Abstract
A key aspect of cognitive flexibility is abstraction, i.e., the ability to represent rules governing appropriate behavior separately from the identities of the particular objects on which those rules presently act. This permits one set of rules to be generalized to different circumstances and inputs. A simple example is the delayed matching task of Miller et al (1991). In this case, the rule requires responding to the second appearance of the object that is presented at the outset of a trial, no matter what that object is. There is evidence that rules are represented in the activity of prefrontal cortical neurons - for instance Shima et al. (2007) reported that many such cells are selective for abstract sequence categories (such as ABAB, AAAA), generalizing over particular actions A and B such as `push' or `pull'. From an algorithmic perspective, the objects are variables in the rules, and generalization is a form of variable substitution.
Despite the critical importance of this form of flexibility, and indeed some connectionist approaches (Touretzkey 1990; Shastri 1999), the more neurally-inspired models of prefrontal cortex and associated areas lack any way of representing this abstraction. Instead, they combine rule structure and stimulus identity inseparably in the weights of their networks. Here, we extend one such model, the bilinear framework of Dayan (2007), to support variables. We demonstrate our new model on an abstracted version of the 12-AX task (Frank et al, 1991), which has become a prime test-case for rule-based models. The original version of this task requires a subject to respond to one of a set of target sequences (letters AX and BY in the original) depending on the context, which itself is indicated by elements of the sequence (digits 1 and 2). We consider an extended version in which the assignment of input stimuli to components of the task is flexible, thus permitting variations such as A2-X1 or DC-21.
The new model specifies a set of gated working memory and action modules controlled by learned, bilinear, forms. The bilinearity offers an ascetic form of hidden unit (Rigotti et al, 2007) or basis function (Poggio, 1990) representation of nonlinear contingencies. It permits the key operation required for variable substitution, namely assessing the match between one quantity stored in a memory module and another quantity presented in the inputs. Given this, switching the stimulus identities becomes as easy as updating the appropriate memory cells. Using matching as a primitive, we provide an algorithm for solving the full task that is couched as a set of rules, only one of which is active at any time. Each such rule is represented as a set of bilinear weights, one per action, acting on the joint vector of inputs and the current state of all memory modules. Rules are triggered by changes in working memory or inputs. We show how to capture these rules in the overall bilinear structure.
