On the Relationship Between the Feedback Control of Digestion and

On the Relationship Between the Feedback Control of Digestion and

Reference: Biol. Bull. 191: 85—9
1. (August, 1996)
On the Relationship Between the Feedback Control of
Digestion and the Temporal Pattern of Food Intake
D. W. STEPHENS
Nebraska Behavioral Biology Group, SchoolofBiologicalSciences, University ofNebraska, Lincoln,
Lincoln,
Nebraska
1983).Indeed, the widespreadand important phenome
non of learning is best viewed (Stephens, 1991) as a
scheme that capitalizes on environmental regularities
that occur on a timescale that is certainly longer than
seconds.
In short, animal feedingbehavior seemsto be sensitive
to a tangle ofdiffering timescales:in someinstancesonly
the very short-term consequencesof behavior seemim
The Time Scales of Foraging Behavior
One ofthe simplest experiments in animal psychology
is a straightforward evaluation of the effectsof time and
amount on animal feeding preferences.At intervals an
animal is offered a choice betweentwo options: one leads
to a small amount of food quickly, while the other leads
to a much larger amount of food after a longer delay. In
these situations, vertebrates abhor delay (I don't know
portant; in others the longer term consequences are
of any studies with invertebrates). One can think of this
clearly important.
pattern of preference as a situation in which increasing
I had this problem in mind when, coincidentally, I was
time devalues amount, and by fitting classic “¿decay―
invited to participate in this symposium in honor of
functions to this kind ofdata we can estimate the power
Vince Dethier. I thought of Dethier's work on the regu
ofthis effect. It is astonishing. For feedingbluejays (stud
lation of ingestion and digestion via feedback mecha
ied in my laboratory, Stephens et al., 1995) the “¿pernisms (Dethier, 1976)and beganto wonder whether we
ceived value―
ofamount decaysby roughly 10%per sec
might be able to understand some of these “¿timescale―
ond (seeKagel et al., 1986 for cogent discussion of the
effectsby looking inside the animal. Indeed, the mathe
quantitative behavior ofthis decayphenomenon).
matical technique of singular perturbation, which is of
This temporal “¿discounting―
of food, together with
ten used to study feedback systems(Murray, 1989; Lo
severalsimilar phenomena studied in the operant labo
gan, 1987),representsa tantalizing possibility: in singu
ratory, paints a picture in which consequencesnow are
lar perturbation one finds a separatecharacterization of
the primary determinants ofanimal feeding preferences. the short- and long-timescale behavior of a system, and
It is asifanimals careonly about the very short term (the
typically these characterizations are quite different.
next few seconds).Ofcourse, a moment's reflection tells
Could this be a clue as to why animals seem to have
us that this can't be the whole story. Many features of
different feeding preferences at short and long time
animal feeding behavior seemto be organized (from an
scales?In the next few pagesI present a preliminary at
evolutionary point ofview) for longer term goals.Clark's
tempt to answerthis question.
nutcrackers (Nucifraga columbiana) diligently harvest
Behavioral ecologistsare surprised when they seeani
and cache piñonseedsin the fall that they consume in
mals preferring a small amount ofimmediately delivered
the spring (Kamil and Balda, 1991). Migratory birds in
food over larger amounts that are delayed(Stephensand
creasetheir food intake and put on weight severalweeks
Krebs, 1986). One way to think about this problem is
aheadofactually beginning to migrate (Carpenter et al.,
shown in Figure 1, in which rate ofingestion is plotted as
a function oftime. To keep things simple, I think of the
time courseofeating aslong gapswhere nothing is eaten
Received 30 November 1995; accepted 15 April 1996.
punctuated byjumps in ingestion rates.Notice that even
This paper was originally presentedat a symposium titled Finding
if we hold the averagerate of intake constant, there are
Food: NeuroethologicalAspectsofForaging. The symposiumwasheld
many possible patterns ranging from small, frequent
at the University ofMassachusetts, Amherst, from 6 to 8 October 1995.
85
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86
D. W. STEPHENS
A LowPulsePeriodInputStream
a(t)
dV1
1@(t)—¿
g( V0 —¿
V2)V1 V, < Vmax
—¿@- 1@-g( V0 - V2)V1
@
V@
Vmax
V2)V1-r
__I I [1
I
We also need two score-keeping equations to keep track
of nutrient extraction (the benefits derived from the in
put stream). First, we consider the absolute amount of
nutrient in the gut at time t, say A(t). If C1is the concen
tration of nutrient in the material flowing from the gut,
then the rate at which nutrient enters the gut is C1g(V0—¿
A HighPulsePeriodInputStream
a(t)
V2)V3 . At any instant
Time
Figure 1: Two input streams give similar overall intake rates but
differ in their temporal organization.
blips in intake
rate to infrequent,
large ones. For a con
stant average rate, these possibilities differ in the period
icity of bouts
of ingestion.
In the following
model
I will
ask whether a simple feedback-regulated digestive sys
tem does better or worse (i.e., extracts nutrient more or
less efficiently) as I vary the “¿pulse
period―but hold the
average intake rate constant.
Figure 2 shows the simplified feedback system that I
will analyze. To keep the terminology concrete, I will call
the first compartment the “¿crop―
and the second the
“¿gut,―
and I will suppose that the crop is simply a storage
organ in which no digestion takes place, and that all di
gestion occurs in the gut. I appreciate, ofcourse, that this
is a vast oversimplification
ofany
real biological
system,
even of Dethier's Phormia after which it is modeled. I
suppose that inputs to the crop are governed by the
square wave type ofprocess mentioned above, and I will
denote this input process as a(t). The volume of matter
in the crop at any instant in time is V@
, and the volume
the concentration
of nutrient
in
the gut is A(t)/ V2(t), and so nutrient is lost with the
effluent at rate r A(t)/ V2(t). The other process that re
moves nutrient from the gut is nutrient extraction. The
process ofnutricnt extraction may be quite complex, and
it may be qualitatively different for different nutrients. I
model extraction as a “¿Fick's
Law―style absorption pro
cess: that is, nutrient is “¿captured―
by diffusing across the
gut wall while some homeostatic process maintains the
concentration of nutrient on the outside of the gut wall
(i.e., changes in the concentration gradient are due only
to changes in the concentration within the gut). This
means that nutrient leaves the gut via extraction at a rate
proportional to the wetted surface area ofthe gut (5) and
the concentration the nutrient in the gut, aSA/V2 . Now,
if we view the gut as a rigid container, a cylinder with
fixed radius, then the wetted surface area will be propor
tional to V2, say S = b V2. This relationship can be com
plicated by elasticity in the gut wall and by allowing more
complex gut geometries. This linear relationship is the
simplest one that captured the simple idea that wetted
surface area increases with gut volume. Putting this all
together, I write an equation for the change in amount of
nutrient in the gut:
dA
A
A
ut
‘¿2
‘¿2
-:i-= C@g(V0- V2)V1-@-r-abV2@-
@= C1g(V0-
V2)V1
-‘Ø-r_icii
where in the second equation I have cancelled V2and
combined the two constants a and b into a single con
controlled by a feedback law; specifically, matter is re
stant k = ab. I remark that ifthe gut is not well-modeled
leased at a rate proportional to the product of (a) the
as a rigid container (say that it is more elastic than rigid)
difference between the actual gut volume and a “¿set then this volume cancelation will not be valid. To keep
point―V0and (b) the actual volume of the crop. I sup
track of the benefit (i.e., nutrient) extracted over time,
pose that there is an upper limit on the amount of matter
say B(t), we would integrate the simple differential equa
the crop can hold Vmax(so if the crop volume reaches
tion dB/dt = kA
Vmax ingestion
cannot
continue).
Finally,
I suppose
that
As a first attempt to understand this system, I analyzed
effluent leaves the gut at constant rate r(therc arc a num
it numerically; that is, I choose parameter values more
or-less arbitrarily and simply integrated the differential
ber of reasons to view this assumption with suspicion,
equations using well-understood numerical techniques
but we withhold judgment for the moment). This leads
(I used a commercial package and 4th order Runge
to two differential equations for the crop-gut system:
in the gut is V2. Releases
from
the crop
to the gut are
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RELATIONSHIP
BETWEEN
DIGESTION
AND
87
FOOD INTAKE
a't)
III
1______i1@
a(t)
t
Crop
Volume=
g(V@-V)V1
@.SetPoint-V0
Gut
Volume= V2
Figure 2: A familiar mathematical caricature of our crop/gut system in which both the crop and gut
are modeledasleaky bucketswith a feedbackcontrol law governingthe output from the crop to the gut.
Kutta integration). I tried a rangeofinput functions a(t),
all ofwhich were “¿square
waves―
that provided the same
averageinput rate but varied in “¿pulse
period.―
The re
lationship that my analysisrevealedbetweennutrient as
similation rate and pulse period is plotted in Figure 3.
This is the relationship we would expect ifdigestive con
straints play a part in animal preferencesfor immediacy:
for example, if more even flows of food yield a higher
assimilation rate than uneven flows with the sameaver
age intake rate, then it follows that it might sometimes
make economic senseto prefer an even flow with a low
rate of intake over an uneven flow with higher rate of
intake. Ofcourse, thesenumerical resultsdon't establish
this effect asa universal result, they simply suggestthat it
can happen in some instances.To understand its gener
ality we needto attack the model analytically.
(i.e., they are dimensionless, or “¿pure,―
numbers). This
is important for two reasons.First, it is always easierto
work with a smaller number of terms. Second,we need
dimensionless numbers to meaningfully evaluate ap
proximate solutions, becausewewant our estimateof the
magnitudes oferrors to be independent ofthe systemof
measurementusedby the experimenters.One way to ap
proach dimensional analysis to pick “¿characteristic
quantities―against which to measure different types of
units in the model. In the present model we have vol
umes (meters cubed), time (seconds), and amounts of
nutrient (moles). Ifwe measurevolumes in units of Vmax
and time in units of l/gVmax(a measureof the time re
quired to drain a full crop), and nutrient quantities by
CiVmax (the
amount
of nutrient
in a full crop),
can rewrite our systemas
di2@J—(Vo—V@)Vi
V1l
Analysis
df
@
A first step in analyzing this system is dimensional
analysis (Stephens and Dunbar, 1993), a technique in
which one considersthe units ofeach ofthe model's vari
ables and parameters and rearrangesthem into a new,
smaller setofparameters and variablesthat haveno units
ta(t)
—¿
(Vo—¿
V2)V1 V < 1
%=(@o-
V@1P
V@)V,-@-A-kA
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then
we
88
D. W. STEPHENS
O.O2@
0.0249
0.0248
@
0.0247
i 0.0246
0.0245
0.0244
4
6
8
10
12
14
16
18
Pulse Period
Figure 3: Results of a preliminary numerical analysis of the crop/gut system showing that nutrient
assimilation rate declines with increasing pulse period, at least for some parameter values. Recall that the
average rate ofintake is held constant as the pulse period is increased.
@=Ak
where “¿hat―
symbols denote that fact that the terms have
been rescaled. In the remainder of this paper, I shall as
sume that all terms have been appropriately resealed,
and so the hat symbols will not be used.
The phase plane
on the null dine (we know that dV2/dt = 0 here, so any
movement on this curve must be parallel to the crop vol
ume V1axis). To decide whether the system is moving
up or down in crop volume, we need to consider the crop
volume equation, dV,/dt.
Since the input stream is a square wave, the crop vol
umc equation takes two forms: (1) a “¿crop
filling―form
@=h—(V0—
V2)V3
As mentioned above, the two volume equations com
pletely determine the system's dynamical behavior,
whereas the two remaining equations are simply mecha
nisms for score keeping. I focus, therefore, on trying to
understand the volume equations. To begin, I consider
the collection of crop ( V1)and gut ( V2)volumes where
dV2/dt equals zero (these points represents the so-called
V2null dine along which gut volume is not changing). If
we plot this collection of points
V2 = V0 in V1 —¿
V2space, we have a hyperbolic curve that ap
proaches V2= —¿
oo as V, approaches zero, and asymptot
ically approaches V0 as V@becomes large. Null dines
help one understand dynamic systems because we know
that the system can only be moving in one dimension
where h is the height of the square wave, or (2) a “¿crop
emptying―form
I@= —¿(V0—
V2)V1
Ofcourse, along the V2null dine, the term —¿(V0
—¿
V2)V1
=—¿r,
sothat
during
emptying
phases
the
crop
volumes
must be going down dV1/t = —¿r
along the null dine; and,
assuming h > r, crop volume must be increasing dV1/t =
h —¿
r during filling phases.
Next considering the null dines for crop volume V1,
we see that for the “¿filling―
equation there is a null-dine
at
V2 = V0 —¿
h/V,
This is exactly same form as the gut volume null dine,
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89
RELATIONSHIP BETWEEN DIGESTION AND FOOD INTAKE
A. Crop Emptying
B. Crop Filling
V2
V2
cr;j@
Null-Cline
V.
Gut Null-dune
Crop Null-dime
Crop Null-dine
V,
il
V,
C. Overall Behavior
V2
V.
-r
il
Figure 4:
(A) Null-dine
V1
analysis ofthe system when the crop is emptying.
The overall is a decrease in
both crop and gut volumes. (B) Analysis when the crop is filling. The trajectories are confined between the
two null dines, while crop and gut volumes move jointly upward toward the maximum crop volume
(which we have taken to be one). (C) Because our system requires that the average intake rate exceed the
outflow rate r, a limit cycle will eventually be reached in which an emptying system stays slightly above the
gut-volume null dine, whereas a filling system changes direction and climbs back to the maximum volume
while staying slight below the gut-volume null dine.
except that h takes the place of r. Since h > r by assump
tion, this null dine @5
crudely parallel and below the gut
volume null dine. Moreover, since the outflow term (V0
—¿
V2)V1 = h along this null dine, gut volume must be
increasing along this null dine, (i.e., dV2/dt = h —¿
r > 0).
For the emptying equation there are two null dines:
V2= V0
J―l= 0
“¿filling―
panels ofFigure 4. We see that when the system
is emptying, thejoint (J―@,
V2)trajectory will be attracted
to the region between the null dines and then will skirt
the gut-volume null dine from above as the crop and gut
volumes jointly decline. Similarly, when the system is
filling, the trajectory will skirt the gut-volume null dine
from below as the volumes jointly increase. I note that
our system is a plausible model only when the overall
rate ofintake exceeds the fixed outflow rate r(simply be
cause an empty gut cannot release material
In both cases, ofcourse, the outflow term ( V0—¿
V2)V, =
0, so the gut volume can only be decreasing dV2/dt =
—¿r.
These facts are put together in the “¿emptying―
and
at constant
rate r). With this assumption, it follows that at some
point, after many cycles offilling and emptying, the crop
volume will be at its maximum value. Once this happens
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90
D. W. STEPHENS
we would expectthe systemto follow a limit-cycle quali
tatively like that shown in Figure 4C: declining from the
maximum crop volume while staying slightly above the
gut-volume null dine, and then increasing to the maxi
mum crop volume while staying slightly below the gut
volume null dine.
Pseudo-equilibria
The qualitative observation that the trajectory skirts
the gut-volume null dine suggestsa major simplification
of the model; the assumption that dV2/dt = 0. Biologi
cally, this is the claim that the gut volume equilibrates
quickly to relatively slowly occuring changesin the crop
volume and input stream. Notice that to claim that the
gut volume is at equilibrium is not the sameasclaiming
that the gut volume is constant; we are simply supposing
that the system'strajectories slide up and down the gut
volume null dine (like a bead on a wire) rather than dir
culating around the gut-volume null dine as we argued
above. The idea of pseudo-equilibrium may be familiar
to studentsofenzyme kinetics; it is part ofthe traditional
derivation of the Michealis-Menten rate law (Murray,
1974).(In addition, one can often build upon a pseudo
equilibrium analysis to construct the “¿two
timescale so
lutions―
that typify the singular perturbation technique.)
Assuming that dV2/dt = 0 changesour systemto
@j@f-r
di
V1l
1,a(t)—r V1<l
V2= V0-@-
dt
@=r— r
r
A-kA
= Ak
Now at any instant, the rate ofchange ofcrop volume is
either —¿r
in an emptying phaseor h —¿
r in a filling phase,
so we expect that the volume ofcrop is either (a) empty
ing linearly at rate r, (b) full and unchanging, or (c) filling
linearly at rate h —¿
r. Our earlier numerical analysescon
firm that the crop-volume function is approximately
piecewise linear. Now consider two square wave input
functions that give the same overall input rate R, one
with a long period (F1) and a large peak input rate (h1),
and the other with a short period (P2) and small peak
input rate (h2)such that
A*=
r
k+
r
r
V0- —¿
V3
Using calculus, it is straightforward to show that this
equilibrium
amount
of nutrient
increases with increas
ing crop volume (the first derivative is alwayspositive) at
a decreasingrate (the second derivative is negative for
all biologically plausible parameter combinations). This
meansthat increasingthe crop volume by one unit hasa
big effecton the extraction rate ifthe crop volume is low,
but a one-unit increasein crop volume has only a small
effect if it is already high. This downward curvature
means that varying crop volumes are a bad thing; this
stems from a basic mathematical result called Jensen's
inequality (seeStephensand Krebs, 1986,chapter 7, for
an elementary discussion).
This conclusion suggeststhat even intake streamspro
vide two economic advantagesover uneven ones. First,
they allow the systemto maintain a higher mean extrac
tion rate because of the truncation effect discussed
above. Second,lower variance in crop volume translates
into higher extraction rates, becausethe amount of nu
trient in the gut increaseswith crop volume at a decreas
ing rate. In light ofthese kinds ofconsiderations one sees
predigestion storageorgans like the crop of Phormia as
devicesto attenuate oscillations in intake. Soperhapsbe
havioral ecologists should not be so surprised that ani
mals have strong preferencesabout the temporal organi
zation offood intake.
Discussion
R=@-@=@
P1
ageinput rate R exceedsthe outflow rate r, as discussed
above. Intuitively, it is easyto seethat the input stream
with the smaller pulse period will, in the long run, pro
duce both a higher mean crop volume and a smaller vari
ance in crop volume. This is essentially a truncation
effect: becauseR > r, the system eventually reachesa
point at which the crop is full V1 = 1 (in our resealed
units). Starting with a full crop, the long pulse-period sys
tem will drain for (P1—¿
w) time units at rate r, reaching
a minimum crop volume of 1 —¿
(P1—¿
w)r, similarly, the
short pulse-period system will drain for the shorter pe
riod P2 —¿
w at the same rate, reaching the larger mini
mum of 1 —¿
(P2—¿
w) r.
Now, imagine for the moment an experiment in which
we hold the crop volume fixed. In this casethe equation
for the amount of nutrient in the gut (which, in turn,
determines the nutrient extraction rate) is strongly at
tracted to the equilibrium amount:
P2
P1> P2and h1> h2. In addition, I assumethat the aver
I have considered a simple caricature of the relation
ship between food intake and digestion. Indeed, I have
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RELATIONSHIP BETWEEN DIGESTION AND FOOD INTAKE
91
only sketched an analysis of this system. Despite its de
Acknowledgments
fects,my analysisshowsa possiblelink betweendigestive
I am grateful for the helpful comments of Peter Bed
regulation and an important outstanding problem in
nekof and Tom Getty, and for the continued financial
feeding ecology—animal preferences for immediate
support of the National Science Foundation (IBN
food reward. Many complications await further analysis:
8958228and IBN-9507668).
What ifmore realistic reaction kinetics are used?What if
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