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BMC BiologyAnimal Behavior & CognitionPeer-reviewed journal article

A responsivity theory of call timing patterns in bat echolocation

Ravi UmadiiD, Uwe FirzlaffiD

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Abstract

Background: The temporal patterning of bat echolocation is extensively characterised, but a general mathematical account linking call timing to acoustic propagation, relative motion, and behaviourally selected information remains incomplete. Results: We present the responsivity framework, which represents each call event as a delayed active-sensing cycle comprising 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. Analytical derivation, call-by-call simulation, internal validation, and comparison with field recordings show how the framework generates hyperbolic call-rate--distance relationships, shifts approach timing with echo-window selection, and places the terminal buzz at the near-target limit of the same timing process. Responsivity and kinematics have separable terminal effects: kr constrains terminal update density and attainable call rate, whereas closing velocity determines the time available within the terminal region. A buzz-readiness state precedes the complete expression of buzz, and transient sonar strobe groups emerge under multi-target anchor allocation without an explicit grouping rule. Field recordings show broad correspondence with simulations in the organisation of movement, call timing and call duration, and illustrate conditional evaluation of acquisition-window and anchor assumptions. Conclusions: Responsivity theory provides a process-based descriptor of echolocation timing and a diagnostic framework for hypothesis generation and empirical testing. The framework identifies the measurements needed to test how selected acoustic information governs subsequent call timing.

In-depth summary

Claude Opus 5.5 · xhigh effort · 5 Nov 2026 · from the full text of version 4 (11 Oct 2026) · CC BY-NC-ND 4.0

Decomposing bat echolocation interpulse intervals into acoustic acquisition and proportional behavioural intervals explains hyperbolic call-rate trajectories, terminal buzz update budgets, buzz readiness, and multi-target sonar strobe groups.

Although temporal call patterns in bat echolocation are extensively documented, a formal mathematical rule linking call emission timing to echo propagation delay, relative closing velocity, and selected echo windows has been lacking. This work introduces a closed-loop responsivity framework that derives diverse calling phases from a single event-level sensorimotor cycle and defines testable diagnostics for field recordings.

Key results

  1. ShownExpanding the selected echo-stream information window shifts interpulse intervals upward and steepens their distance dependence while primarily adding a constant offset to call duration.

    Across 4039 simulated 3D calls with one moving target at fixed responsivity (kr = 5), increasing the echo-window fraction phi from 0 to 0.5 and 1 increased fitted interpulse interval slopes from 34.36 to 37.62 and 40.42 ms m^-1 (p ~ 0, R^2 = 0.997), whereas call-duration slopes remained 1.09, 1.08, and 1.01 ms m^-1 with intercepts rising from 0.33 to 0.55 and 0.94 ms. · 3.1 Echo-window choice shifts distance-dependent…; Figure legends

  2. ShownThe responsivity coefficient selectively scales the behavioural interval rather than the acoustic acquisition interval across target distance.

    In parameter sweeps across kr = 3, 5, 7, and 9, fitted distance slopes for the acoustic acquisition interval (Ta) remained nearly constant at 5.99 to 6.05 ms m^-1, whereas behavioural interval (Tb) slopes increased from 17.96 to 30.23, 42.38, and 54.43 ms m^-1. · 3.2 Distance dependence of acoustic acquisition…; Figure legends

  3. ShownResponsivity sets the attainable terminal call-rate ceiling and update density, whereas radial closing velocity controls terminal buzz duration.

    At a stop distance of 0.15 m, increasing kr from 3 to 5, 7, and 9 raised minimum IPIs from 3.50 to 5.25, 7.00, and 8.75 ms and lowered maximum call-rate ceilings from 285.83 to 190.56, 142.92, and 114.33 Hz; decreasing radial closing velocity from 3.0 to 0.1 m s^-1 increased time in the 75%-100% Cr,max terminal zone from 0.017 to 0.501 s and varied terminal call counts from 1.71 to 123.50. · 3.3 Terminal buzz length emerges from the…; Figure legends

  4. ShownFree-flying bat field recordings share the simulated movement-timing and reciprocal call-duration structure in inter-call displacement space, but exhibit longer interpulse intervals within matched velocity bins.

    Across 4427 filtered field calls from 496 sequences velocity-matched to 3D simulations (median velocity 6.461 vs 6.460 m s^-1, p = 0.999985), hyperbolic call-duration scale terms matched (29.57 vs 29.62 ms Hz) and 94.1% (IQR 93.1-95.0%) of field durations fell in the 5-95% band, while conditional profiling at kr = 5 minimized quantile discrepancies at phi = 1.35 (0.362 m gap) and phi = 1.20 (16.83 ms Tb gap). · 3.4 Field timing shares the predicted movement…; Figure legends

  5. ShownTransient sonar strobe groups emerge during multi-target random anchor switching without a dedicated group-generating rule.

    Out of 141,112 simulated calls across six target and anchoring conditions, 1067 calls formed 516 sonar strobe group events (482 doublets, 33 triplets, and 1 quadruplet) exclusively in two-target random-switching conditions C3 and C6, with 100% (C3) and 98.4% (C6) of calls within each group committed to a single anchor. · 3.6 Sonar strobe groups emerge under…; Figure legends

Methods, data and reproducibility

Formulated an analytical delayed active-sensing model partitioning each onset-to-onset interpulse interval into a motion-aware acoustic acquisition interval (Ta) and a proportional behavioural interval (Tb = kr * Ta), and built a 1D/3D call-by-call simulator in MATLAB. Validated parameter sweeps for machine precision and compared trajectories in displacement space against 4-channel microphone-array field recordings.

  • Four-channel microphone array field recordings of free-foraging Pipistrellus and Myotis bats in Weihenstephan, Freising, Germany · Pipistrellus and Myotis bats · n = 6360 calls across 496 sequences (4427 retained for IPI, 4399 for call duration) · https://doi.org/10.5281/zenodo.21834638
  • Synthetic 1D and 3D call-by-call approach trajectories across single-target and multi-target simulation sweeps · n = 4039 calls (echo-window), 1600 sequences (buzz-readiness), 141,112 calls (SSG) · https://doi.org/10.5281/zenodo.21834638
Methods & tools
MATLAB, Knowles SPU0410LR5H-QB microphones
Software & environment
MATLAB; Zenodo archive release version V1.1.0
Reproducibility gaps
The Zenodo archive provides the processed field dataset and analysis code but does not include an automated pipeline to reconstruct the manually curated dataset from raw multichannel audio files.
Limitations (3)
  • The simulator condenses biological processing into one effective responsivity parameter, leaving out hearing thresholds, directional beam/pinna filters, echo strength, attention, breathing coupling, and vocal-motor mechanics. · stated by the authors
  • Field recordings pool unidentified Pipistrellus and Myotis sequences and lack direct prey trajectories and call-resolved echo anchors, restricting empirical comparisons to inter-call displacement. · stated by the authors
  • Conditional field parameter profiles rely on a fixed responsivity coefficient (kr = 5) and assumed starting distance (d0 = 5 m), where kr, echo-window fraction, initial distance, anchor identity, and unmodelled lag can partially trade off against one another. · stated by the authors

Read this if you model active sensing or animal vocal-motor control and want an analytical and simulation framework linking call timing to target kinematics and echo selection.

Citations
6
FWCI
37.46
Field percentile
top 1%

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