Self-Organized Cognitive Maps

Self-Organized Cognitive Maps

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Robert Lloyd

The University of South Carolina

This paper argues cognitive mapping is a learning process that can be simulated by a self-organizing neural network.The learning of city locations was considered in two studies. One study focused on the learning of city locations onfour continents. Results indicated the neural network aligned the cities producing systematic errors similar to thosein human cognitive maps. A second study had a neural network learn a biased sample of city locations in the UnitedStates. Results indicated a non-linear relationship between cognitive and physical distances. Self-organized cognitivemaps naturally produce this non-linear relationship when information from more than one scale is mapped into onespace.

Key Words: learning, cognitive maps, neural networks, self-organized.

In thinking, and in subconscious information process-ing, there is a general tendency to compress informa-tion by forming

reduced representations

of themost relevant facts, without loss of knowledge abouttheir interrelationships. The purpose of intelligent in-formation processing seems in general to be creation ofsimplified images of the observable world at variouslevels of abstractions, in relation to a particular subsetof received data.

—Kohonen (1989, 119)

Introduction

ur experiences are transformed into ourmemories. Our memories affect our be-

havior. Our behavior provides new experi-ences. Learning is the mechanism that initiatesthis cycle and keeps it refreshed. The first sen-tence of this paper is the simple statement thatinspired the central question for this research.How is information about spatial locations oncartographic maps projected into cognitivemaps in our memories?

The human brain was not designed for pre-

cision when it comes to encoding spatial infor-mation. Although we start encoding the spatiallocations of important objects very early in life(Walton et al. 1997), we appear to never accu-rately encode the details. Our inexactness is notcaused by exposure to inaccurate informationor memory loss, but rather by the normallearning processes used to encode informationinto memory (Tversky 1981). These normallearning processes appear to produce system-atic errors as we learn locations on a map orlandmarks in an urban environment (Lloyd1989; Kitchin 1996).

The spatial structures that we encode intomemory have generally been referred to ascognitive maps (Tolman 1948). This paper usesthe term cognitive map in the broadest sense.Any internal representation of a set of geo-graphic locations that has been learned, i.e., en-coded into memory, is considered a cognitivemap. The learner might be a person with abiological-based memory or an artificial neuralnetwork simulating human memory in a com-

O

Professional Geographer, 52(3) 2000, pages 517–531 © Copyright 2000 by Association of American Geographers.Initial submission, November 1998; revised submission, May 1999; final acceptance, May 1999.

Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, and 108 Cowley Road, Oxford, OX4 1JF, UK.

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puter. The geographic locations are specified ascoordinates in some physical space. Theymight refer to any objects noticeable on a mapor in an environment.

A number of factors have made it difficult tostudy the nature of cognitive maps. First, cog-nitive maps are learned over time. Any serialprocess that causes someone to encounter a se-quence of objects in an environment or scanfrom object to object on a cartographic map isan example of learning taking place over time.If the environment or map is visited again an-other round of learning occurs. One might saythese cognitive maps are never actually com-pleted, but periodically updated with new in-formation that might impact the nature of thecognitive map. A person’s cognitive map at onepoint in time, therefore, is not necessarily thesame at another point in time (Golledge et al.1975).

Second, individuals who encode these worksin progress have some shared experiences withothers, but also have many unique learning ex-periences. Although individuals might be try-ing to learn about the same city or read thesame map, they have different navigation orscanning patterns, goals, and values, and expe-rience locations from different perspectives.Lloyd and Heivly (1987) have shown that thecognitive maps for people living in differentneighborhoods in a city were systematicallyaffected by the location of the home neigh-borhood within the city. Lloyd (1989) hasshown that a group of individuals who learnedspatial locations from a common cartographicmap were much more consistent and accuratethan another group who learned the samelocations by freely navigating in an urbanenvironment.

Third, although cognitive maps are learnedthrough a dynamic process, most previousstudies used techniques that focused on differ-ences between the current representations ofcognitive maps and cartographic maps. Thisputs the focus of most cognitive mappingstudies on evaluating the success of the learn-ing rather than on the process used for learningmaps or environments. Although the nature oferrors in cognitive maps has been well docu-mented, a complete explanation of why system-atic errors occur remains as an important goalfor cognitive mapping researchers. The focusof the research presented here is not on

what

systematic errors occur in cognitive maps, buton

why

those errors occur.An ideal cognitive mapping study should be

able to provide identical experiences to all sub-jects as they learn the same spatial information.The ideal study should also explicitly considerthe learning process used by subjects and beable to model that process. People usually learnspatial information without supervision. Al-though there may be some reinforcement fromthe successes and failures that come from usingour cognitive maps, there is usually no “truthgiver” who monitors our progress and correctsour mistakes (Nigrin 1993). Unsupervisedlearning models would, therefore, appear to re-flect actual learning experiences better than su-pervised models that learn by minimizing dif-ferences between current knowledge and thetruth that is explicitly defined for the model.

The purpose of this paper is to consider thelearning of map locations as an encoding of in-formation into a neural network. The basicproblems considered use a Kohonen neuralnetwork to learn configurations of real spatiallocations (Kohonen 1989, 1993). The Ko-honen neural network model was thought to bean ideal choice for learning spatial informationbecause it learns without supervision. This iswhy the network is said to create self-organizedmaps. Learning is based on the recognition ofpatterns in the input data rather than knowl-edge of the truth. Kohonen neural networksare also inherently spatial in nature. A basicprescript of the learning process is based on adistance decay principle that results in similarinput patterns being mapped into neighboringlocations in the self-organized map. This ap-proach models learning processes that trans-form cartographic maps into cognitive maps.Maps learned by the Kohonen networks canthen easily be compared to the actual maps orto cognitive maps learned by human subjects.

What Do We Know about

Cognitive Maps?

Cognitive maps are internal memory struc-tures. Most cognitive mapping studies focus on

where

objects are located in space (Golledge etal. 1982; Lloyd 1989). These might be called

projection

studies because they basically consid-ered the underlying structure of the geographicspace as it appears to be represented in mem-


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ory. One could think of a cognitive mapper ashaving the same problem as a cartographer.Locations (the truth) need to be projected intoanother space (e.g., a Mercator map projectionor your memory) for storage and representa-tion. Just like cartographic projections, cogni-tive mapping projections produce systematicerrors that have been well documented.

Reference Points

A number of researchers have argued that im-portant locations or objects serve as cognitivereference points around which other locationsor objects can be organized (Rosch 1975; Tver-sky 1977; Hirtle and Jonides 1985; Couclelis etal. 1987; Ferguson and Hegarty 1994). Thespace near the reference point tends to be ex-panded relative to other locations in the spacewhile space far from the reference point tendsto be contracted. Holyoak and Mah (1982) re-ported this for cities in the U.S. with the refer-ence point being the Atlantic or PacificOceans. Lloyd (1989) reported this effect forlandmarks within a city with reference pointsbeing either shopping malls or a university.Lloyd and Gilmartin (1991) reported this effectfor world cities with the reference point beingVictoria, Canada.

Spatial Simplifications

When complex spatial information is learned,it tends to be encoded into memory as simplerand more symmetrical than it is in reality.Howard and Kerst (1981, 502) reported therectangular properties of a distribution of cam-pus landmarks were “squared up” in the ob-server’s long-term memory and tended to clus-ter into groups. Tversky (1981) reportedevidence from multiple experiments that sug-gested people tend to simplify cognitive mapsby aligning landmark locations to have a morelinear structure and rotating spaces towardnorth-south or east-west axes. She reportedthat subjects move the South American conti-nent in their world cognitive maps to positionit directly south of North American and movethe European and African continents south toposition them directly east of North and SouthAmerica. Lloyd and Heivly (1987) reported ev-idence that residence of urban neighborhoodsrotated their cognitive maps to align majortransportation arteries with a canonical axis.

When people are asked to sketch the streets

in a city or the outline of a political region theytend to simplify the shapes and make themmore symmetrical than they are in reality.Byrne (1979) reported subjects rememberedangles of road intersections to be 90

8

even whenthe true angles varied considerably from a rightangle. Tversky (1981) reported similar resultsfor sketch maps. Tversky and Schiano (1989)found that sketches of lines representing riverson maps were remembered as more symmetri-cal than they actually were. Lloyd (1997) hadstudents who did not live in Texas draw a sketchof the outline of Texas. The drawings appearedto preserve the basic categorical features (e.g.,the panhandle) but details of the changes in theboundary were not remembered. He arguedthat similar looking outline maps could be cre-ated from the detailed outline by a line simpli-fication process.

Cognitive Distances

The most commonly reported cognitive map-ping error is that subjects generally overesti-mate shorter distances and underestimatelonger distances (Holyoak and Mah 1982;Lloyd 1989; Poulton 1989). When cognitivedistances are plotted on the vertical axis in ascatter diagram and actual distances are plottedon the horizontal axis, most studies have re-ported a nonlinear relationship between thetwo variables (Montello 1991; Golledge andStimson 1997). This pattern could result froman expansion of local space around some im-portant reference point, e.g., a person’s homelocation, and a contraction of space at the edgeof the space far from the reference point. It alsohas been argued that additional distortions arecaused by unique environmental characteris-tics. For example, barriers (Newcombe andLiben 1982), clutter (Thorndyke 1981), thenumber of turns in a route (Sadalla and Magel1980), and the number of nodes (Sadalla andStaplin 1980a) all cause an overestimationof distances.

Parts of a Larger System

The human brain contains hundreds of billionsof neurons that can be connected and orga-nized in many different ways. It is important todistinguish between a separate

what

system forencoding characteristics of objects and a

where

system for encoding locations (Ungerleider

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and Mishkin 1982; Landau and Jakendoff1993). An electrophysiological study of theworking memory used to encode location,color, and shape has indicated separate sub-systems for encoding spatial information andobject information (Martin-Loeches and Rubia1997). Another study, conducted with mon-keys, identified some neurons that respondedonly to

what

information and others that re-sponded only to

where

information (Rao et al.1997). This study also reported finding a thirdtype of neuron that responded to both

what

and

where

information. It was speculated that theseneurons might be needed to link the

what

and

where

information for the purpose of guidingbehavior.

The current research focuses only on learn-ing

where

information related to relatively largespaces. Small-scale maps are frequently used toacquire this type of information. For the learn-ing simulations, it was assumed that a learnerhad no prior knowledge of the geographicspace as the learning process begins. Some re-searchers argue that learning frequently in-volves using prior knowledge, e.g., schema ormental models, to more easily organize new in-formation into meaningful cognitive represen-tations (Bruner 1990, 1996). Cognitive mapsmay serve as prior knowledge for learning andunderstanding other types of spatially distrib-uted information. Understanding how thesefundamental structures are learned and the sys-tematic errors that will naturally be part oftheir organizations should be important issuesin geographic education. Connectionist models,such as the neural networks used in this study,assume learning is simply the adjustment ofthe weights connecting neurons in a network(McClelland and Rumelhart 1986; Rumelhartand McClelland 1986). For this study, the stageof learning is assumed to be so primitive thatit is reasonable to assume no prior knowl-edge exists.

Self-Organizing Neural Networks

as Cognitive Maps

In their seminal work on cognitive mappingsystems in the brain O’Keefe and Nadel (1978,374) argued “the hippocampus both constructsand stores cognitive maps.” Most of their dataon learning were based on rats moving aroundin an environment. They argued for the “exist-

ence of place-coded neurons in the hippocam-pus of the freely moving rat” (O’Keefe andNadel 1979, 487). They also argued the hip-pocampus is concerned with the storage of spa-tial information to represent absolute spacesrather than egocentric relative spaces. It is suchabsolute spaces that are considered here.

Two studies are reported here that use thesame type of neural network model to learn thelocations of points in a space. The studies con-sider spaces (e.g., continents and countries)that are too large for individuals to learn bynavigation. Most people would learn where ob-jects are located in such spaces by encoding co-ordinate information from small-scale maps. Itis this coordinate information that was learnedby the neural networks.

The Structure of the Model

The neural network models used in the follow-ing studies consisted of layers of neurons thatwere connected to each other (Fig. 1). Therewere always two neurons in the input layer thatwere set to digital values (e.g., latitude and lon-gitude) for locations of objects (e.g., cities) inthe space to be learned (e.g., a region of theworld). During the learning phase of the pro-cess these neurons were set to a series of valuesto be learned and the information passedthrough the connections to a square array ofneurons usually referred to as the Kohonenlayer (Kohonen 1989). The Kohonen layer al-ways had a fixed number of neurons that weredistributed in a square array (e.g., 10 by 10) asillustrated in Figure 1. Each input neuron wasconnected with a weight to each neuron in theKohonen layer. There were no connectionsamong the N by N neurons in the Kohonenlayer or among the neurons in the input layerof the network. When the network is initial-ized, the weights are set randomly to somesmall positive number. These initialized ran-dom weights represent a subject who has noprior knowledge. When studying a primitivelearning process, it is useful to have subjectswho can begin the learning process with no bi-ases or previously learned information. Since itis difficult to find human subjects with no priorknowledge of the locations of familiar worldcities, artificial subjects represented by neuralnetworks are a good alternative. Each of the

i

neurons in the Kohonen layer is activated (

A

i

)when a new trial (geographic coordinates rep-


521

resented by I

1

and I

2

) is presented to the net-work through the

j

input neurons by a simplelinear equation:

(1)

If the weights (

w

i,j

) connecting a neuron in theKohonen layer with the input neurons haverelatively high values, then that neuron shouldhave a relatively high activation. The neuron inthe Kohonen layer with the highest activationand its neighbors get to adjust their weightsconnecting them to the input neurons. Theusual procedure is to have a broader definitionof neighborhood initially and to decrease thisover time until only the winning neuron is

Ai Wi, jj 1=

2

∑ Ij=

being adjusted at the end of the learning pro-cess (Kohonen 1993). This winner-takes-allstrategy causes trials with similar coordinatesin physical space to activate neurons located inthe same region of the Kohonen space. The se-lected neurons adjust their weights using thefollowing equation:

(2)

The weight (

w

i, j

) connecting the

i

th neuronin the Kohonen layer with the

j

th input neuronis adjusted proportional to a learning coeffi-cient,

b

, multiplied by the difference betweenthe input value and the current weight. Aftermany learning trials that allow all the inputsmultiple presentations and adjustments to theweights, the needed adjustments become verysmall and the network stabilizes with the sameneurons winning each time for each physicalcoordinate. Locations that are close together inphysical space have similar coordinates andshould also be close together in the Kohonenlayer. Conversely, locations that are far apart inphysical space should also be far apart in theKohonen layer. The set of physical coordinatesbeing learned are “projected” into the Ko-honen layer in the same sense that the standardMercator equations transforms latitude andlongitude into rectangular X,Y coordinates.Once all the weights are fixed, an X,Y coordi-nate (Fig. 1) can be computed for any latitudeand longitude input to the network using aweighted average of the locations of the threeneurons in the Kohonen layer producing thehighest activations (NeuralWare 1996).

It is theoretically possible to use any rectan-gular array of neurons in the Kohonen layer(Kohonen 1989). Given that the typical humanbrain contains several hundred billion neurons,it would take a relatively small allocation of re-sources to learn a cognitive map (McNaughtonand Nadel 1990). It was assumed for these sim-ulations that these learning resources were asmall network of neurons. Two characteristicsof the neural network could be the basis of sys-tematic distortions. First, simplifications wouldoccur if an array of neurons (the Kohonenlayer) were used to encode physical locationswhose natural distribution did not conform to arectangular distribution. Knowing the appro-priate number of neurons and their optimaldistribution in rows and columns requires prior

DWi, j b Ij Wi, j–( )=

Longitude

Latitude

X

Y

Figure 1: The basic Kohonen neural networkmodel used for this research had two inputneurons, e.g., latitude and longitude. The truelocations in geographic space were presentedthrough these neurons. The input neurons wereboth connected to each neuron in the Kohonenlayer of the model. This layer is represented as a 10by 10 array of neurons in the illustration. The X,Ycoordinate is the location of the geographiclocation in the cognitive map represented in theKohonen layer.

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knowledge of what is to be learned. An artificialor biological network experiencing one loca-tion at a time would have no knowledge of thewhole distribution until all locations have beenencountered. Second, additional simplifica-tions could occur if the number of neuronsdevoted to encoding the cognitive map doesnot match the number needed to completelyencode the details in the physical map. Thereare a number of studies that have argued thathumans have a very limited capacity for work-ing memory (Miller 1956; Shiffrin and Nos-ofsky 1994). The simplest assumption, andthe one used here, is to specify the Kohonenlayer as a symmetrical array with a relativelysmall and equal numbers of neurons in therows and columns.

Hypotheses

Given what is known about typical errors inhuman cognitive maps and the assumptionsused to specify the Kohonen neural networkmodels, it is possible to form two hypotheses.

Hypothesis 1: Like human cognitive mappers,self-organized cognitive maps learned by Ko-honen neural networks will simplify spatial dis-tributions by aligning locations.

Hypothesis 2: Like human cognitive mappers,self-organized cognitive maps learned by Ko-honen neural networks will exaggerate shorterdistances and shrink longer distances.

Learning Simulations

Cities on Four Continents

The first study considers 12 cities located onfour continents that rim the Atlantic Ocean(Fig. 2). Three well-known cities were selectedfrom each continent and their latitude and lon-gitude used as a set of input data for a Kohonenneural network structured as described above(Fig. 1). The network was initialized with ran-dom weights and run for 2,000 learning trials.This number of trials was determined experi-mentally in preliminary training runs that de-termined the amount of training needed toproduce stable weights. Once the weights havestabilized, additional training does not changethe weights and the learning process is com-pleted unless new information becomes avail-able. After training was completed, the weights

were fixed and data for the same 12 cities wereused to compute X,Y coordinates for each cityin the Kohonen layer. This procedure was re-peated 29 more times so that a reliable meanresponse could be computed. Since the randomweights were different for each of the 30 simu-lations, the solutions were slightly differenteach time. This is the nature of neural net-works. A somewhat unique solution is learnedeach time the network is run. Models that donot produce completely consistent results mayappear to be undesirable, but one must con-sider that individual human learners would alsoproduce somewhat unique cognitive maps.The solutions for the 30 artificial neural net-work simulations were averaged to produce atypical response. Since the neural networkmodel had no sense that North should be at thetop of the map, it was necessary to rotate and/or reflect the axes of individual models to alignthem all to the correct cardinal axes beforeaggregation.

A simulated self-organized cognitive map(Fig. 3) that represents the average locations ofthe 12 cities in the Kohonen layers of the neu-ral networks shows some interesting similari-ties and differences when compared with theactual locations in geographic space (Fig. 2).Some of the errors of simplification typicallyreported for human cognitive maps are clearlypresent in the self-organized cognitive maplearned by the Kohonen neural network. The

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Figure 2: Cities on four continents were se-lected for the study. The latitude and longitudevalues associated with the cities were used as theinformation to be learned by the Kohonen neuralnetwork.


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continents are aligned with South America po-sitioned under North America and Europe andAfrica positioned east of North and SouthAmerica. Given that the Kohonen neural net-work was only given information on the 12 citylocations and did not have a “home bias,” it isnot surprising that the cities in the four conti-nents would be treated as equally important in-formation. For example, the areas for the trian-gles made by connecting the three points foreach continent is approximately the same onthe simulated cognitive map (Fig. 3). This isclearly not true for the same triangles on thecartographic map (Fig. 2).

Location data for the same 12 cities werecollected from 15 geography teachers as part ofa class exercise. Geography teachers representhuman subjects that should be more familiarwith world maps and city locations than thetypical human subjects used in cognitive map-ping studies. It was, however, expected thatthey would still have the usual systematic dis-tortions in their cognitive maps because accu-racy is a function of the learning process ratherthan amount of experience. Subjects wereseated in front of monitors that displayed aMercator map projection of the world. Themap was similar to Figure 4, but was blank ex-

cept for the latitude and longitude lines and la-bels. The 12 cities and their countries werenamed one at a time in a different random or-der for each subject and the subjects simplymarked where they thought each city was lo-cated on the blank map with a mouse. The mapremained blank throughout the process so thateach city was an independent trial with thesame background information available.

The locations marked by the subjects wereaggregated and are represented in Figure 4 bythe grey ellipses connected to the actual loca-tions (black dots). The center of the ellipses in-dicate the average locations marked for the 12cities and the length of horizontal and verticalradii of the ellipses indicate one standard devi-ation around this central estimate. The usualdistortions can easily be identified. Subjectsgenerally expanded the space around NorthAmerica by moving cities to the east and south.The cities are aligned for the American conti-nents by shifting New York and Miami to theeast while shifting the South American cities tothe west. All cities in Europe and Africa wereshifted south to align them with cities on theAmerican continents. The relative sizes of theellipses indicate subjects were most consistentmarking the cities in North America and leastconsistent marking cities in Africa. This islikely a bias based on the home location of thesubjects.

Saarinen et al. (1996) measured the sizes ofcontinents drawn on sketch maps by subjectsfrom 22 different home sites around the world.They reported the home continent was gener-ally drawn at an exaggerated size and that Eu-rope was always exaggerated and Africa was al-ways diminished in size. Since each continentin the current study has three cities, the areas ofthe triangles made by connecting the pointsrepresenting the three cities were computed torepresent a physical space on each continent.For the triangles made for the actual locations(black dots in Fig. 4) the areas should reflect ac-curacy as it is represented visually in a Merca-tor projection. For the triangles made from thelocations estimated by the subjects (grey el-lipses in Fig. 4) the areas should reflect theaccuracy of the subjects’ aggregate cognitivemap as it is represented in the same Mercatorprojection.

The triangular areas were summed and theproportion of the total area represented by the

Los Angeles New York

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Figure 3: The locations of the 12 cities in theKohonen layer of the neural network aggregatedover 30 simulations. Learning errors appear to besimilar to those typically made by human subjects.The continental locations appear to be aligned withSouth America south of North America and Europeand Africa east of North and South America.

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three cities in a continent was used to constructpie charts (Fig. 5). Based on the representationof the cities in a Mercator projection, the carto-graphic truth (Fig. 5a) indicates the African tri-angle is the largest. South and North Americantriangles are second and third largest and theEuropean triangle is the smallest. Trianglesbased on the human subjects’ estimates (Fig.5b) appear to agree with Sarrinen et al.’s (1996)sketch map results. North America is exagger-ated to be the largest area. Europe is also in-creased in size, while Africa is substantially de-creased in size.

A similar procedure was used to constructtriangles and compute areas for the locations inthe aggregate self-organized cognitive map(Fig. 3). The pie chart for these continents(Fig. 5c) indicates a bias in the same directionas the human subjects, but not as pronounced.North America is larger than it should be, Eu-rope is larger than it should be, and Africa issmaller than it should be. Although this

matches the expected bias found for humansubjects, it appears to just reflect the neuralnetwork’s equal treatment of the cities. Thecity locations were simplified in the Kohonenlayer so they were evenly spaced. This exagger-ated the smaller areas and reduced the sizes oflarger areas. The human subject data appearsto enhance this distortion by further exaggerat-ing the size of the area in North America anddiminishing the area for Africa.

Cities in the United States Learnedfrom a Reference Point

Although the self-organized cognitive mapsimulated for the 12 world cities did show analignment bias, it did not appear to have a biasthat expanded the space around any one loca-tion in the space (Fig. 3). The geographyteachers’ cognitive map appeared to expand thespace around North America (Fig. 4). A num-ber of authors have related this effect to theamount information stored in memory that re-

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Figure 4: The 12 cities used in the simulation represented in a Mercator Projection (black dots) and thelocations marked by geography teachers (grey ellipses). The size and shape of the ellipses representvariation among the teachers.


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lates to particular locations. Milgram (1973)suggested that space expands around regions ofa cognitive map in proportion to increasedamounts of information a person might haveencoded about the spatial properties of those

region. Sadalla and Staplin (1980b) extendedthis idea to encoding route information andshowed that the cognitive representation ofdistances along routes were significantly re-lated to the amount of information storedabout the route. Holyoak and Mah (1982) ar-gued that people generally need to make finerdistinctions between nearby locations thanfaraway locations. This need is accommodatedby “an ecologically important flexibility inpeople’s ability to distribute a limited discrimi-natory capacity over a given magnitude range”(Holyoak and Mah 1982, 348). They also arguethat the density of known locations will bemuch greater near a reference point. Thiswould certainly be true for home locations.

A number of these arguments can be directlyrelated to self-organized cognitive mapslearned by Kohonen neural networks. The net-work has a limited capacity based on the num-ber of neurons in the Kohonen layer. If thedensity of locations that need to be mappedinto the Kohonen layer decreases with distancefrom a home reference point, then a conflictwill occur. This conflict also happens when hu-man cognitive mappers are forced to map loca-tion information available at different scalesand details into a common space. The U.S. cit-ies used in the second study provide an exampleof this conflict. The sample of cities is meant torepresent a collection of cities that someoneliving in Columbia, SC might know. The fre-quency of known cities decreases with distance.

Starting with all the cities in the U.S. in a da-tabase, cities were collected for the sample instages. Each stage set a maximum distance acity could be away from Columbia, SC and aminimum population size for the city. The firststage collected only large cities (more than 1million population) that were at least 2,500miles from Columbia. This is based on the no-tion that cities known by people that are farfrom their homes are more likely to be largercities. Each subsequent stage reduced the max-imum distance a selected city could be fromColumbia and lowered the accepted range ofpopulation size. The final stage collected onlysmall cities (less than 15,000 population), butthey needed to be near Columbia (within 100miles). This is based on the notion that thesmaller cities that people know are more likelyto be near their home. This process resulted in232 cities being selected for the sample.

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ica

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pe

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Figure 5: The proportions of the total area in thefour triangles accounted for by a triangle repre-senting a continent. The cartographic truth basedon a Mercator projection (a), human subject esti-mates (b), and the self-organized cognitive map (c)are represented in pie charts.

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One solution to mapping these cities (Fig. 6)illustrates the basic problem facing cartogra-phers and cognitive mappers. Mapping the cit-ies at a national scale positions the cities nearthe reference point are so close together thatthey cannot be discriminated. Plotting all thecities so that local scale details can be seenwould require a very large piece of paper. Whatis a cognitive mapper to do? The hypothesis ex-amined here is that we map these cities into aspace that can accommodate the neededchange in scale. This accommodation, how-ever, has a significant cost. Distance in thespace must be nonlinear.

The sample of U.S. cities was learned by aKohonen neural network structured similar tothe one illustrated in Figure 1 and used for theprevious simulations. Map coordinates (Fig. 6)measured in state plane coordinates were pro-vided to the two input neurons for each of the232 cities. Columbia was one of the cities, but itwas treated exactly like all the other cities to belearned. The network had no special indicationthat the Columbia observation had any specialmeaning in the learning process. Columbia’sserved as a reference point because proximityto its location was used to select the sample.The 232 coordinate locations were projectedinto the 10 by 10 neuron Kohonen layer duringthe learning stage. After 10,000 learning trialsthe network weights had stabilized. Theweights were then fixed and all the cities werepassed through the network to compute the

X,Y position of each city in the Kohonen layer.The self-organized cognitive map for the firstsimulation (Fig. 7) shows Columbia in thelower right quadrant of the map. The spacewould appear to be expanded around the refer-ence point with the faraway places near theedge of the space. This procedure was repeated29 more times with the initial weight set tosmall random numbers each time. Thirty sim-ulations were computed so that reliable meanscould be computed for distances between Co-lumbia and other cities.

Linear distances were computed betweenColumbia and each of the 232 cities for all 30self-organized cognitive maps and then aggre-gated. The distances in miles between Colum-bia and 232 cities were also computed in physi-cal space. Distances in the self-organizedcognitive maps were computed in spaces whoseaxes were constrained by the simulation pro-gram to range between

1

1 and

2

1. Z-scoreswere computed for both distance measures sothey could be more easily compared. A plot ofthe physical distances on the horizontal axisand the aggregated distances in the self-orga-nized cognitive map on the vertical axis indi-cate a remarkably regular nonlinear relation-

Figure 6: Cartographic map of the 232 cities usedfor the learning simulations. The frequency of thecities selected decreases with distance. At thescale represented in the map the cities locatednear the reference point (Columbia, SouthCarolina) cannot be visually discriminated.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0Chicago

Los Angeles

Atlanta

Columbia

MiamiCharleston

Figure 7: Plot of the 232 cities in the Kohonenlayer for the first simulation. The self-organizedcognitive map shows the expansion of space nearthe home reference point (Columbia) and acontraction of space far from the reference point.Some dots overlap.


527

ship (Fig. 8). The pattern is the one frequentlyfound for human subjects (Golledge and Stim-son 1997). As physical distance increases, cog-nitive distance increases at a decreasing rate. Alog-linear model explaining cognitive distancewas significant (r

2

5

0.87, F

5

1,489.8, P

.

F

5

0.000) as was a quadratic model (r

2

5 0.91, F 51,112.1, P . F 5 0.000). The Kohonen neuralnetworks learned the city locations andmapped them into the self-organized maps sodistances to locations near Columbia would beoverestimated and distances to locations farfrom Columbia would be underestimated.

Discussion

The self-organized maps produced by Ko-honen neural networks are based on very sim-ple learning principles: 1) the winning neuron’sconnection weights try to take on values equalto the information being learned and 2) obser-vations with similar input values should beneighbors in the self-organized cognitive map.Both of these principles help construct the self-organized map in the Kohonen layer whengeographic locations are being learned. In ef-fect, the connection weights are equations thatproject physical locations into another space—

the self-organized cognitive map. Unlike equa-tions for cartographic projections, the connec-tion weights are not based on geometric prin-ciples, but are learned. The weights areadjusted as they respond to the patterns in theinformation being presented to the network.This is important for cognitive mapping re-search if people learn basic spatial locations in asimilar way (O’Keefe and Nadel 1978; Ko-honen 1989). Distortions in human cognitivemaps may simply be a function of the way neu-ral networks (biological and artificial) learn(Martindale 1991; Kosslyn and Koenig 1992).

The type of learning studied here took placeover time. In the first study of world cities, neu-ral networks with no prior knowledge, i.e., ini-tial random weights, processed geographiclocations one at a time. The initial randomweights combined with a randomly selectedgeographic location to start the process. Thefirst winning neuron with the highest activa-tion (Equation 1) started the organization ofthe cognitive map being learned. All of theneurons in the neighborhood of the winnerchanged their weights to look more like the in-put information (Equation 2). When this sameobservation or one with similar values is pro-cessed in the future, this region of the Kohonenlayer is more likely to respond with higher acti-vation levels than other regions in the layer.Conversely the region will not respond to dis-similar input values, but some other neuron inthe Kohonen layer will win and be allowed,along with its neighbors, to change weights. Asthe process continues all the input values haveto find some neuron(s) that responds to themwith a high activation. When this is true for allobservations (cities) in the learning set, adjust-ments to the weights become small and the net-work stabilizes. Learning is then completeunless new information is presented to the neu-ral network.

The hypothesis related to the study of worldcities stated that both humans and Kohonenneural networks would encode cognitive mapswith alignment distortions. The alignment oflocations to simplify spatial structures had beenreported in the literature by both Tversky(1981) and Lloyd and Heivly (1987). The sim-ulation of the learning of 12 world cities sug-gested the same effect occurred when a Ko-honen neural network learned geographiclocations. The organization process generally

0

1

2

3

4

Physical Distance as Z-Scores

Dis

tance i

n K

ohonen L

ayer

as Z

-Score

s

6

5

0 1 2 3 4 5 6

Figure 8: Plot of the 232 cities as their distancefrom Columbia, South Carolina in physical spaceon the horizontal axis and in the Kohonen layer ofthe neural network on the vertical axis. Bothdistances are represented in standardized units.

528 Volume 52, Number 3, August 2000

positioned the sets of three cities for each of thefour continents in one corner of the squarespace defined by the 10 by 10 array of neuronsin the Kohonen layer of the neural network(Fig. 3). This simplified and aligned the cities.One could infer that other cities on the samecontinents would follow the same pattern. Forexample, if the latitude and longitude for St.Louis, MO were inputted, it would activateneurons near the center of the triangle made byLos Angeles, Miami, and New York in the self-organized cognitive map. The alignment sim-ply happens because the information is beingprojected into a square space with fixed outerlimits. As the cities are learned, unlike citiesmove away from each other and are con-strained by the edges of the space until equilib-rium is reached. This naturally produces analignment of the locations. This explanationworks for humans only if one assumes biologi-cal neural networks allocate resources in thesame way. The limited capacity of workingmemory for humans is well documented (Shif-frin and Nosofsky 1994). Why should the arraybe square? One simple argument for this is thatthe resources must be allocated before theshape of the space being mapped is known.Symmetrical arrays are generally useful undersuch conditions of ignorance.

The second study considered the knownlearning bias caused by reference points andthe nonlinear relationship between cognitivedistance and physical distance. This study con-sidered a biased sample of cities that might belearned by someone living in Columbia, SouthCarolina. It was assumed the hypothetical per-son would know some larger faraway cities andmany smaller nearby cities. The density ofknown cities decreased with distance awayfrom the reference point. Mapping the loca-tions of both near and far cities into a singlespace is a problem for a cartographer (Fig. 6). Avery large display page or several normal sizedcartographic maps with different geographicscales would be needed to visibly display theavailable details in the information. Humansand Kohonen neural networks solve the prob-lem in a different way than cartographers. Allthe information is learned by projecting it intoa single space, but allowing the space to warp.Locations farthest from the reference point arepushed away until they are activating the neu-rons at the edge of the self-organizing cogni-

tive map. The many locations near the re-ference point also need to activate neurons inthe self-organizing cognitive map that allowtheir differences to be recorded. This expandsthe space near the reference point relative tothe space near the edge of the cognitive map.The nonlinear relationships frequently re-ported between cognitive distance and physicaldistance are simply caused by this process thatadjusts the relative positions of all the loca-tions being learned subject to non-uniformdensities around the reference point and the fi-nite limits of the spatial array of neurons stor-ing the information.

The hypothesis related to the study of U.S.cities learned by a hypothetical person living inColumbia, SC stated that self-organized cogni-tive maps learned by Kohonen neural networkswould mimic humans by exaggerating shorterdistances and shrinking longer distances. Theresults supported this hypothesis. Individualcognitive maps appeared to expand the spacenear Columbia and contract space far fromColumbia (Fig. 7). Cognitive distances fromColumbia to 232 cities, aggregated over 30simulations, had a significant non-linear re-lationship with physical distances (Fig. 8). Inthe first study the human subjects expanded thespace around North America (home) butthe neural network did not. This difference islikely to be caused by the additional knowledgethat the human subjects have about locations inNorth America. The non-linear mapping ap-pears not to occur unless multiple scales are in-volved with more information existing at thelocal scale.

Conclusions

A Kohonen neural network was used to learngeographic information for two related studies.The first study considered just a small numberof cities evenly distributed on four continentson a world map. The second study had thesame type of network learn the distribution of alarger number of cities in the U.S. that werespatially biased. In both studies the unsuper-vised Kohonen neural networks learned thespatial information with some distortions.These distortions appeared to have some pat-tern that could be described as systematic.Since the information being learned wasknown to be accurate when presented to the

Self-Organized Cognitive Maps 529

networks, it is not possible to argue that thedistortions were caused by the acquisition ofinaccurate information. The distortions ap-peared to be related to the process used to learnthe information, rather than the particular in-formation being learned. As the serial learningprocess proceeded, the coordinate informationfrom the cartographic maps was projected intothe self-organized cognitive map through acompetition among the neurons in the Ko-honen layer of the neural network. This com-petition projects the physical locations into theself-organized map so that differences amongneighbors can be discriminated. The learningprocess is practical and intuitive rather thanprecise and geometric. One conclusion is thatKohonen models can learn spatial informationand that the learning process functions like avery fuzzy and inconsistent map projection. Amore general conclusion is that connectionistmodels (neural networks) appear to representspatial learning very well. Lloyd (1994) also re-ported that a neural network and humans ap-peared to learn map prototypes in a similar way.

Another conclusion is that the model’s faultsas a map projection make it a very good mimicof human cognitive mappers. People producethe same systematic errors when they encodecognitive maps. Two important error patternsthat appear to be shared by human cognitivemappers and Kohonen self-organized maps are1) alignment of locations that result in a simpli-fication of spatial information and 2) expansionof space in regions that contain more spatial in-formation. This latter effect explains why cog-nitive maps do not usually have a constant scaleand why distance in cognitive maps are fre-quently reported as being a non-linear functionof physical distance. It also explains why peoplefrequently overestimate short distances andunderestimate long distances.

Another conclusion is that the learning pro-cess can only partially be blamed for distortionsin cognitive maps. Biased samples of informa-tion are also partially responsible for the detailsof the distortions. Replicating the second studyusing Salt Lake City as the reference point in-stead of Columbia would probably result in thesame type of distortions, but they would showan expansion of space around Salt Lake City.Distances to cities near Salt Lake City wouldbe overestimated and distances to East Coastcities would be underestimated.

Future studies should consider more simula-tions that extend the current research findings.The current studies have considered distor-tions caused by varying amounts of informa-tion available for different regions of a carto-graphic map. Simulations that have three (ormore) input neurons might investigate othercommon distortion in cognitive maps, (e.g.,barrier). One could have the model learn notonly the physical coordinate for a location, butalso the categorical side of a river barrier (21or 11) for a geographic location. Would thisinformation expand the space around the bar-rier as it seems to in humans (Newcombe andLieben 1982)?

Additional input neurons besides physicallocation could also be investigated to study theconnections between what and where informa-tion (Rao et al. 1997). Some studies have sug-gested that liking a place seems to shorten itscognitive distance (Stea 1969; Golledge andZannaras 1973). If preferences for places weremeasured for a sample of human subjects, pref-erence values could be learned by a Kohonenneural network through an additional inputneuron. Learning self-organized cognitive mapswith and without this extra preference inputneuron would enable one to assess the influ-ence that affect has on cognitive distance.

Both studies reported here considered coor-dinate information as inputs for the model tolearn. Spatial information represented in alter-nate forms should also be considered. Self-organizing models are also very good at learn-ing information expressed as visual images.Lloyd (1997) reports success in training a Ko-honen network to recognize map symbols thatdiffered in shape, color and orientation. Filippi(1998) was able to train another type of Ko-honen network with satellite imagery to recog-nize types of vegetation.

Cognitive issues, related to learning spatialinformation, are difficult to study directly. Sim-ulations, such as the ones reported here, enableexperimenters to control characteristics of thelearner, such as having no prior knowledge.One can also be very precise about what infor-mation is being learned with an artificial neuralnetwork. This paper has reported studies thatcompared the results of learning simulationswith the truth and with the results of learningexperiences for human subjects. This approachshould help geographers untangle the complex

530 Volume 52, Number 3, August 2000

processes that occur when humans transfer ex-periences into memories.

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ROBERT LLOYD is Professor of Geography at theUniversity of South Carolina, Columbia, SC 29208.His research interests include spatial cognition,learning theory, and cartography.

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