## Abstract

Fish must orient in three dimensions as they navigate through space, but it is unknown whether they are assisted by a sense of depth. In principle, depth can be estimated directly from hydrostatic pressure, but although teleost fish are exquisitely sensitive to changes in pressure, they appear unable to measure absolute pressure. Teleosts sense changes in pressure via changes in the volume of their gas-filled swim-bladder, but because the amount of gas it contains is varied to regulate buoyancy, this cannot act as a long-term steady reference for inferring absolute pressure. In consequence, it is generally thought that teleosts are unable to sense depth using hydrostatic pressure. Here, we overturn this received wisdom by showing from a theoretical physical perspective that absolute depth could be estimated during fast, steady vertical displacements by combining a measurement of vertical speed with a measurement of the fractional rate of change of swim-bladder volume. This mechanism works even if the amount of gas in the swim-bladder varies, provided that this variation occurs over much longer time scales than changes in volume during displacements. There is therefore no *a priori* physical justification for assuming that teleost fish cannot sense absolute depth by using hydrostatic pressure cues.

## 1. Introduction

A fish navigating between two points in space is required to orient in three dimensions, rather than the two with which we as terrestrial animals are familiar. A sense of depth would therefore be exceedingly useful in localizing position—for example, when navigating to feeding grounds at a specific depth—but is there any plausible physical mechanism by which a fish could sense its depth? At first glance, it might seem obvious that this question should be answered in the affirmative: hydrostatic pressure provides a global cue that varies linearly with depth, so it is straightforward in principle to determine absolute depth from a measurement of absolute pressure. This, indeed, is the physical mechanism by which a scuba diver's depth gauge operates, but teleost fish are thought to lack the sense of absolute pressure that this mechanism of static depth sensing requires (Bone & Moore 2008).

Teleosts are acutely sensitive to changes in pressure (Qutob 1962; Blaxter & Tytler 1972; Tytler & Blaxter 1973), which they sense primarily by registering changes in the volume of their gas-filled swim-bladder. Those teleosts that have lost the swim-bladder do appear to be able to sense changes in hydrostatic pressure, but with rather lower acuity than other teleosts. The site of pressure reception in teleosts lacking a swim-bladder has not been identified, but in elasmobranchs (which also lack a swim-bladder) vestibular afferents are known to be sensitive to hydrostatic pressure (Fraser & Shelmerdine 2002). In this paper, we focus exclusively upon the physics of a possible mechanism of depth sensing in teleosts possessing a swim-bladder. The central problem that we address is that the molar amount of gas that the swim-bladder contains is varied in order to regulate buoyancy, such that it cannot act as the long-term steady reference that is needed to infer absolute pressure. Consequently, it has generally been assumed that teleost fish cannot sense depth using hydrostatic pressure cues (e.g. Bone & Moore 2008). Here, we overturn this entrenched assumption by deriving an alternative physical mechanism through which changes in swim-bladder volume could be used to infer absolute depth.

## 2. A physical mechanism of dynamic depth sensing

The volume (*V*) of gas in the swim-bladder can be written using the ideal gas law as
2.1
where *n* is the number of moles of gas, *R* is the universal gas constant, *T* is the absolute temperature and *P* is the total pressure inside the swim-bladder. This equation applies to any contiguous volume of gas, whether bounded or unbounded, and assumes only that the contents of that volume behave as an ideal gas.

The total pressure inside the swim-bladder (*P*) can be written as the sum of four components: the hydrostatic pressure owing to the weight of the column of water above (*P*_{H}), the static atmospheric pressure at the surface (*P*_{A}), the excess internal pressure owing to tension in the swim-bladder wall (*P*_{T}) and the total integrated dynamic pressure owing to movement of the fish through the water (*Q*). We, therefore, have
2.2
in which *P*_{H} = *g ρ D* is the hydrostatic pressure, and where

*ρ*is the density of water,

*g*is gravitational acceleration and

*D*is depth. Equation (2.2) makes no detailed assumptions about the form of the swim-bladder, as it applies to any volume of gas bounded by elastic walls.

Substituting equation (2.2) into equation (2.1), the volume of gas in the swim-bladder is 2.3 which can be rearranged for depth as 2.4

Now, as *g*, *ρ* and *R* are the only physical constants in this equation, it is clear by inspection of the first term on the right-hand side of equation (2.4) that depth (*D*) could be estimated from a measurement of swim-bladder volume (*V*) only if the molar amount of gas in the swim-bladder (*n*) and its absolute temperature (*T*) were either known or approximately constant. Hence, as the molar amount of gas in the swim-bladder (*n*) is varied to control buoyancy and cannot be measured directly, it follows that the physics of static depth sensing do not provide a viable mechanism by which teleost fish could sense absolute depth.

So far, we have merely formalized the physical argument against static depth sensing that was made informally in §1. This formalization is useful in clarifying that any alternative physical mechanism of absolute depth sensing must eliminate the need to know how the molar amount of gas in the swim-bladder varies. We can do this by invoking a separation of time scales among the various causes of changes in swim-bladder volume. Differentiating equation (2.4) with respect to time yields

2.5where we have used Newton's dot notation for time derivatives and have used equation (2.1) to make the substitution *p* = *nRT*/*V*. Equation (2.5) is arranged so that the time derivatives on the right-hand side are expressed as fractions of the total quantities of which they are derivatives. They therefore have the units of 1 over time, so are inversely related to the characteristic time scales over which these quantities vary. For sufficiently fast vertical displacements, it is reasonable to assume that the swim-bladder volume (*V*) changes over a much shorter characteristic time scale than the molar amount of gas in the swim-bladder (*n*), the absolute temperature (*T*), the atmospheric pressure (*P*_{A}) and the excess internal pressure (*P*_{T}). This being so, their derivatives may be neglected in equation (2.5). Furthermore, if we assume a steady displacement, then the total integrated dynamic pressure (*Q*) is constant, and hence .

These assumptions are justified fully in §3, but for now we may use them to approximate equation (2.5) as
2.6
which assumes in effect that the only significant cause of changes in swim-bladder volume during fast, steady vertical displacements is the change in hydrostatic pressure. Equation (2.6) can now be combined with equation (2.4) to write a new equation for depth (*D*),
2.7
from which we have eliminated the molar amount of gas in the swim-bladder (*n*). This equation therefore encapsulates the physics of a possible mechanism of dynamic depth sensing. We must now attend in detail to the terms on the right-hand side.

The first term on the right-hand side of equation (2.7) can be computed by taking the ratio of a measurement of the vertical speed () and a measurement of the fractional rate of change of swim-bladder volume with respect to time (). This ratio is linearly related to depth, which is best explained by noting that its inverse, , is equivalent to the fractional rate of change of swim-bladder volume with respect to depth. This quantity is an inverse function of depth (figure 1*b*) because the volume of the swim-bladder is proportional to the total pressure (equation (2.1)), and the change in pressure resulting from a given vertical displacement represents a smaller fraction of the total at greater depth (figure 1*a*). It follows that it is not necessary to know how the molar amount of gas in the swim-bladder varies in order to estimate depth during fast, steady vertical displacements. Contrary to the received wisdom (e.g. Bone & Moore 2008), there is therefore no *a priori* physical justification for assuming that teleost fish cannot sense absolute depth by using hydrostatic pressure cues.

The second term on the right-hand side of equation (2.7) is the same as appeared earlier in equation (2.4) describing the physics of static depth sensing, and is probably best treated as a constant plus a small fluctuating error term. Of the three pressure components contained in its numerator, the atmospheric pressure is typically the largest, of order *P*_{A} ∼ 10^{5} N m^{−2}. The largest measured excess internal pressures in fish are an order of magnitude smaller, being of the order of *P*_{T} ∼ 10^{2} to *P*_{T} ∼ 10^{4} N m^{−2} for Cypriniformes, in which swim-bladder tension is important for the functioning of the Weberian ossicles (Alexander 1959*a*, 1961); a companion paper by the same author (Alexander 1959*b*) found that the excess internal pressure was too small to measure in the 17 species investigated from other orders. The maximum local dynamic pressure (i.e. stagnation pressure) is likely to be no more than *Q* ∼ 10^{3} to *Q* ∼ 10^{4} N m^{−2} for most species of fish with a swim-bladder, and the total integrated dynamic pressure over a streamlined body will be many times smaller again. It follows that the total magnitude of these pressure terms is expected to be of the order of 10^{5} N m^{−2}, which after dividing through by their denominator implies that the second term on the right-hand side of equation (2.7) will be of the order of 10 m. Given that the dominant atmospheric pressure component of this term fluctuates by rather less than 10 per cent, we may effectively replace it with a constant (*K*) of the order of *K* ∼ 10 m plus an error term (*ε*) of the order of *ε* ∼ 1 m or less. Clearly, a fluctuating error of this magnitude is trivial in navigational terms.

## 3. Discussion

The derivation of our physical mechanism for dynamic depth sensing in equation (2.7) rests upon a minimal set of assumptions, which may be summarized as follows.

—The walls of the swim-bladder are elastic and contain a single contiguous volume of ideal gas.

—The fish is undergoing a steady vertical displacement that is fast enough that the accompanying changes in hydrostatic pressure are the only significant cause of changes in swim-bladder volume.

We emphasize again that we have made no detailed assumptions about the form of the swim-bladder. The first of these assumptions is uncontroversial; the second requires a little more justification. It is clear from equation (2.3) that changes in swim-bladder volume could in principle result from changes in the molar amount of gas in the swim-bladder (*n*), the absolute temperature (*T*), the atmospheric pressure (*P*_{A}), the excess internal pressure (*P*_{T}) and the total integrated dynamic pressure (*Q*). We will consider each in turn.

Changes in total integrated dynamic pressure (*Q*) are immediately excluded by assuming that the fish is undergoing a steady displacement, such that its velocity—and therefore the dynamic pressure—does not change. Passive changes in the excess internal pressure (*P*_{T}) might result from changes in Young's modulus of the swim-bladder wall during stretching, but these are likely to be negligible because the total excess internal pressure is small in comparison with the other pressure components. Atmospheric pressure (*P*_{A}) changes over characteristically long time scales, so its effect on changes in swim-bladder volume will be negligible during fast vertical displacements. Changes in ambient temperature occur quite rapidly when moving through the water column, but the thermal inertia of the fish will ensure that the absolute temperature of the gas in the swim-bladder (*T*) changes only slowly. Changes in the molar amount of gas in the swim-bladder (*n*) are invariably slow in physoclist teleosts, which have a closed swim-bladder whose contents can only be varied through the slow physical and chemical processes of gas resorption and secretion. Physostome teleosts are able to release air at depth through a connection between the swim-bladder and the gut. However, while this gas-spitting reflex (‘Gasspuckenflex’; Franz 1937) can occur quite quickly in response to changes in pressure (Alexander 1959*a*), it can only occur *after* a perceptible change in swim-bladder volume has occurred. We conclude that it is indeed reasonable to assume that changes in hydrostatic pressure are the only significant cause of changes in swim-bladder volume during fast, steady vertical displacements.

So far, we have argued only for the physical plausibility of dynamic depth sensing in teleost fish. Is it plausible that their sensory systems could make the physiological measurements needed to exploit the physical mechanism that we have proposed? Actual use of this mechanism would be possible if two kinds of physiological measurements were combined: a measurement of the fractional rate of change of swim-bladder volume with respect to time (), and a measurement of the vertical speed of the fish (). Vertical speed might in principle be measured by flow sensors in the vertically oriented canals of the lateral line system, such as the pre-opercular canal, while changes in swim-bladder volume might in principle be detected by sensing stretching of the swim-bladder wall (Jones & Marshall 1953; Qutob 1962; Nilsson 1972, 1980, 2009; McLean & Nilsson 1981; Wahlqvist 1985; Finney *et al*. 2006)—at least in fish that keep the swim-bladder wall under tension (Alexander 1966). In fact, there is no need to keep track directly of the swim-bladder volume, because the fractional rate of change of swim-bladder volume is expected to be proportional to the rate of linear extension of a representative wall element divided by the total length of that element. This could be computed quite simply by combining the output of a phasic and a tonic stretch receptor. Hence, although too little is known of the neurophysiology of the swim-bladder to draw any concrete conclusions, it does not seem implausible that teleost fish could make the physiological measurements needed to exploit the physical mechanism of dynamic depth sensing that we propose.

An important corollary of dynamic depth sensing is that any fish making use of this mechanism could only use it to infer depth when moving vertically in the water column. Such movements need not be large to be effective, however. Behavioural experiments have shown that fish are capable of detecting pressure changes equivalent to 0.05 m or less of water pressure at the surface (Qutob 1962; Blaxter & Tytler 1972; Tytler & Blaxter 1973). The same fractional change in pressure would result from a vertical displacement of approximately 0.1 m at a depth of 10 m, and from a vertical displacement of approximately 0.5 m at a depth of 100 m. It follows that the vertical movements needed to drive the pressure sensors of a fish above threshold are small in comparison with the resolution that is likely to be required of a depth measurement system for navigation. We conclude that there is no longer any *a priori* justification for assuming that teleost fish are unable to sense their absolute depth. Behavioural and physiological studies are now required to test whether teleost fish do indeed make use of the physical mechanism that we have proposed.

## Acknowledgements

G.K.T. was supported by a Royal Society University Research Fellowship during the course of this research and currently holds an RCUK Academic Fellowship. T.B.P. is supported by a Dorothy Hodgkin Royal Society Research Fellowship. R.I.H. acknowledges the Biotechnology and Biological Sciences Research Council for a Doctoral Training Award. We thank Adrian Thomas and Shane Windsor for helpful comments.

## Footnotes

- Received November 30, 2009.
- Accepted February 4, 2010.

- © 2010 The Royal Society