For a given frequency, the system response can be characterized by a point on the complex plane, αe^iφ, where the magnitude of α is the gain of the system and φ is the phase shift, with positive phase measured counter clockwise from the positive real axis. The circle of unit magnitude (representing unity gain) and the positive real axis (representing zero phase shift) are denoted in blue. Each trial yields one estimate for the system response at each frequency. (A) Phasor distribution. A Gaussian probability density function was fitted in the complex plane at each frequency; to illustrate this, the 95% covariance ellipse for single (red) and sum of sines (black) is shown for 2.05 Hz. (B) Phase 95% confidence. The phase confidence interval is the conic region over which the probability density function (PDF) integrates to 0.95. (C) Magnitude 95% confidence. Similarly, the magnitude confidence interval (95%) of the estimate of the mean is the annulus over which the PDF integrates to 0.95.

A velocity-scaled repetition of a sum-of-sines trial with fish and refuge trajectories depicted in blue and red, respectively. The individual sinusoidal components have an amplitude of (A) 0.8 cm s^-1 and (B) 1.2 cm s^-1.

(A) The magnitudes of the input (refuge) and output (fish) fast Fourier transforms (FFTs). (B) The coherence between refuge and fish trajectories (black) and the magnitude of the refuge trajectory FFT. The near unity coherence suggests fish and refuge trajectories are related by a linear dynamical system.

The Bode plot describes the system response (gain and phase shift) over a range of frequencies. Bode plots for stimuli of different velocity amplitudes are compared for (A) single-sine trials and (B) sum-of-sines trials. (C) Average responses (collapsed across scaling), comparing sum-of-sines (black) and single-sine (red) trials, are depicted with confidence intervals calculated as described in Fig. 2. For a linear system, these estimates would agree.

An example demonstrating that frequency response functions generalize better within stimulus class than across classes. Assuming superposition holds, the frequency response functions (FRFs) in Fig. 5C generated from four fish, can be used to predict the response to an arbitrary input for a fifth fish. (A) 10 s of a sum-of-sines stimulus (blue) and the fish's response (green). (B) A comparison of predictions made by different FRF models. The sum-of-sines prediction (black) closely matches the fish's performance (green). The single-sine prediction (red) is worse than for the sum-of-sines FRF. (C) The difference between the single-sine and sum-of-sines prediction errors. Negative values (in red) indicate time intervals for which the single-sine FRF model has greater error than the sum-of-sines model. Predominantly, the sum-of-sines model better predicts the fish's actual response. If the system were linear, an assay of single-sine experiments would be sufficient for predicting the response to the sum-of-sines stimulus.

In switched stimulus trials, fish (N=3 corresponding to the rows above) adapt to a change from sum-of-sines to single-sine stimuli. On the left are the mean of eight trials; sum-of-sines stimuli (interval shaded in grey) were designed so that the mean input would be purely sinusoidal. On the right, the instantaneous phase estimate is calculated from the averaged response using the argument of the analytic signal (the Hilbert phase, shown in green). A coarser phase estimate is shown for non-overlapping windows of 5 s (as black error bars). The asymptotic phase over each interval was calculated using the Fourier transform method used to generate Fig. 5.

(A) Frequency response phasors shift to decrease tracking error in the transition from sum-of-sines to single-sine stimuli. (B) For a frequency response characterized by gain a and phase lag φ, denoted by the red dot above, the magnitude of the sensory slip e is the distance from the frequency response point to the perfect tracking point 1+0i. For a fixed phase φ, the minimal error e*=sin(φ) for φ∈(-π/2, π/2) is achieved by a gain of a*=min(cos(φ), 0) (depicted as the green dot). For φ∉(-π/2, π/2) the minimum achievable error is 1. The locus of minimum-error responses given fixed phase lag (or lead) is denoted by a circle with diameter equal to one and centered at the point 0.5 + 0i; within this circle (shaded green), there is a trade-off between increased error and savings in expended energy.