The role of mechanical resonance in the neural control of swimming in fishes

Abstract

The bodies of many fishes are flexible, elastic structures; if you bend them, they spring back. Therefore, they should have a resonant frequency: a bending frequency at which the output amplitude is maximized for a particular input. Previous groups have hypothesized that swimming at this resonant frequency could maximize efficiency, and that a neural circuit called the central pattern generator might be able to entrain to a mechanical resonance. However, fishes swim in water, which may potentially damp out many resonant effects. Additionally, their bodies are elongated, which means that bending can occur in complicated ways along the length of the body. We review previous studies of the mechanical properties of fish bodies, and then present new data that demonstrate complex bending properties of elongated fish bodies. Resonant peaks in amplitude exist, but there may be many of them depending on the body wavelength. Additionally, they may not correspond to the maximum swimming speed. Next, we describe experiments using a closed-loop preparation of the lamprey, in which a preparation of the spinal cord is linked to a real-time simulation of the muscle and body properties, allowing us to examine resonance entrainment as we vary the simulated resonant frequency. We find that resonance entrainment does occur, but is rare. Gain had a significant, though weak, effect, and a nonlinear muscle model produced resonance entrainment more often than a linear filter. We speculate that resonance may not be a critical effect for efficient swimming in elongate, anguilliform swimmers, though it may be more important for stiffer carangiform and thunniform fishes.

Keywords: Central pattern generator; Damping; Entrainment; Resonance; Stiffness.

Fig. 1

Hydrodynamic damping can eliminate a resonant peak. Plots show the gain and phase for an underdamped harmonic oscillator (ζ = 0.1) with fluid forces. (A) Gain of the output position with respect to the input force for three different values of the hydrodynamic damping coefficient C (solid line: C = 0; dashed: C = 10; dotted: C = 0.1). (B) Phase of the output position relative to the input force.

Fig. 2

Complex resonance-like effects in an elongated fish body. Data are computed using the computational fluid dynamic simulation from Tytell et al. (2010). (A) Midline traces for one bending cycle at a range of frequencies for: (case 1) passive bending when the head region (first 30%) oscillates up and down over 1 cm and most of the body is free to move; (case 2) passive bending when most of the body oscillates over 1 cm and only the last 30% is free to bend; (case 3) free swimming with a traveling muscular activation wave at a range of frequencies. (B) Amplitude of the resulting tail motion. (C) Normalized body wavelength for the first passive trial (case 1) and active swimming (case 3). Wavelengths were too short to be estimated in case 2. (D) Absolute and nondimensional swimming speed for the active swimming case (case 3; left axis: absolute speed, filled diamonds; right axis: nondimensional speed, open squares). (E) Cost of transport for the active swimming trials (case 3).

Fig. 3

Schematic of the connections between a generic central pattern generator (CPG) and proprioceptors, based on the lamprey. (A) Lines ending with circles or bars represent inhibitory or excitatory connections, respectively. The CPG is shown as two mutually inhibitory half centers (labeled R and L). There are two classes of proprioceptors: a crossed inhibitory class, labeled s_i, and an ipsilateral excitatory class, s_e. Right and left motor neurons (mn) are shown with arrows. (B) Example of the effect of a bend toward the right. The left proprioceptors are excited by the curvature, and are shown with open symbols.

Fig. 4

Diagram of the closed-loop preparation. Motor output from a physiological preparation of the lamprey spinal cord is sampled and used in a real-time simulation of the muscle and body, which computes a bend angle x. The bend angle is fed back to the preparation using a motor that bends the spinal cord and notochord.

Fig. 5

Examples of different effects from closed-loop stimulation. In each panel, the top black trace shows the bend angle, the middle raster plot shows the spike times from three motor nerve recordings, labeled according to segment number, and the bottom plot shows the frequency of the bursts (circles) and the bending (black line). The blue and green channels are recorded close to the point of bending, while the red channel is further away. The dashed line shows the resonant frequency. Filled symbols indicate when the stimulus is on. (A) Resonance entrainment. Burst frequency (circles) and stimulus frequency (thick line) match the resonant frequency (dashed line). (B) Change in frequency and stabilization of the burst rhythm, but burst and stimulus frequency are below the resonant frequency. Variability in frequency is reduced after the stimulus is turned on (compare open and closed circles). (C) Destabilization of a stable rhythm. Six seconds after the beginning of the trial, the bursts become quite irregular and do not match either the stimulus frequency (solid line) or the resonant frequency (dashed line).

Fig. 6

Summary of the number of trials with different effects. Clear resonance entrainment is separated from “match”, when the system oscillated at the resonant frequency, but the resonant frequency was not sufficiently different from the baseline. “Change freq.” represents the trials when the system changed frequencies, but not to match the resonant frequency. “Destabilization” means that the rhythm was not stable.

Fig. 7

Effects of different simulation parameters on resonance entrainment. In all cases, the largest fraction of trials did not produce resonance entrainment (black bars). The fractions of trials that produced resonance entrainment above or below the baseline frequency are shown in red and blue, respectively. (A) Effect of the muscle model, either a linear filter or a Hill-type model. (B) Effect of gain. Each bar represents a bin of several different gain values. (C) Effect of the intrinsic baseline frequency. Bars represent frequency bins.

Fig. 8

Differences in open- and closed-loop entrainment ranges among individuals. For each individual, a filled box plot shows the range of closed-loop frequencies that were not entrained to the resonant frequency, where the box ranges from the 25 to 75 percentile and the white line is at the median. Outliers, shown as diamonds, are outside 1.5 the inter-quartile range. Red and blue points indicate trials with resonance entrainment above or below the baseline, respectively. Open boxes indicate the open-loop entrainment range.