perceptions as attractor basins

[Martin_Taylor2]

[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

[Bill_Powers1]

[From Bill Powers (2004.05.09.0430 MST)]

Martin Taylor 2004.05.08.2121 --

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.

But this alone does not produce an attractor basin. If you consider
uncontrolled perceptions, this is like considering any kind of uncontrolled
variable, including the physical variables corresponding to the perception.
Given the value of the variable and the values of all its derivatives with
unlimited accuracy, one can indeed, mathematically, describe the entire
future history of the variable. But that concept fails when there is any
degree of uncertainty in the system, because derivatives amplify noise
relative to signal, and the future history becomes very rapidly
unpredictable as the time of prediction increases. It is this fact that put
the final quietus on the concept of a deterministic universe, as I
understand the arguments.

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.

An attractor basin with orbits can exist only when there is a central force
acting, described by something like a second-order differential equation,
which is not the case for uncontrolled perceptions. A closed orbit
describes the very special and unusual case of a periodic oscillator.

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.

There's something wrong with this reasoning. I could easily draw, in phase
space, an arbitrary curve that crossed previous parts of the curve as many
times as I pleased. This curve would obviously describe some time-plot of
the variable, though there would be no regular period of oscillation
involved. The case where one magnitude and rate of change implies a closed
curve is the very special case of an exactly periodic oscillator, which
exemplifies the behavior of very few physical or perceptual variables. The
picture you present involves no derivatives higher than the first.

I suppose you are going to say that introducing the higher derivatives
introduces, potentially at least, the phenomena of chaos. But let's not
forget that we're talking about perception here, not control, and that
without control, perceptions simply represent the state of the perceived
natural world and do not, ordinarily, oscillate. Look around you -- how
many things do you see in oscillation, whether periodic or almost-periodic?
At the moment, I see none at all. I see changes, of course, but I seldom
see any repeating combinations of position and velocity.What I see is a
world that remains close to one state for a while, as I type (with the
forms on the screen going through no periodic patterns that I can see), and
then undergoes large changes as I move to the kitchen for another cup of
tea and come back to typing again. If there is anything oscillatory or even
chaotic going on in all this, it is far from obvious to my eye.

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.

This is all obvious, but the question here is not whether perceptions are
brought toward reference levels in some dynamical fashion, but the nature
of the perceptual variables that are made to behave in this way. Is the
perception itself created by some sort of almost-periodic process? This is
what Peter Small appears to be saying. You're speaking of the behavior of a
perceptual variable through time, which of course does not explain how that
perceptual variable is produced, but only describes how it behaves. If you
want to speak of the behavior of a perceptual variable in relation to a
fixed reference level in terms of basins of attraction, I have no problem
with that. All controlled variables can be described in that way, although
a description involving higher derivatives would be more accurate in most
cases. But is any particular VALUE of the perceptual variable an
oscillatory phenomenon in itself? Is an uncontrolled perception, whose
orbits can cross in any conceivable manner, to be described that way?

I think the crux of the matter here is whether we are talking merely about
the behavior of a perceptual variable whose existence is taken for
granted, or about the processes by which one perceptual variable is
generated as a function of other variables: the signals, or the transfer
functions. It seems to me that in these discussions, particularly those of
Peter Small, this distinction is blurred almost out of existence.

Best,

Bill P.

[Martin_Taylor2]

[Martin Taylor 2004.05.09.10.15]

[From Bill Powers (2004.05.09.0430 MST)]

Martin Taylor 2004.05.08.2121 --

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.

But this alone does not produce an attractor basin.

I see that we have some prior misconceptions to unravel in areas
where I thought we already had a common understanding. I wrote the
above, and most of the message on which you are commenting, not as a
tutorial, but as a prelude to a tutorial.

If you consider
uncontrolled perceptions, this is like considering any kind of uncontrolled
variable, including the physical variables corresponding to the perception.

The concept of attractors and their basins has no connection with
whether the variable in question is controlled. It's a description of
the _system_ dynamic from the viewpoint of the variable being
described. If it's a control system, then it's the control system's
dynamic that is being probed. If not, provided you have access to ALL
the physical influences on the variable, you can still describe the
curve in question. You know the mechanism and its state, so you know
what it will do.

One curve does not describe an attractor basin, indeed. But the
totality of all possible curves that the mechanism can generate does.

Given the value of the variable and the values of all its derivatives with
unlimited accuracy, one can indeed, mathematically, describe the entire
future history of the variable. But that concept fails when there is any
degree of uncertainty in the system,

We aren't talking about a variable and all its derivatives. We are
talking about the current state of the variable and the mechanisms
that influence it. Yes, if there's uncertainty, the infinite future
will be uncertainly described. However, this matters only in the case
of chaotic systems, which I said were important, but intended to
leave until later, once the non-chaotic concepts were clear.

In a non-chaotic system (and in many chaotic systems, too), the
effects of initial uncertainty diminish over time, since the orbits
from neighbouring starting points approach one another (except for
the special case in which the neighbours are on opposite sides of an
attractor basin boundary).

An attractor basin with orbits can exist only when there is a central force
acting, described by something like a second-order differential equation,
which is not the case for uncontrolled perceptions. A closed orbit
describes the very special and unusual case of a periodic oscillator.

Not so. This statement really does suggest a major misconception that
has to be cleared up. You think of an "attractor" as a physical
central force. It doesn't mean that. It means a path through the
phase space toward which orbits tend over time. How that tendency
arises is unspecified. To know it, you have to know the mechanism
that is being described by the dynamic that includes the attractor in
question.

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.

There's something wrong with this reasoning. I could easily draw, in phase
space, an arbitrary curve that crossed previous parts of the curve as many
times as I pleased.

You could, but each time you did that, you would be describing the
behaviour of a different mechanism. Either that, or the system you
are describing isn't closed--it has influences from outside itself.
Remember, the behaviour of the curve is determined by the mechanism
you are describing, not by any instantaneous observation of state and
derivatives. So long as you don't change the mechanism and its
parameters, and there aren't any outside disturbances, no two curves
can cross, because the entire future of the curve is determined by
its present state and the mechanism that affects its state.

I intend to talk about apparently self-crossing orbital patterns in a
future posting. In fact, I already did, in my "Digression" in my
message [Martin Taylor 2004.05.08.1259]. It partially explains what I
think Peter is talking about when he says thing about jumping between
attractor basins.

The
picture you present involves no derivatives higher than the first.

I suppose you are going to say that introducing the higher derivatives
introduces, potentially at least, the phenomena of chaos.

No, you are going a long way from my thoughts. Chaos has nothing to
do with higher derivatives, and higher derivatives have nothing to do
with the attractor basin concept or with the future path of an orbit.

(Parenthetic interpolation: Chaos has to do with nonlinearity and
feedback, not with uncertainty or high-order derivatives).

But let's not
forget that we're talking about perception here, not control,

No. I used perception as an example signal in a conceptual universe I
know to be of interest to you. Perhaps that was a didactic mistake,
as it seems to bring in all sorts of things you know about real
perceptions. They just muddy the waters. We are talking now about
variables in any fully known mechanistic system.

and that
without control, perceptions simply represent the state of the perceived
natural world and do not, ordinarily, oscillate.

Yes, the perceptual signals you get are in some way influenced by
events outside the system being described. We call those
"disturbances", which were specifically excluded in the part of my
discussion you have so far alluded to in your comments. I dealt with
disturbances later in the same message.

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.

This is all obvious, but the question here is not whether perceptions are
brought toward reference levels in some dynamical fashion, but the nature
of the perceptual variables that are made to behave in this way.

I think the word "here" ought to be substituted by "in my mind"
(where "my" means BP). That is a question toward which this intended
series of messages is aiming, and I don't think it will be fully
answered. It's not the question I am addressing "here" (yet).

What I'm really trying to do is to provide some background against
which what Peter has to say will become more intelligible. I'm not
trying to pre-empt him. He can talk for himself. But if you don't
follow his language, or know some of the background that he assumes,
you won't know whether what he is saying has any value to you.

Is the
perception itself created by some sort of almost-periodic process? This is
what Peter Small appears to be saying. You're speaking of the behavior of a
perceptual variable through time, which of course does not explain how that
perceptual variable is produced, but only describes how it behaves.

You are mixing two things here. I'm talking about the behaviour of a
variable over time. It so happens that the variable I've been using
as an example is the perceptual signal in an elementary control unit
that has no disturbance input (though later in my earlier message I
considered the effect of disturbance). Peter is talking about
perceptions. The perception is seen not as being a variable following
an orbit so much as the orbit itself (or its attractor). It's
conceptually not unlike in the way you used the contents of a shift
register in your artificial cerebellum to represent at any one moment
the recent history of a waveform.

You are quite right, though, in saying it "does not explain how that
perceptual variable is produced, but only describes how it behaves".
I thought I had emphasized that point in my earlier message.

To describe how it behaves is not worthless, however. The topology,
or if you can determine it, the topography of the system dynamic says
quite a lot about the kinds of mechanism that could underly the
behavioour described. It works two ways, doesn't it? If you connect
up a set of active and passive entities (amplifiers, resistors,
sensors), you want to describe its behaviour. If you observe a
mechanism behaving, you want to know how the mechanism works.

You started by observing behaviour (in youself or in others) that
could be mimicked by control systems, and as you observed further,
certain kinds of elaborated control structures seemd to produce
behaviour that correspond to other observations. Then you considered
various how the imagined mechanism might actually behave, in more
detail. You simulated and measured. You used mathematical
descriptions such as Laplace transforms that (if you understand them)
provide parametric limits on useful behaviour, such as under what
conditions the system goes into an exponentially increasing
oscillation (approaches the attractor "point at infinity").

Peter has pointed out repeatedly that the if the underlying
biological mechanism is presumed to be the neural interactions in the
brain, then there are other observations that a correct functional
description must accommodate.

Those observations are that one frequently finds modular parts of the
brain behaving in what seems to be a chaotic manner, with strange
attractors, and that there seem to be abrupt shifts in the behaviour
of modules from orbiting in one attractor basin to another. He thinks
(as I do) that much of this behaviour has to do with the generation
of perceptions. If this is correct, then it's an observation that a
full model has to accommodate. It says something about the possible
functional (as opposed to physical) mechanism.

I think the crux of the matter here is whether we are talking merely about
the behavior of a perceptual variable whose existence is taken for
granted, or about the processes by which one perceptual variable is
generated as a function of other variables: the signals, or the transfer
functions. It seems to me that in these discussions, particularly those of
Peter Small, this distinction is blurred almost out of existence.

It's a distinction that is easy to lose. So far, I'm talking only
about the behaviour of a variable, which might be perceptual. The
generation mechanism determines the behaviour, the topology and
topography of the attractor basins describe it.

It's very easy to think the other way, that the attractor basins
generate the behaviour, but that's a bit like saying that maps
determine where the roads are. (As I found out on several occasions
in graduate school, this last can be shown to be false by direct
observation

Martin

[Richard_Marken2]

[From Rick Marken (2004.05.09.1000)]

Martin Taylor (2004.05.09.10.15)--

Bill Powers (2004.05.09.0430 MST)

I think the crux of the matter here is whether we are talking merely
about
the behavior of a perceptual variable whose existence is taken for
granted, or about the processes by which one perceptual variable is
generated as a function of other variables: the signals, or the
transfer
functions. It seems to me that in these discussions, particularly
those of
Peter Small, this distinction is blurred almost out of existence.

It's a distinction that is easy to lose. So far, I'm talking only
about the behaviour of a variable, which might be perceptual.

But the behavior of the perceptual (and other) signals is explained
quite well by control theory. Of course, the most important thing, I
think, about the dynamic behavior of the perceptual signal is that it
follows, to a close approximation, temporal variations in the reference
signal. I don't see what the notion of dynamic attractors can
contribute to our understanding of the dynamics of the perceptual (or
any other variables) in a control that goes beyond this.

I think there _might_ be something dynamic attractor theory can
contribute to our understanding of the functions that generates one
perceptual variable as a function of others - the perceptual function.
But I think the behavior of perceptual variables themselves is pretty
well handled by PCT.

Best

Rick

···

--
Richard S. Marken
marken@mindreadings.com
Home 310 474-0313
Cell 310 729-1400

[Boss_Man]

[From Boss Man (2004.05.09)]

Rick Marken (2004.05.09.1000)

I think there _might_ be something dynamic attractor theory can
contribute to our understanding of the functions that generates one
perceptual variable as a function of others - the perceptual function.
But I think the behavior of perceptual variables themselves is pretty
well handled by PCT.

Try the following on for size. You are visiting a friend. While he is making drinks in the kitchen, he
calls out to you. "Please let the cat out." You've never seen his cat. How do you recognise that
something is a cat in order to let it out? Is there one decisive test? More than one? How do you
know?