A drift-diffusion model of interval timing in the peak procedure

Drift-diffusion models (DDMs) are a popular framework for explaining response times in decision-making tasks. Recently, the DDM architecture has been used to model interval timing. The Time-adaptive DDM (TDDM) is a physiologically plausible mechanism that adapts in real-time to different time intervals while preserving timescale invariance. One key open question is how the TDDM could deal with situations where reward is omitted, as in the peak procedure-a benchmark in the timing literature. When reward is omitted, there is a consistent pattern of correlations between the times at which animals start and stop responding. Here we develop a formulation of the TDDM's stationary properties that allows for the derivation of such correlations analytically. Using this simplified formulation we show that a TDDM with two thresholds-one to mark the start of responding and another the stop-can reproduce the same pattern of correlations observed in the data, as long as the start threshold is allowed to be noisy. We confirm this by running simulations with the standard TDDM formulation and show that the simplified formulation approximates well the full model under steady-state conditions. Moreover, we show that this simplified version of the TDDM is formally equivalent to Scalar Expectancy Theory (SET) under stationary behaviours, the most prominent theory of interval timing. This equivalence establishes the TDDM as a more complete drift-diffusion based theory with SET as a special case under steady-state conditions.