Electroreception and the compass sense of sharks

J. theor. Biol. (1995) 174, 325–339

Electroreception and the Compass Sense of Sharks M G. P Department of Zoology, University of Otago, New Zealand (Received on 1 July 1994, Accepted in revised form on 21 November 1994)

Elasmobranchs have an electric sense that is sensitive enough to detect electric fields as weak as those induced through their bodies as they swim through the earth’s magnetic field. Because the intensity and direction of these fields are related to the speed and direction of the movements that cause them, elasmobranchs could use their electric sense in navigation. There is evidence that they do. According to a current theory, elasmobranchs can determine their direction of motion in an Earth-fixed frame using a computation involving electroreceptor voltages, swimming speed and the local geomagnetic field vector. However, this theory is inconsistent with physical and biological constraints, notably that elasmobranch electroreceptors can not measure d.c. voltages, and that a voltage due to water flow in the ocean is not uniquely interpretable in terms of the speed and direction of flow at the point where the electrical measurement is made. This paper presents a new theory that explains how an elasmobranch could use its electric sense to determine a compass bearing as it swims. According to this theory, the direction cue is the directional asymmetry of the change in induced electroreceptor voltage during turns. A neural network could use this cue to determine swimming direction by comparing vestibular and electrosensory signals.

elasmobranchs should be able to use their electric sense for navigation†. There is circumstantial evidence that they do. Kalmijn (1982) has shown that rays can orient in electric fields as weak as those that should be induced by their normal swimming movements in the geomagnetic field. Carey & Scharold (1990) have tracked migrating blue sharks off the northeastern coast of the United States, and found that they can maintain quite straight courses for hundreds of kilometers over many days. The geomagnetic field would seem to be the only continuously available cue that could be used to achieve this (Kalmijn, 1990). Klimley (1993) tracked hammerhead sharks off the Californian coast. These sharks follow a route that correlates with the pattern of magnetic anomalies on the sea floor. Elasmobranchs have two modes of navigation using their electric sense, according to Kalmijn (1981, 1984). In active mode, the shark measures voltage gradients that develop through the body due to its swimming movements. It applies the rule that the receptor voltage induced by moving in a magnetic field

1. Introduction Elasmobranchs (sharks, skates and rays) have an electric sense that enables them to detect electric fields as weak as 5 nanovolts per centimeter (Murray, 1962; Kalmijn, 1966, 1982). In more familiar units, this is 1.5 volts over 3000 kilometers. Marine fish and invertebrates produce bioelectric fields—which elasmobranches can detect from tens of centimeters away—in the surrounding seawater, mainly from ionic current flows across gut and gill epithelia. Many elasmobranchs use these electric fields to detect and strike at their prey (Kalmijn, 1982). The electric sense is so acute that elasmobranchs should be able to detect electric fields induced by their own movements and by bulk water movements through the geomagnetic field. Because the strength and direction of motion-induced fields depend on the speed and direction of the movements that cause them, † Faraday did not foresee commercial applications of magnetoelectric induction, but did note that it might explain how fish navigate in the ocean (Faraday, 1832; referenced in Kalmijn, 1981). 0022–5193/95/110325+15 $08.00/0

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is proportional to the speed of movement and the sine of the angle u between the direction of movement and the magnetic field vector. Knowing its speed and the constant of proportionality, the shark solves for sin u. This gives two possible swimming directions. For example, swimming northeast and southeast at a given speed both generate the same receptor voltages. The shark determines the correct direction by turning, and comparing the direction of change in receptor voltage amplitude with the direction of the turn. As the shark turns away from the magnetic field lines the receptor voltages increase. For example, if the shark is swimming northeast then the amplitude of induced receptor voltages will increase when it turns to the right and decrease when it turns to the left. This is an elegant and attractive theory: the shark can use one simple rule to determine its swimming direction, anywhere in the ocean, at any time. There is also a passive mode of navigation in Kalmijn’s theory. In passive mode, the shark measures voltage gradients that develop through its body as a result of electric fields in the environment. These fields are largely due to induction in moving water. The strength and direction of such gradients is related to the speed and direction of the water flows that cause them, but the relationship is not simple. Kalmijn (1984: 530) notes various factors in addition to major ocean flows—surface waves, internal waves, upwellings and tidal flows—that contribute to the complexity of ocean electric fields. Electric field geometry also depends on the field boundary conditions, geomagnetic variations, self induction and mutual induction. There is no simple, universal rule that a shark could employ to navigate using environmental electric fields. As Kalmijn (ibid.) puts it: ‘‘Interpretation of the various fields would require considerable expertise on the part of the animals’’. The elegance of active electronavigation is somewhat undermined by its dependence on passive electronavigation. In order to measure voltages due to its own movement through the geomagnetic field, a shark must first apply its ‘‘considerable expertise’’ to compensate for voltages due to environmental fields. The existing theory does not include an explanation of how this could be done, and therefore is incomplete at best. The transduction properties of elasmobranch electroreceptors also pose a difficulty for the existing theory. The theory requires that they detect d.c. voltages, but neurophysiological measurements show that they do not. In Section 4, I introduce a new theory of the role of the electric sense in elasmobranch navigation. The key to the new approach is the observation that directional

information is captured in receptor voltage modulations that occur during turns. Sharks could estimate their compass bearing by comparing the waveform of these modulations with the waveform of head movement. They could do this with uncalibrated voltage measurements, without knowing their swimming speed, and without interference from electric fields in the environment. In the final section of the paper I discuss how a shark could compute directional representations using neurons. The existing theory seems to imply that the shark’s nervous system contains signals representing electric field strength and swimming velocity, and has neural circuitry that can divide one by the other. Neural circuits implementing other mathematical operators would also seem to be required. Given our present limited understanding of neural computation, almost nothing can be ruled out, by the existing theory is not expressed in a computational form that seems suitable for translation into neural circuitry. On the other hand, the computations required by the new theory can be carried out using simple combinations of simple rules that seem naturally suited to implementation using neuron-like computing elements. I shall explain how this might be done.

2. Electromagnetic Theory During my reconsideration of how elasmobranchs may use their electric sense in navigation, I have found some explanations of magnetic-electric phenomena in the literature that are unnecessarily complicated and potentially misleading. Given the difficulties faced by biologists who may not have a strong background in electromagnetism but want to understand the mechanisms that may be employed by elasmobranchs to navigate with the aid of their electric sense, it seems useful to begin this paper with a brief review of the relevant physics. Many undergraduate physics texts cover this material, for example Serway (1990) or Hammond (1986).       A particle with charge q moving at velocity v in a magnetic field B experiences a force 2.1.

F=qv×B.

(1)

Consider a uniform, vertically oriented rod of length l moving with velocity v horizontally in air, perpendicular to a horizontal magnetic field B. Free charges in the rod experience a force qvB, and move to the end(s) of the rod. As they do so, charge separation induces an electric field E that exerts an opposing force

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qE on each charge. Equilibrium occurs when qE+qvB=0, i.e. when E=−vB [Fig. 1(a), (b)]. The induced electric field is uniform within the rod, over distance l. This implies that there is a potential difference

The circuit current I equals this voltage divided by the total circuit resistance,

=V ==El=vBl

The current induces a resistive field E=−rJ at each point in the circuit, where r is the resistivity and J is the current density at that point. The potential difference Vs over a segment of s of the circuit is the integrated sum of the motional field v×B and the resistive field −rJ,

(2)

between the ends of the rod. If the ends of the rod are now electrically connected through a stationary conductor, current I flows through the rod in the direction of F, and returns via the stationary current path [Fig. 1(c)]. The electromotive force induced in the circuit is

g

emf= vB dl=vBl.

I=

V R

g

Vs= (v×B−rJ) · ds.

(4)

(5)

s

(3)

The emf, despite its name, is the work done in the circuit. It is measured in volts and is numerically equal to the potential difference across the ends of the rod.

In the stationary part of the circuit v×B=0 and across a segment s with resistance r, eqn (5) gives Vs =

−rV =−Ir. R

(6)

F. 1. (a) Force on charged particles moving in a magnetic field. The force F is perpendicular to the motion v and to the magnetic field vector B. (b) Conducting rod moving in air. Charges accumulate at the ends of rod. A steady state is reached when the attraction between oppositely charged particles counterbalances the motional force that pushes them apart. The induced electric field E opposes the force F. (c) Moving rod with stationary return circuit. The rod slides within a stationary frame that provides a low resistance path between its ends. The magnetic field vector B is directed into the page. Current I flows when the rod moves. Given B, the total circuit resistance and the voltage observed across a known resistor in the stationary part of the circuit, it is possible to compute the bar’s velocity, v. The dark bar at the top of the moving rod represents the receptor, a high-resistance element. (d) Laterally insulated rod moving in a conducting medium in a magnetic field. The stationary environment provides a current path between the ends of the rod. From inside the rod, this situation is indistinguishable from that in (c). The only relevant parameter of the environment is its total resistance measured between the ends of the rod. (e) Ampullary organ in an elasmobranch. The organ is a conductive rod with insulated sides. The main effect of the body is to impede current flow outside the organ, adding to the total resistance of the return path. (f) Equivalent circuit. For the purpose of analysing voltage and current flow across the receptor, this circuit applies to the situations in (d) and (e). Total circuit resistance is R, receptor resistance is r, and emf induced in the canal (rod) is V.

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If the segment is inside the rod, on the other hand, eqn (5) gives Vs=V

0 1

h r − , l R

(7)

where h is the height of the segment. Over the entire rod, h=l and Vs=V(1−r/R) with r/RQ1. Thus, voltage always decreases in the direction of current flow outside the rod, but generally increases in the direction of current flow within it. Outside the rod, the magnitude of the voltage across a resistor is simply equal to the circuit emf times the ratio of its resistance to the circuit resistance. Suppose that there is a thin, resistive element across the top of the moving bar. For reasons that will become obvious, if they are not already, I call this the receptor. The height of the receptor is small but its resistance is not, h r  . l R

(8)

That is, a negligible proportion of the circuit’s emf is generated within the receptor, but a significant portion of the circuit’s resistance occurs across it. Then, from (7), the voltage across the receptor is Vr=−Vr/R.

(9)

This has the same form as (6); for purposes of analysing potential differences that occur across the receptor as the rod moves, the receptor can be treated like a resistor in the stationary part of the circuit [Fig. 1(f)].      Consider a conductive rod with insulated walls, moving in still seawater [Fig. 1(d)]. Electrical current flows along the rod and back through the fluid. It may be difficult to calculate potentials and current densities in the fluid, but from the rod’s point of view the fluid is a passive load with fixed resistance. From inside the rod, this situation looks no different from that in Fig. 1(c). Now suppose that one end of the rod is capped by a receptor and the rod is embedded in a less conductive body. One end of the rod forms a pore open to the seawater while the other, capped by the receptor, is in the interior of the body [Fig. 1(e)]. What voltage occurs across the receptor when the fish moves in a magnetic field? This is difficult to answer precisely, but features of the system make it reducible to a simple equivalent circuit that allows a good approximate analysis. Elasmobranch ampullary canals contain a mucopolysaccharide with a low resistivity, similar to that of seawater, while their walls 2.2.

have extremely high resistivity. The resistivity of elasmobranch body tissues is about ten times higher than the resistivity of seawater and the conductive core of the canals (Murray, 1967; Bennett, 1971; Kalmijn, 1974). Consider current flow in an arbitrary circuit through the canal, like that shown in Fig. 1(e). The segment of the circuit through the seawater forms a fixed passive resistance. The segment in the body contributes an emf proportional to its vertical extent, and has resistance proportional to its length. Because the body resistivity is high relative to that of the canal lumen, the body segment’s contribution to the circuit current is relatively small. The main effect of the body segment is to increase the circuit resistance. Therefore, current flow is largest in circuits that take the shortest path from the receptor the body surface. Receptors tend to be placed close to the body surface, so, to a first approximation, the effect of the body is simply to increase the total resistance of return paths for current generated in the canal. More detailed analysis would undoubtedly show that the body makes some active contribution to the receptor voltage. However, such an effect will be small and systematic. It may affect the details, but not the principles, of what follows. The voltage induced across the receptor of an ampullary organ oriented nearly perpendicular to the velocity vector v and the magnetic field vector B is, to a good approximation, proportional to the shark’s velocity perpendicular to B, V=lvB sin u,

(10)

where v and B are the magnitudes of v and B respectively, u is the angle between these two vectors, and l is a constant that depends on the vertical extent of the canal and the relative resistance of the receptor in circuits through the canal. The equivalent circuit shown in Fig. 1(f) can be used to analyse the model in Fig. 1(e).        Electric fields as large as several hundred microvolts per centimeter are induced by water movements in the ocean (Von Arx, 1962). Because elasmobranchs can detect and orient in fields somewhat smaller than this, these fields provide both a potential navigation cue and a potential jamming signal for navigation. To avoid confusion, I refer to electrical currents as currents, and water currents as flows. Consider a surface flow moving with velocity v in deep, still water, crossing a magnetic field B. If B is horizontal then charges are forced to move vertically through the flow, but the surface boundary condition J · z=Jz=0 (current can not cross the surface) forces them to move 2.3

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F. 2. (a) Example of electrical field induced in an ocean surface stream. The stream flows into the page. The component of the geomagnetic field vector perpendicular to the flow is (in this example) directed upwards and to the left. The net force on charges in the stream is perpendicular to this, upwards and to the right. Charge movements are constrained by the surface boundary condition, so that near the surface, electrical current flows parallel to the surface from left to right. A counter-rotating eddy current may occur on the left edge of the stream. An electric field is induced, whose direction is opposite to that of the current flow. (b) Electrical lensing by a relatively insulating body in a conducting medium. Current flowing down the page is forced around the relatively insulating body. Field equipotentials, which are always perpendicular to the current lines, are drawn into the body, so that voltage gradients in the body are steeper than they would be in the medium without the body. Conceptually, the ampullary canals in elasmobranchs are not sensory accessory structures that amplify external field gradients, as they are usually described. They are functional holes in a sensory accessory structure—the fish’s body—that surrounds them. The resistive tissues around the ampullary canal amplify the receptor voltage in the presence of an external field. Conversely, the canals are accessory structures for measuring motion-induced voltages.

horizontally near the surface. If B is vertical then charges are forced across the flow. In general the geomagnetic field has both horizontal and vertical components, resulting in an asymmetric current pattern [Fig. 2(a)]. This current induces an electric field in the opposite direction. The induced electric field is a consequence of, and therefore contains information about, the flow kinematics. In particular uniform, wide, shallow surface flows near the equator are likely to contain electric fields whose strength and direction are uniform over much of their extent. Similar analyses could be performed for more complex situations. Electric field geometry in the ocean depends on flow kinematics, the electrical parameters of the water and the boundary conditions. There is a consistent relationship between these things and the electric field at any point. This relationship can be expressed using partial differential equations but is not reducible to a simple rule. You cannot use a globally valid rule to compute the flow at a point in a stream using only the local electric field vector.    Because an elasmobranch’s body has higher resistivity than seawater, currents tend to flow around it rather than through it. The equipotentials of the induced field are always perpendicular to the current lines, and are therefore compressed or focussed near the electrical poles of the body [Fig. 2(b)]. As a result of this focussing effect, the voltage drop through the fish’s body exceeds the voltage drop that would occur over the same distance if the fish were not present (Pickard, 1988; Kalmijn, 1974).

When measuring imposed electric fields, the fish’s body is a sensory accessory structure. Without the body the canal would have a negligible effect on the receptor voltage, because its conductivity is similar to that of seawater. When measuring an imposed field, the receptor can be modelled as a voltmeter that measures the voltage drop through the body parallel to the canal. The canal forms a passive, low-resistance lead for this voltmeter.

3. Existing Theory I shall briefly summarize and then critically evaluate the existing theory of how elasmobranchs navigate using magneto-electric induction (Kalmijn, 1981; Kalmijn, 1984). According to this theory, elasmobranchs use two modes of electromagnetic orientation. In the active mode, the animal estimates its magnetic compass heading by measuring receptor voltages induced by its own swimming movements. In the passive mode, the animal measures electric fields induced by its drift and by water movements.

2.4.

  In analysing active electronavigation, Kalmijn models the ampullary organs as voltmeters spanning parallel resistive paths through the body (e.g. figure 2a in Kalmijn, 1984). That is, he approaches the analysis as if the receptor voltage was due to an external field imposed across the fish. The velocity–sine rule (10) can be applied to determine the receptor voltage in an Earth-fixed reference frame. Assuming negligible resistance on the return path, the resistive field −rJ 3.1.

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induced in the body balances the emf generated in the body because the potential difference must return to zero in a complete circuit. Kalmijn argues that while a stationary voltmeter would measure zero volts, this voltmeter is moving—a moving voltmeter spanning the body generates the same emf in its leads as does the path through the body; the reading on the voltmeter is the same as if it were measuring the voltage drop through the body; therefore the receptor measures the voltage drop through the body. The measured voltage is proportional to v sin u, with a constant of proportionality that depends on the strength of the horizontal component of the magnetic field and the vertical extent of the canal. The shark can solve for sin u if it knows this constant and the swimming speed. This gives two possible values for u, with a north–south ambiguity. For example, northeast can not be distinguished from southeast and northwest can not be distinguished from southwest. The shark can determine which of the two possibilities is correct by checking what happens when it turns. For example, if it is swimming northeast, then the induced voltage will increase when it turns to the right. Thus, swimming direction is computed in two logical stages. The first stage requires knowledge of the swimming speed and the local magnetic field strength. It provides a quantitative value but (except in special cases where the shark is swimming due west or due east) this is consistent with two possible swimming directions. The ambiguity is resolved in the second stage using information gathered during turns. 3.2.   According to Kalmijn (1984), the main cues available for passive orientation are horizontal electric fields in surface flows. Because of the large volume, low-resistance path available for return current outside such a flow, induced fields are small outside the flow, and the field within the flow approximately balances the motion-induced emf. Kalmijn treats this situation as if the flow is in effect a giant fish. He puts forward a variation of the same argument given in the previous section to explain how a fish within the flow can see the resistive field despite the (assumed) fact that the net field in the flow is zero. The argument is that the animal generates an emf when it is drifting with the flow, and on closed paths that go through the fish and return within the flow this cancels the flow emf. Thus, the emf induced within the fish when it drifts with the flow unmasks the field in the environment and the animal ‘‘experiences the −rJ field’’ (Kalmijn, 1984). In general, when the fish is swimming in a flow, the effect of the flow is to add a d.c. offset to the receptor voltage that the fish would detect if it were moving at the same

Earth-referenced velocity in still water. The fish can measure this offset by drifting with the flow, and make the appropriate adjustment for navigating when it swims. Unfortunately, electric fields in ocean flows depend not only on local flow velocity and the local geomagnetic field, but on flow kinematics, electrical parameters of the water and boundary conditions. The electrical parameters and boundary conditions are unique to each flow. While the field at a point in a flow is a function of the flow velocity at that point, the function will be different at different points, even within the same flow. Oceanographic measurements show that the induced field can vary between 16% and 90% of v×B in different locations (Kalmijn, 1974). It follows that a shark can not in fact determine the local flow velocity by measuring the electric field as it drifts. Some kind of local knowledge or a calibration mechanism is needed in order to use drift-induced fields in navigation. Kalmijn (1984) comments that a shark would violate special relativity if it could measure its velocity in a particular reference frame using only electromagnetic measurements. He observes that a shark ‘‘must know the relation between v×B and −rJ in the desired frame of reference’’ and suggests that ‘‘To acquire the information needed to interpret −rJ correctly, the animals might occasionally dive to the bottom, or at least well below the [flow]’’. Exactly what information the shark should gather on these excursions, and how the information should be used to calibrate the shark’s navigation system, is not stated. Kalmijn (1984: 530) simply notes that ‘‘Interpretation of the various fields would require great expertise on the part of the animals’’. 3.3.  When analysing active electronavigation, it is not correct to model the ampullary organs as voltmeters spanning the body resistance parallel to the canal. While it is true that the voltage induced in the parallel body path equals the voltage measured by the ampullary organ, this is because the ampullary canal and the body path parallel to it have, by necessity, the same height in the magnetic field. From Kalmijn’s own argument and the associated equivalent circuit diagram (Kalmijn, 1984, figure 2) it is apparent that the source of receptor voltage in this situation is magneto-electric induction in the canal. It has nothing to do with the emf or resistive field in the body. The receptor does not see a voltage drop across the body, it sees a voltage of approximately the same magnitude that would be present even if the body was not. Kalmijn’s relativistic approach does give approximately the correct magnitude of receptor voltage

   induced by movements. However, his analysis is incorrect and rather more complex and esoteric than necessary. Special relativity is often important in analysing electromagnetic problems, but it is spurious to introduce it in this context. It is true that velocities, electric fields and magnetic fields are only defined with respect to particular reference frames. However, the geomagnetic field is magnetic in Earth-fixed frames and it is these frames in which the shark wants to navigate. Equation (5) can be applied directly without violating special relativity. Kalmijn’s juxtaposition of the suggestion that sharks may calibrate electric field measurements in a flow to a particular reference frame by swimming below the flow, with the observation that electric fields tend to be unique in particular flows (Kalmijn, 1984: 529), gives the impression that such a calibration solves the problem of how to determine flow velocity from electric field measurements. It does not. The shark’s inability to calculate flow using electrical measurements is not a problem of relativity, it is a problem of not knowing the parameters and boundary conditions that determine the electric field. From the shark’s point of view, the voltage across a receptor can be regarded as coming from three sources: induction in the canal due to drift, induction in the canal due to the shark’s active swimming movement and a d.c. offset due to induction in the flow. The last effect depends on the particular location in the particular flow. If the shark could measure the voltage due to drift then it could compute the flow velocity. Unfortunately, this is not possible because of the unknown d.c. offset. What can the shark learn by swimming below the flow? If it could find a region of still water containing a negligible electric field, then it could compute its Earth-referenced velocity as it swims in that region. However, this is of no value in calibrating measurements within the flow unless the shark can maintain its velocity in the earth frame as it swims back into the flow. At least, the shark would need to know its velocity in the flow relative to its velocity outside the flow. That is, the ‘‘information needed to interpret the receptor voltage correctly’’, supposedly gathered during this maneuver, is effectively the flow velocity. If a shark could gather the information needed to compute the flow velocity from electric field measurements in the way suggested by Kalmijn, it wouldn’t need to. Another problem for the existing theory is that elasmobranch electroreceptors do not make d.c. measurements. Mathematical models predict that elasmobranch ampullary receptors could detect d.c. voltages (Waltman, 1966; Kalmijn, 1974), but these models only deal with the electrical characteristics of

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the canal and do not take into account receptor and afferent dynamics. Recordings from electrosensory neurons show that d.c. voltages are not transduced. The sensory neuron response adapts with a (high-pass) time constant of the order of about 10 seconds (Montgomery, 1984). Ocean flows and accompanying fields occur on such a scale that a shark, swimming at speeds in the order of tens to perhaps hundreds of centimetres per second, will experience field variations that are too slow to be detected by the electric sense. The same is true for the self-induced or ‘‘active’’ field. If a shark swims at constant speed across the geomagnetic field it will induce receptor voltages proportional to v sin u. But these voltages, being constant, will not be transduced. Sharks can not make the measurements that would be required to navigate by solving eqn (10) in the manner suggested by Kalmijn (1981: 1115). Kalmijn (1984: 525) notes that his theory provides the physical insights necessary to understand how elasmobranchs could navigate using their electric sense. ‘‘Future research’’, he comments, ‘‘will require a more formal, mathematical approach’’. From Kalmijn’s work it seems likely that elasmobranchs can and do make use of electrical cues in navigation. However, his theory is incomplete at best. The crucial parts of the theory, namely how an elasmobranch could calibrate the scale and d.c. offsets of electric field measurements in the way required by the theory, are subsumed under the comment that navigation using the electric sense would require ‘‘considerable expertise on the part of the animal’’. Even if this is true, it falls somewhat short of what might be called a theory of navigation using the electric sense. 4. A Theory of Navigation Using the Electric Sense In this section, I show that it is possible for an elasmobranch to determine its heading direction using uncalibrated measurements of motion-induced electric fields, without knowing its swimming speed and without knowing the strength of the local geomagnetic field vector. Furthermore, it can do this in a way that is not affected by environmental electric fields.       Consider a shark swimming so that its head moves through the water at constant speed v. The horizontal component of the local geomagnetic field vector, which points north, is N. The shark swims horizontally and undulates horizontally about a straight line called the swimming path. The swimming path forms an angle u, the heading, measured clockwise from N. In still water the heading is the direction that the shark is moving

4.1.

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with respect to north. The angle that the velocity vector of the head, v, makes with the swimming path varies sinusoidally with frequency v and maximum amplitude a [Fig. 4(c)]. The following theory does not depend on whether this model of shark head kinematics is accurate. It depends only on the fact that a shark’s head moves back and forth across the swimming path as the shark swims. This particular model has been chosen because it is easy to analyse quantitatively. If a shark swims in still water, its head velocity vector has constant amplitude and forms an angle c=a sin(vt)+u with respect to N. The receptor voltage induced in an ampullary organ is proportional to the sine of this angle, V=lvB sin(a sin(vt)+u).

(11)

Some special cases are easy to analyse. For example, when the shark swims due north (u=0) with small a, then since sin(x)1x for small x, V1alvB sin(vt).

(12)

When it swims south, V1−alvB sin(vt),

(13)

the receptor voltage varies sinusoidally at the swimming frequency, in phase with head movements when the shark swims north and 180° out of phase when it swims south. The effects of swimming due east or west can also be analysed. While swimming due east (u=p/2) with small a, V1lvB(1−a 2/2 sin 2vt). Substituting sin 2vt=12 (1−cos 2vt), shows that there is a constant (d.c. offset) term lvB(1−a 2/4) and an undulation term, V=−l

vBa 2 cos 2vt. 4

(14)

The constant term is not transduced. Thus, while swimming due east the shark observes a signal that varies at twice the undulation frequency. Similarly, while swimming due west the observed signal is V=l

vBa 2 cos 2vt. 4

(15)

This has the same form as the signal seen while swimming east, but is 180° out of phase with that signal.     How large are these signals relative to the sensitivity of the elasmobranch electrosensory system? In this section I show that receptor voltages induced during swimming are easily large enough to be measured by the elasmobranch electrosensory system. The peak amplitude of the a.c. component of emf per 4.2.

unit vertical extent of a canal induced while swimming north or south is avB, and while swimming west or east it is vBa 2/4. In other swimming directions, the peak amplitude of the induced emf lies between these values. The strength of the horizontal component of the geomagnetic field is about 40 mT near the equator, falling to about half that value at latitude 245° (Parkinson, 1983). Consider a shark moving at 0.5 m sec−1 near Dunedin, New Zealand (45°S). With a=p/8 this gives a peak of V180 nV per vertical centimeter of canal while swimming north or south, and V116 nV per vertical centimeter of canal while swimming east or west. When measuring voltage gradients through the body, the linear dynamic range of elasmobranch ampullary electroreceptors is about 1 mV cm−1 with a threshold below 5 nV cm−1 (Montgomery, 1984; Kalmijn, 1982). As discussed above, receptor voltages caused by canal emfs are roughly equal to those that would be caused by emfs or equivalent voltage gradients in the body parallel to the sense organ. Therefore, a small, slow moving shark far from the equator will be able to observe receptor voltage modulations due to swimming undulations rather easily, and with good resolution. It will see the whole waveform not just the peaks. Larger or faster sharks, or those closer to the equator, will find the measurement even easier to make.       Receptor voltage modulations due to swimming undulations fall within the linear dynamic range and frequency bandwidth of the electrosensory system. These modulations provide directional information, at least in terms of distinguishing between north, south, east or west. Do they provide information that would enable a shark to determine its heading with greater resolution? To answer this question, I have calculated receptor voltage waveforms numerically from eqn (11). The results for selected headings are plotted in Fig. 3(b). This shows receptor voltage waveforms while swimming with different headings, from −90° (west) to 90° (east). For headings between 90° (east) and 270° (west), the waveforms are 180° out of phase with those shown in the figure. Consider the quadrant between north and east, i.e. 0EuEp/2. The receptor voltage waveform is symmetric when the fish swims due north. I shall call this in phase with head movement; receptor voltage is in phase if it increases in phase with head turns to the right. When the fish heads to one side or the other of due north the waveform flattens on the side away from north. Therefore, if the receptor voltage is in phase and

4.3.

   flattened on the right, the fish is heading between north and east. The larger the heading, the greater the flattening. If the heading is so large that u+aqp/2, then the head velocity vector crosses the normal to north (i.e. crosses due east) during the undulatory cycle. In this case, the receptor voltage waveform inflects producing a ‘‘ripple’’ in the waveform at twice the undulation frequency [Fig. 3(b)]. When the swimming path parallels the normal to the magnetic field line (u=p/2) the receptor voltage waveform is

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again symmetric but varying at twice the undulation frequency. Similar analyses are possible for the other quadrants. Between north and west, receptor voltage modulation is in phase and flattened on the left. The flattening becomes a ripple near due west. Between west and south, the modulation is 180° out of phase and flattened or rippled on the right. Between south and east, the modulation is 180° out of phase and flattened or rippled on the left. There is a quantitative

F. 3. Receptor voltage waveforms generated while swimming through the geomagnetic field. (a) Head rotation velocity waveform while swimming, calculated from the model in Section 4.1. (b) Receptor voltage waveforms in an ampullary organ of vertical extent l, based on the assumption that circuit resistance is negligible except across the receptor, calculated from equation 11. Waveforms are shown for various headings between due west and due east. N=0°, NE=+45°, E=+90°, NW=−45°, E=−90°. Note that the waveform flattens on the side away from N, and ‘‘ripples’’ if the head tangent crosses the normal to the magnetic field line. (c) Comparison of horizontal semicircular canal and vertical electroreceptor afferent firing rate modulations during swimming NNE (22.5°), with normalized gains. The vestibular signal is independent of heading, but the electrosensory signal changes systematically as a function of heading. A simple mechanism based on comparing these two signals could compute the heading.

334

. . 

F. 4. Surface/contour plots of sensory afferent modulations due to head movements during swimming. (a) The height of the surface at distance t from the center of the plot in direction u represents the motion-induced receptor voltage at time t when the fish is swimming with heading u, according to the model of Section 4.1. (b) The corresponding point on this plot represents head rotational velocity at the same time. (c) Swimming trajectory. N=north, H=vector in heading direction, v=velocity vector at time t, u= heading.

relationship between the phase and asymmetry of the modulation and the heading. Figure 4(a) shows a surface plot and contour plot of receptor voltage modulation as a function of heading, for a fish swimming with head movements as described in Section 4.1. The traces of Fig. 3(b), minus the d.c. level, are radial slices of this surface. Magneto-electric navigation in the ocean is possible because the surface is rotationally antisymmetric; no two directions give the same pattern of modulation. Figure 4(b) shows a

surface plot and a contour plot of head velocity as a function of heading. This surface is rotationally symmetric; all directions give the same pattern of modulation.   The computation described in the previous section will not be disrupted by vertical electric fields because the angle between the canals and the vertical does not change as the fish swims in the horizontal plane. 4.4.

   However, electric fields in the ocean, especially near the surface, tend to be largest in the horizontal plane. This is because of the surface boundary condition and because oceans and ocean flows tend to be much wider and longer than they are deep. In passive mode, an ampullary electroreceptor organ measures the potential difference that develops through the fish’s body parallel to the canal. The receptor voltage is proportional to the cosine of the angle between the canal and the field lines. Therefore, a horizontal electric field in the environment will cause receptor voltages to be modulated during turns. The modulations will have the same form as those due to magneto-electric induction in the canal, except that flattening and rippling will depend on the orientation of the canal with respect to the electric field vector, not the velocity of the fish with respect to the geomagnetic field vector. To navigate in the presence of an electric field, a shark must be able to separate receptor voltage modulations due to turning in the electric field from modulations of identical form due to varying velocity in the geomagnetic field. Moving in the horizontal plane induces vertical voltage gradients, while turning in the electric field causes modulations of horizontal voltage gradients. Electroreceptor organs with vertically oriented canals will detect changes in motioninduced voltages while those with horizontally oriented canals will detect changes in orientation with respect to the environmental electric field. Thus, the effects can be separated by a combination of behaviour (horizontal swimming) and peripheral anatomy (canal orientation). In individual receptors, there is undoubtedly interference caused by vertical components of environmental fields and the geomagnetic field, and cross-talk between magnetically induced and environmental voltages—for several reasons. First, even if a subset of electroreceptors are specialized for navigation, their canals may not be oriented precisely. Second, a canal may provide a low-resistance shunt path for a current generated in the body even if the force causing the charge movement is perpendicular to the canal. Third, elasmobranchs do turn in a vertical plane (change pitch) during swimming, and they vary their swimming speed. Central processing is probably required to isolate the relevant receptor voltage modulations by combining inputs from populations of receptors with different orientations. There is evidence that a neural analog of an adaptive array filter, capable of performing this kind of signal separation, is present at an early stage of electrosensory processing in the elasmobranch hindbrain (Bodznick et al., 1992; Paulin & Nelson, 1993; Bodznick, 1993).

335 4.5.

  

It is possible for sharks to measure their orientation with respect to the geomagnetic field as they swim, despite the jamming effect of electric fields induced by ocean flows. However, the direction in which a shark moves within a flow is usually not the same as its heading, the direction in which it is pointing. How could a shark measure and compensate for drift? This is a central question in Kalmijn’s theory. According to that theory, a shark cannot obtain any directional information using its electric sense without making quantitative corrections for the effects of water movement. The need to engage the electric sense in measuring and compensating for these effects is a problem for the theory (especially since it would appear to be impossible) but it may not be a problem for sharks. Given a compass sense a shark does not need to compute real trajectories in order to navigate. All it needs to know, for example, is that it can get from point A to point B by heading in direction u at speed v. We might see that the shark is actually moving in a different direction at a different speed, perhaps even in a giant arc, but this perspective is irrelevant, indeed meaningless, to the shark. Similarly, the fact that the shark generally would not get from point B to point A by swimming in the opposite direction at the same speed for the same time, does not mean that sharks will get lost if they don’t correct for drift. It simply means that a sharp would be unwise to apply a rule like ‘‘Head in the opposite direction’’ if it wants to get back to where it came from. To navigate in its world, a shark needs a set of rules about how to use available cues (about places, directions, distances, velocities and times) to travel between locations in its environment. At least, it needs a structure that contains information from which such rules can be generated, i.e. a cognitive map or spatial memory. A cognitive map of a region does not have to resemble a little picture of that region, like the paper maps that we are familiar with. This can be deduced from the fact that a set of navigation rules is itself a cognitive map—a structure from which navigation rules can be deduced—that could be implemented as a randomly arranged set of modules each representing one rule. I am not arguing that cognitive maps are ever implemented this way, but merely pointing this out as a counter-example to a too-literal interpretation of the term ‘‘cognitive map’’. It is not necessary for a shark to calculate its trajectories in an inertial reference frame and then correct for the effect of water movement when it tries to move along that trajectory. It seems more natural to assume that an animal that lacks the opposable

. . 

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thumbs necessary to use pencil and paper effectively would evolve spatial memory appropriate for the environment in which it evolved, rather than adopt the abstract and comparatively recent human conventions of Gallilean–Newtonian relativity. The problem of how to correct for drift is meaningless to a shark. Its problem is how to develop useful cognitive maps. This is not a problem that is unique to elasmobranchs. It is a problem for any animal, on land, in water or in air, with or without an electric sense. It is a problem about neural representations of space and movements, and it is beyond the scope of the present work. Uncovering the role of electroreception in elasmobranch navigation does not solve the problem of how elasmobranchs navigate, only how elasmobranchs obtain important navigational information using their electric sense. By analogy, a compass is useful but neither necessary nor sufficient for navigation. The suggestion herein is that electroreception provides a compass sense for some elasmobranchs. 5. Algorithm and Implementation: A Model The waveform of receptor voltage modulations during swimming contains information about the swimming direction. How could the nervous system extract this information from the waveform? I shall present a simple algorithm for transforming the electrosensory waveform into a representation of swimming direction, and explain how this could be implemented using neurons. The swimming direction cue is the asymmetry of the waveform of receptor voltage modulations. While heading due north the receptor voltage modulation is sinusoidal and in phase with head rotation velocity. If the animal heads a little to the west, the waveform is compressed during turns to the left. If it heads a little to the east, the waveform is compressed during turns to the right. The heading can be estimated by quantifying this effect. The effects are the same while heading south except that the phases are reversed. The frequency-doubling or ripple effect provides a conspicuous marker for swimming near due east or due west. Some possible mechanisms for measuring the asymmetry require either long- or short-term memory. For example, the firing rate of electrosensory neurons during a turn could be compared with permanently stored expected firing rates, or with firing rates † Firing rate modulation in afferents from the right horizontal semicircular canals is in phase with head turns to the right, while modulation in afferents from the left is in phase with head turns to the left.

remembered from recent, oppositely directed turns. Any such mechanism would require calibration to enable comparisons of signal levels at different times. This cannot be ruled out, but I shall focus on what seems to me to be an elegant and credible alternative hypothesis, that the quantification of the waveform asymmetry involves a comparison of electrosensory waveforms with symmetric reference waveforms derived from the vestibular system. This electrosensory–vestibular comparison hypothesis has the important feature that it leads to novel predictions that will allow it to be eliminated quickly if it is wrong. Firing rates of afferent neurons from the horizontal semicircular canals of the vestibular apparatus encode horizontal head rotations. In elasmobranchs, as in other vertebrates, this firing rate modulation is proportional to and in phase† with head rotational velocity at head rotation frequencies that predominate during natural movements (Montgomery, 1980). These vestibular afferents therefore provide a head rotation velocity reference signal that is independent of swimming direction. Figure 4(b) shows surface and contour plots of head velocity for a fish swimming according to the model in Section 4.1. These plots are rotationally symmetric; the fish’s heading makes no difference. Asymmetry of the waveform of receptor voltage modulations could be quantified by comparing responses of neurons responding to changes in motion-induced electroreceptor voltage with responses of horizontal semicircular canal afferents. Suppose, for example, that a particular semicircular canal unit tends to fire at a particular point in the locomotory cycle as the head swings to the right. Now suppose that, while swimming due north, a particular electrosensory unit tends to fire at the same time. If the fish alters course towards the east then firing of the electrosensory neuron will be delayed, because the receptor voltage modulation is smaller during right turns. This is a special case of a general phenomenon: small changes in heading will result in small changes in firing times of certain electrosensory neurons without corresponding changes in the firing times of the vestibular neurons. It is then rather easy to imagine how a neural compass—a network that could transform electrosensory and vestibular signals into a representation of swimming direction—could be set up. For example, an array of coincidence-detector neurons could each establish connections with pairs of vestibular and electrosensory neurons that have a tendency to fire synchronously over certain intervals of time. That is, a period of synchronous firing reinforces synaptic strength. Because sharks have a tendency to swim with

   a certain heading for a number of locomotor cycles followed by relatively brief turns (Harris, 1965), this rule automatically selects members of the correct functional classes within each sensory modality. Only neurons encoding the correct cue signals will show the requisite inter-modality synchrony. When the swimming direction changes, some electrosensory–vestibular pairs will move into synchrony while others will move out of synchrony. The array of coincidence detectors then forms a map of swimming direction. A high proportion of elasmobranch vestibulo– cerebellar Purkinje cells that respond during horizontal sinusoidal head rotation respond at double the head oscillation frequency. Montgomery (1980) reported that 19% of vestibulo–cerebellar Purkinje cells in the dogfish Scyliorhinus respond in this manner. Such neurons could provide reference signals for estimating directions near due east or due west, where the motion-induced electroreceptor voltage modulations occur at twice the head oscillation frequency. The ripple effect provides a very sensitive marker for when the swimming path grazes (is tangent to) or crosses the normal to the geomagnetic field vector. This could be very useful for navigating. The direction map is not necessarily ‘‘orientopic’’: i.e. such that neighbouring neurons encode similar directions. However, an advantage in having similar neighbours is that errors can be recognized and treated. By adding a synaptic learning rule that encourages neighbourhood uniformity, any unit that displays oddball behaviour can be disconnected from its faulty inputs and reconnected to valid inputs at hand. Thus, an orientopic map could be self-correcting as well as self-organizing. Although this model is described as if the orientation map were constructed of secondary sensory neurons, receiving inputs directly from the vestibular and electrosensory periphery, this is not a necessary feature. On the contrary, it seems likely that there is some central filtering of the sensory inputs before they are brought together to construct the directional map. The relevant signals are the horizontal turn-related modulation of the vertical component of electric field and the horizontal rotation velocity of the head. Each of these signals can be reconstructed centrally with greater accuracy than is available in the raw sensory data, by taking into account other peripheral and central signals that are correlated to them. Brainstem and cerebellar structures may be involved in such ‘‘clean up’’ operations on incoming sense data (Paulin & Nelson, 1993; Bodznick & Montgomery, 1992). Thus, if there is electrosensory–vestibular convergence to an orientation map in elasmobranchs, it is likely to

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be at a level higher than the secondary brainstem nuclei. 6. Discussion It seems likely that elasmobranchs can and do use their electric sense in navigation. According to Kalmijn, elasmobranchs (i) use their electric sense to determine which direction they are moving in an Earth-fixed reference frame, and (ii) do this by making calibrated measurements of the amplitude of motion-induced electroreceptor voltages, taking into account their swimming speed and the strength of the local geomagnetic field. I have suggested instead that (i) electroreception is used to provide a compass bearing, and (ii) this is done by quantifying asymmetry in turn-related modulations of electroreceptor voltages. The critical distinction between the two theories can be understood by comparing Kalmijn’s (1981: 1115) calculation of whether the elasmobranch electric sense is sensitive enough to be used in navigation, with my analogous calculation in Section 4.2 above. At that point, Kalmijn explicitly states that the variables that determine whether or not the induced receptor voltage will be large enough to be measured by an ampullary organ—and used to determine swimming direction— are the forward velocity of the shark and the strength of the geomagnetic field vector. In contrast, I have argued that it is not the size of the induced voltage that matters, but the amplitude of modulations of that voltage within the electroreceptor bandwidth. This depends not only on the forward velocity of the shark but also on the turn frequency and maximum turn angle [cf. eqns (12) and (14)]. Kalmijn’s model requires only that the direction of such modulations be detectable. His subsequent comment that ‘‘The dynamics of magnetic field detection are of great importance as the sense organs are not true dc but very low frequency ac receptors with a time constant of several seconds’’ (ibid.) is difficult to interpret in this context, as it implies that the calculation immediately preceding it is incorrect. Indeed, it was this inconsistency in Kalmijn’s theory that led me to develop the new theory. The main difficulty in the earlier theory is that it does not explain how a shark can distinguish receptor voltages due to water movement from those due to its own swimming movement. These problems are entangled in that theory because of the assumption that the shark uses its electric sense to determine its direction of movement in an earth-fixed reference frame. In contrast, I suggest that it is not possible to compute the Earth-referenced direction using electrical

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measurements alone. I suggest that electrical measurements are used within a broader computational scheme for navigation. Electrical measurements could be used to determine a compass bearing and to determine the strength and direction of any local electric field gradient. This information is neither sufficient nor necessary for navigation, as shown by the fact that many marine species lacking an electric sense have navigation skills as impressive as those of elasmobranchs. A general computational scheme for navigation, able to exploit whatever navigation cues may be available, may be shared by many marine vertebrates including elasmobranchs. The neural network model of how the elasmobranch central nervous system might compute a direction map is clearly speculative. Its importance lies not in the possibility that it might be true (although it might), but in the fact that it shows explicitly how a physical device could extract directional information from electroreceptor inputs given the physical and biological constraints. In contrast, Kalmijn has failed to explain even in principle how this could be done under the assumptions of his theory. The new theory makes a number of testable predictions. There may be ampullary organs specialized for measuring motion-induced voltages. According to my analysis, the body is a hinderance to measuring these voltages, not an accessory structure as in Kalmijn’s analysis. These ampullary organs should be vertically oriented and laterally placed. This would seem to implicate the hyoid group of electroreceptors in navigation. The path from the basal face of the receptors to the body surface should have a low resistance compared to the corresponding resistance in organs specialized for detecting external fields. The hyoid receptors are placed dorso- and ventro-laterally, caudally on the head, well behind the mouth (Bodznick & Boord, 1986), a position that would seem unsuitable for prey detection and electrical guidance of strikes. The manta ray, Manta birostris, a pelagic planktivore, has only hyoid electroreceptors. It has already been noted that this may indicate a specific role for these receptors in navigation by the manta (ibid.). In contrast, the swell shark Cephaloscyllium ventriosum a benthic shark that feeds on buried and drifting prey at the sea floor using its electric sense and seems to remain in a restricted area (Tricas, 1982), lacks hyoid electroreceptors. The pattern may be general. That is, the extent and development of the hyoid group of receptors in elasmobranchs should be correlated to the animals’ tendency to move long distances through open water, not to their feeding mode, and independent of phylogeny. Specific functional blocking of the hyoid receptors should

cause serious short-term impairment of navigation ability, and may cause some observable long-term impairment. Feeding and navigation may involve distinct, as well as shared, central nervous system structures. Projections from the hyoid group of electroreceptors to the central nervous system may be notably different from those that come from other groups of electroreceptors. These differences are likely to be most evident in the midbrain and telencephalon, where computations for spatial mapping and navigation probably occur. I have suggested that the horizontal semicircular canals of the vestibular system may provide a reference signal for analysing modulations of motion-induced receptor voltages, to determine a compass bearing. If this is true, then there must be a central convergence site for horizontal semicircular canal and hyoid electrosensory inputs in migratory elasmobranchs. Thanks to Sara Metcalf, John Montgomery, David Bodznick, Mark Nelson and Patricia Langhorne for useful criticism and discussions of this manuscript. Supported by NZ MoRST:RST-08-2-1-c:SP.

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