Learning Polymorphic Neural ODEs With Time-Evolving Mixture.

Neural ordinary differential equations (NODE) present a new way of considering a deep residual network as a continuous structure by layer depth. However, it fails to overcome its representational limits, where it cannot learn all possible homeomorphisms of input data space, and therefore quickly saturates in terms of performance even as the number of layers increases. Here, we show that simply stacking Neural ODE blocks could easily improve performance by alleviating this issue. Furthermore, we suggest a more effective way of training neural ODE by using a time-evolving mixture weight on multiple ODE functions that also evolves with a separate neural ODE. We provide empirical results that are suggestive of improved performance over stacked as well as vanilla neural ODEs where we also confirm our approach can be orthogonally combined with recent advances in neural ODEs.

Saccade vigor and the subjective economic value of visual stimuli.

Decisions are made based on the subjective value that the brain assigns to options. However, subjective value is a mathematical construct that cannot be measured directly, but rather is inferred from choices. Recent results have demonstrated that reaction time, amplitude, and velocity of movements are modulated by reward, raising the possibility that there is a link between how the brain evaluates an option and how it controls movements toward that option. Here, we asked people to choose among risky options represented by abstract stimuli, some associated with gain (points in a game), and others with loss. From their choices we estimated the subjective value that they assigned to each stimulus. In probe trials, a single stimulus appeared at center, instructing subjects to make a saccade to a peripheral target. We found that the reaction time, peak velocity, and amplitude of the peripherally directed saccade varied roughly linearly with the subjective value that the participant had assigned to the central stimulus: reaction time was shorter, velocity was higher, and amplitude was larger for stimuli that the participant valued more. Naturally, participants differed in how much they valued a given stimulus. Remarkably, those who valued a stimulus more, as evidenced by their choices in decision trials, tended to move with shorter reaction time and greater velocity in response to that stimulus in probe trials. Overall, the reaction time of the saccade in response to a stimulus partly predicted the subjective value that the brain assigned to that stimulus.NEW & NOTEWORTHY Behavioral economics relies on subjective evaluation, an abstract quantity that cannot be measured directly but must be inferred by fitting decision models to the choice patterns. Here, we present a new approach to estimate subjective value: with nothing to fit, we show that it is possible to estimate subjective value based on movement kinematics, providing a modest ability to predict a participant's preferences without prior measurement of their choice patterns.

Movement Vigor as a Reflection of Subjective Economic Utility.

To understand subjective evaluation of an option, various disciplines have quantified the interaction between reward and effort during decision making, producing an estimate of economic utility, namely the subjective 'goodness' of an option. However, variables that affect utility of an option also influence the vigor of movements toward that option. For example, expectation of reward increases speed of saccadic eye movements, whereas expectation of effort decreases this speed. These results imply that vigor may serve as a new, real-time metric with which to quantify subjective utility, and that the control of movements may be an implicit reflection of the brain's economic evaluation of the expected outcome.

Control of movement vigor and decision making during foraging.

During foraging, animals decide how long to stay at a patch and harvest reward, and then, they move with certain vigor to another location. How does the brain decide when to leave, and how does it determine the speed of the ensuing movement? Here, we considered the possibility that both the decision-making and the motor control problems aimed to maximize a single normative utility: the sum of all rewards acquired minus all efforts expended divided by total time. This optimization could be achieved if the brain compared a local measure of utility with its history. To test the theory, we examined behavior of people as they gazed at images: they chose how long to look at the image (harvesting information) and then moved their eyes to another image, controlling saccade speed. We varied reward via image content and effort via image eccentricity, and then, we measured how these changes affected decision making (gaze duration) and motor control (saccade speed). After a history of low rewards, people increased gaze duration and decreased saccade speed. In anticipation of future effort, they lowered saccade speed and increased gaze duration. After a history of high effort, they elevated their saccade speed and increased gaze duration. Therefore, the theory presented a principled way with which the brain may control two aspects of behavior: movement speed and harvest duration. Our experiments confirmed many (but not all) of the predictions, suggesting that harvest duration and movement speed, fundamental aspects of behavior during foraging, may be governed by a shared principle of control.