262 
ON TROPISTIC PROCESSING AND ITS APPLICATIONS 
Manuel F. FernRndez 
General Electric Advanced Technology Laboratories 
Syracuse, New York 13221 
ABSTRACT 
The interaction of a set of tropisms is sufficient in many 
cases to explain the seemingly complex behavioral responses 
exhibited by varied classes of biological systems to combinations of 
stimuli. It can be shown that a straightforward generalization of 
the tropism phenomenon allows the efficient implementation of 
effective algorithms which appear to respond "intelligently" to 
changing environmental conditions. Examples of the utilization of 
troplstic processing techniques will be presented in this paper in 
applications entailing simulated behavior synthesis, path-planning, 
pattern analysis (clustering), and engineering design optimization. 
INTRODUCTION 
The goal of this paper is to present an intuitive overview of 
a general unsupervised procedure for addressing a variety of system 
control and cost minimization problems. This procedure is based on 
the idea of utilizing "stimuli" produced by the environment in which 
the systems are designed to operate as basis for dynamically 
providing the necessary system parameter updates. 
This is by no means a new idea: countless examples of this 
approach abound in nature, where innate reactions to specific 
stimuli ("tropisms" or "taxis" --not to be confused with 
"instincts") provide organisms with built-in first-order control 
laws for triggering varied responses [8]. (It is hypothesized that 
"knowledge" obtained through evolution/adaptation or through 
learning then refines or suppresses most of these primal reactions). 
Several examples of the implicit utilization of this approach 
can also be found in the literature, in applications ranging from 
behavior modeling to pattern analysis. We very briefly depict some 
these applications, underlining a common pattern in their 
formulation and generalizing it through the use of basic field 
theory concepts and representations. A more rigorous and detailed 
exposition --regarding both mathematic and 
application/implementation aspects-- is presently under preparation 
and should be ready for publication sometime next year ([6]). 
TROPI SMS 
Tropisms can be defined in general as class-invariant systemic 
responses to specific sets of stimuli [6]. All time-invariant 
systems can thus be viewed as tropistic provided that we allow all 
possible stimuli to form part of our set of inputs. In most 
tropistic systems, however, response- (or time-) invariance applies 
only to specific inputs: green plants, for example, twist and grow 
in the direction of light (phototropism), some birds  flight 
patterns follow changes in the Earth's magnetic field 
(magnetotropism), various organisms react to gravitational field 
@ American Institute of Physics 1988 
263 
variations (geotropism), etc. 
Tropism/stimuli interactions can be portrayed in term of the 
superposition of scalar (e.g., potential) or vector (e.g., force) 
fields exhibiting properties paralleling those of the suitably 
constrained "reactions" we wish to model [1],[6]. The resulting 
field can then be used as a basis for assessing the intrinsic cost 
of pursuing any given path of action, and standard techniques (e.g., 
gradient-following in the case of scalar fields or divergence 
computation in the case of vector fields) utilized in determining a 
response*. In addition, the global view of the situation provided by 
field representations suggest that a basic theory of tropistic 
behavior can also be formulated in terms of energy expenditure 
minimization (Euler-Lagrange equations). This formulation would 
yield integral-based representations (Feynman path integrals 
[4],[11]) satisfying the observation that tropistic processes 
typically obey the principle of least action. 
Alternatively, fields may also be collapsed into "attractors" 
(points of a given "mass" or "charge" in cost space) through laws 
defining the relationships that are to exist among these 
"attractors" and the other particles traveling through the space. 
This provides the simplification that when updating dynamically 
changing situations only the effects caused by the interaction of 
the attractors with the particles of interest --rather than the 
whole cost field-- may have to be recalculated. 
For example, appropriately positioned point charges exerting 
on each other an electrostatic force inversely proportional to the 
square of their distance can be used to represent the effects of a 
coulombic-type cost potential field. A particle traveling through 
this field would now be affected by the combination of forces 
ensuing from the interaction of the attractors  charges with its 
own. If this particle were then to passively follow the composite of 
the effects of these forces it would be following the gradient of 
the cost field (i.e., the vector resulting from the superposition of 
the forces acting on the particle would point in the direction of 
steepest change in potential). 
Finally, other representations of tropism/stimuli interactions 
(e.g., Value-Driven Decision Theory approaches) entail associating 
"profit" functions (usually sigmoidal) with each tropism, modeling 
the relative desirability of triggering a reaction as a function of 
the time since it was last activated [9]. These representations are 
* In order to bring extra insight into tropism/stimuli 
interactions and simplify their formulation, one may exchange vector 
and scalar field representations through the utilization of 
appropriately selected mappings. Some of the most important of such 
mappings are the gradient operator (particularly so because the 
gradient of a scalar --potential-- field is proportional to a 
"force" --vector-- field), the divergence (which may be thought of 
as performing in vector fields a function analogous to that 
performed in scalar fields by the gradient), and their combinations 
(e.g., the Laplacian, a scalar-to-scalar mapping which can be 
visualized as performing on potential fields the equivalent of a 
second derivative operation. 
264 
 Model fly as a positive geotropistic point of mass M. 
 Model fence stakes as negative geotropistic points 
with masses m t, m  ..... m . 
 At each update time compute sum of forces acting on 
frog: 
F-k 
d I N 
i-I d m 
m I 
Compute frog's heading and acceleration based on 
the ensuing force: then update frog's position, 
Figure 1: Attractor-based representation of a frog-fence-fly 
scenario (see [1] for a vector-field representation). The objective 
is to model a frog's path-planning decision-making process when 
approaching a fly in the presence of obstacles. (The picket fence is 
represented by the elliptical outline with an opening in the back, 
the fly --inside the fenced space-- is represented by a "+" sign, 
and arrows are used to indicate the direction of a frog's trajectory 
into and out of fenced area). 
265 
particularly amenable to neural-net implementations [6]. 
TROPISTIC PROCESSING 
Tropistic processing entails building into systems tropisms 
appropriate for the environment in which these systems are expected 
to operate. This allows taking advantage of environment-produced 
"stimuli" for providing the required control for the systems' 
behavior. 
The idea of tropistic processing has been utilized with good 
results in a variety of applications. Arbib et.al., for example, 
have implicitly utilized tropistic processing to describe a 
batrachian's reaction to its environment in terms of what may be 
visualized as magnetic (vector) fields' interactions [1]. 
Watanabe [12] devised for pattern analysis purposes an 
interaction of tropisms ("geotropisms") in which pattern "atoms" are 
attracted to each other, and hence "clustered", subject to a 
squared-inverse-distance ("feature distance") law similiar to that 
from gravitational mechanics. It can be seen that if each pattern 
atom were considered an "organism", its behavior would not be 
conceptually different from that exhibited by Arbibian frogs: in 
both cases organisms passively follow the force vectors resulting 
from the interaction of the environmental stimuli with the 
organisms' tropisms. It is interesting, though, to note that the 
"organisms'" behavior will nonetheless appear "intelligent" to the 
casual observer. 
The ability of tropistic processes to emulate seemingly 
rational behavior is now begining to be explored and utilized in the 
development of synthetic-psychological models and experiments. 
Braitenberg, for example, has placed tropisms as the primal building 
block from which his models for cognition, reason, and emotions 
evolve [3]**; Barto [2] has suggested the possibility of combining 
tropisms and associative (reinforced) learning, with aims at 
enabling the automatic triggering of behavioral responses by 
previously experienced situations; and Fernandez [6] has used 
CROBOTS [10], a virtual multiprocessor emulator, as laboratory for 
evaluating the effects of modifying tropistic responses on the basis 
of their projected future consequences. 
Other applications of tropistic processing presently being 
investigated include path-planning and engineering design 
optimization [6]. For example, consider an air-reconnaissance 
mission deep behind enemy lines; as the mission progresses and 
unexpected SAM sites are discovered, contingency flight paths may be 
developed in real time simply by modeling each SAM or interdiction 
site as a mass point towards which the aircraft exhibits negative 
geotropistic tendencies (i.e., gravitational forces repel it), and 
modeling the objective as a positive geotropistic point. A path to 
** Of particular interest within the sole context of Tropistic 
Processing is Dewdney's [5] commented version of the first chapters 
of Braitenbergts book [3], in which the "behavior" of mechanically 
very simple cars, provided with "eyes" and phototropism-supporting 
connections (including Ledley-type "neurons" [4]), is "analyzed". 
266 
 
 
 
Figure 2 (Geotropistic clustering [12]): The problem being 
 ............ portrayed 
here is t at of clustering dots distributed in [x,y]-space as shown 
and uniformly in color ([red,blue,green]). The approach followed is 
that outlined in Figure 1, with the differences that normalized 
(Mahalanobis) distances are used and when merges occur, conservation 
of momentum is observed. Tags are also kept --specifying with which 
dots and in what order merges occur-- to a].low drawing cluster 
boundaries in the original data set. (Efficient implementation of 
this clustering technique entails using a ring of processors, each 
of which is assigned the "features" of one or more "dots" and the 
task of carrying out computations with respect to these features. If 
the features of each dot are then transmitted through the ring, all 
the forces imposed on it by the rest will have been determined upon 
completion of the circuit). 
267 
the target will then be automatically drawn by the interaction of 
the tropisms with the gravitational [orces. (Once the mission has 
been completed, the'target and its effects can be eliminated, 
leaving active only the repulsive forces, which will then "guide" 
the airplane out of the danger zone). 
In engineering design applications such as lens modeling and 
design, lenses (gradient-index type, for example) can be modeled in 
terms of photons attempting to reach an objective plane through a 
three-dimensional scalar field of refraction indices; modeling the 
process tropistically (in a manner analogous to that of the 
air-reconnaissance example above) would yield the least-action paths 
that the individual photons would follow. Similarly, in 
"surface-of-revolution" fuselage design ("Newton's Problem"), the 
characteristics of the interaction of forces acting within a sheet 
of metal foil when external forces (collisions with a fluid's 
molecules) are applied can be modeled in terms of tropistic 
reactions which will tend to reconfigure the sheet so as to make it 
present the least resistance to friction when traversing a fluid. 
Additional applications of tropistic processing include target 
tracking and multisensor fusion (both can be considered instances of 
"clustering") 6], resource allocation and game theory (both closely 
related to path-planning) [9], and an assortment of other 
cost-minimization functions. Overall, however, one of the most 
important applications of tropistic processing may be in the 
modeling and understanding of analog processes 6], the imitation of 
which may in turn lead to the development of effective strategies 
PAST EXPERIENCE 
(e.g. MEMORY MAPS) 
RESPONSE 
FUNCTION 
PREDICTED (i.e. MODELLED) 
OUTCOME 
 RESPONSE 
TROPISM-BASED SYSTEM 
Figure 3: The combination of tropisms and associative (reinforced) 
learning-can be used to enable the automatic triggering of 
behavioral responses by previously experienced situations [2]. Also, 
the modeled projection of the future consequences of a tropistic 
decision can be utilized in the modification of such decision |6]. 
(Note analogy to filtering problem in which past history and 
predicted behavior are used to smooth present observations). 
268 
/" 
Figure 4: Simplified representation of air-reconnaissance mission 
example (see text): objective is at center of coordinate axis, thick 
dots represent SAM sites, and arrows denote airp]anes direction of 
flight (airplane's maximum attainable speed and acceleration are 
constrained). All portrayed scenarios are identical except for 
tropistic control-law parameters (mainly objective to SAM-sites mass 
ratios in the first three scenarios). Varying the masses of the 
objective and SAM sites can be interpreted as trading off the 
relative importance of the mission vs. the aircrafts safety, and 
can produce dramatically differing flight paths, induce chaotic 
behavior (bottom-left scenario), or render the system unstable. The 
bottom-right scenario portrays the situation in which a tropistic 
decision is projected into the future and, if not meeting some 
criterion, modified (altering the direction of flight --e.g., 
following an isokline--, re-evaluating the mission's relative 
importance --revising masses--, changing the update rate, etc.). 
269 
for taking full advantage of parallel architectures [11]***. It is 
thus expected that the flexibility of tropistic processes to adapt 
to changing environmental conditions will prove highly valuable to 
the advancement of areas such as robotics, parallel processing and 
artificial intelligence, where at the very least they will provide 
some decision-making capabilities whenever unforeseen circumstances 
are encountered. 
ACKNOWLEDGEMENTS 
Special thanks to D. P. Bray for the ideas provided in our 
many discussions and for the development of the finely detailed 
simulations that have enabled the visualization of unexpected 
aspects of our work. 
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*** Optical Fourier transform operations, for instance, can be 
modeled in high-granularity machines through a procedure analogous 
to the gradient-index lens simulation example, with processors 
representing diffraction-grating "atoms" [6]. 
