A zero-bounded model of operant demand

Contemporary approaches for evaluating the demand for reinforcers use either the Exponential or the Exponentiated model of operant demand, both derived from the framework of Hursh and Silberberg (2008). This report summarizes the strengths and complications of this framework and proposes a novel implementation. This novel implementation incorporates earlier strengths and resolves existing shortcomings that are due to the use of a logarithmic scale for consumption. The Inverse Hyperbolic Sine (IHS) transformation is reviewed and evaluated as a replacement for the logarithmic scale in models of operant demand. Modeling consumption in the “log₁₀-like” IHS scale reflects relative changes in consumption (as with a log scale) and accommodates a true zero bound (i.e., zero consumption values). The presence of a zero bound obviates the need for a separate span parameter (i.e., k) and the span of the model may be more simply defined by maximum demand at zero price (i.e., Q₀). Further, this reformulated model serves to decouple the exponential rate constant (i.e., α) from variations in span, thus normalizing the rate constant to the span of consumption in IHS units and permitting comparisons when spans vary. This model, called the Zero-bounded Exponential (ZBE), is evaluated using simulated and real-world data. The direct reinstatement ZBE model showed strong correspondence with empirical indicators of demand and with a normalization of α (ZBEn) across empirical data that varied in reinforcing efficacy (dose, time to onset of peak effects). Future directions in demand curve analysis are discussed with recommendations for additional replication and exploration of scales beyond the logarithm when accommodating zero consumption data.     

Prototypes,Location, and Associative Networks (PLAN): Towards a Unified Theory of Cognitive Mapping

An integrated representation of large-scale space, or cognitive map, colled PLAN, is presented that attempts to address a broader spectrum of issues than has been previously attempted in a single model. Rather than examining way-finding as a process separate from the rest of cognition, one or the fundamental goals of this work is to examine how the wayfinding process is integrated into general cognition. One result of this approach is that the model is “heads-up,” or scene-based, because it takes advantage of the properties of the human visual system and, particularly, the visual system's split into two pathways. The emphasis on the human location or “where” system is new to cognitive mapping and is port of an attempt to synthesize prototype theory, associative networks and location together in a connectionist system. Not all of PLAN is new, however. Many of its parts have analogues in one or another preexisting theory. What makes PLAN unique is integrating the various components into a coherent whole, and the capacity of this resulting system to speak to a wide range of constraints. Our approach emphasizes adaptiveness; thus, our focus on such issues as ease of use and efficiency of learning. The result is a model that has a stronger relationship both to the environment, and to the ways that humans interact with it, compared with previous models. The resulting model is examined in some detail and compared to other systems.