Multilayer connectionist models of memory based on the encoder model using the backpropagation learning rule are evaluated. The models are applied to standard recognition memory procedures in which items are studied sequentially and then tested for retention. Sequential learning in these models leads to 2 major problems. First, well-learned information is forgotten rapidly as new information is learned. Second, discrimination between studied items and new items either decreases or is nonmonotonic as a function of learning. To address these problems, manipulations of the network within the multilayer model and several variants of the multilayer model were examined, including a model with prelearned memory and a context model, but none solved the problems. The problems discussed provide limitations on connectionist models applied to human memory and in tasks where information to be learned is not all available during learning. The first stage of the connectionist revolution in psychology is reaching maturity and perhaps drawing to an end. This stage has been concerned with the exploration of classes of models, and the criteria that have been used to evaluate the success of an application have been necessarily loose. In the early stages of development of a new approach, lax acceptability criteria are appropriate because of the large range of models to be examined. However, there comes a second stage when the models serve as competitors to existing models developed within other theoretical frameworks, and they have to be competitively evaluated according to more stringent criteria. A few notable connectionist models have reached these standards, whereas others have not. The second stage of development also requires that the connectionist models be evaluated in areas where their potential for success is not immediately obvious. One such area is recognition memory. The work presented in this article evaluates several variants of the multilayer connectionist model as accounts of empirical results in this area. I mainly discuss multilayer models using the error-correcting backpropagation algorithm and do not address other architectures such as adaptive resonance schemes (Carpenter & Grossberg, 1987) . Before launching into the modeling of recognition memory, I need to specify the aims and rules under which this project was carried out. This is important in a new area of inquiry because there are many divergent views about what needs to be This research was supported by National Science Foundation (NSF) Grant 85-10361 and National Institute of Mental Health Grant MH 44640 to Roger Ratcliffand NSF Grant 85-16350 to Gall McKoon. I thank Gail McKoon, Neal Cohen, Douglas Hintzman, Michael McCloskey and Richard Shiffrin for extremely valuable discussion of the research presented in this article and Douglas Medin, Bennet Murdock, Charles Rosenberg, and two anonymous reviewers for comments on the manuscript. Correspondence concerning the article should be addressed to Roger Ratcliff, Psychology Department, Northwestern University, Evanston, Illinois 60208. 285 done to advance theory. In a systematic application of connectionist models to an empirical domain, examining the whole range (possibly infinite) of potential models is not possible. There is always the suggestion that "if you only try... "' This response to the failure of a particular model can be shortsighted. Although a new fix might deal with one ditt~culty, it will probably not deal with the range of phenomena that need to be explained, unless the implications are thoroughly understood. Of course, a variant that handles the failures I discuss (as well as other empirical phenomena in recognition memory) is the outcome of this work that is most desirable. The approach I take in this article is to examine the more obvious variants of a class of models and apply them systematically to empirical results in recognition memory. The aim is to provide a cumulative study of the models, examining both their strong and weak points, and to establish recognition memory as an empirical area for further theoretical development using connectionist models. The focus on different variants across classes of models allows general strengths and weaknesses that apply across the classes to be examined, and so this approach makes inappropriate the if-you-only-try response. An important corollary to this approach concerns the application of the models to domains that are similar or related to those in which the models have previously failed. This kind of application should be viewed with suspicion. For example, a multilayer model that has problems with learning and forgetting in recognition memory should not be applied in related domains (e.g., frequency judgment or categorization) for which recognition can be shown to be a special case unless the previous failures can be adequately addressed. The presentation in this article includes considerable detail in some sections. For a more rapid reading, summaries or previews are provided for some of the main conclusions of the detailed sections. The neurally inspired connectionist models have recently become major challengers to the more traditional information-286 ROGER RATCLIFF processing models in cognitive psychology. Although there has been work on such neurally inspired models for several years (e.g., J. A. Anderson, 1972 Anderson, , 1973 J. A. Anderson & Hinton, 1981; J. A. Anderson, Silverstein, Ritz, & Jones, 1977; Grossberg, 1980 Grossberg, , 1988 Kohonen, 1978) , recent interest has been fueled by the finding that earlier limitations of simple onelayer models, such as their inability to compute various functions (Minsky & Papert, 1969) , do not hold when there is more than one layer and when nonlinearity is introduced into processing (Rumelhart & McClelland, 1986) . In addition, a body of successful applications of these connectionist architectures to several domains of study in cognitive psychology has been produced. In this first stage of the connectionist revolution, the emphasis has been (properly) on accounting for major findings within various areas. Individual models have been quite successful, but the success has brought some skepticism. The focus of the skepticism concerns the issue of constraints on the models. No matter what the successes of the models, there is an additional need for information about the limitations of the models and about phenomena that the models cannot explain. One major goal of this article is to use the memory domain to investigate limitations on the multilayer connectionist architecture, and in so doing provide results that will help to develop an understanding of general constraints on the models. The particular area I address is recognition memory. Recognition memory is an important area of study because it is a fundamental aspect of memory and one that has an extensive data base. Equally important, responses in recognition tests are rapid (for single words, typically 500-800 ms), so that the recognition process is less susceptible to the effects of conscious strategies than other, slower tasks. Thus, it can be argued that the processes underlying recognition memory belong in the category of the microstructure of cognition, the term that has been used to describe the basic parallel processing approach to the elementary processes underlying cognition (Rumelhart & McClelland, 1986) . Memory is also an area in which there are substantive quantitative models that can be used as benchmarks for these investigations (Gillund & Shiffrin, 1984; Hintzman, 1986; Murdock, 1982; Ratcliff & McKoon, 1988) . These global memory models have been developed to explain a range of experimental data across a range of experimental paradigms. The model of GiUund and Shiffrin (1984) was designed to explain both recall and recognition performance as a function of a number of independent variables. Hintzman (1986, 1988) has applied his model to recognition, frequency judgments, and data from categorization procedures. Murdock (1982 Murdock ( , 1983 , Eich (1982 Eich ( , 1985 , and Pike (1984) have applied their models to subsets of data from recognition, serial recall, cued recall, and free recall. All of these models are primarily concerned with explaining patterns of data across experimental paradigms, with a major effort toward capturing quantitative relationships. The initial aim of the research described in this article was to develop a multilayer connectionist model and apply it to the domain of memory phenomena as a competitor for the global memory models. Besides providing a connectionist account of recognition memory, the model would be useful in providing competitive and contrasting comparisons. However, the more obvious connectionist models failed, as did several variants. Thus, in this article, I document these failures and attempt to explain in detail the properties of the models that led to the failures. The empirical phenomena to be examined in this article concern learning and forgetting in the standard recognition memory paradigm. In a typical experiment, subjects study a list of words, and then they are given a test list of words for recognition. The test list contains half old words that were on the study list and half new words that were not on the study list. The old and new words are presented in random order, and the subject's task is to decide for each word whether it is old or new. For example, with a study list of 16 words, the test list would contain 32 words, half old and half new. Typical measures of performance are accuracy in terms of proportion correct and discriminability between old items and new items (another important measure is reaction time, but I do not consider it here; see Ratcliff, 1978 Ratcliff, , 1988 . Learning in this context is defined as improvement in discriminability (d') between old and new test items as a function of amount of study. Forgetting is defined as the decrement in accuracy as a function of intervening information presented between study and test of the item. The first and main model I evaluate is the encoder network (Ackley, Hinton, & Sejnowski, 1985), using the backpropagation learning algorithm. The encoder model was originally devised as a test of learning algorithms; the model reproduces any one of a previously trained set of input vectors at its output layer, with a narrow channel (fewer hidden units than input or output units) between the input and output layers (see Figure   1 ). As the model is implemented in the work described in this article, items entered into the system are assumed to be vectors of features, typically of length 32 (although some initial demonstrations use vectors of length 4). The system is made up of an input layer of this number of units, a hidden layer of fewer units (1/2 to 3/4 the number of units in the input layer), and an output layer of the same size as the input layer. The hidden units have no direct connections to or from the external input or output. Instead, the hidden units project to units at the input and output layers so that each unit in the hidden layer is connected to each unit in the input layer and also to each unit in the output layer. The task of the system is to reproduce an input signal as an output signal. During learning, an item vector is presented at the input layer, and activation is transmitted through the network, resulting in a set of values at the units of the output layer. The differences between the values of the input and the values at the output layer are used as the training signals to modify the weights in the network. Weights are changed in proportion to the difference between the actual output value and the required output value for a particular unit. In this model, the location of memory can be conceptualized as being distributed across the interconnections between the input and hidden layers and the hidden and output layers. The tests of the model that follow move from examination of a simple system of four vectors to examination of larger systems for which statistical properties are averaged over large numbers of Monte Carlo simulations. The simple system was studied to examine the behavior of individual elements within a vector as a function of learning and forgetting. The larger systems were studied as potentially viable models of recognition memory in CONNECTIONIST MODELS OF MEMORY 287 Hidden Units Input Units Figure 1 . The multilayer encoder model. which statistical properties of the data could be examined. To model recognition, a set of vectors is trained (the old items), and then at test, these old vectors and some new vectors (randomly chosen with the same statistical properties as the old vectors) are presented to the system. The backpropagation algorithm takes an input vector and produces an output vector with elements that vary between 0 and 1. For the encoder model, the desired output is the same as the input, and to quantify the degree of match between the output and the input, a match value based on the dot product of the two vectors is used. From the match values for tests of old and new vectors and the variability in these values (over the Monte Carlo trials), the discriminability between old and new items is computed. The input vectors used in the simulations are random vectors with elements either 0 or 1, and this is a potential limitation. In many tasks, information is organized and structured, and issues of representation are of major concern. Some of the problems encountered in the simulations might disappear if reasonable representations were used instead of random vectors. In the evaluation of variants of the encoder model, some representation issues are examined (e.g., learning as increments to prelearned memory, and stimulus vectors as variants on a prototype), but these examples are by no means exhaustive. Until further theoretical development of feature representations, this is probably the best that can be done (see also the representation assumptions made by McCIoskey & Cohen, 1989, that produce results similar to those presented later). Model of Recognition Memory The multilayer model was chosen for study because it is an architecture that is currently being applied to many different domains in psychology. In addition, it allows computation of functions that linear models (single or multiple layer) are unable to compute (Rumelhart, Hinton, & Williams, 1986 ), a problem that was used to argue against neural models (Minsky & Papert, 1969) . A further reason for choosing the nonlinear backpropagation algorithm in contrast to linear models is that some linear models cannot account for performance in recognition memory because they predict that discrimination between studied items and nonstudied items remains constant as a function of amount of study (J. A. Anderson, 1973; Murdock, 1982) . In the multilayer model, items are represented as vectors, with each feature or element having a value between 0 and 1. An input vector with elements oj and the weights between the nodes in the input and hidden layers are used to compute the input to the hidden layer: neh = ~ wijo j. (1) J This net input to the hidden layer is transformed to activation (the output of the hidden layer) by a logistic sigmoid transformation that transforms neti values from a range of-oo to +oo to the range 0 to 1: ( 2) The activation oi and the set of weights between the nodes in the hidden and output layers are used to produce the net input to the output layer by Equation 1. This net input is transformed with the logistic transformation to activation at the output layer, which is the final output from the system. Once activation values are obtained for the output layer, they are used in conjunction with the desired or target output values to modify the weights in the system. The weights are modified to minimize the sum of the squared error between the value of each target element and the corresponding element of the obtained output. This is analogous to minimizing the energy in a physical system, where energy is defined as the sum of squared differences between the desired target vector and the output vector produced by the system. With the logistic function (Equation 2), the error signal at an output element is given by where ti is the teaching signal (or desired output) for element i . Then this error value is used to change the weights between the hidden and output layers with where n is a learning rate parameter. Once the weights between the output and hidden layer are modified, the weights from the hidden layer to the input layer have to be modified. The error signal is computed by propagating the error from the output layer to the hidden layer from The change in the weights between the hidden and input layers is given by Equation 4 with ~i from Equation 5. Weights were updated after each presentation of a vector to the system during learning but not during testing. There are two other factors to consider before proceeding to simulations. First, if the sum of squares or energy space is highly variable as a function of parameters and ifa high learning rate is used (large ~ in Equation 4 leading to large changes in weights), the system can oscillate and so produce slow convergence to an adequate solution. To avoid this problem, weight modification is based on a linear combination of the current computed weight change and the prior weight change. The component of weight change that is based on earlier weight changes is termed momentum and refers to the fact that prior weight modifications as well as the newly computed changes guide weight modification (see . In many of the simulations presented later, a value of momentum set to 0.5 was used, but to check the generality of the results, the qualita-288 ROGER RATCLIFF tive behavior of the model was compared to the case for which the value of momentum was set to 0 (i.e., weight changes based only on current computation). Results showed few differences (except in the case of mixed vs. pure list presentation; see specific examples presented later). The second factor to consider is called symmetry breaking by , and the term refers to the fact that if weights start with equal values, modifications can only keep the weights equal because the hidden units will all receive the same error signal back from the output layer. To overcome this problem, small random unequal values are assigned as initial values to the weights (e.g., from a uniform distribution in the range -0.3 to +0.3). Again, variation in the size of these initial values was examined, and the effects are noted later in the article. One additional factor sometimes used in the multilayer model is bias on the nodes. This adds a term in the exponent of Equation 2 (see and is used to capture the average match across all nodes. No biases on the weights were used in these simulations. Last, a measure of the degree of recognition for a test vector is required. With the encoder model, the appropriate measure is the dot product between the input and output vectors (transformed to the range -1 to + 1 and divided by the vector length). The model attempts to reproduce the input at the output, and the dot product measures how well this is accomplished. To form the dot product, each element is transformed to a new value 2 times the old value minus 1; this provides values of the elements in the range -1 to + 1. The dot product between the transformed vectors is then divided by the number of elements to provide an overall value of match between -1 and + 1. With several Monte Carlo replications, the mean in the match value and its standard deviation can be obtained for both old and new items and be used to calculate the standard discriminability statistic between old and new items: d' = (mean of old matchmean of new match)/SD of new match. Model: Serial Presentation A major focus of the research reported here involves a departure from the usual training method for stimuli in a multilayer connectionist system. In the usual system, the vectors to be learned are trained in sweeps through the set of all the vectors to be learned. Thus, all vectors are available to the network throughout training. For example, if four vectors, A, B, C, and D, were to be learned, the system would be presented with the sequence ABCDABCDABCDABCD • • • until some learning criterion for the joint solution was reached. I define this as remote rehearsal because after item D is presented, the remote item A is brought back for another presentation. In domains such as perception, one can imagine repeated exposures to a set of stimuli to be learned that would approximate this remote sequence (in fact, some recent models such as Seidenberg & McClelland's, 1989 , rotate training through stimuli according to their frequency of occurrence in English). In contrast, in recognition memory, when the items in a study list are presented at fast presentation rates, items are learned largely in the order in which they are presented, with little opportunity for remote rehearsal. For example, it is unlikely that a subject could retain all the items in a 16-or 32-item list in order to sweep through the whole list. Note that for remote rehearsal, the memory system that retained the whole list would need to be modeled within a connectionist framework, and this would lead to redundancy in memory systems (one to retain it for the other to learn it). In addition, it is possible to use study materials of a kind that cannot be easily rehearsed throughout presentation of the study list (e.g., pictures presented at fast rates). Because of these considerations, models in the class I examine use only local rehearsals; that is, only the item presented last is rehearsed, or only the last small group of presented items is rehearsed, where the number of items in the group is typically set to 4 (e.g., see the rehearsal scheme presented in Gillund & Shiffrin, 1984). A second departure from many connectionist models is that only positive instances are presented to the system. Only old items that require a positive match from the system are presented at study; new items (requiring a negative response) are not presented for training. This means that the system receives no discriminative training between old and new items, only discriminative training among positive instances. Yet, the primary requirement for the system is to discriminate new items from old items. The first set of studies I discuss explores forgetting in systems with small vectors followed by large vectors, and the second set of studies investigates amount of learning as a function of the number of learning trials. Interference Effects The first set of simulations uses the encoder model with small vectors to examine the feature-by-feature behavior of the system before and after forgetting. There are four orthogonal vectors to be learned (1 0 0 0, 0 1 0 0, 00 1 0, and 000 1) and a multilayer system with four input units, four output units, and three hidden units (cf. Ackley et al., 1985). When all four vectors are learned as an ensemble (remote rehearsal) by repeatedly sweeping through the whole set, the system successfully learns them all. The learning criterion typically used is to continue sweeping until the output values are close to the input values, that is, within some small percentage of the input values, for example, 6%. Interference is examined by observing what happens when three of the vectors are learned to criterion in sweeps (e.g., Modifying all weights. In the first implementation, all weights were allowed to change during training of the first three vectors, and all weights were also allowed to change during training of the fourth vector. each of the four vectors after the fourth was learned and shows that there were serious distortions in the reproductions of the first three vectors. For each of these three output vectors, there was an entry for the feature trained for the fourth vector. So, for example, the response to 1 0 0 0 after training on the fourth vector (0 0 0 1) was 0.77, 0.01, 0.01, and 0.90. These results indicate that weights in the system were altered by the training of the fourth vector so that almost any test vector would evoke a response in the fourth feature position. Thus, any output is a blend of the correct output and the vector that was trained last; an item presented earlier for test retrieves the last-studied item modulated b~, the earlier item. The general conclusion is that training new vectors after training an earlier set, with no repetitions of the earlier set, will result in loss of the ability of the network to reproduce the earlier set. (Also included in Table 1 are the results for testing two other vectors, 1 1 1 1 and 0 1 1 0. These are included to show what happens when a new item is tested; in this example, 'the last feature is turned on for these new items as well as for the old items.) Modify only low weights. After the items in a set have been learned to criterion, the weights on the connections between units will vary in how much they have been modified from the initial values. For the system to learn more items, perhaps the weights that were modified least in original learning should be used for new learning, leaving the other weights unaltered (the notion would be that altered weights become more resistant to change, or less plastic). To implement this scheme, three items were learned to criterion (as in the previous simulation). I then empirically determined a cutoff value of weight strength so that approximately half the weights were large (either positively or negatively); these were not modified during training of the fourth item. The other half of the weights were small and could be modified. Table 2 shows results parallel to those in Table 1 for the output vectors produced when each of six input vectors are tested. The results show somewhat better reproduction of the input vectors than when all weights are modified (cf. Table  1 ), but the improvement is not large and does not change the general conclusion that later learning distorts earlier learning. Note that reproductions of the diagonal elements for the first three patterns after training the fourth vector were quite variable (0. 85-0.19). This is a joint effect of dividing the weights into two classes, fixed and adjustable, and of the variability in starting values of weights. Ifa different set of randomly selected initial weights are chosen, the diagonal elements are just as varied, but instead of the second vector reproducing itself well, the first or third vector might. Add new hidden units. If neural systems are taken seriously as the inspiration for connectionist models, then the notion that uncommitted nodes and connections can become committed during learning is worth examination. Bringing in uncommitted nodes is simulated by assuming that even after considerable training, there are still uncommitted nodes available to be recruited during further training. By this model, learning new information is viewed not as reorganizing already existing memory but as training new resources alone or as modifying old memory plus new resources. These two possibilities are examined in the next simulations. The first simulation adds one extra hidden unit for training the fourth vector, and all the weights in the network are allowed to change. One might expect that much of the new learning would be accomplished through the new connections. However, although the results (Table 3) show that recovery of the first three vectors is better than in the simulations described earlier, performance is not perfect. For example, after learning 0 0 0 l, responses in the fourth position of the first three vectors (originally 0) varied from 0.46 to 0.50, indicating considerable interference. In the simulation just described, the network as a whole (the old connections plus the new connec- Table 4 , and performance is similar to that shown in Table 3 . The main difference is that the fourth item did not reproduce itself as well at final test because training did not include the whole network. However, the three initial vectors reproduced themselves better than in the earlier simulations. These two results suggest a tradeoff." The last vector can be reproduced better at the expense of earlier vectors or the earlier vectors can be reproduced better at the expense of the last vector. More than one interfering item. In the simulations described so far, three items were trained to criterion first, and then a fourth was trained to criterion. In the next simulation, fewer items were trained initially, and more items were trained subsequently to determine whether this would result in poorer reproduction of the initially trained items. First, the vectors 1 0 0 0 and 0 1 0 0 were trained to criterion, and then two more vectors, 0 0 1 0 and 0 0 0 1, were trained to criterion. The results for the test items are shown in Table 5 . The disruption to reproduction of the initial two items after training the last two is large. In fact, comparing the response to the two new vectors tested (1 1 1 I and 0 1 1 0), there is little information in the output to identify the initially trained vectors. The conclusion from this simulation is that the greater the number of vectors trained in a group after original training, the less likely the earlier trained vectors are to be accurately reproduced. Each of the simulations showed that when items or sets of items are trained sequentially, training on the later items disrupts learning of the earlier items, The simulations used small vectors of length 4, with one set of items trained to criterion and then a second set trained to criterion. Results showed that a test of an item from the first set after training the second set resulted in massive interference: The output vector was more similar to one of the items from the second set than it was to the test item. Attempts to reduce interference by modifying only some weights, adding hidden units, and coding the two training sets by different hidden urfits reduced interference a little but did not solve the interference problem. McCIoskey and Cohen (1989) have obtained similar results for two different tasks: learning arithmetic facts (using a coarsecoding vector representation) and the Barnes and Underwood (1959) A-B, A-C interference paradigm. McCloskey and Cohen conceptualized the rapid forgetting caused by interference in terms of a multidimensional space (an energy or sum-ofsquares space), where each dimension in the space is a connection weight and where the learning algorithm searches through this space for a minimum energy. The learning algorithm adjusts the weights to bring the sum of the differences between the values of each output feature and its training signal to a minimum (e.g., an energy minimum). Thus, learning a set of items involves gradient descent in a multidimensional space where each dimension or weight is gradually adjusted to produce the energy minimum. If the number of hidden units is large, then there will be several (if not very many, e.g., 10 9) solutions; that is, there will not be a unique minimum (Pavel & Moore, 1988) . Then, the particular solution obtained for one set of items will serve as the starting point for training of subsequent items (see Grossberg, 1987) , and the new solution will be near the old solution in the multidimensional space. This new solution will not generally be a solution that maximally satisfies both the set of constraints on the initially learned items and the constraints for the subsequently trained items; rather, it will satisfy only the last items trained. Thus, reproduction of the earlier items will be degraded. The initially trained items will be reproduced to some extent because the solution to the new set of items is not too far from the solution for the earlier learning. In contrast, when a set of items is to be learned all at once, the method of sweeps leads to an iterative path through the parameter space to converge on a joint solution. In the simulations I have described, attempts to constrain the space of possible solutions by allowing only a subset of the weights to be modified were of limited success. The reason for this is that the network is highly interconnected, and learning is distributed across the whole network so that all weights, both small and large, are carrying part of the solution. The same conditions hold in the simulations in which extra hidden units and connections are added. The solution is still carried by the whole network, and despite adding new resources, there is still movement of the whole system through the solution space to a new point. The only difference between these latter simulations and the earlier simulations is that reproduction of old vectors is somewhat better because, with extra resources, a solution can be found nearer to the starting point (the prior solution). The simulations with a small set of short, orthogonal input vectors allowed the effects of new learning on prior learning to be observed easily and conceptualized feature by feature within a vector. However, a limitation of these simulations might be that such short vectors do not provide a complex or rich enough system to acceptably mimic the human processing system. Most vector models of memory use vectors of length greater than l0 (because long vectors would be needed to represent all the words in a mental dictionary, for example) and assume that properties of the solutions scale up; that is, they remain qualitatively the same as vector length is increased. For the next set of studies of forgetting, I used a system with input and output vectors of 32 units and a hidden layer of 16 units. Only a maximum of 12 items were trained so that rapid forgetting could not be attributed to lack of resources in the network (i.e., a vector length too short for the number of items to be learned). In addition, the vectors for the items to be trained were not orthogonal; values for each feature for each item were selected randomly so that random correlations between items would be present as they would be with real items in memory. The simulations presented in this section parallel the manipulations examined in the first section with simulations in which all weights are modified, subsets of weights are modified, or extra hidden units and weights are added to the system for each new item to be learned. In these simulations, the system was first presented with four vectors in a group, with the elements of each vector chosen randomly to be either 0 or 1. Training sweeps through the whole group continued until the vectors were learned to criterion. Then, eight random vectors were trained individually. After each of the items reached criterion, that item and all previously trained items were presented for test. The match between each of the trained items and its output was computed, yielding forgetting functions for the four initially trained items as well as the successively presented single items. The items were learned to a criterion of 0.94 (unless otherwise noted), and a typical range of training trials was 50-120. Twenty replications of this basic simulation were performed to generate estimates of variability in performance; the results reported are based on means over these replications. For the first set of simulations, all connections were available for modification during each training trial (both for the initial four items and for each successive new item). The results are presented in Figure 2 and show that there was rapid forgetting for the initially learned four items as a function of successive learning trials on the single items. The match value fell from 0.94 to 0.51 after training one intervening item. By six or seven intervening items, the match had fallen to 0.23. For singly presented items, after one intervening item, the drop in match was worse, falling from the initial value of 0.94 to 0.29, and after six or seven further items had been learned, the match dropped to 0.10. This last value is close to the match for new items (random vectors with the same statistical properties as the trained vectors). The match for new items was 0.125 when they were tested immediately after the four initially trained items and 0.046 +_ 0.046 when they were tested after the successive single items. The decrease in the match value for new items can be understood by considering how well the new items match the trained items. For a new item tested immediately after the initial four, its output will be like the item of the first four that is most similar to it: The new item is trying to reproduce the item most like it. The output will match the test item to the extent that both are similar. For new items tested after single items are trained, the last item trained (or with lower probability, the item before) will be reproduced to some degree, and this will match the last item learned best (or to a smaller extent, the item before). This will lower the average match relative to testing after the initial four are learned because by chance, one of the four will match better than only the last one (or two). Overall, from this set of results, two conclusions can be drawn: First, as with the short four-element vectors, items learned singly are poorly reproduced after several intervening learning trials. Second, items encoded in groups are significantly more resistant to forgetting than items learned individually. In the second set of simulations, the items learned individually (after the initial four items) were trained with only the lowvalued weights altered. To accomplish this, the weights were divided in half after examination of the results of training the four initial vectors. The smaller weights were allowed to change during training, but the larger weights were not. The cutoff value dividing the large and small weights was fixed for subsequent learning of individual items so that when any weight exceeded this value, it was not allowed to change during training of any further items. Occasionally, the system would fail to reach the learning criterion for one of the later individual items. In this case, 1,000 training sweeps of the item were completed to ensure that training asymptoted before proceeding to the next item. Results presented in Figure 3 show that the asymptote is higher than the corresponding asymptote in Figure 2 , both for items learned in the group of four and items learned singly. For new random vectors tested immediately after the first four, the mean match was 0.125, and for new items tested later, the mean match was 0.045 _+ 0.039. Although performance was better than in the first study, there was still rapid forgetting of highly learned information as a function of later learning. The third study was designed to examine the issue of adding resources to the system: Two new hidden units and their connections were added for each additional item to be learned. Each additional item was trained by modifying only the weights to and from the new hidden units, but connections to and from all the hidden units determined the output during training. Items in the initial group were retained better than in the first set of simulations (Figure 4) because the weights to and from the 16 hidden units were not altered in later training and they dominated over the effects of new learning. The single items learned subsequently were retained much more poorly for the same reason. By the time seven or eight subsequent items had been trained, the match for these items was close to 0. In the fourth study, two new hidden units were added for each single item to be learned. Only connections to and from these two new hidden units were altered during training, and only the new hidden units were used to determine the output during training, so that training of the single items was carried out independently of the rest of the network (this contrasts with the previous simulation, in which connections to and from both the old and new hidden units were used to produce the output dur-+60 E ing training). At test, all connections were included to produce outputs. As in the example in Table 3 for the four-element vectors, immediate tests of the single items learned last did not give high matches because previously set weights were involved at test, but these weights had not been involved in training the item. Also, as in the two studies discussed earlier, there was a rapid drop in the match for the initial four items as a function of subsequent learning (see Figure 5) . However, compared with the first study, the match was better for both the initial items and the single items because the effects of interference were not in learning but at test, where subnetworks (the different hidden units and weights used in learning the different items) were combined. To show that this result is due to keeping learning separate and combining only at retrieval and not due to simply providing more hidden units, a further simulation was carried out with all weights modified and more hidden units provided (32 instead of 16). Figure 6 shows the results. Values of match for items encoded in the 4-item set were only a little below those in Figure  5 when the new hidden units were trained in isolation, but the single-item match was approximately half that of the values shown in Figure 5 . In fact, performance is comparable to that in Figure 2 , showing that extra resources have little effect on performance. Results showed that if a set of items is trained and then another item is trained without retraining items from the first set, then there is a huge drop in recognition performance on the first set. In addition, recognition performance for a single item, trained to criterion after the first set, drops even more after another single item is trained to criterion. There was little improvement when low weights were trained while keeping larger weights fixed. There was a reduction in the amount of interference when extra hidden units were added to the system for each item trained, but only when the rest of the network was held fixed while each new item was trained. When all weights were allowed to change, increasing resources (numbers of hidden units and connections) did not improve performance.
Catastrophic forgetting is a well-known phenomenon arising in the problem of lifelong/continual learning REF ).
Connectionist models of recognition memory: Constraints imposed by learning and forgetting functions
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