TODAM, developed by Bennet Murdock (1982, 1993), is a distributed memory model in which all items and associations are superimposed in a single high-dimensional memory vector. Encoding uses vector addition and convolution, while retrieval uses correlation (the approximate inverse of convolution). TODAM provides an elegant algebraic framework for understanding storage and retrieval in both item recognition and associative recall.
Representational Framework
Each item is represented as a vector of N random values (typically drawn from a normal distribution with mean 0 and variance 1/N). Items are stored by simply adding their vectors to a common memory vector M. Associations between items are stored by adding the convolution of their vectors. Convolution produces a new vector that is orthogonal to both of its constituents but from which either constituent can be approximately recovered.
where fᵢ are item vectors, fᵢ * gⱼ denotes convolution of paired items, and α and β are encoding strength parameters.
Convolution and Correlation
The convolution of two N-dimensional vectors f and g produces a new N-dimensional vector c where each element is c(k) = Σⱼ f(j) · g(k−j). This operation acts as a binding mechanism, creating a representation of the association that is distinct from either constituent. To retrieve an associate, the memory vector is correlated with the cue, which is the approximate inverse of convolution:
where # denotes correlation. The retrieved vector g' is a noisy approximation of the original associate g, with noise from all the other items and associations superimposed in M.
Item and Associative Information
TODAM elegantly separates item information (the individual vectors added to M) from associative information (the convolutions added to M). Item recognition is modeled as the dot product of the test probe with M: if M · f exceeds a criterion, the item is judged as old. Associative recall uses correlation to extract a noisy copy of the paired associate. This separation explains the empirical finding that item and associative recognition can be independently manipulated.
Serial Order and Chunking
Murdock extended TODAM to handle serial order by using successive convolutions to encode position. The serial order of items f₁, f₂, f₃ is stored as f₁ + f₁*f₂ + f₁*f₂*f₃ + ..., creating a chained representation that supports forward recall. Later versions (TODAM2) incorporated chunking operations to handle hierarchical structure in memory.
Convolution can be approximately implemented by tensor products in neural networks. The appeal of TODAM is that it shows how a mathematically principled binding operation, combined with simple superposition, can account for the core phenomena of human associative memory without requiring separate storage locations for individual memories.