Biosonar Responsivity Sets the Stage for the Terminal Buzz

Abstract

The temporal patterning of bat echolocation has been extensively characterised and used to interpret behavioural regulation during active foraging. Yet a general mathematical account linking these patterns to acoustic propagation, relative motion, and the timing of behaviourally selected information remains incomplete. Here we present the responsivity framework, which represents each call event as a delayed active-sensing cycle composed of an acoustic acquisition interval and a subsequent behavioural interval. Their proportional relation is described by the responsivity coefficient, kr, connecting call timing to effective anchor distance, relative velocity, call duration, and the selected echo window. We evaluate the framework through analytical derivation, call-by-call simulation, internal validation, and comparison with field recordings from free-flying bats. The framework generates the characteristic hyperbolic relation between call rate and anchor distance, explains how echo-window selection shifts approach timing, and places the terminal buzz at the near-target limit of the same timing process rather than requiring a separate buzz-specific rule. Responsivity and kinematics had separable terminal effects: kr strongly constrained terminal update density and the attainable call rate, whereas closing velocity determined the time available within the terminal region and hence potential buzz duration. The framework also identified a buzz-readiness state preceding complete buzz expression, while transient sonar strobe groups emerged under multi-target anchor allocation without an explicit grouping rule. Application to field recordings showed broad correspondence between empirical and simulated movement-timing and call-duration organisation, and illustrated how the framework can be used to evaluate latent acquisition-window and anchor assumptions against observed timing. Together, these results support responsivity as a process-based descriptor of echolocation timing and highlight its potential as a diagnostic and analytical framework for hypothesis generation and empirical testing.