SUMMARY

Weakly electric fish generate an electric field with their electric organ to navigate in space, detect objects and communicate with conspecifics. Several studies have examined how electric fish identify objects with their electroreceptors and use electric images for electrolocation. It has been argued that sensor readings from electroreceptors along the rostrocaudal line allow fish to determine the location of a target object. It is well known that the ratio between the maximal slope and the maximal amplitude of the electric image can allow the discrimination of object distances, regardless of object size and conductivity. In order to understand the temporal pattern of electric images, we used a model of electric field perturbation. Using the model, we suggest that the temporal pattern generated at an electrosensor during tail bending is another cue that can be used by the fish to discriminate object distances. The time course of electric sensor signals from a specific electroreceptor when tail-bending movements are applied can provide information about the lateral distance of a target object.

INTRODUCTION

Weakly electric fishes are equipped with a highly specialized electrosensory system, and there is great interest in understanding how these fish detect prey using an electric field. Electric fishes often use active electrolocation in which the electric field is created by an electric organ (Heiligenberg, 1991; Nelson and MacIver, 2006). This discharge organ is typically located in the tail, and a large number of electroreceptors are distributed on the skin surface. To capture their prey, weakly electric fish use active body movements that do not seem related to locomotion, but the role of these movements in prey detection are unknown.

In this paper, we investigate the electrolocation mechanism of a weakly electric fish, the black ghost knifefish, Apteronotus albifrons (Linnaeus 1766). Weakly electric fish sense electric field perturbations caused by neighbor objects in a self-generated electric field. Electroreceptors of weakly electric fish are found in pores on the surface of the whole body; these electroreceptors are sensitive to changes in the electric field. Perturbations in the electric field allow the fish to estimate object distances and even discriminate objects, for example, cubes from spheres (von der Emde et al., 1998). A collection of sensor readings from electroreceptors is used to generate an electric image. These electric images are then used to identify nearby objects. Furthermore, electric images can provide localization information for a target object, and there is also a correlation between sensory image features and physical object features. Apteronotus albifrons has electroreceptors called tuberous receptors that can detect weak electric fields in the range of 0.1–1 μV cm–1 (Bennett, 1971; Bennett and Obara, 1986; Kramer, 1996; Nelson and MacIver, 1999). Weakly electric fish have the highest density of electroreceptors at the rostral region and approximately 14,000 tuberous receptors are distributed across the body surface.

Rasnow studied the perturbation of transdermal voltage in the weakly electric fish Apteronotus leptorhynchus according to object features (Rasnow, 1996). He claimed that there is a certain relationship between electric image features and object features; namely, the peak location of the electric image indicates the rostrocaudal location of the object, and the amplitude of the image depends on its distance. Von der Emde et al. examined whether weakly electric fish are able to classify objects at various distances from their bodies (von der Emde et al., 1998). They proposed a function to detect the distance of a target object, regardless of object size and electrical properties. They used the ratio of the maximal slope to the maximal amplitude of the electric image, which can be called the relative slope, and verified that this function relies only on the location and the shape of a target object. The same electrolocation results have subsequently been verified by simulation (Sicardi et al., 2000).

To determine the position of a target object in three-dimensional space, we need to estimate the rostrocaudal, dorsoventral and lateral distance of the object from a reference point (for example, the center of the body). The electrolocation rule of the slope-to-amplitude rate can be applied to the two-dimensional electric image. In three-dimensional space, weakly electric fish can initially identify the location of a target object in the rostrocaudal–dorsoventral plane (Engelmann et al., 2008), and the sensory image has peak amplitude at the sensor closest to the object. The lateral distance as another dimension in three-dimensional space is not simply determined but it may be estimated with the relative slope measure described above, that is, the slope-to-amplitude rate of the electric image (von der Emde et al., 1998; von der Emde, 1999; Schwarz and von der Emde, 2001). However, the slope-to-amplitude rate varies depending on the three-axis (rostrocaudal, lateral and dorsoventral) position of a target object. Thus, how weakly electric fish organize the sensory map for object localization and process the electrolocation is still undetermined.

An electric image is affected by the distance, size, shape and conductivity of an object, and the maximum amplitude varies depending on these properties. The amplitude of an electric image cannot provide a direct measure of lateral distance, but the relative slope provides an effective measure of lateral distance. Rasnow suggested an electrolocation rule of relative width, which is the width of the electric image at a particular amplitude relative to the peak (Rasnow, 1996). Similarly, Chen et al. defined the full-width at half-maximum (FWHM) as the width of the electric image at the point where the amplitude is half the maximum value (Chen et al., 2005). This measure was developed for temporal sensor readings of an electrosensor when an object moves along the rostrocaudal line. The FWHM can serve as a distance measure for a target object. Later, Babineau et al. used a new electrolocation rule based on bimodal electric images (Babineau et al., 2006). On a bimodal curve, an object's rostrocaudal position can be determined by using the location of the peak whose amplitude is the maximum, and its lateral distance by the distance between the rostral and caudal peaks. Detailed behavioral studies, however, are required to validate these proposed electrolocation rules.

If a weakly electric fish swims forward or backward, the electrosensors experience transient changes in transdermal potential over a static object. Temporal sensor readings from an electrosensor, while the fish swims forward or backward over a static object at its side, can be interpreted as spatial sensor readings from a series of electrosensors distributed along the rostrocaudal axis. Weakly electric fish can also experience changes in transdermal potential during tail bending and other body movements, and it has been reported that tail-bending movements of electric fish are strongly related to prey capture (Heiligenberg, 1975; Lannoo and Lannoo, 1993; MacIver, 2001). Tail-bending movements cause large distortions of the electric image (Bastian, 1995; Bastian, 1999, Caputi, 2004; Engelmann et al., 2008). However, no study has investigated whether the tail-bending action of weakly electric fish can serve as a localization procedure for a target object near the fish body.

In this modeling study, we quantify the temporal information provided by simulated electrosensors during a tail-bending cycle in A. albifrons and investigate whether this information could potentially be used for distance discrimination. We argue that tail bending triggers the same temporal pattern of electric images at a given electroreceptor for an object at a fixed distance independent of the size of the object and its conductivity. Based on this observation, we suggest a new electrolocation rule of slope ratio based on temporal electric image patterns during tail bending.

MATERIALS AND METHODS

Tail-bending movements of weakly electric fish

To model the electroreception process of A. albifrons, we simulated a collection of electrosensors on the skin surface in our model when a tail-bending movement was applied, using MATLAB version R2009b (The MathWorks, Inc., Natick, MA, USA). For the simulation, we set the body length to 21 cm and the length of the electric organ to 15.47 cm, and fixed the density of electric poles to 10 poles cm–1 following the body model used in previous studies (Chen et al., 2005; Babineau et al., 2006). The number of electric poles was approximately 155, and these poles were located along the midline of the fish. All poles were positive except for the negative pole at the end of the tail.

When a weakly electric fish bends its tail, approximately 65% of the body length can bend whereas the rostral portion of the body maintains a straight line (Chen et al., 2005). The bending angle is defined by the angle between the body axis and the line from a certain pivot point to the end of the tail. In our simulation, the tail-bending angle ranges from –45 to 45 deg. We assumed that the caudal portion of the tail draws a circular arc around the pivot and the bending angle changes linearly with time. A radius R of this curvature is given by R=L/2θ, where θ is the bending angle and L is the bent proportion of the body length (Chen et al., 2005). The center point or rotational axis of the curvature is at a distance R from the pivot point of the body. For our simulation, we assumed that the object was near the fish body and that the fish bent its tail rhythmically from side to side at a speed of 90 deg s–1. Each electroreceptor measures perturbations in the transdermal potential caused by an object near the fish body. Sphere objects with varying sizes were used as target objects and these were placed at various positions from the body of the fish in the simulation experiments.

Modeling of the electric field

The electric potential φ can be derived easily when the electric organ is considered to be a collection of poles. We assumed that poles were uniformly distributed over the electric organ. The electric potential can then be calculated as the sum of the potential induced by each pole (Rasnow and Bower, 1996; Chen et al., 2005) as given below:
formula
(1)
where is an arbitrary position for the electric potential in space. Assume that there are m positive poles and one negative pole among the total of n poles (m=n–1). is the location of the ith positive pole, indicates the position of the last negative pole and q is the normalized magnitude of the potential. The total sum of charges of poles of the electric organ should be equal to zero. The charge of each positive pole is q/m and the negative charge is –q. The value of q generally ranges from 8 to 20 mV (Chen et al., 2005). The electric field is defined as the gradient of the electric potential and we can estimate the electric field at the point as follows:
formula
(2)
We need to know the transdermal potential difference , projected onto the normal vector of the skin, to estimate the appropriate electrosensor readings where is an observation point on the skin surface of the fish, i.e. the position of an electroreceptor. The electric field with a normal vector of an observation point can be derived as:
formula
(3)
where ρs is the resistivity of the skin and ρw is the resistivity of water. The resistivity ratio, ρsw, is approximately 0.1–0.5. Rasnow examined how much perturbation of an electric field occurs when a sphere object is found in the electric field (Rasnow, 1996). The perturbation potential at point is:
formula
(4)
where is the centre of an object, a is the radius of the object and χ is the electrical contrast that varies from 1 for a perfect conductor to –0.5 for a perfect insulator. Accordingly, the change in transdermal potential at a location due to a sphere object, , is:
formula
(5)

We assumed that this amount of change in transdermal potential is measured by the electroreceptors.

Relative slope

The relative slope is defined as the ratio between the maximal slope and the maximal amplitude of electric image over a series of electroreceptors. The relative slope can be represented as:
formula
(6)
where I(xi) is the electric perturbation caused by a target object at the sensor point, xi. A series of sensors {x1, x2,..., xn} are positioned along the rostrocaudal line or dorsoventral line of the skin surface. For object localization, the location of the peak amplitude provides the rostrocaudal position of an object whereas the relative slope provides the lateral distance (von der Emde et al., 1998). Here, this measure will be evaluated with a sweep of tail-bending movements.

RESULTS

Normalized temporal pattern of sensor readings associated with tail-bending movements

We present several different views of the sensor reading patterns of electroreceptors on the skin surface and a new analysis of the representation. Most electrolocation rules have focused on the sensory pattern over a distribution of electroreceptors (Rasnow, 1996; von der Emde et al., 1998; Babineau et al., 2006). We investigate the temporal change of sensor readings when tail-bending movements are applied.

Fig. 1 shows the temporal sequence of transdermal potential acquired from tail-bending movements for various lateral distances of a target object. Each potential curve shows temporal variation according to the tail-bending phase. The tail-bending angle first increased from –45 to 45 deg and then returned to –45 deg, generating a symmetrical potential curve. When the tail was bent towards the target object, the electric potential increased. The potential amplitude peaked when the tail moved closest to the target object. A smaller distance between the target object and the body trunk can therefore produce a larger electric field potential. As the next step, we monitored changes in the transdermal potential caused by a target object, ΔVtd in Eqn 5, which is the difference between the transdermal potential in the presence of an object and the baseline transdermal potential.

Fig. 1.

(A) Relative positions of objects (dotted circles) at varying lateral distances to an electroreceptor. (B) Transdermal potential (Vtd) when the tail bends from left to right and from right to left (with an electroreceptor at the rostrocaudal position of 8 cm from the head). A sphere with a radius of 0.8 cm was tested as a target object at the rostrocaudal position of 7 cm from the head, and each curve represents the lateral distance of a target object; the potential depends on the distance (the solid line represents the transdermal potential without an object).

Fig. 1.

(A) Relative positions of objects (dotted circles) at varying lateral distances to an electroreceptor. (B) Transdermal potential (Vtd) when the tail bends from left to right and from right to left (with an electroreceptor at the rostrocaudal position of 8 cm from the head). A sphere with a radius of 0.8 cm was tested as a target object at the rostrocaudal position of 7 cm from the head, and each curve represents the lateral distance of a target object; the potential depends on the distance (the solid line represents the transdermal potential without an object).

Fig. 2.

Dependence of the temporal pattern on lateral distances. A sphere with a radius of 0.8 cm was located at the rostrocaudal position of 7 cm from the head (obtained with an electroreceptor at the rostrocaudal position of 7 cm from the head).

Fig. 2.

Dependence of the temporal pattern on lateral distances. A sphere with a radius of 0.8 cm was located at the rostrocaudal position of 7 cm from the head (obtained with an electroreceptor at the rostrocaudal position of 7 cm from the head).

During active sensing, the tail bends from side to side and a rhythmic temporal pattern can be observed at each electroreceptor. When the electric organ at the tail comes closer to a target object, a large change in the transdermal potential often occurs. Fig. 2 shows the normalized pattern of transdermal potential changes for a single electroreceptor. Changes in the lateral distance of the object significantly altered the temporal pattern. We also evaluated what other factors affected the temporal pattern. Interestingly, we found that object size did not influence the temporal pattern. The potential perturbation itself was affected by the object size and large-sized objects increased the perturbation level, but the normalized temporal pattern remained unchanged for objects of various sizes at a fixed lateral distance (data not shown here). Furthermore, the conductivity of the object had no effect on the normalized temporal pattern obtained by a tail sweep (see Figs 3, 4).

Fig. 3 shows the temporal pattern of electric perturbation at an electroreceptor with varying conductivities of objects when a sweep of tail-bending movements is applied. Here, the conductivities are estimated with the electrical contrasts of objects (Rasnow, 1996). The perturbation level directly depends on the conductivity of the target object. However, the normalized temporal pattern does not change with varying positive conductivities (Fig. 4). Objects with a negative conductivity only changed the orientation of the temporal pattern, reversing it. The patterns obtained for objects with a positive and negative conductivity were mirror images of each other.

There are a number of electroreceptors along the rostrocaudal axis of a fish. The rostrocaudal position of an electroreceptor can strongly influence the temporal patterns during a tail sweep, although information about the lateral distance can be provided by a single sensor. Fig. 5 shows variations in the temporal pattern according to the rostrocaudal position of a target object. The relative position of a target object to an electrosensor influences the temporal pattern of electric perturbation. The temporal variation measure described above provides information about the position of a target object. The measure is independent of object size and conductivity, but is affected by the rostrocaudal position of the object and its lateral distance from the midline of an electric fish. However, the rostrocaudal position of a target object is determined easily from the peak amplitude position in the electric image. If the rostrocaudal position of a target object is determined in advance, the temporal pattern can be used to infer the lateral distance of the object.

Fig. 3.

Electric perturbation (ΔVtd) generated by objects with various conductivities, obtained from electrosensors at the rostrocaudal positions of (A) 2 cm, (B) 5 cm and (C) 8 cm from the head. Spheres with a radius of 0.8 cm were tested at a fixed lateral distance of 3 cm and at the rostrocaudal position of 8 cm.

Fig. 3.

Electric perturbation (ΔVtd) generated by objects with various conductivities, obtained from electrosensors at the rostrocaudal positions of (A) 2 cm, (B) 5 cm and (C) 8 cm from the head. Spheres with a radius of 0.8 cm were tested at a fixed lateral distance of 3 cm and at the rostrocaudal position of 8 cm.

Fig. 4.

Normalized temporal patterns (ΔVtd) produced by objects with various conductivities, obtained from electrosensors at the rostrocaudal positions of (A) 2 cm, (B) 5 cm and (C) 8 cm from the head. Spheres with a radius of 0.8 cm were tested at a fixed lateral distance of 3 cm and at the rostrocaudal position of 8 cm.

Fig. 4.

Normalized temporal patterns (ΔVtd) produced by objects with various conductivities, obtained from electrosensors at the rostrocaudal positions of (A) 2 cm, (B) 5 cm and (C) 8 cm from the head. Spheres with a radius of 0.8 cm were tested at a fixed lateral distance of 3 cm and at the rostrocaudal position of 8 cm.

When we monitored the responses of an array of sensors in the rostrocaudal line to a target object at various lateral distances, a monotonic rising curve was often observed when the object was positioned closer to the tail than one of the electrosensors. Generally, the relative position of the target object and the sensor from the tail organ had a significant influence on the temporal pattern generated during the tail sweep. As shown in Fig. 5, the perturbation signal did not have a regular form. However, the electrolocation rule with the normalized complex patterns of transdermal potential changes could be applied as described above. The normalized temporal pattern did not change when the size of the target object increased or the electrical constant varied. Therefore, the temporal variation at a fixed electroreceptor during tail bending allows localization of a target object.

Fig. 5.

Temporal patterns (ΔVtd) obtained for an object at various rostrocaudal positions; electrosensors were located at a distance of (A) 2 cm, (B) 5 cm or (C) 8 cm from the head. A sphere with a radius of 0.8 cm was placed at rostrocaudal distances of 2.4 cm (circle), 3.6 cm (cross), 4.8 cm (triangle) and 6.0 cm (square) from the head, but at a fixed lateral distance of 3 cm. (D) Objects (dotted circles) at varying rostrocaudal positions with three electroreceptors and a single sphere object tested to obtain each temporal pattern.

Fig. 5.

Temporal patterns (ΔVtd) obtained for an object at various rostrocaudal positions; electrosensors were located at a distance of (A) 2 cm, (B) 5 cm or (C) 8 cm from the head. A sphere with a radius of 0.8 cm was placed at rostrocaudal distances of 2.4 cm (circle), 3.6 cm (cross), 4.8 cm (triangle) and 6.0 cm (square) from the head, but at a fixed lateral distance of 3 cm. (D) Objects (dotted circles) at varying rostrocaudal positions with three electroreceptors and a single sphere object tested to obtain each temporal pattern.

Fig. 6.

Diagram for the derivation of slope ratio. A sphere object with a radius of 1 cm was tested at the rostrocaudal position 7.5 cm from the head and at the lateral distance of 4 cm and the electroreceptor was positioned 6 cm away from the head.

Fig. 6.

Diagram for the derivation of slope ratio. A sphere object with a radius of 1 cm was tested at the rostrocaudal position 7.5 cm from the head and at the lateral distance of 4 cm and the electroreceptor was positioned 6 cm away from the head.

The slope ratio of the temporal pattern

In this study, we observed the transdermal potential changes at an electroreceptor during a tail sweep and found that this temporal pattern can be used to determine the lateral distance of an object from the fish body. However, it may not be realistic to assume that an electric fish memorizes many different types of temporal patterns. Thus, we suggest a new electrolocation measure for object distances. If the fish keeps its body straight without any bends, the tail is at a neutral position. During a tail-bending movement, the tail bends towards a target object (positive bending angle) or retracts away from the object (negative bending angle); this active movement allows the measurement of the temporal slope of potential change at a specific electrosensor. Slopes can be measured at each bending angle, positive and negative. The slope ratio is defined as the ratio of the slope for a positive-angle movement to that of the slope for a negative-angle movement. A pair of slopes can therefore be measured at positive and negative small bending angles near the neutral position of the tail during a bending movement cycle. Fig. 6 shows the schematic diagram for the slope ratio. The slope ratio can be represented as:
formula
(7)
where ΔynVtd(tn+1)–ΔVtd(tn) is the difference of the electric perturbation at the tn+1 and tn and Δt2t1=t3t2 time intervals. At t2, a weakly electric fish has its tail at the neutral position, at time t1, the tail is bent to the left, and at time t3, the tail is bent to the right.

Even for a small bending angle, a slope ratio can be determined from the change in potential. We suggest that weakly electric fish use this slope ratio to determine the lateral distance of a target object. This measure remains constant regardless of the size of the target object (Fig. 7). It also holds for other electrical contrasts of objects. If the target object is near the electroreceptor and is positioned closer to the tail, the slope ratio has a monotonically decreasing curve when the lateral distance increases (see Fig. 7C). The slope ratio curve changes depending on the rostrocaudal position of a target object, because the temporal pattern is influenced by the rostrocaudal position of a target object. Depending on the position of a target object relative to the electroreceptor, the slope ratio may not have a monotonic pattern.

Fig. 7.

Transdermal potential change (ΔVtd) and slope ratio for objects of various sizes at different lateral distances (d; with the electroreceptor located at the rostrocaudal position 6 cm from the head); the sphere objects were placed at the rostrocaudal position of 7.5 cm from the head. (A) Transdermal potential change according to lateral distance when the tail bends from –45 to 45 deg (a sphere object with a radius of 1 cm was tested). (B) Transdermal potential change according to the radius (r) of the sphere. (C) Slope ratio with varying sizes of objects (different radii were tested).

Fig. 7.

Transdermal potential change (ΔVtd) and slope ratio for objects of various sizes at different lateral distances (d; with the electroreceptor located at the rostrocaudal position 6 cm from the head); the sphere objects were placed at the rostrocaudal position of 7.5 cm from the head. (A) Transdermal potential change according to lateral distance when the tail bends from –45 to 45 deg (a sphere object with a radius of 1 cm was tested). (B) Transdermal potential change according to the radius (r) of the sphere. (C) Slope ratio with varying sizes of objects (different radii were tested).

Fig. 8.

Electric images generated by various tail-bending angles. Solid lines, lateral distance 2.4 cm; dotted lines, lateral distance 2.8 cm; open circle, –45 deg, cross, 0 deg; triangle, 45 deg. (A) Fish diagram with sensor distribution, (B) transdermal potential change (ΔVtd) along the rostrocaudal axis and (C) transdermal potential change along the dorsoventral axis (a sphere with a radius of 0.8 cm was placed at the rostrocaudal distance of 8 cm from the head and at the middle position along the dorsoventral axis).

Fig. 8.

Electric images generated by various tail-bending angles. Solid lines, lateral distance 2.4 cm; dotted lines, lateral distance 2.8 cm; open circle, –45 deg, cross, 0 deg; triangle, 45 deg. (A) Fish diagram with sensor distribution, (B) transdermal potential change (ΔVtd) along the rostrocaudal axis and (C) transdermal potential change along the dorsoventral axis (a sphere with a radius of 0.8 cm was placed at the rostrocaudal distance of 8 cm from the head and at the middle position along the dorsoventral axis).

Relative slope and tail-bending movements

The electric images obtained from a series of sensors along the rostrocaudal line showed similar bell-shaped patterns when the tail was bent at a negative or positive angle (Fig. 8). The same was true for a series of sensors along the dorsoventral axis. Surprisingly, the relative slope curve was similar regardless of the bending angle (Fig. 9). Only a slight deviation from the normal relative slope (when the tail was in the neutral position) was found. These findings indicate that the electrolocation rule based on the relative slope is not much affected by the tail-bending movement.

The relative slope is measured based on the spatial distribution of electroreceptors and provides information regarding the location of an object. The relative slope over accumulated sensor readings rather than the relative slope at one instant in time could provide more accurate information concerning object location. Another hypothesis is that, because each electrosensor can accumulate sensor readings over a cycle of tail-bending movements, integration of these measurements may allow low-pass filtering over sensor readings, which leads to accurate estimation of the distance.

We examined the localization of a fixed target object based on accumulated sensor readings over a cycle of tail bending. When the size and lateral distance of a target object changed, the integrated sensor readings of each electroreceptor also changed in intensity. The integration readings increased when the size of the target object increased or the lateral distance decreased. We calculated the relative slope measure over the accumulated sensor readings during the tail-bending movement. The readings from a series of sensors on the rostrocaudal and dorsoventral lines formed bell-shaped curves. Each axis line can provide the lateral distance based on the relative slope measure. As shown in Fig. 10, there was almost no difference between the relative slope over the accumulated sensor readings and the normal relative slope obtained when there was no tail-bending movement. The lateral distance of a target object can therefore be estimated from sensor readings integrated during a cycle of tail-bending movement. The method would be useful to obtain accurate localization in the environment with noisy sensor readings. This measure is also independent of the size and conductivity of a target object.

Fig. 9.

Relative slope with tail-bending phases, determined from a series of sensors along the (A) rostrocaudal and (B) dorsoventral axes. Open circle, –45 deg; cross, 0 deg; triangle, 45 deg. A sphere with a radius of 0.8 cm was placed at the rostrocaudal distance of 8 cm from the head.

Fig. 9.

Relative slope with tail-bending phases, determined from a series of sensors along the (A) rostrocaudal and (B) dorsoventral axes. Open circle, –45 deg; cross, 0 deg; triangle, 45 deg. A sphere with a radius of 0.8 cm was placed at the rostrocaudal distance of 8 cm from the head.

Fig. 10.

Relative slope from integrated sensor readings during tail bending, determined from a series of sensors along the (A) rostrocaudal and (B) dorsoventral axes. Open circle, integration; cross, original relative slope.

Fig. 10.

Relative slope from integrated sensor readings during tail bending, determined from a series of sensors along the (A) rostrocaudal and (B) dorsoventral axes. Open circle, integration; cross, original relative slope.

DISCUSSION

Numerous electrolocation studies have focused on the sensor readings from a collection of electroreceptors along the rostrocaudal axis of weakly electric fish. We determined the characteristics of temporal sensor readings from an electrosensor during active tail-bending movements in the weakly electric fish A. albifrons and, based on our findings, suggest new electrolocation rules that fish may use to identify object distance. In this paper, we tested three different electrolocation rules with a tail sweep: the normalized temporal pattern of sensor readings, slope ratio and relative slope with accumulated sensor readings.

The peak amplitude in the electric image becomes smaller when the object moves away from the fish body. However, the amplitude is affected not only by the distance between the fish and the object, but also by the conductivity of the target. When the size of a target object increases, the electric perturbation also increases. How does the fish determine whether there is a large-sized object far from its body or a small-sized object close to its body? It is known that the relative slope or FWHM makes it possible to estimate the lateral distance of a target object, regardless of its size and conductivity (von der Emde et al., 1998; Sicardi et al., 2000; Chen et al., 2005; von der Emde, 2006). In this paper, we suggest that tail-bending movements allow the extraction of another type of localization information. Similar to the relative slope measure, the normalized temporal patterns at an electroreceptor can provide information about object distance, irrespective of the size and conductivity of the target object. Interestingly, a single sensor with temporal pattern over a cycle of tail bending is sufficient to tell the distance information, whereas the relative slope needs a collection of sensor readings along the rostrocaudal line or dorsoventral line.

The relative slope varies depending on the position of a target object in the rostrocaudal or dorsoventral axis. Similarly, the normalized temporal pattern relies upon the position of a target object in three-dimensional space, and thus requires a map of temporal patterns for different rostrocaudal and dorsoventral positions, as does the relative slope. It is an open question whether there exists a simpler electrolocation rule that does not depend on the object position.

Biological experiments with weakly electric fish show that fish spent approximately 50% of the prey-search time in forward swimming, 25% in backward swimming and 25% in side-scanning movements, but that backward swimming and side-scanning time become more common near the edges of the water tank (Nelson and MacIver, 1999). The tail-bending movements of electric fish are often involved with prey capture (Heiligenberg, 1975; Lannoo and Lannoo, 1993; MacIver, 2001). It has been reported that the foraging patterns such as tail bending might help to enhance the electric image for electrolocation (Nanjappa et al., 2000; Bacher, 1983). It was also argued that some electric fish show tail-bending movements related not to locomotion, but to exploration of nearby novel objects (Heiligenberg, 1975; Assad et al., 1999). Our electrolocation rules based on tail bending might be connected to those biological reports.

The slope ratio was suggested in this paper as a new electrolocation rule to consider the temporal aspect of sensor readings associated with tail bending. The slope ratio is the ratio of temporal changes when weakly electric fish bend the tail from the left to mid-line and from mid-line to the right with the same time intervals. Similar to the normalized temporal pattern, the measure is also independent of the size and conductivity of a target object and can be used as a distance measure. It is still unknown whether weakly electric fish obtain the object localization or some electrolocation rule with tail bending, but the biological behaviors mentioned above suggest a possibility using the temporal aspect of sensor readings accompanied by tail-bending movements. This paper shows that electrolocation can be obtained with only one or two electrosensors of the electrosensory system and many electrosensors are not necessary for estimating the lateral distance of an object, if the tail-bending movement is applied.

Our method requires the estimation of the difference in transdermal potential caused by an object. A large part of the transdermal potential comes from the tail-bending movement itself. It is believed that weakly electric fish compensate for the electrosensory reafference in their brains (Bastian, 1995). Electrosensory reafference caused by tail bending and its cancellation process may be required to handle the modulation signal. Reafference was introduced by von Holst and Mittelstaedt as the sensory input generated by movements or actions (von Holst and Mittelstaedt, 1950). The pyramidal cells in weakly electric fish remove reafferent signals and both repetitive and predictable electrosensory input signals are cancelled (Bastian, 1986; Bastian, 1995; Bastian, 1996).

The sensory patterns associated with tail bending are considered as the reafference signals. In our experiments, the sensory pattern associated with tail bending depends on the perturbation caused by a target object. We assumed that the predictive temporal signal with the proprioceptive input is obtained without any nearby object when the tail bending movement is applied. Then the pure perturbation potential of a target object can be estimated by the reafference cancellation, i.e. subtraction of the predictive signal from the reafferent signal obtained with a target object in the tail-bending phase.

Based on our findings, we consider another aspect of tail bending movement. If sensor readings from multiple sensors on the rostrocaudal or dorsoventral line are accumulated for a cycle of tail bending, more accurate target localization could be obtained with a low-pass filter effect. Tail bending produces large modulations in the transdermal potential because of movement of the electric organ located at the tail. However, it is still an open question how electric fish use the temporal sequence of signals during a cycle of tail-bending movements. We suggest that spatiotemporal information from electroreceptors and a combination of information from both temporal and spatial features of an electric image can help the fish to identify a target object more effectively. Here, we do not argue that one measure is more effective than the others at discriminating object distances; several measures could be used by the electric fish to determine the distance of a target object.

To determine the validity of our electrolocation rule, detailed behavioral and physiological studies are required. Our modeling approach allowed us to gain insight into the relevant electrolocation behavior and nature of complex electrosensory environments of weakly electric fish.

FOOTNOTES

This study was supported by the Mid-career Researcher Program through a National Research Foundation grant funded by the Ministry of Education, Science and Technology in South Korea (no. 2010-0000460).

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