E31 Position and location

Topic

This localization process involves inferring the absolute or relative position of an organism—or its prey—based on physical signals; the primary physical methods for achieving this include triangulation via signal arrival-time differences across multiple sensors, signal strength, and dead reckoning (integrating velocity over time)—a form of odometry that a moving organism effectively has built into its vestibular system.

Triangulation based on arrival-time differences converts the time lag between two separated sensors into a hyperbolic curve representing possible locations for the signal; the intersection of two or more such curves ultimately pinpoints the signal source. The human binaural auditory system employs both interaural time differences and interaural level differences between the ears to achieve this, thereby locating a sound source with considerable angular precision within the horizontal plane surrounding the individual. The fact that the interaural time difference is easier to detect results in greater precision when localizing the sound source.

The Kalman filter—the same optimal estimator for linear systems with Gaussian noise discussed elsewhere in this set of models—is applied here to this localization task: it combines predictions from the organism's internal dynamic model with corrections derived from new sensory measurements, weighting each source according to its respective uncertainty. Consequently, numerous biological neural systems implement computations analogous to this Kalman filter to integrate signals from the vestibular system—which establishes another model within this set—with visual signals (forming yet another distinct model) and the organism's own proprioceptive signals. If the uncertainty associated with one of these information sources is lower than that of the others, the Kalman filter dictates that the organism's final localization will rely more heavily on that low-uncertainty source than on the other available sources.

The Cramér-Rao bound—the same bound established by another model in this set—defines the lower limit of the variance for the best possible estimator applied to this time-difference-of-arrival triangulation; thus, a wider signal bandwidth, combined with a superior signal-to-noise ratio, improves the final localization of the signal's source. Consequently, dolphins and bats utilize wide-bandwidth pulses specifically to maximize the Fisher information defined by that other model in the set, thereby enhancing the precision of their echolocation when tracking moving prey—a process described in greater detail by yet another distinct model in the set. The fact that the bat's echolocation pulse has a wider bandwidth means that, according to the Cramér-Rao bound, the final precision of the prey's localization is further improved. Consequently, this model—utilizing the time-difference triangulation and Kalman filter with which it was initiated—enables a probabilistic occupancy map to accumulate sensory evidence over time, thereby constructing a complete spatial representation of the environment; each cell in the map accumulates the probability of being occupied through successive Bayesian updates, following the exact same Bayes' rule established by another model in this set. This approach proves robust against both the noise associated with another model in the set and momentary signal occlusion, ultimately serving as the foundation for both autonomous robot navigation and, likely, the active exploration of an environment by a mammal. As the organism accumulates sensory evidence about the environment over a longer period, the resulting probabilistic occupancy map becomes increasingly precise and complete.

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