  Task based neuroimaging tools for the study of cognitive neuroscience provide
insight into understanding how the brain responds to increasing cognitive
demand. Theoretical models of neural-cognitive relationships have been
developed based on observations of linear and non-linear increases in brain
activity. Neural efficiency and capacity are two parameters of current
theoretical models. These two theoretical parameters describe the rate of
increase of brain activity and the upper limits of the increases, respectively.
The current work demonstrates that a quadratic model of increasing brain
activity in response to the n-back task is a solution to a differential
equation model. This reinterpretation of a standard approach to analyzing a
common cognitive task provides a wealth of new insight. The results include
brain wide measures of neural efficiency and capacity. The quantification of
neural-cognitive relationships provides evidence to support current cognitive
neuroscience theories. In addition, the methods provide a framework for
understanding the neural mechanisms of working memory. This allows estimation
of the effects of experimental manipulations within a conceptual research
framework. The proposed methods were applied to twenty-one healthy young adults
while engaging in four levels of the n-back task. All methods are easily
applicable using standard current software packages for neuroimaging.
