Control of amount of food in the gut

[From Bruce Abbott (960104.1430)]

Well, Florida weather was terrible, so I used some of the time I might have
spent on the beach to work on a control system model. Here are some of the
results.

···

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The following is a simple control-system model for control of the amount of
food in the gut, which I am using as a simple "stand-in" for the more
complex control systems that would regulate nutrient level. Picture a rat
in an operant chamber, pressing a lever to earn food pellets on a ratio
schedule. At any given time there are F grams of food in the rat's gut. In
the typical operant study the rat has been food-deprived so that the initial
amount in the gut at the beginning of the session, F0, is 0 grams (i.e., the
gut is empty). If the reference level Fr is 10 grams, then the error e is
Fr-F0 = 10 grams. This error times the output sensitivity g drives the
output, which is the rate of lever-pressing behavior B in responses/second.
Through the environment function (schedule of reinforcement) this
lever-pressing rate is converted into a rate of pellet delivery R, which is
then multiplied by the amount of food per pellet to give the rate of food
repletion in grams/sec. (For simplicity I assume that the food is somehow
delivered to the gut; I will flesh out this part of the model at a later
date to include ingestive behaviors.) Food repletion adds to the initial
amount of food, while food depletion at rate M grams/sec (due to digestive
processing) decreases the food amount; the difference between repletion and
depletion rates determines whether the gut fills, empties, or stays the
same. For simplicity I assume that the percepual signal p = F.

Here is a list of the variables and constants, followed by the model equations:

R pellets/sec rate of reinforcement
m grams/pellet magnitude of reinforcement
F grams amount of food in gut
Fr grams reference level for food in gut
F0 grams initial amount of food in gut
p grams perception of amount of food in gut
e grams error
g resp/gram output sensitivity
M grams/sec rate of food depletion
n resp/reinf ratio schedule requirement
B resp/sec rate of behavior

Model equations:

(1) R = B/n environment function (ratio schedule)
(2) F = F0 + (m*R - M)*t food amount at time t
(3) p = F perceptual signal
(4) e = Fr - F error signal
(5) B = g * e output function

System equations:

         F0 + (g*m*t/n)*Fr - Mt
(6) F = ----------------------
              1 + g*m*t/n

         g(Fr -F0 + Mt)
(7) B = --------------
          1 + g*m*t/n

We can extract a common factor (g*m*t/n)/(1 + g*m*t/n) from both equations:

           g*m*t/n n
(6a) F = ----------- * [Fr + ----- * (F0 - Mt)]
         1 + g*m*t/n g*m*t

           g*m*t/n n
(7a) B = ----------- * --- * (Fr - F0 + Mt)
         1 + g*m*t/n m

The factor g*m*t/n is the negative feedback loop gain. Note that this gain
increases with the size of the food pellet (m) AND with the time since the
start of the session (t). As t grows, the ratio of loop gain/(1 + loop
gain) quickly approaches 1.0.

Assume the following constants:

g = 10 resp/gram
m = 0.045 gram
t = 600 seconds [ten minutes into session]
F0 = 0 gram [empty gut]
Fr = 10 gram
M = 0 grams/sec [depletion rate would be negligible compared to repletion]

We can calculate R from B using the ratio schedule function:

R = B/n

At t = 600 seconds and n = 1, the negative feedback loop gain will be
g*m*t/n = 270 and observed values will have reached 270/(1 + 270) = 270/271
= 0.9963, or 99.63% of their final values. At n = 20 the loop gain will be
13.5 and the values will be at 13.5/14.5 = 0.9310 or 93.1 % of their final
values.

Here is a table showing how F, R, and B will vary with the ratio size, n, at
ten minutes into the session:

n F e R B
--------------------------------------
1 9.96 0.04 0.37 0.37
2 9.93 0.04 0.37 0.74
5 9.82 0.18 0.36 1.82
10 9.64 0.36 0.36 3.57
20 9.31 0.69 0.34 6.90
--------------------------------------

These numbers are a bit unrealistic, because the model does not include a
limit on response rate and thus on rate of repletion. The initial error (10
g) produces an excessively high response rate. This could be corrected in
the model by imposing a limit on max response rate of, e.g., 5 resp/sec.

As the session continues and t increases, all rows converge on the same
stable values, although the time required to approach these values increases
with the ratio requirement (by reducing the loop gain):

          F e R B
      --------------------------------
      10.00 0.00 0.00 0.00

If we set F0 = Fr and make the rate of depletion M = 0.0003 grams/sec
(roughly 1 gram per hour), and t = 1800 sec (30 min) we get the following
values:

n F e R B
--------------------------------------
1 9.99 0.01 0.12 0.12
2 9.98 0.02 0.12 0.25
5 9.94 0.06 0.12 0.61
10 9.88 0.12 0.12 1.22
20 9.76 0.24 0.12 2.41
--------------------------------------

These are still somewhat higher than steady-state values. Response rates
(B) in the steady-state will be those required to exactly balance the
depletion rate (mR = M):

R = M/m = 0.0003/0.045 = 0.007 pellets/sec = about 24 pellets/hr

The rate at which the system converges on the steady-state depends on the
loop gain and could be increased in the simulation by setting g higher,
e.g., 100. At 24 pellets/hr and a ratio n = 20 resp/reinf, the rat would be
responding at an average rate of 20 * 24 = 480 resp/hr to maintain itself.
If the model included a threshold level of error before responding
commenced, the rat would eat in a series of spaced "meals."

Comments?

Regards,

Bruce

[From Rick Marken (960104.1210)]

Bruce Abbott (960104.1430) --

Well, Florida weather was terrible, so I used some of the time I might have
spent on the beach to work on a control system model.

Well, it was gorgeous here in LA LA Land so I continued to get nothing done.
Maybe I should move to Florida?

The following is a simple control-system model for control of the amount of
food in the gut ...

Comments?

Looks very nice; a perceptual control system, if ever there was one.

This isn't the bombshell, is it? If so, I hope this is the only kind of
bombshell our troops can expect to encounter in Bosnia.

Best

Rick