Antagonistic Redundancy - A Theory of Error-Correcting Information Transfer in Organisms

225
J. W. Dietrich, B. O. Boehm
Abteilung Innere Medizin I, Medizinische Klinik und Poliklinik, UniversitŠtsklinikum Ulm,
Robert-Koch-Str. 8, D-89081 Ulm, F. R. Germany
[johannes.dietrich | bernhard.boehm]@medizin.uni-ulm.de
Abstract
Living organisms are exposed to numerous
influencing factors. This holds also true for
their infrastructures that are processing and
transducing information like endocrine
networks or nerval channels. Therefore, the
ability to compensate for noise is crucial for
survival. An efficient mechanism to neutralise
disturbances is instantiated in form of parallel
complementary communication channels
exerting antagonistic effects at their common
receivers. Different signal processing types
share the ability to suppress noise, to widen
the system's regulation capacity, and to
provide for variable gains while leaving the
transferred signal to a large extent unchanged.
1. Introduction
Redundancy is a common phenomenon in living
systems. Its high costs in terms of energy, space
consumption and ontogenetic complexity are balanced
by a significant gain of robustness against damages
[Kitano 2001, Krakauer and Plotkin 2002]. Redundant
structures can also be found at the level of processing
circuits. While they obviously contribute to fault
tolerance it is also conceivable that this kind of
parallelity has some impact on the quality of
transmitted signals.
The large number of variables that are able to irritate
processing and transduction structures has its origin
both outside and inside the organism. While known
load factors can be compensated for by means of
feedforward mechanisms and efference copies [von
Holst and Mittelstaedt, 1950], the situation is different
for unknown influences like noise signals.
The following study aims to evaluate how
complementary antagonistic processing structures
influence the impact of noise on the quality of signal
transfer.
As a starting point the control of plasma glucose by the
two polar hormones glucagon and insulin Ð a fragment
of a more complex network Ð may be considered. Both
hormones are antidromicly controlled by the plasma
glucose level and act mutually oppositionally on the
production and degradation of glucose (Fig. 1).
Plasma
Glucose
Level
Insulin
Glucagon
Tissue Glucose Level
b-Cells
a-Cells
Pancreas
Adrenal
Medulla
Adrenal
Cortex
Hypothalamus
Pituitary
CRH
ACTH
Cortisol
Epinephrine
+
+ +
+–
– –
+
+
+
Figure 1: Control of plasma glucose level by several
complementary hormones. Mutual influences of insulin on
glucagon secretion and vice versa and of epinephrine on
insulin and glucagon secretion are not shown.
2. Models
2 .1 Simplif ied mo dels
A first very plain model (Figure 2) may start with an
original signal a encoded in the subtractively
complementary channels b1 and b2 that are given with

b1 = a (1)
and

b2 = – a
. (2)
Then the output signal c is defined with
c =
b1 – b2
2 = a
. (3)
The two channels b1 and b2 can be subsumed as
mediation vector b with
b = Gij a (4)
Antagonistic Redundancy Ð A Theory of Error-Correcting
Information Transfer in Organisms
CYBERNETICS AND SYSTEMS
2004

226
and
c = Hij b
, (5)
where G ij denotes a dispatcher matrix and H ij a
collector matrix performing a summation of the
mediation vector‘s components. In the example
depicted above the assumption is
i = 1; j = 2, (6)
but the vector form allows for the formulation of even
multiple transmission channels.
+ +a c
b
b
0,5
1
2
Figure 2: Subtractive antagonistic redundancy. See the
appendix for an explanation of symbols.
Similarly, in a divisive model from eqn. (1) and
b2 = 1a (7)
the output follows with

c =
b1
b2 = a
. (8)
a c
b
b
1
2
Figure 3: Divisive antagonistic redundancy
Logarithmic transformation of the mediation variables
bi allows for using the dispatcher and collector matrices
of equations (4) and (5) for the divisive model, too.
2 .2 Micha elis-Menten-H ill K inet ics
In organisms signal transduction is usually
implemented as Michaelis-Menten-Hill (MMH)
kinetics
ya = GxeD + xe
. (9)
According to this Langmuir-equation we get the
agonistic channel b1 with
b1 = Gb1aDb1 + a (10)
and the antagonistic channel as result of a non-
competitive inhibition process [Zech and Domagk
1986, Dietrich 2002, Dietrich et al. 2002] with
b2 = 1
1 + Gb2aDb2 + a (11)
(Figure 4).
a c
b
b
+
+
+
Gb1
Db1
1
+
Hb1
Db2 +
Hb2 +
1
1
2
Eb1
Eb2
Gb2
Figure 4: Antagonistic Redundancy with Michaelis-Menten-
Hill kinetics and non-competitive inhibition.
From
c =
Hb1b1
(Eb1 + b1)(1+ Hb2b2Eb2 + b2 ) (12)
the output signal follows as a function of a with

c = Gb1Hb1a /

[(Db1 + a)(Eb1 + Gb1 aDb1 + a ) (13)

(1+ Hb2(Db2 + a)Db2 + a + Eb2a + Db2Eb2 + Eb2Gb2a )] .
2 .3 Inhibit ing subsy stems
Subsystems with inhibiting effect have generally the
same formal structure as depicted above, except for an
inverted output stage (Figure 5). Especially in
endocrine intercausal networks MMH kinetics with
inverted structure are frequently to be found as
components of feedback control circuits.
a c
b
b
+
+
+
Gb1
Db1
1
+
Hb1
Db2 +
Hb2
1
1
2
Eb1
Eb2
Gb2
+
Figure 5: Antagonistic Redundancy using Michaelis-Menten-
Hill kinetics with inverting characteristics.

227
Similar to equation (12) c is defined by
c =
Hb2b2
(Eb2 + b2)(1+ Hb1b1Eb1 + b1 )
. (14)
Then in dependence of a c is given with

c = Hb2 / (15)

1 + Gb2aDb2 + a

Eb2 + 1
1 + Gb2aDb2 + a

1 + Hb1Gb1a
Db1 + a Eb1 + Gb1aDb1 + a
3. Results
Assuming that in the simple subtractive model exactly
one channel is disturbed with an additively
superimposed noise signal z with

b1 = a + z (16)
then in the resulting signal
c = a + z2 (17)
the noise influence is halved. In the case of both
channels being disturbed with the same interfering
signal the perturbation will be entirely eliminated.
With the additive noise signal of eqn (16) c is in the
divisive model given with
c = a (a + z)
. (18)
If both signals are simultaneously disturbed with eqn
(16) and
b2 = 1a + z (19)
then c follows with

c =
a (a + z)
1 + az (20)
As shown in Figs. 6 and 7 in the cases of both one and
two channels being additively disturbed the influence
of erroneous signals is significantly diminished in a
system with Antagonistic Redundancy (AR) compared
to transmission with single channels.
5 10 15 20
z
10
15
20
25
c
Single
Channel
Antagonistic
Redundancy
Figure 6: Comparison of the effects of additive noise (z)
impacting one channel only on the output signal of
unichannel transmission and on a system with divisive
antagonistic redundancy.
2 4 6 8 10
z
2
4
6
8
10
12
14
c
Single
Channels
Antagonistic Redundancy
Figure 7: The two additively disturbed channels b1 and b2
and the resulting signal c in divisive antagonistic redundancy.
If both signals are disturbed in a multiplicative manner
then with
c =
a z
z
a
= a
(21)
the noise is completely eliminated.
Models with Michaelis-Menten-Hill Kinetics show a
more complex behaviour. Nevertheless, they are also
capable of reducing the dependency of c from the error
signal z significantly (Figure 8 and Figure 9).
In a similar way, in inverted MMH kinetics AR is able
to reduce the influence of noise signals on the output
signal, too (Figs. 10 and 11).
2 4 6 8 10
z
1
2
3
4
c
MMH Kinetics
AR with single sided error
AR with single sided error
AR with error on both channels
Figure 8: Dependency between z and c for a single channel
with Michaelis-Menten-Hill kinetics and for AR with the
disturbance signal influencing only the agonistic or the
antagonistic channel, respectively, and for AR with both
channels disturbed.

228
0.6
0.8
1
1.2
1.4
z
0
2
4
6
8
10
a
0
1
2
3

0.6
0.8
1
1.2
1.4
z
0
2
4
6
8
10
a
0
0.5
1
1.5
2

Figure 9: The relation between a, z and c in transfer systems
with MMH kinetics shows the reduction of the influence of z
on c by antagonistic redundance (bottom) when compared to
a single channel (top). Note the different scales of the vertical
axis.
0.2 0.4 0.6 0.8 1 1.2 1.4
z
0.2
0.4
0.6
0.8
1
c
b only2
b only1
AR
0.2 0.4 0.6 0.8 1 1.2 1.4
z
0.1
0.2
0.3
0.4
0.5
0.6
c
b only2b only1
AR
Figure 10: Inhibiting MMH kinetics with AR: Relation
between z and c for a=1 (top) and a=10 (bottom).
0.005 0.01 0.015 0.02 0.025 0.03
5. 10
-10
1. 10
-9
1.5 10
-9
2. 10
-9
Plasma Glucose
Vi
rtu
al
O
ut
pu
t S
ig
na
l
Insulin
Glucagon
AR
0.0005 0.001 0.0015 0.002 0.0025 0.003
Glucose
2
4
6
8
Glucagon
Inverted Insulin
0.6 0.8 1.2 1.4
z
4. 10
-10
5. 10
-10
6. 10
-10
7. 10
-10 Glucagon
Insulin AR
AR
Figure 12: Dependency of insulin, glucagon and a unified
output from glucose (top) and an error signal (bottom). Inset
in the upper panel: Comparison of glucagon and a virtual
signal derived from inverted insulin. Note that the glucagon
concentration has been multiplied with 1013 to make its
curves visible.
The extent of error extinction in coupled MMH
kinetics depends from the parameters of the respective
network. As shown in Figure 12 the subsystem of
glucagon and insulin controlling the glucose level in
blood plasma shows only a slight error reduction
[Parameters from Cascieri et al. 1999, Cypess et al.
1999, Hjorth et al. 1998, Hoefig et al. 1996,
Kruszynska et al. 1998, Kurstjens et al. 1990,
Lundquist and Panagiotidis 1992, Lynch et al. 1996,
Marks and Botelho 1986 and Rizza et al. 1981]. This
restriction is a consequence of the high nonlinear
amplification Gb2 of the input stage of the glucagon
path resulting in very low glucagon concentrations (see
Figures 5 and 12, where the glucagon concentration
had to be multiplied with the factor 1013 to be visible).
In vivo, this asymmetry is adjusted by the fact that
insulin has multiple additional antagonists like
catecholamines and glucocorticoids. In any case, each
hormone compensates for the limitations in the effect
of the respective other hormone. Where the glucagon-
level is very small near the threshold of undetectability
insulin is in the range of high concentrations.
Conversely, in the situation of low plasma glucose
glucagon shows an intense response (inset of Figure
12).

229
0.6
0.8
1
1.2
1.4
z
0
2
4
6
8
10
a
0.25
0.3
0.35
0.4

0.6
0.8
1
1.2
1.4
z
0
2
4
6
8
10
a
0.25
0.5
0.75
1
1.25

0.6
0.8
1
1.2
1.4
z
0
2
4
6
8
10
a
0.1
0.2
0.3

Figure 11: Effect of input a and error z on the output signal c
in single channel transfer with b1 (top) or b2 (middle),
respectively, compared to antagonistic redundancy (bottom)
in a system with inverting MMH kinetics.
4. Discussion
Information processing structures with antagonistic
redundance – e. g. in the autonomous nervous system –
have been known for decades without being described
mathematically. In 1969 Manfred Clynes reported on
an organizational principle compensating for
unidirectional rate sensitivity in neural signal
transmission [Clynes 1969] which he called rein
control. A functional – albeit not quantitative –
description has been formulated in the early 1970s
when Sachsse [1971] described bipolar control
structures in the autonomous and central nervous
system. A special case of antagonistic organization in
integral feedback control systems has been illustrated
by Saunders et al. [1998 and 2000].
These models have significantly contributed to our
understanding of parallel complementary signal-
transduction even though they abstained from
systematically analysing the influence of error signals
on the quality of the transmitted information.
AR can be found in different types of biological
information transfer. Besides the autonomous nervous
system where centrifugal control signals are
transmitted via the two complementary sympathetic
and parasympathetic systems antagonistic redundance
is common in the somatic nervous system, too. Due to
mechanical necessity scelettal muscles are arranged in
an antagonistic manner. This finds its reflection in the
information processing structure of the corresponding
nerval organization. Other examples of nerval encoded
AR are the implementation of temperature perception
with different receptor types for cold and warm stimuli
and the joint control system for respiration and acid-
base balance which is not only equipped with distinct
receptors for e.g. O2 and acidity but that is also
antagonistically organised on the level of control in the
medulla oblongata.
Numerous endocrine transfer systems share
antagonistic organization. Important examples to be
mentioned are the complementary effects (and
regulation) of growth-hormone RH and somatostatin,
TRH and somatostatin, leptin and ghrelin, and
parathyroid hormone and calcitonin. Apart from
glucagon insulin has multiple other antagonists like
catecholamines and cortisol.
Endocrine control systems are easily disturbed by
exogenous and endogenous interfering factors, first of
all variations in the body fluid balance. Intake of water
after fluid restriction rapidly decreases the
concentrations of all hormones. However, its impact on
the controlled system is considerably reduced, as all
antagonistic channels are disturbed in the same extent.
In pathological situations the function of one channel
can be impaired or delayed. As seen from the examples
this results in higher sensitivity to interference and,
depending from the affected information processing
structure, narrowed control range.
For decades feedback control systems have been
known as universal compensation mechanisms for
various interference factors in organisms, and their
characterisation has been one of the classic objects of
biological cybernetics. The reafference principle [von
Holst and Mittelstaedt, 1950] has been described as a
first adjustment circuit in form of feedforward control.
Reafferences balance the efferent output of the nervous
system that would otherwise disturbe the interpretation
of sensory perception. Unlike efference copies
compensating for pre-known disturbance factors
antagonistic redundancy provides for elimination or at
least attenuation of a priori unknown interfering

230
signals like the effect of blood plasma’s dilution on the
concentration of hormones.
It may be expected that further examples of
antagonistic redundancy will be found in the hormonal
and the nervous system and even in the cell. To
evaluate this structure’s consequences on physiology
and pathology of several information processing
systems is a task that remains for the future.
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Appendix
Legend for IPS symbols:
x1
x2
y
Multiplication Point
x2y =

x1
Filled sectors denote inversion
(here: division).
Empty input signals are neutral
(1 for muliplication and division).
+
x1
x2
y
Addition Point
y = x2 – x1
Filled sectors denote inversion
(here: subtraction).
Empty input signals are neutral
(0 for addition and subtraction).
G yx
Constant Gain
y = Gx
yx
Special Function
y = sqrt(x )
Symbols used (see the OEP-site http://open-site.org/Science/
Mathematics/Applied/Cybernetics/K1_and_K2_-_General_
Cybernetics/Systems_Science/Information_Processing_Struct
ures/)