Modeling multi-sensory feedback control of zebrafish in a flow

Dipole- and vortex sheet-based models of fish swimming

Elucidating the hydrodynamics of fish swimming is critical to identifying the processes underlying fish orientation and schooling. Due to their mathematical tractability, models based on potential flow are preferred in the study of bidirectional interactions of fish with their surroundings. Dipole-based models that assimilate fish to pairs of vortices are particularly enticing, but yet to be thoroughly validated. Here, we embark on a computational fluid dynamics (CFD) campaign informed by experimental data to validate the accuracy of dipole-based models. The locomotory patterns of a fish undergoing carangiform swimming are reconstructed from existing experimental data, which are used as inputs to CFD simulations of a fish swimming in a channel flow. We demonstrate that dipole-based models are accurate in capturing key features of the fluid flow, but cannot predict the elongated flow streamlines around the fish that are evident in CFD. To address this issue, we propose an alternative model that replaces each vortex in the pair with a sheet along the fish length. Using a pair of vortex sheets that span approximately 80% of the fish body length with a separation distance of approximately 50% of the body width, the model is successful in predicting the fluid flow around the swimming fish for a range of background flow speeds and channel widths. The proposed model shows improved accuracy at the cost of a mildly increased computational effort, thereby constituting an ideal basis for research on fish hydrodynamics.

Hydrodynamic model of fish orientation in a channel flow

For over a century, scientists have sought to understand how fish orient against an incoming flow, even without visual and flow cues. Here, we elucidate a potential hydrodynamic mechanism of rheotaxis through the study of the bidirectional coupling between fish and the surrounding fluid. By modeling a fish as a vortex dipole in an infinite channel with an imposed background flow, we establish a planar dynamical system for the cross-stream coordinate and orientation. The system dynamics captures the existence of a critical flow speed for fish to successfully orient while performing cross-stream, periodic sweeping movements. Model predictions are examined in the context of experimental observations in the literature on the rheotactic behavior of fish deprived of visual and lateral line cues. The crucial role of bidirectional hydrodynamic interactions unveiled by this model points at an overlooked limitation of existing experimental paradigms to study rheotaxis in the laboratory.

One fascinating and perplexing fact about fish is that they tend to orient themselves and swim against the flow, rather than with it. This phenomenon is called rheotaxis, and it has countless examples, from salmon migrating upstream to lay their eggs to trout drift-foraging in a current. Yet, despite over a century of experimental studies, the mechanisms underlying rheotaxis remain poorly understood. There is general consensus that fish rely on water- and body-motion cues to vision, vestibular, tactile, and other senses. However, several questions remain unanswered, including how blind fish can perform rheotaxis or whether a passive hydrodynamic mechanism can support the phenomenon. One aspect that has been overlooked in studies of rheotaxis is the bidirectional hydrodynamic interaction between the fish and the surrounding flow, that is, how the presence of the fish alters the flow, which, in turn, affects the fish.

To address these open questions about rheotaxis, Porfiri, Zhang and Peterson wanted to develop a mathematical model of fish swimming, one that could help understand the passive hydrodynamic pathway that leads to swimming against a flow. Unlike experiments on live animals, a mathematical model offers the ability to remove cues to certain senses without interfering with animal behavior.

Porfiri, Zhang and Peterson modeled a fish as a pair of vortices located infinitely close to each other, rotating in opposite directions with the same strength. The vortex pair could freely move through an infinitely long channel with an imposed background flow, devoid of all sensory information expect of that accessed through the lateral line. Analyzing the resulting system revealed that there is a critical speed for the background flow above which the fish successfully orients itself against the flow, resulting in rheotaxis. This critical speed depends on the width of the channel the fish is swimming in. Depriving the fish of sensory information received through the lateral line does not preclude rheotaxis, indicating that rheotaxis could emerge in a completely passive manner.

The finding that the critical speed for rheotaxis depends on channel width could improve the design of experiments studying the phenomenon, since this effect could confound experiments where fish are confined in narrow channels. In this vein, Porfiri, Zhang and Peterson’s model could assist biologists in designing experiments detailing the multisensory nature of rheotaxis. Evidence of the importance of bidirectional hydrodynamic interactions on fish orientation may also inform modeling research on fish behavior.