A Different View

[From Bruce Abbott (950614.1510 EST)]

In my post of 950611.1635 EST I suggested something to the effect that
ECOLI4a could be viewed as a form of adaptive controller something like the
one Hans Blom has been describing. I've tried to make this view more
visible in the diagrams below. The first diagram is that of an e. coli-type
control system:

                   dN/dt > 0
                      > error
              ----->[ ]-------------->-----------------
             > >
             > >
             > >
       dN/dt | |
      [Input Function] [Output Function]
             > >
             > TUMBLE
             > N |
             > direction
             > >

This is a one-way control system in which an error develops if the perceived
nutrient rate of change is not positive. Unlike the standard e. coli
control model, however, the error signal is a two-state, logical variable
rather than a continuous one. If this variable is True (no error), the
output function produces a tumble rate determined by PTS+; if it is false,
the output function produces a tumble rate determined by PTS-. If PTS+ is
greater than PTS- the result is negative feedback. In the ideal case, PTS+
= 0 and PTS- = 1.0. In that case, tumbling will never occur when dN/dt is
positive and will always occur so long as dN/dt is not positive.

The second part of the system supplies the adaptation, and looks more or
less as diagramed below:

                      dS/dt = 0
                          > error
              --------->[ ]----------------
      dS/dt | dN/dt = 0 |
             > > S+ or S- |
             > -- [ ]------->------- |
             > > S = dN/dt | V
         [Input Function] [Parameters]--->[Output Function]
             > > >
             > ------<--------TUMBLE
             > N |
             > direction
             > >

This diagram isn't quite right, as I am not sure how to represent some of
the relationships. The input function provides both the rate of nutrient
change, S (dN/dt) and the change in nutrient rate, dS/dt. The change in
nutrient rate will always be zero except following a tumble. If the is
change is non-zero, one of the two output function parameters is adjusted.
If S+ was present prior to the tumble (dN/dt positive), then PTS+ is
changed; if S- was present (dN/dt negative) then PTS- is changed. The
direction of these changes is determined by the error signal relating to
dS/dt: if dS/dt is positive (improvement), P is incremented; if negative, P
is decremented.

The problem I have with the diagram is in representing the fact that the
value of dN/dt which applies when adjusting parameters is the value
immediately prior to a tumble rather than its value at the time adjustment
of parameters takes place. In effect there is a delay or lag that should be
inserted between dN/dt and the connection to the parameter adjustment
mechanism. This amounts to a brief storage or memory of the pre-tumble
value. The diagram also does not make explicit the fact that dS/dt is
computed across a tumble episode by comparing dN/dt before and after a
tumble. I've added the line from TUMBLE to the parameter adjustment
mechanism to suggest that parameter adjustment takes place only following
each tumble, although I don't find this representation very satisfactory.
Bill P., perhaps you could suggest a better way to construct this diagram?

Despite these problems, I think the diagram does highlight the parallel with
Hans Blom's adaptive controller: feedback from the consequences of a tumble
is used as a means of adjusting the low-level nutrient-rate control system's
parameters so that control over nutrient rate is achieved.

As a suggestion, I think a "cleaner" example of this concept could be
developed based on the standard e. coli control model, with the adaptation
mechanism operating on the control system gain factor. To function
properly, the adaptation control loop must be sluggish compared to the
system it adjusts.