“Lateral Inhibition” in a Fully Distributed Connectionist Architecture
Simon D. Levy (levys@wlu.edu)
Department of Computer Science, Washington and Lee University
Lexington, VA 24450 USA
Ross W. Gayler (r.gayler@gmail.com)
Department of Psychology, La Trobe University
Victoria 3086 Australia
Abstract
We present a fully distributed connectionist architecture sup-
porting lateral inhibition / winner-takes all competition. All
items (individuals, relations, and structures) are represented by
high-dimensional distributed vectors, and (multi)sets of items
as the sum of such vectors. The architecture uses a neurally
plausible permutation circuit to support a multiset intersec-
tion operation without decomposing the summed vector into
its constituent items or requiring more hardware for more com-
plex representations. Iterating this operation produces a vector
in which an initially slightly favored item comes to dominate
the others. This result (1) challenges the view that lateral in-
hibition calls for localist representation; and (2) points toward
a neural implementation where more complex representations
do not require more complex hardware.
Keywords: Lateral inhibition; winner-takes-all; connection-
ism; distributed representation; Vector Symbolic Architecture
Introduction
Connectionist representations are typically classified as local-
ist, distributed, or some combination of both. In a localist rep-
resentation each node corresponds to a single item or concept.
In a distributed representation each node participates in the
representation of every concept, and each concept is “spread
out” (distributed) among every node. Proponents of local-
ist representation cite simplicity and transparency as benefits
of localist coding. Proponents of distributed representations
argue that the robustness of such representations in the pres-
ence of noise makes them more plausible and appealing, and
cite related impressive work on modeling neuropsychological
disorders using distributed connectionist representations. For
a review see Olson & Humphreys (1997). A comprehensive
argument for distributed representations is of course beyond
the scope of this article. We will focus here instead on a par-
ticular capability that appears to be exclusive to localist rep-
resentations, and will provide an alternative analysis using a
distributed representation.
In a 2000 target article in Behavioral and Brain Sciences,
Page (2000) argues for a “generalized localist model” with a
localist representation on one layer and general (distributed)
o representations on the others. Each node in the localist layer
is associated with a category, and a lateral inhibition (winner-
takes-all competition) function is used, allowing the localist
layer to act as a classifier for (distributed) patterns on an in-
coming layer. Indeed, the ability of localist representations
to support competitive classification seems to be the main ap-
peal of localism, as suggested by the remarks of the commen-
tators who supported Page’s position (e.g. Phaf & Wolters,
2000).
In this article we will argue that localist representations
are not necessary to support winner-takes-all competition or
lateral inhibition in general. We will present a fully dis-
tributed connectionist architecture supporting lateral inhibi-
tion / winner-takes all behavior, in which all items (indi-
viduals, relations, and structures) are represented by high-
dimensional distributed vectors, and (multi)sets of items as
the sum of such vectors. Unlike a localist representation, such
representations are based on a fixed neural architecture that
does not need to grow as new representational categories are
added.
Problems with Localism
The greatest challenge to connectionist accounts of cognition
continues to be the problem of compostionality, that is, the
problem of how to put simpler items like words and concepts
together to make more complex structures like sentences and
propositions (Fodor & Pylyshyn, 1988; Jackendoff, 2002).
Localist connectionism addresses this challenge by assigning
one neuron or pool of neurons to each item, and employing
additional (pools of) neurons as higher-order elements for or-
ganizing the simpler items via physical connections or tem-
poral synchrony. For example, the Neural Blackboard Archi-
tecture of van der Velde (2006) builds sentences out of words
via “structure assemblies” corresponding to traditional syn-
tactic categories like Noun Phrase and Verb Phrase. Hummel
and Holyoak’s LISA model of analogical mapping (Hummel
& Holyoak, 1997) uses higher-order assemblies to represent
the bindings of individuals to semantic roles like agent and
patient.
In a forthcoming article, Stewart and Eliasmith (Stewart
& Eliasmith, forthcoming) provide a detailed analysis of the
computational complexity entailed by localist accounts of
composition. This analysis suggests that the need to have
physical connections between all pairs of items causes lo-
calist representations lead to a combinatorial explosion when
applied to realistically-sized item inventories, such as the vo-
cabularies of natural languages. An alternative approach,
which dates back to the work of Pollack (1990) and others,
attempts to encode structures of arbitrary complexity on a
fixed-size connectionist architecture.1 Commenting on Pol-
1A serious limitation of Pollack’s Recursive Auto-Associative
(RAAM) network was the need to learn representations (via back-
propagation). The work presented here avoids the need for learning,
318

lack’s results, Hammerton (1998) notes that it is important to
consider whether the representations produced by such archi-
tectures can be manipulated holistically, or whether they re-
quire “functional localism”, such as serial extraction of com-
ponents, in order to support useful computations. Even if it
is neurally plausible, which seems unlikely, functional local-
ism strikes us as essentially an implementation of classical
symbol processing (cf. Fodor & Pylyshyn, 1988), foregoing
much of the appeal of connectionism.
Another problem with localist implementation of lateral in-
hibition is that the system can only implement winner-takes-
all; that is, the result is the choice of 1 out of k alternatives.
For some problems, however, it would be more appropriate to
have a set of answers returned. It is not clear how this would
be achievable with localist winner-take-all implementation.
What is needed is a new type of network that exhibits the
attractor dynamics of localist winner-takes-all networks, but
which can converge simultaneously to a set of items, rather
than a single item.
The work presented here addresses these issues, providing
a holistic implementation of an operation previously thought
to require localist coding.
Vector Symbolic Architectures
Vector Symbolic Architecture is a name that we coined
to describe a class of connectionist models that use high-
dimensional vectors (with as few as 1000 dimensions, but
more typically around 10,000) of low-precision numbers to
encode structured information as distributed representations.
That is, VSAs can represent complex entities such as trees
and graphs; and every such entity, no matter how simple
or complex, is represented by a pattern of activation dis-
tributed over all the elements of the vector. This general
class of architectures traces its origins to the tensor prod-
uct work of Smolensky (1990), but avoids the exponential
growth in dimensionality of tensor products. The currently
available VSAs employ three types of operation on vectors: a
multiplication-like operator, an addition-like operator, and a
permutation-like operator. The multiplication-like operation
is used to associate or bind vectors. The addition-like oper-
ation is used to superpose vectors or add them to a set. The
permutation-like operation is used to quote or protect vectors
from the other operations.
The use of hyperdimensional vectors to represent sym-
bols and their combinations provides a number of mathe-
matically desirable and biologically realistic features. A hy-
perdimensional vector space can contain as many mutually
orthogonal vectors as there are dimensions, and exponen-
tially many almost-orthogonal vectors (Hecht-Nielsen, 1994),
thereby supporting the representation of astronomically large
numbers of distinct items. Such representations are also
highly robust to noise: a significant fraction of the values in
a vector can be randomly changed before it becomes more
by relying on fixed, constant-time mechanisms for associating and
composing vector representations.
similar to another vector than to its original form. To cite a
result from a forthcoming paper by Kanerva (in press): When
meaningful entities are represented by 10,000-[element] vec-
tors, many of the bits can be changed more than a third by
natural variation in stimulus and by random errors and noise,
and the resulting vector can still be identified with the correct
one, in that it is closer to the original “error-free” vector than
to any unrelated vector chosen so far, with near certainty. It
is also possible to implement such vectors in a spiking neu-
ron model (Eliasmith, 2005), lending them a further degree
of biological plausibility.
The main difference among types of VSAs is in the kind
of numbers used as vector elements and the related choice of
multiplication-like operation. Holographic Reduced Repre-
sentations (Plate, 2003) use real numbers and circular con-
volution. Binary Spatter Codes (Kanerva, 1994) use binary
(Boolean) values and elementwise exclusive-or. MAP (Mul-
tiply, Add, Permute) coding (Gayler, 1998) uses bipolar (-
1/+1) values and elementwise multiplication. A useful feature
of BSC and MAP is that every vector is its own multiplicative
inverse: multiplying a vector by itself elementwise yields the
multiplicative identity vector (A A = 1 = B B, where 1 is
the identity vector, but A+A = 2A). As in ordinary algebra,
multiplication and addition are associative and commutative,
and multiplication distributes over addition.
We used MAP in the work described here. In MAP, prop-
erties are accumulated through vector addition; hence, it is
trivial to have multiple, self-reinforcing copies of the same
property (vector) in a single representation. For example,
given a vector representation A of the property affluent and
a vector representation B of the property brave, the represen-
tation A+A+B = 2A+B could represent being very affluent
and somewhat brave. Second, the association of two rep-
resentations through elementwise multiplication produces a
third representation that is completely dissimilar from both
elements. If C represents an individual, say, Charlie, the
proposition that Charlie is brave could be represented as BC,
whose similarity (vector cosine) with both B and C is close to
zero. Together, these facts mean that a given entity can be
associated with a large number of properties (and vice versa):
C  (2A+B), etc.
Without an additional mechanism, self-cancellation would
pose a challenge when copies of structures are embedded in
themselves recursively. For example, if D1 and D2 repre-
sented the semantic roles doubter and (thing) doubted, then
one possible way to represent the proposition Bill doubted
that Charlie doubted that Ed is affluent as
D1 B+D2  (D1 C+D2 AE)
Without further modification, the two copies of D2 would
have the undesired affect of canceling each other out. As
mentioned above and discussed at length in (Levy, to appear),
the permutation operator of the MAP architecture provides a
neurally plausible mechanism for quoting or protecting vec-
tors in these situations.
319

As an example of holistic computation in MAP, consider
the common task of retrieving a set of items associated with
a given property. We imagine three individuals: A and B
having property P and C having property Q. In a MAP en-
coding, each individual and property would be encoded in a
hyper-dimensional vector, and the association of properties
with individuals would be the vector sum of the elementwise
products between each individual and its property:
V = AP+BP+C Q
To retrieve the set of individuals having property P, we
multiply the “knowledge-base” vector V by P. The self-
inverse property of MAP produces a representation of the in-
dividuals A and B, as well as a “noise” component not cor-
responding to any individual or property “known” to the sys-
tem:
PV =
P (AP+BP+C Q) =
APP + BPP + C QP =
A + B + C QP =
A + B + noise
Comparing this resulting vector to the vectors for each of
the individuals will yield a high similarity (dot product, co-
sine) between the result vector and both A and B, but not C. In
other words, a single holistic computation on two vectors (P
and V ) has retrieved structurally sensitive information about
distinct individuals, without (1) the need for explicit phys-
ical connections among the individuals (and the concomi-
tant additional representational hardware) or (2) a function-
ally localist decomposition. This power comes at the cost of
of noise in the retrieved representation, which is not a deal-
breaker for this example. If noise becomes a problem (as it
can in recurrent circuits like the analogy-mapping circuit de-
scribed below, where noise accumulates), the noise can be
removed from the result vector by passing the vector through
a “cleanup memory” that stores only the meaningful items, or
vector directions: here, A, B, and C.
The issue of noise in VSA is rather subtle. In a localist
representation there are distinguished directions in the vec-
tor space that correspond to the individual units, because in-
dividual units represent individual concepts. In VSA there
are no inherently distinguished directions. For example, the
vector X might represent the concept A, but it could just as
well represent A+B or C D+E, etc. The functional equiv-
alent of distinguished directions is provided by the contents
of the cleanup memory, which are initialized for a particu-
lar problem. Noise is then any pattern which is not stored
in cleanup memory. Unlike localist representations, which
require reconfiguring the “hardware” for each new problem,
VSA reuses the same fixed cleanup hardware (e.g. an autoas-
sociative Hofpield network) for every problem.
Lateral Inhibition as Self-Intersection
Consider a situation in which three categories A and B, and C
are competing with one another on a given neural layer L2, to
classify input patterns on an incoming layer L1. An example
localist implementation is shown in Figure 1. Each node in L2
has an inhibitory connection to every other node in that layer.
The connections from L1 to L2 can be interpreted as setting
the state of L2 to reflect the initial evidence for each of the
categories. The inhibitory connections within L2 implement
a recurrent process that increases the differences between the
most supported category and the other categories.
We can interpret the state of L2 as a multiset - a set of
weighted elements. Each category Xi (here, A, B, or C) is
weighted by a non-negative real-valued coefficient ki that re-
flects the importance of Xi in the multiset, with 0  ki  1.
Given this interpretation of the L2 state as a multiset we need
a multiset operation that increases the differences between
the most supported category and the other categories. We
do this with multiset intersection (multiplication of the corre-
sponding category weights) and normalization (constraining
the sum of the category weights to be constant).
To see what we mean by multiset intersection, con-
sider multisets X = fk1A;k2B;k3Cg;Y = fk4A;k5B;k6Cg.
Intersecting X and Y would would yield a multiset
fk1k4A;k2k5B;k3k6Cg. Intersecting X with itself would yield
(k1)2A + (k2)2B + (k3)2C, magnifying the differences be-
tween the ki. Normalization of the result forces the smaller
ki towards zero. The repeated application of self-intersection
with normalization yields a similar dynamic to lateral inhibi-
tion thereby implementing winner-takes-all competition.
A B CL1L2
Figure 1: Lateral inhibition in a localist network
In a localist network like the one in Figure 1, the multiset
coefficients ki correspond to the activations of the nodes in
the L2 layer. In VSA, a multiset is represented as a single
vector, for example, k1A + k2B + k3C where A, B, and C are
hyperdimensional vectors and k1, k2, and k3 are non-negative
scalars. Note that this network can only represent a choice
between A, B, and C. No other category can be considered
without modifying the physical structure of the network.
How are we to perform the multiset intersection of two
such vectors? Because of the self-cancellation property of
the MAP architecture, simple elementwise multiplication (the
standard MAP product operator) of the two vectors will not
implement this operation. We could extract the ki by iterating
through each of the vectors A, B, and C and dividing x and
y elementwise by each mapping, but this is the very kind of
functionally localist approach that we are trying to avoid.
320

To implement this intersection operator in a holistic, dis-
tributed manner we exploit the third component of the MAP
architecture: permutation. For explanatory purposes we can
conceive of our solution as a simple register-based machine,
where (as in a traditional von Neumann architecture), each
register holds a temporary stage of the computation. (In our
version, of course, the register contents are hyperdimensional
vectors.). As depicted in Figure 2, our solution works as fol-
lows: 1: and 2: are registers loaded with the vectors repre-
senting the multisets to be intersected. P1() computes some
fixed permutation of the vector in 1:, and P2() computes a dif-
ferent fixed permutation of the vector in 2: (randomly chosen
permutations are sufficient). Register 3: contains the prod-
uct (via elementwise multiplication) of these permuted vec-
tors. Register 4: is another variety of “cleanup” memory (a
constant vector value) pre-loaded with each of the principal
vectors transformed by multiplying it with permutations of
itself; i.e., 4 := Sni=1Xi P1(Xi)P2(Xi). In other words, reg-
ister 4: indicates the items of interest to the system and is
functionally analogous to the L2 units in the localist network;
however, the contents of the register can be changed at any
time without modifying the underlying hardware. Note so
that each of these registers contains a high-dimensional vec-
tor representing an arbitrarily complex multiset, and each ar-
row in Figure 2 represents the transfer of a high-dimensional
vector.
P2()1:2: P1() 3: 4: 5:* *
Figure 2: A neural circuit for vector intersection.
In brief, the circuit in Figure 2 works by guaranteeing that
the permutations will cancel for only the subset of Xi present
in both input registers, with the other Xi being rendered as
random noise. In order to improve noise-reduction it is nec-
essary to take the sum over several such intersection circuits,
each based on different permutations. This sum over permu-
tations has a natural interpretation in the synaptic connections
between neural layers of sigma-pi units. Each unit (neuron) in
one layer calculates the sum over many products of a few in-
puts from units in the prior layer. The apparent complexity of
Figure 2 is a consequence of drawing it for explanatory clar-
ity rather than computational complexity. The intersection
network could be implemented in a single layer of sigma-pi
units.
To see how this circuit implements intersection, consider
again the simple case of a system with three meaningful vec-
tors A, B, and C where we want to compute the intersection
of x = k1A + k2B + k3C with y = k4A + k5B + k6C. The
vector x is loaded into register 1:, y is loaded into 2:, and the
sum
AP1(A)P2(A)+BP1(B)P2(B)+C P1(C)P2(C)
is loaded into 4:. After passing the register contents through
their respective permutations and multiplying the results, reg-
ister 3: will contain
P1(k1A+ k2B+ k3C)P2(k4A+ k5B+ k6C) =
(k1P1(A)+k2P1(B)+k3P1(C)) (k4P2(A)+k5P2(B)+k6P2(C)) =
k1k4P1(A)P2(A)+ k2k5P1(B)P2(B)+ k3k6P1(C)P2(C)+
noise
where noise represents terms not corresponding to a mean-
ingful component of the intersection. Multiplying this sum
in register 3: by the contents of register 4: will then result in
the desired intersection (plus additional noise), via the self-
cancellation property:
[k1k4P1(A)P2(A)+ k2k5P1(B)P2(B)+ k3k6P1(C)P2(C)]
[AP1(A)P2(A)+BP1(B)P2(B)+C P1(C)P2(C)] =
k1k4P1(A)P2(A)AP1(A)P2(A)+
k1k4P1(A)P2(A)BP1(B)P2(B)+
k1k4P1(A)P2(A)C P1(C)P2(C)+
k2k5P1(B)P2(B)AP1(A)P2(A)+
k2k5P1(B)P2(B)BP1(B)P2(B)+
k2k5P1(B)P2(B)C P1(C)P2(C)+
k3k6P1(C)P2(C)AP1(A)P2(A)+
k3k6P1(C)P2(C)BP1(B)P2(B)+
k3k6P1(C)P2(C)C P1(C)P2(C) =
k1k4A+
k1k4P1(A)P2(A)BP1(B)P2(B)+
k1k4P1(A)P2(A)C P1(C)P2(C)+
k2k5P1(B)P2(B)AP1(A)P2(A)+
k2k5B+
k2k5P1(B)P2(B)C P1(C)P2(C)+
k3k6P1(C)P2(C)AP1(A)P2(A)+
k3k6P1(C)P2(C)BP1(B)P2(B)+
k3k6C =
k1k4A+ k2k5B+ k3k6C+noise
Note that this apparently complex calculation is actually
a single elementwise vector product operation. The circuit
does not “see” the complexity of the vectors it operates on.
The same holds true for the normalizing operator mentioned
above: normalization is implemented as a scalar multiplier
applied to the entire vector to keep the sum of the element
activations approximately constant.
Experimental Results
As a proof-of-concept for our distributed lateral inhibition
architecture, we ran several experimental trials using the
circuit from Figure 2. We started with an initial vector
x0 = 1=NåNi=1 kiXi, with k j = 1:02 for one arbitrarily cho-
sen j and ki = 1 for i 6= j. We then iterated the operation
xt+1 = normalize(xt ^ xt), where ^ is the intersection opera-
tor in Figure 2, and normalize(x) = x=maxl(jxl j). (The initial
conditions thus represent a temporary violation of the con-
straints given above for ki that are immediately rectified by
the normalizing operation.) We stopped iterating when the
Euclidean distance between xt and xt1 fell below 0.01.
Figure 3 shows a typical result, for N = 3 a vector x of
2000 dimensions, and 100 permutations. The system quickly
converges to an x in which a single Xi dominates. We have
reproduced these results for larger values of N, using vectors
321

Figure 3: Winner-takes all in VSA implementation
with more realistically large dimensions and more permuta-
tions (connectivity). It is important to emphasize that all rep-
resentations and operations in this and the next experiment
are fully distributed. The figure was produced by serial ex-
traction of the strengths of the three principal vectors of this
system, but this was done only for purposes of illustration.
There is nothing in the system that requires the intervention
of a localist “homunculus” at any stage.
Application to Analogical Mapping
Analogical mapping has long been a focus of efforts in cog-
nitive modeling. There are several successful connection-
ist cognitive models of analogy (Holyoak & Thagard, 1989;
Hummel & Holyoak, 1997; Eliasmith & Thagard, 2001).
These models vary in their theoretical emphases and the de-
tails of their connectionist implementations. However, they
all share a problem in the scalability of the amount of com-
putational resources or effort required to construct the con-
nectionist mapping network. We contend that this is a conse-
quence of using localist connectionist representations or us-
ing distributed representations in a localist manner.
To address this issue, we have recently developed a model
that treats analogical mapping as a special case of graph iso-
morphism; that is, the solution of finding an optimal map-
ping between two structures (graphs) consisting of individ-
uals (vertices) and their relations (edges). For example, in
the simple graphs in Figure 4, the maximal isomorphism is
fA=P, B=Q, C=R, D=Sg or fA=P, B=Q, C=S, D=Rg. Our
model builds on the work of Pelillo (1999), who uses repli-
cator dynamics (originally developed in evolutionary game
theory) to solve the problem with a localist representation. In
Pelillo’s solution, iterated multiplication of a localist edge-
consistency matrix w by a localist vertex-mapping vector x
produces a localist “payoff” vector p expressing the quality
of the solution. Elementwise multiplication of x with p pro-
duces an updated x representing an improved set of vertex
mappings. This elementwise multiplication can be construed
as a multiset intersection.
In our VSA implementation of this model, all entities
(vertices, edges, and w, x, and p) are represented as high-
dimensional MAP vectors. Vertex mappings in x are rep-
resented as the sums of the corresponding pairwise edge-
mapping products (A*P + A*Q + ... + C*S), and the winner-
takes-all intersection circuit of Figure 2 supports competi-
tion among mutually inconsistent mappings (C=R, D=S vs.
C=S, D=R), without decomposing x into its constituent edge
mappings. As shown in Figure 5, the VSA implementation
can exhibit dynamic convergence to a solution in a way that
is qualitatively similar to the localist implementation. Here,
each curve corresponds to the level of support for a specific
node mapping; e.g., AP represents the support for the corre-
spondence between nodes A and P. Notice that the compo-
nents corresponding to the correct node mappings compete
with and suppress the components corresponding to incorrect
node mappings. As in the previous experiment, the conver-
gence takes place without decomposition into localist compo-
nents, the figure being a localist presentation for illustration
only.
A B CD P Q SR
Figure 4: A simple graph isomorphism problem
Conclusion
We have presented a fully distributed connectionist architec-
ture supporting lateral inhibition / winner-takes all competi-
tion. The architecture uses a neurally plausible permutation
circuit to support a multiset intersection operation without de-
composing the summed vector into its constituent items. This
approach compares favorably with a localist approach when
applied to the task of analogical mapping. Our results thus
challenge the commonly-accepted view that lateral inhibition
calls for localist representation. More profoundly, our model
points toward a neural implementation where more complex
representations do not require more complex or dynamically
rewired hardware, a long-standing goal of connectionist cog-
nitive modeling.
Acknowledgments
The authors thank two anonymous reviewers for the most
helpful comments.
Software Download
Matlab code implementing the experiments described in this
paper can be downloaded from tinyurl.com/lidemo.
322

Figure 5: Convergence of localist (top) and VSA (bottom)
implementations of graph isomorphism.
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