Slowdown and saturation of internal time according to the statistics of information input: a minimal model of response systems

TL;DR

This paper models how internal perception of time in response systems slows down and saturates based on input statistics, revealing how internal time diverges from physical time over long periods.

Contribution

It introduces a minimal model linking information input statistics to internal time progression, including saturation effects and a weighted extension for information content.

Findings

Internal time slows and saturates with finite code types

Closed-form relation between internal and physical time for uniform codes

Quantification of uncertainty and information content in observed codes

Abstract

We consider a response system that updates its internal state in accordance with information input arriving from outside. In this paper, we define as internal time the ``number of kinds'' of codes that have been observed at least once up to a given time, and analyze how the way internal time advances is determined by the statistics of information input (arrival rate and code distribution). When arrivals follow a Poisson process, the average advancing speed of internal time decreases monotonically with time, and if the number of kinds of codes is finite, it eventually approaches an upper limit and saturates. As a result, on long time scales, internal time becomes relatively shorter than physical time. For a uniform code distribution, we provide a closed form for the correspondence between internal time and physical time, and show that the physical time required to ``advance internal time by one step'' increases in later stages. As an ancillary quantity, we quantify by conditional entropy the remaining uncertainty of ``which codes have been observed'' when only internal time is known, and we give unimodality and the maximization time in the uniform case, and upper bounds, equality conditions, and expressions of the difference from the upper bound in the non-uniform case. Finally, we also present a generalization that assigns weights (description lengths) to each code so that internal time is ticked according to the amount of information in the input.