Effect of inter-movement interval on movement duration under various temporal discounting regimes. Effect of inter-movement interval on movement duration under various temporal discounting regimes. A, Top, We consider an arbitrary class of movements for which probability of success (acquisition of reward) increases with movement duration (green line). Hyperbolic temporal discounting, plotted here by the red line, is the function . τ is movement duration, and δ is inter-movement interval (here assumed to be 0.5 s). The blue line is the multiplication of probability of reward with the temporal discount function (Eq. 1). The movement duration that maximizes the discounted reward is noted by the dashed line. In this case, the discounted reward corresponds to reward rate. B, C, Top, Corresponding plots for an exponential discount function with linear exponents exp(−k(τ + δ)), and an exponential discount function with squared exponents exp(−k(τ + δ)2). All discount functions are scaled to be equal 1 at 0.5 s and parameters for the exponential discount functions were adjusted to predict the same optimal movement duration as rate of reward for δ = 0.5 s. Bottom, The effect of increasing the inter-movement interval δ to 1 s. For hyperbolic discounting, as this delay is increased the optimum movement duration becomes longer, i.e., the movement vigor decreases. For an exponential temporal discount function with linear exponents there is no change in the optimum movement duration as inter-movement intervals are changed. For an exponential discount function with quadratic exponents, movement duration decreases as the inter-movement interval increases. Adrian M. Haith et al. J. Neurosci. 2012;32:11727-11736 ©2012 by Society for Neuroscience