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Many perceptual processes and neural computations, such as speech recognition, motor control and learning, depend on the ability to measure and mark the passage of time. However, the processes that make such temporal judgements possible are unknown. A number of different hypothetical mechanisms have been advanced, all of which depend on the known, temporally predictable evolution of a neural or psychological state, possibly through oscillations or the gradual decay of a memory trace. Alternatively, judgements of elapsed time might be based on observations of temporally structured, but stochastic processes. Such processes need not be specific to the sense of time; typical neural and sensory processes contain at least some statistical structure across a range of time scales. Here, we investigate the statistical properties of an estimator of elapsed time which is based on a simple family of stochastic process.