1832 MST)]
I'm mystified by this concept of perceptions as basins of attraction.
Yes, it's not easy to come to terms with. It took me as long to come
to terms with that as with PCT, and you know from experience how long
that took me
I started with a perceptron-based background, the same as you. I
think I tried my first perceptron-like simulation on a machine called
a Royal-McBee LGP-30 or something like that, around 40 years ago. So
I have been intellectually committed to the one-line one-signal kind
of thinking for a long time. Thinking in terms of system dynamics
needs a whole different mind-set, just as does thinking about
perceptual control when you've spent your life looking at the
responses to stimuli.
It
seems to me that this might apply in some relatively rare situations, and
to perceptions that are categorical (it's either a "bump" or a "dent" but
nothing in between). But how would you create a perception that varies
smoothly from one state to another under feedback control? That wouldn't
seem amenable to treatment as something being attracted to a basin, or
jumping from one basin into another. And what about perceptions of a
logical nature? If A is to the left of B (on a flat surface) and B is to
the left of C, then B must be "between" A and C. We perceive that this is a
logical fact, yet I don't see now an attractor basic could represent
this fact.
OK. I think Peter has been muddying the waters somewhat by using
imprecise language.
I'm going to say things I know that you already know very well, but I
have to repeat them to make sure we are starting from the same
premises.
Attractor basins are not a property of the behaviour of anything at
any moment. They are a property of the dynamic possibilities inherent
in a system, any kind of system that isn't totally static. A simple
control system has a dynamic. In the absence of any disturbance, what
will the perceptual signal be doing, given that it started at
such-and-such a value, the reference value was thus-and-so, and the
derivative of the perceptual signal was this much? You can describe a
curve that will give the entire future history of the perceptual
signal value.
When you plot the future history of the perceptual signal in a space
of its value and its time derivative, the line you get is called an
"orbit". The orbit may go round and round repetitively. That's a
periodic oscillator. It may go smootly toward a point and stay there.
The point is called a fixed-point attractor. It may spiral inward if
the perceptual signal shows a damped oscillation, arriving at a
central point after an infinite time. That point also is a
fixed-point attractor. It may spiral outward to infinity, in an ever
growing oscillation. The "point at infinity" (in all directions (!))
is a fixed-point attractor, too, though that seems odd since the
signal is getting infinitely big.
So, we have an orbit that starts with the perceptual signal being at
a point in the phase space and follows some path through the phase
space as we wait an infinite time to see where it will go. Let's make
another orbit, by starting the perceptual signal at som other place
in the phase space. One thing we can say for sure: this new orbit
will NEVER cross the first one. Why is that? because the entire
behaviour of the signal is specified by its location in the phase
space, and if the new orbit had one point in common with the old one,
it would follow the old path for all future time.
If the new orbit eventually arrives at the same attractor as the old
one, whether the attractor be a fixed point or an osciallator, then
the two orbits are in the same basin of attraction. If they arrive at
different attractors, they are in different basins of attraction.
When you have traced orbits starting from all possible points in the
phase space, some will be in one basin of attraction, some in
another. The totality of orbits specifies the dynamical possibilities
of the system, but rather than trying to keep track of an uncountable
infinity of orbits, it is easier to keep track of a (usually) finite
number of attractors and their attractor basins.
You will note that I haven't mentioned chaos, fractals, or strange
attractors yet. They enter the picture MUCH later, but they are
important.
Now let's think about the questions you asked. We can't answer them
all at this stage, but we can make a start.
(1) But how would you create a perception that varies smoothly from
one state to another under feedback control?
This question practically answers itself, in that the "perception" is
just a value, which varies, smoothly or in jumps, depending on the
nature of the underlying dynamic. If it's a continuous variable, it
varies smoothly. If the system you are analyzing includes the entire
control loop, but not the disturbance source, then the orbits within
the attractor basin describe the way the control system brings the
perceptual signal to its reference value (which in this case would be
the value of the fixed point attractor to which the basin belongs).
Move the reference signal value, and the whole system of orbits
changes so that they approach the new attractor. The reference value
IS the attractor, for the simple control system.
Now add a disturbance. What does this do? It moves the perceptual
signal along some path that has nothing to do with the system of
orbits that define the attractor basin, but the dynamics of the
system still work to move the signal back toward its attractor, and
would do so along one of the orbits, were the disturbance to cease.
So long as the disturbance continues, so the perceptual signal state
will be represented by a point in the attractor basin, but it won't
be moving along the orbits that define the basin.
Perhaps more plainly, under the influence of the disturbance, the
perceptual signal does not change its nature. It just changes its
value.
Now let's think of a more complex dynamic, in which there are, say,
two attractor basins (two ECUs in conflict, for example, in which the
gain functions were appropriately nonlinear so that one or the other
would win if it gained sufficient initial advantage). So long as the
disturbance is small enough or appropriately directed, the signal
will stay in the sam attractor basin, but a big enough disturbance
that moves it in the direction of the boundary of the other basin
might bump it completely out of its original basin of attraction and
into the other.
There's no need for the dynamic to be describing a control system. As
I said before, any system that isn't inherently static will do. For
the two-attractor example, a flip-flop would serve very well. One
basin of attraction would be represented by the flip-flop being
committed to a "zero" state, the other to a "one" state. And that
leads to the possibility of identifying different basins of
attraction with different categorical perceptions.
It looks to me as though the attractor concept of perception was created to
explain certain special cases of perception, but that it leaves most
perceptual phenomena unexplained.
I don't know whether I written so much as to confuse or whether it
helps. But I hope you can see the glimmerings of starlight at the end
of the tunnel? None of these constructs "explain" perceptual
phenomena. The mechanisms do the explaining. Dynamical descriptions,
though, often provide conceptually useful ways of looking at
phenomena, because a lot of the dynamical phenomena have common
properties across many different kinds of situation. What you learn
in one situation can then be applied in another, and not just
metaphorically.
What I expect is not yet clear from what I have written is under what
circumstances a description of the system dynamics is likely to be
useful. The answer to that question really depends on your fluency
and ease with that kind of description. The more familiar you are
with it, the more circumstances there will be in which you prefer it.
If you find it an awkward mode of description, you won't find it
useful very often. But no matter how familiar or unfamiliar you are
with it, there will be times when it's the only reasonable way of
looking at the situation--rather like using Fourier or laplace
transforms, as opposed to using the time domain. Sometimes, they are
the only reasonable way.
Martin