Effect of Addiction Modeling Reinforcement Schedules on Delay Discounting

Elevated delay discounting, in which delayed rewards quickly lose value as a function of time, is associated with substance use and abuse. Currently, the direction of causation is unclear: while some research indicates that elevated delay discounting leads to future substance use, it is also possible that chronic substance use and specifically the rate of reinforcement associated with drug use, leads to elevated delay discounting. This project aims to examine the latter possibility. 47 participants completed ten 30-minute daily sessions of a visual attention task, and were reinforced at a rate intended to model drug use (fixed ratio 1) or drug abstinence (fixed ratio 10). Baseline and post-training rates of delay discounting were assessed for hypothetical $50 and $1000. Area under the curve of the indifference points as a function of delay was calculated. A greater area under the curve suggests more self-control, whereas a lower value represents more impulsiveness. Results at the monetary value of both $50 and $1000 showed increased impulsivity in relation to the control for both the FR1 and FR10 groups indicating that the two schedules may both model drug use.

Hierarchical Reconstruction Method for Solving Ill-posed Linear Inverse Problems

We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method successively extracts information from the previous hierarchical residual to the current hierarchical term at a finer hierarchical scale. As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of finitely many hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.