Low carb diets

OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
Elzinator wrote:
"OmegaZero2003" wrote...

This is very similar to the issues facing cancer
researchers. Three very different mechanisms/theories
using separate processes all interacting to produce the
endpoint.

Biological systems are more complex than most realize:
feedback loops, negative and positive regulators,
redundant and overlapping pathways, etc.

And, they're all nonlinear.

Well - theyt are not *all* non-linear!

Actually, if you want to argue mathematics, they are *all*
nonlinear, because linearity is such a special case that is
never achieved in practice. 8-p

I won't argue the rest here, except to say that my statement
stands: the threshold and saturation phenomena so common in
biological systems are related to the robustness of their
operation.

--
-Wayne

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
Elzinator wrote:
"OmegaZero2003" wrote...

This is very similar to the issues facing cancer
researchers. Three very different mechanisms/theories
using separate processes all interacting to produce the
endpoint.

Biological systems are more complex than most realize:
feedback loops, negative and positive regulators,
redundant and overlapping pathways, etc.

And, they're all nonlinear.

Well - theyt are not *all* non-linear!

Actually, if you want to argue mathematics, they are *all*
nonlinear, because linearity is such a special case that is
never achieved in practice. 8-p

AND I LOVE to argue or debate or discuss Mathematics. Why, me and my trusty
Mathematica app have been through many wars together. Akk Steven Wolfram
about what that might mean.

You must be dreaming of another dimension.

Lineararity and non-linearity are different concepts *in principle*.

Qualitatively.

In practice as you say, given that measurement is an approximation, and
given that linearity lay on one extreme of a spectrum and total (what ever
that can mean), the other extreme, it may be the case that all of nature
exhibits non-linearity in the various processes that constitute its form and
function.

However, given category logic, one can see that at one point some distance
off the non-linear extreme to the extreme, would constitute "non-linearity"
in a given context. Ditto linearity.

That is where the principles play a part - in determining where to place the
points and what to consider in placing thouse points Now, it is likely a
tuplpe of considering complexity. Indeed, in practice, if you are
considering very low level desciptions (in terms of particle physics), one
need only look at the Lagrangian, for a complex system, and visualize that
alongside several other system-characteristic_describing "equations", and
one has some work to do!



I won't argue the rest here, except to say that my statement
stands: the threshold and saturation phenomena so common in
biological systems are related to the robustness of their
operation.

Related perhaps - but correlation DNE cause or a particularly close
relationship in any dimension. But I also note that you say it is the
robustness of the system is relarted to some "threshold and saturation
phenomena".
That is different than your first postulation. Which was:

"And, they're [biological systems in nature] all nonlinear. That is, they
are rife with
thresholds and saturation effects. This makes them very,
very (very) complicated, but has a lot to do with their
effectiveness and robustness."

That non-linearity itself has a lot to do with thier effectiveness and
robustness.

Perhaps you can elaborate. I would like to know what you thin thresholds
and saturation effects have to do with linearity such that they help
constitute a property or process of robustness and effectiveness. Note that
specifying the system/domian will help establish criteris with which to
robustness and effectiveness can be defined and measured.




thresholds and saturation effects. This makes them very,
very (very) complicated, but has a lot to do with their
effectiveness and robustness

I agree with your implicate approval of Elzi's take on such systems in
general though.





--
-Wayne

OmegaZero2003 wrote:

"Wayne S. Hill" wrote:

Actually, if you want to argue mathematics, they are *all*
nonlinear, because linearity is such a special case that is
never achieved in practice. 8-p

AND I LOVE to argue or debate or discuss Mathematics. Why,
me and my trusty Mathematica app have been through many wars
together. Akk Steven Wolfram about what that might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

You must be dreaming of another dimension.

I see dead dimensions.

Lineararity and non-linearity are different concepts *in
principle*.

Qualitatively.

In practice as you say, given that measurement is an
approximation, and given that linearity lay on one extreme
of a spectrum and total (what ever that can mean), the other
extreme, it may be the case that all of nature exhibits
non-linearity in the various processes that constitute its
form and function.

However, given category logic, one can see that at one point
some distance off the non-linear extreme to the extreme,
would constitute "non-linearity" in a given context. Ditto
linearity.

OK, you can *sometimes* view a complex system as quasi-linear
around an operating point (but in some systems this is
literally useless), but even such systems can only be viewed
as piecewise linear. Ultimately, the system changes as you
move away from the operating condition, so what has
linearization taught you?

I won't argue the rest here, except to say that my
statement stands: the threshold and saturation phenomena
so common in biological systems are related to the
robustness of their operation.

Related perhaps - but correlation DNE cause or a
particularly close relationship in any dimension. But I
also note that you say it is the robustness of the system is
relarted to some "threshold and saturation phenomena".
That is different than your first postulation. Which was:

"And, they're [biological systems in nature] all nonlinear.
That is, they are rife with
thresholds and saturation effects. This makes them very,
very (very) complicated, but has a lot to do with their
effectiveness and robustness."

No, you're misreading me. I said the same thing both times.

That non-linearity itself has a lot to do with thier
effectiveness and robustness.

It does, but the nature of the nonlinearity has a lot to do
with it.

Perhaps you can elaborate. I would like to know what you
thin thresholds and saturation effects have to do with
linearity such that they help constitute a property or
process of robustness and effectiveness.

I really don't want to get into this too deeply (not why I
come here), but threshold and saturation phenomena remap an
infinite range of possibilities into a modest finite range.
Since a biological system can only act within such a range,
this permits the system to respond to very broad ranges of
environments. The system does this by employing different
mechanisms or strategies in different ranges of external
influence (with each mechanism triggered by its own threshold,
and limited by saturation). For example, for a room-
temperature environment, the body maintains core temperature
using different strategies than in very cold or very hot
conditions.

Note that this is what makes neural networks into
computational engines. Without threshold and saturation
phenomena, a NN would be useless.

--
-Wayne

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote:

Actually, if you want to argue mathematics, they are *all*
nonlinear, because linearity is such a special case that is
never achieved in practice. 8-p

AND I LOVE to argue or debate or discuss Mathematics. Why,
me and my trusty Mathematica app have been through many wars
together. Akk Steven Wolfram about what that might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

I would not argue with Steve; he is onto something. He put it all together
and formulated another view of reality consistent with certain other current
views, yet enabling a look at complexity_from_simplicity that has heretofore
not been appreciated in its scope of applicability.


You must be dreaming of another dimension.

I see dead dimensions.

Dimensions see multiple dead yous.


Lineararity and non-linearity are different concepts *in
principle*.

Qualitatively.

In practice as you say, given that measurement is an
approximation, and given that linearity lay on one extreme
of a spectrum and total (what ever that can mean), the other
extreme, it may be the case that all of nature exhibits
non-linearity in the various processes that constitute its
form and function.

However, given category logic, one can see that at one point
some distance off the non-linear extreme to the extreme,
would constitute "non-linearity" in a given context. Ditto
linearity.

OK, you can *sometimes* view a complex system as quasi-linear
around an operating point (but in some systems this is
literally useless), but even such systems can only be viewed
as piecewise linear. Ultimately, the system changes as you
move away from the operating condition, so what has
linearization taught you?

I won't argue the rest here, except to say that my
statement stands: the threshold and saturation phenomena
so common in biological systems are related to the
robustness of their operation.

Related perhaps - but correlation DNE cause or a
particularly close relationship in any dimension. But I
also note that you say it is the robustness of the system is
relarted to some "threshold and saturation phenomena".
That is different than your first postulation. Which was:

"And, they're [biological systems in nature] all nonlinear.
That is, they are rife with
thresholds and saturation effects. This makes them very,
very (very) complicated, but has a lot to do with their
effectiveness and robustness."

No, you're misreading me. I said the same thing both times.

I copied and pasted your original statement.


That non-linearity itself has a lot to do with thier
effectiveness and robustness.

It does, but the nature of the nonlinearity has a lot to do
with it.

What does that mean?


Perhaps you can elaborate. I would like to know what you
thin thresholds and saturation effects have to do with
linearity such that they help constitute a property or
process of robustness and effectiveness.

I really don't want to get into this too deeply (not why I
come here), but threshold and saturation phenomena remap an
infinite range of possibilities into a modest finite range.
Since a biological system can only act within such a range,
this permits the system to respond to very broad ranges of
environments.

The possible system states have little to do with whether a system is linear
or non-linear.

However, complextity is all about such.

The system does this by employing different
mechanisms or strategies in different ranges of external
influence (with each mechanism triggered by its own threshold,

I agree with this. But how does that (threshold and saturation) affect
robustness and saturation directly. They are parameters constraining
response yes and I get your point here, but a response to a perturbation
using, say, Green's Theroem to determine such (where the result of solved
SPDEs will eventually converge to zero - meaning the system will reach a
minima on a mapping - energy/complexity/activity/etc), in terms of its
robustness to that perturbation (ability to so converge/relax), will not
have threshold and saturation terms in those equations. Similarly for the
effectiveness parameter(s) (again, in tems of? meeting a goal (if an
intensional system), surviving an environment?) . If what you mean *is* a
system's effectiveness in surviving perturbations of an environment without
becoming unstable, there are aharmonic mutivibrator-characterized systems
that can tend to chaos or to stable systems with zippo to do with. There are
many other complex systems that do not reach such extrema (saturation) in
their response, nor are they especially threshold-based system. For example,
the brain can detect one photon of light (via the VC) when such impinges
upon a photoreceptor. That is the smallest threshold one can imagine - a
pseudo-infinitely-small threshold in the *sense* that it is representative
of the quanta of em energy. No telling if any brain has actually detected
*only* one photon at a "time" of course, but the point is one of
threshold-based systems. You have to make a quantum leap to get to that
threshold arr, arr!

There are also discontinuous processes that "jump" right over "thresholds".



Can you point me to a ref. where you are reading/getting this relationship
from?


and limited by saturation). For example, for a room-
temperature environment, the body maintains core temperature
using different strategies than in very cold or very hot
conditions.

Note that this is what makes neural networks into
computational engines.

That is one level of description - or -one view of what brain does among
several. I have a bit of experience constructing ANNs for process control
and there are levels of description of brain that are not also
characterizable as a TME (Turing Machine Equivalent).



Without threshold and saturation
phenomena, a NN would be useless.

Threshold is apparent in the neuronal characterization of all-or-nothing
firings (which itself is a function of humongous complexity); however, that
one aspect of the messenger processes (first or second) of the brain.

I cannot see where it has the import ascribed WRT robustness or
effectiveness (towards a goal for example).

Saturation is an example of an extrema - a perturbation causing a behavior
point, and subsequent behavior points that are the same or similar magnatude
until the system relaxes. The system simply has no differential response to
continuing stimula.

Again, this is orthogonal to robustness and effectiveness of a system (in
terms of - we have not defined except as my intial take on what each means
earlier.

Here is another thought. Man-made complex systems are engineered, usually,
to clamp to a safe value(s), all those parameters that may compromise safety
or efficiency/waste-control. That, and the other characteristics I mentioned
(no single point of failure, graceful degradation etc.) make a system robust
(in the face of error or failure). Threshold and saturation are not part of
that consideration except as knowledge that can be employed to determine
startpopints (states), endpoints (end states), and the PID coefficients
affecting operation. When a P/I/D/PI/PD/PID process goes awry, the PID and
any cascaded processes/points to which it is related/connected get reset to
some clamp value(s) and a good system will transfer control to simple
LL-based controllers and/or simple interlocks completely divorced from the
other control system (isolatability is another aspect of robustness).

A good ref on all of this is the classic N. Weiner's Cybernetics Second
Edition: or the Control and Communication in the Animal and the Machine

Any good book on control systems theory incorporating the good ole PID
controller strategy should give more insight into the parameters affecting
system control, especially systems with feedback.



--
-Wayne

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote:

Actually, if you want to argue mathematics, they are *all*
nonlinear, because linearity is such a special case that is
never achieved in practice. 8-p

AND I LOVE to argue or debate or discuss Mathematics. Why,
me and my trusty Mathematica app have been through many wars
together. Akk Steven Wolfram about what that might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

You must be dreaming of another dimension.

I see dead dimensions.

Lineararity and non-linearity are different concepts *in
principle*.

Qualitatively.

In practice as you say, given that measurement is an
approximation, and given that linearity lay on one extreme
of a spectrum and total (what ever that can mean), the other
extreme, it may be the case that all of nature exhibits
non-linearity in the various processes that constitute its
form and function.

However, given category logic, one can see that at one point
some distance off the non-linear extreme to the extreme,
would constitute "non-linearity" in a given context. Ditto
linearity.

OK, you can *sometimes* view a complex system as quasi-linear
around an operating point (but in some systems this is
literally useless), but even such systems can only be viewed
as piecewise linear.


BTW, as a PS to my other answer post, here are some linear systems.

- those characterizable by linear algebra. there are lots of these!
- Hamiltonian oscillators and like systems. (the direction field
specifically)
- continuous-time systems like electrical networks, many mechanical systems
- any discrete system with a transfer function whose input, response and
output functions depend on one variable
- any systems preserving homogeneity (output proportional to input) and
superposition (a way of combining linear functions such that the result is a
linear function)

Even non-deterministic systems can be modeled using statistics for linear
dynamics.

But the main point to not be belabored is that there are linear systems on
nature and manmade (1)

Most systems *are* non-linear but some of those are characterizable using
linear methods to some degree of accuracy; you did make something like this
point.

(1) Schwarz, Ralph J. and Friedland, Bernard, 1965, Linear Systems, New
York: McGraw-Hill Book Company, 521 pp.

On Sun, 21 Dec 2003 06:29:45 GMT, "OmegaZero2003"
wrote:

BTW, as a PS to my other answer post, here are some linear systems.

- those characterizable by linear algebra. there are lots of these!
- Hamiltonian oscillators and like systems. (the direction field
specifically)
- continuous-time systems like electrical networks, many mechanical systems

Only simple RLC electrical networks fall into this category. And even
then, it's just a theoretical assumption over the useful operating
range. Too much current or voltage or flux will flux up your circuit.
Linear electrical networks only exist on paper.

- any discrete system with a transfer function whose input, response and
output functions depend on one variable
- any systems preserving homogeneity (output proportional to input) and
superposition (a way of combining linear functions such that the result is a
linear function)

Even non-deterministic systems can be modeled using statistics for linear
dynamics.

But the main point to not be belabored is that there are linear systems on
nature and manmade (1)

Most systems *are* non-linear but some of those are characterizable using
linear methods to some degree of accuracy; you did make something like this
point.

(1) Schwarz, Ralph J. and Friedland, Bernard, 1965, Linear Systems, New
York: McGraw-Hill Book Company, 521 pp.

---
Proton Soup

"If I drink water I will have to go to the bathroom and
how can I use the bathroom when my people are in bondage?"
-Saddam Hussein

Proton Soup wrote:

"OmegaZero2003" wrote:

BTW, as a PS to my other answer post, here are some linear
systems.

- those characterizable by linear algebra. there are lots
of these! - Hamiltonian oscillators and like systems. (the
direction field specifically) - continuous-time systems like
electrical networks, many mechanical systems

Only simple RLC electrical networks

and their analogs in other domains

fall into this category.
And even then, it's just a theoretical assumption over the
useful operating range. Too much current or voltage or flux
will flux up your circuit. Linear electrical networks only
exist on paper.

Exactly. They're (essentially) linear in a linear range.

--
-Wayne

OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
OmegaZero2003 wrote:

AND I LOVE to argue or debate or discuss Mathematics.
Why, me and my trusty Mathematica app have been through
many wars together. Akk Steven Wolfram about what that
might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

I would not argue with Steve; he is onto something. He put
it all together and formulated another view of reality
consistent with certain other current views, yet enabling a
look at complexity_from_simplicity that has heretofore not
been appreciated in its scope of applicability.

That's not clear to me.

No, you're misreading me. I said the same thing both
times.

I copied and pasted your original statement.

I must have been unclear the first time, because I intended
the same meaning both times.

That non-linearity itself has a lot to do with thier
effectiveness and robustness.

It does, but the nature of the nonlinearity has a lot to do
with it.

What does that mean?

Nonlinearity can arise in many different forms. Aside from
quadratic/cubic forms, which you might call "local"
nonlinearities (because the "slope" of the interaction varies
locally), the global behaviors of threshold and saturation
phenomena are common themes in biological systems.

Perhaps you can elaborate. I would like to know what you
thin thresholds and saturation effects have to do with
linearity such that they help constitute a property or
process of robustness and effectiveness.

I really don't want to get into this too deeply (not why I
come here), but threshold and saturation phenomena remap an
infinite range of possibilities into a modest finite range.
Since a biological system can only act within such a range,
this permits the system to respond to very broad ranges of
environments.

The possible system states have little to do with whether a
system is linear or non-linear.

Au contraire. If a system is linear, it must accommodate an
infinite range of input variables linearly. Thus, the output
range has to be of infinite extent, and cannot exhibit
different types of states.

However, complextity is all about such.

You've got to be careful here. I take it you're referring to
complex dynamical systems that exhibit self-organizing so-
called emergent behaviors. A mass of nitrogen molecules is a
counter example: it never does anything "emergent", and so
doesn't (normally) have distinguishably different states.
That is, given N molecules in a box of size V and temperature
T, it exerts a pressure P. This varies in a simple and smooth
manner from above the boiling point to the neighborhood of
dissociation. The difference between a "simple" complex
system and one capable of self-organization is the way it
approaches equilibrium in the face of large disequilibrium.

The system does this by employing different
mechanisms or strategies in different ranges of external
influence (with each mechanism triggered by its own
threshold,

I agree with this. But how does that (threshold and
saturation) affect robustness and saturation directly. They
are parameters constraining response yes and I get your
point here, but a response to a perturbation using, say,
Green's Theroem to determine such (where the result of
solved SPDEs will eventually converge to zero - meaning the
system will reach a minima on a mapping -
energy/complexity/activity/etc), in terms of its robustness
to that perturbation (ability to so converge/relax), will
not have threshold and saturation terms in those equations.
Similarly for the effectiveness parameter(s) (again, in tems
of? meeting a goal (if an intensional system), surviving an
environment?) . If what you mean *is* a system's
effectiveness in surviving perturbations of an environment
without becoming unstable, there are aharmonic
mutivibrator-characterized systems that can tend to chaos or
to stable systems with zippo to do with. There are many
other complex systems that do not reach such extrema
(saturation) in their response, nor are they especially
threshold-based system. For example, the brain can detect
one photon of light (via the VC) when such impinges upon a
photoreceptor. That is the smallest threshold one can
imagine - a pseudo-infinitely-small threshold in the *sense*
that it is representative of the quanta of em energy. No
telling if any brain has actually detected *only* one photon
at a "time" of course, but the point is one of
threshold-based systems. You have to make a quantum leap to
get to that threshold arr, arr!

There are also discontinuous processes that "jump" right
over "thresholds".

True.

Can you point me to a ref. where you are reading/getting
this relationship from?

Sorry, I just made it up (but it happens to be true).
Remember, I see dead dimensions.

and limited by saturation). For example, for a room-
temperature environment, the body maintains core
temperature using different strategies than in very cold or
very hot conditions.

Note that this is what makes neural networks into
computational engines.

That is one level of description - or -one view of what
brain does among several. I have a bit of experience
constructing ANNs for process control and there are levels
of description of brain that are not also characterizable as
a TME (Turing Machine Equivalent).

True, but I'm referring to much simpler ensembles of neurons.
The computational capability of a NN is directly traceable to
the threshold and saturation characteristics of the neurons.

Without threshold and saturation phenomena, a NN would be
useless.

Threshold is apparent in the neuronal characterization of
all-or-nothing firings (which itself is a function of
humongous complexity); however, that one aspect of the
messenger processes (first or second) of the brain.

I cannot see where it has the import ascribed WRT
robustness or effectiveness (towards a goal for example).

Then open your eyes.

I think we're arguing past each other, something that wouldn't
happen if we actually discussed this in person.

Saturation is an example of an extrema - a perturbation
causing a behavior point, and subsequent behavior points
that are the same or similar magnatude until the system
relaxes. The system simply has no differential response to
continuing stimula.

Right, and this is really important: beyond a narrow range,
the cost of responding linearly to external stimuli would be
too taxing to the organism. Consequently, the organism lets
that mechanism saturate, and turns on a different one.

Again, this is orthogonal to robustness and effectiveness of
a system (in terms of - we have not defined except as my
intial take on what each means earlier.

No, it's key.

Here is another thought. Man-made complex systems are
engineered, usually, to clamp to a safe value(s), all those
parameters that may compromise safety or
efficiency/waste-control.

This is a simple form of saturation.

That, and the other
characteristics I mentioned (no single point of failure,
graceful degradation etc.) make a system robust (in the face
of error or failure). Threshold and saturation are not part
of that consideration except as knowledge that can be
employed to determine startpopints (states), endpoints (end
states), and the PID coefficients affecting operation. When
a P/I/D/PI/PD/PID process goes awry, the PID and any
cascaded processes/points to which it is related/connected
get reset to some clamp value(s) and a good system will
transfer control to simple LL-based controllers and/or
simple interlocks completely divorced from the other control
system (isolatability is another aspect of robustness).

You don't see the analogy?

A good ref on all of this is the classic N. Weiner's
Cybernetics Second Edition: or the Control and Communication
in the Animal and the Machine

Any good book on control systems theory incorporating the
good ole PID controller strategy should give more insight
into the parameters affecting system control, especially
systems with feedback.

Well, yeah, but they provide little insight into profound
nonlinearity.

--
-Wayne

OmegaZero2003 wrote:

BTW, as a PS to my other answer post, here are some linear
systems.

- those characterizable by linear algebra. there are lots
of these!

'ang on there, when we refer to linearity in such systems,
we're referring to linearity of the dynamics, i.e.,

x-dot = A * x + b

This omits simple algebraic behaviors (because x-dot = 0).

- Hamiltonian oscillators and like systems. (the
direction field specifically)

Not of great interest in biological systems, except as a
backdrop.

- continuous-time systems like electrical networks, many
mechanical systems

Aside from the fact that linearity is an approximation in all
such systems, they all have their nonlinear limits. The
nonlinearity is whatever it is that keeps the systems
operating in their linear range.

- any discrete system with a transfer
function whose input, response and output functions depend
on one variable

Huh? The most common form of discrete systems is the
iteration (strobing based on time or state) of a continuous
system. A single-variable system would be of a form

x-dot = f(x)

Here, f can be (generally is) a nonlinear function of x, so
the system will show nonlinear behavior, both continuously and
discretely.

- any systems preserving homogeneity (output
proportional to input) and superposition (a way of combining
linear functions such that the result is a linear function)

In other words, linear systems.

Even non-deterministic systems can be modeled using
statistics for linear dynamics.

Not in their dynamics. This is the classic engineer's mistake
of characterizing in statistical terms what is not understood.
When you delve into the NLD of such systems, you gain true
insight into what makes them work. I can give examples.

But the main point to not be belabored is that there are
linear systems on nature and manmade (1)

Most systems *are* non-linear but some of those are
characterizable using linear methods to some degree of
accuracy; you did make something like this point.

That is an approximation that people sometimes find useful to
make. That doesn't make it so, especially when you take the
system outside of the limited context in which you placed it
for convenience.

--
-Wayne

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
OmegaZero2003 wrote:

AND I LOVE to argue or debate or discuss Mathematics.
Why, me and my trusty Mathematica app have been through
many wars together. Akk Steven Wolfram about what that
might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

I would not argue with Steve; he is onto something. He put
it all together and formulated another view of reality
consistent with certain other current views, yet enabling a
look at complexity_from_simplicity that has heretofore not
been appreciated in its scope of applicability.

That's not clear to me.

Have you read his book?


No, you're misreading me. I said the same thing both
times.

I copied and pasted your original statement.

I must have been unclear the first time, because I intended
the same meaning both times.

That non-linearity itself has a lot to do with thier
effectiveness and robustness.

It does, but the nature of the nonlinearity has a lot to do
with it.

What does that mean?

Nonlinearity can arise in many different forms. Aside from
quadratic/cubic forms, which you might call "local"
nonlinearities (because the "slope" of the interaction varies
locally),

It does not ahve to; the form and whether it is local or non-local are
orthogonal.

the global behaviors of threshold and saturation
phenomena are common themes in biological systems.

Sure.

My point is that there are linear systems.


Perhaps you can elaborate. I would like to know what you
thin thresholds and saturation effects have to do with
linearity such that they help constitute a property or
process of robustness and effectiveness.

I really don't want to get into this too deeply (not why I
come here), but threshold and saturation phenomena remap an
infinite range of possibilities into a modest finite range.
Since a biological system can only act within such a range,
this permits the system to respond to very broad ranges of
environments.

The possible system states have little to do with whether a
system is linear or non-linear.

Au contraire. If a system is linear, it must accommodate an
infinite range of input variables linearly. Thus, the output
range has to be of infinite extent, and cannot exhibit
different types of states.

This makes little sense. It is the complexity of a system that determines
the breadth and depth of a system_state tree.


However, complextity is all about such.

You've got to be careful here. I take it you're referring to
complex dynamical systems that exhibit self-organizing so-
called emergent behaviors.

A complex system, or a dynamical system need not exhibit emergent phenomena.
The systems that do exhibt emergent phenomena however, are usually complex
dynamical sytems.


A mass of nitrogen molecules is a
counter example: it never does anything "emergent",

That is what I said just above.

and so
doesn't (normally) have distinguishably different states.
That is, given N molecules in a box of size V and temperature
T, it exerts a pressure P. This varies in a simple and smooth
manner from above the boiling point to the neighborhood of
dissociation.

That something varies smoothly (not descrete steps I presume you mean), does
not mean it does not have distinguisable states!!!!!! That is what
intergation and differentiation are all about.

Not only that but ther are clever theories purporting to show:

a) everything is quatal/descrete to the finest level of description
b) everything is analog/no_quantal_states to the finest level of
description.

Both positions are far from established given our level of instrumentality.


The difference between a "simple" complex
system and one capable of self-organization is the way it
approaches equilibrium in the face of large disequilibrium.

Or VV!! Chaotic systems far from equilibrium. See Prigogine.


The system does this by employing different
mechanisms or strategies in different ranges of external
influence (with each mechanism triggered by its own
threshold,

I agree with this. But how does that (threshold and
saturation) affect robustness and saturation directly. They
are parameters constraining response yes and I get your
point here, but a response to a perturbation using, say,
Green's Theroem to determine such (where the result of
solved SPDEs will eventually converge to zero - meaning the
system will reach a minima on a mapping -
energy/complexity/activity/etc), in terms of its robustness
to that perturbation (ability to so converge/relax), will
not have threshold and saturation terms in those equations.
Similarly for the effectiveness parameter(s) (again, in tems
of? meeting a goal (if an intensional system), surviving an
environment?) . If what you mean *is* a system's
effectiveness in surviving perturbations of an environment
without becoming unstable, there are aharmonic
mutivibrator-characterized systems that can tend to chaos or
to stable systems with zippo to do with. There are many
other complex systems that do not reach such extrema
(saturation) in their response, nor are they especially
threshold-based system. For example, the brain can detect
one photon of light (via the VC) when such impinges upon a
photoreceptor. That is the smallest threshold one can
imagine - a pseudo-infinitely-small threshold in the *sense*
that it is representative of the quanta of em energy. No
telling if any brain has actually detected *only* one photon
at a "time" of course, but the point is one of
threshold-based systems. You have to make a quantum leap to
get to that threshold arr, arr!

There are also discontinuous processes that "jump" right
over "thresholds".

True.

Can you point me to a ref. where you are reading/getting
this relationship from?

Sorry, I just made it up (but it happens to be true).

Well - I think it is mostly true as I said. But my original point is that
there are linear systems (hell - that is what LP is for!!).

But - in that case...

I think we should have been discussing the (a)/(b) dichotomy I mentioned
above. Whether nature is discontinuoous or continuous.

Instead!

;^)


Remember, I see dead dimensions.

and limited by saturation). For example, for a room-
temperature environment, the body maintains core
temperature using different strategies than in very cold or
very hot conditions.

Note that this is what makes neural networks into
computational engines.

That is one level of description - or -one view of what
brain does among several. I have a bit of experience
constructing ANNs for process control and there are levels
of description of brain that are not also characterizable as
a TME (Turing Machine Equivalent).

True, but I'm referring to much simpler ensembles of neurons.
The computational capability of a NN is directly traceable to
the threshold and saturation characteristics of the neurons.

Well -err - not necessarily. The computational capability of brain or any
subset depends on how yu think it characterizes, or *represents*
information!!

IFF it is based on the go/no-go neuronal firing model, and IFF one
establishes as a premise that such is representative of 1 "bit" of
information (which is problematic itself, since the binary system is one of
*represenation* of higher-order/more-complex information represenation
schemes - that is, it is a mapping itself!!!), then one can estimate the
computational extent of that group of neurons (assuming further that one can
charaterize the NN architecture (its connection scheme) in sufficient
detail.


A little bit goes a long way - haha!



Without threshold and saturation phenomena, a NN would be
useless.

Threshold is apparent in the neuronal characterization of
all-or-nothing firings (which itself is a function of
humongous complexity); however, that one aspect of the
messenger processes (first or second) of the brain.

I cannot see where it has the import ascribed WRT
robustness or effectiveness (towards a goal for example).

Then open your eyes.

I think we're arguing past each other, something that wouldn't
happen if we actually discussed this in person.

More than likely! I like to waves hands and draw on boards!!



Saturation is an example of an extrema - a perturbation
causing a behavior point, and subsequent behavior points
that are the same or similar magnatude until the system
relaxes. The system simply has no differential response to
continuing stimula.

Right, and this is really important: beyond a narrow range,
the cost of responding linearly to external stimuli would be
too taxing to the organism. Consequently, the organism lets
that mechanism saturate, and turns on a different one.

Or goes crazy! (Becomes chaotic) There are examples of systems that
persist in their output without switching. This kind of system exists on an
organismic level and in vary large systems - like socail systems and
economics.

It all depends on the feedback mechanisms (FF/FB) and *where* in the input
stream the FB occurs, and *whether* that point in the input stream has the
capability to clamp or switch its input-processing algorithm.



Again, this is orthogonal to robustness and effectiveness of
a system (in terms of - we have not defined except as my
intial take on what each means earlier.

No, it's key.

I don't see as strong a link as you do evidently. There are far more
important aspects that affect robustness and effectiveness of a system than
saturation and threshold.

I guess it depends on how you have learned to view system sciences, control
systems, and biological systems etc.


Here is another thought. Man-made complex systems are
engineered, usually, to clamp to a safe value(s), all those
parameters that may compromise safety or
efficiency/waste-control.

This is a simple form of saturation.

Yo can clamp well-before a saturation level!


That, and the other
characteristics I mentioned (no single point of failure,
graceful degradation etc.) make a system robust (in the face
of error or failure). Threshold and saturation are not part
of that consideration except as knowledge that can be
employed to determine startpopints (states), endpoints (end
states), and the PID coefficients affecting operation. When
a P/I/D/PI/PD/PID process goes awry, the PID and any
cascaded processes/points to which it is related/connected
get reset to some clamp value(s) and a good system will
transfer control to simple LL-based controllers and/or
simple interlocks completely divorced from the other control
system (isolatability is another aspect of robustness).

You don't see the analogy?

Huh? Between what and what?


A good ref on all of this is the classic N. Weiner's
Cybernetics Second Edition: or the Control and Communication
in the Animal and the Machine

Any good book on control systems theory incorporating the
good ole PID controller strategy should give more insight
into the parameters affecting system control, especially
systems with feedback.

Well, yeah, but they provide little insight into profound
nonlinearity.

Weiner's book does. ANd if you learn about how control systems are
engineered to deal with "profound non-linearities", then you will see what
is important and what is not (from a control standpoint that is.)

The books/people that do provide the "insight" however include another I
mentioned - Alwyn Scott's work.


--
-Wayne