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First published online August 30, 2006
Journal of Experimental Biology 209, 3636-3651 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02403
Modeling the electric field of weakly electric fish
1 Department of Physics
2 Department of Biology, University of Ottawa, Ottawa, Ontario K1N 6N5,
Canada
* Author for correspondence (e-mail: jlewis{at}uottawa.ca)
Accepted 22 June 2006
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: electric image, electrolocation, finite-element-model, Apteronotus leptorhynchus
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). This
sense relies on an electric organ discharge (EOD) emitted from the fish's
electric organ (EO). The EOD can either be continuous and quasi-sinusoidal
(`wave-type' species), or discrete and pulse-like (`pulse-type' species). The
species studied in this paper, Apteronotus leptorhynchus, is a
wave-type species with an EOD frequency of approximately 1 kHz. The electric
field time course generated by this fish is complex with tri-phasic regions
close to the skin, but becomes approximately dipolar further away
(Knudsen, 1975;
Assad et al., 1999). Objects
whose electrical properties differ from those of the surrounding water perturb
the fish's electric field (Lissman and
Machin, 1958) and are sensed by specialized receptors located
within the fish's epidermis. Many studies have shown that these
electroreceptors respond to transdermal potential differences rather than to
transdermal current (e.g. Bennett,
1971; Migliaro et al.,
2005). The rostro-caudal, dorso-ventral profile of the transdermal
potential differences is commonly referred to as the `electric image'. Because
these images are the primary source of electrosensory input, a detailed
knowledge of the electric field and image formation is required in order to
understand the neural mechanisms of electrosensory information processing.
In recent years, different modeling techniques have been used to
characterize electric images under various natural scenarios. Two- and
three-dimensional numerical and analytical models have been developed, each
with its specific advantages and disadvantages. Analytical models (e.g.
Bacher, 1983;
Chen et al., 2005), while
being computationally fast and accurate in some cases, are limited to the
study of simplified object geometries and also do not allow for a thorough
study of the effects of the fish's body parameters. Numerical models (e.g.
Hoshimiya et al., 1980;
Assad, 1997;
Rother et al., 2003) can mimic
the fish's body in a more realistic way, yet are considered more
computationally time-demanding. The two main methods used in numerical
modeling are the finite element method (FEM) and the boundary element method
(BEM). The BEM is advantageous because only the solution at the boundaries is
necessary (e.g. at the fish-water interface), therefore reducing the number of
calculations and allowing for three-dimensional models
(Assad, 1997;
Rother et al., 2003).
Three-dimensional FEM models are currently not practical in general due to the
large number of elements required to accurately represent, for example, the
crucial thin skin layer. On the other hand, the BEM requires additional
calculations to find potentials on non-boundary regions and is harder to
implement than the FEM (e.g. boundary integrals are more difficult to evaluate
and not all linear problems can be treated)
(Yamashita, 1990;
Assad, 1997).
The goal of this paper was to study the fish in a realistic and accurate
manner, using a fast and easily-implementable model. Taking advantage of
improved computational power and software, a morphologically realistic
two-dimensional FEM model, which is both computationally fast and easy to
implement, was created. In addition to the realistic model created, two
geometrically simple models were also created in order to independently study
the effects of different fish body geometries and electrical properties for
the first time. Using the three distinct models, an analysis of the fish's
mid-planar field (view from above) and of the electric images caused by
objects of different sizes and locations is presented. This study complements
other recent studies that have begun to characterize the fish's electric field
and electric image formation using different approaches
(Caputi and Budelli, 1995;
Caputi et al., 1998;
Rother et al., 2003;
Migliaro et al., 2005;
Chen et al., 2005). In
addition, our approach allows for future modeling efforts involving realistic
electrosensory landscapes.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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;
Assad, 1997), the problem was
treated as an electrostatics boundary-value problem, governed by Poisson's
equation,
 | (1) |
where 
is the potential (in V), j is the current source
density (in A m-3) and 
is the conductivity (in S
m-1) (each defined for all points in the two-dimensional plane).
This equation was solved using finite element method (FEM) software, COMSOL
Multiphysics (formerly known as FEMLAB) on an IBM computer with a 3.2 GHz
Intel Xeon processor. The current source density units are in A m-3
because COMSOL Multiphysics treats the problem as a three-dimensional problem
in which the model has a 1 m thickness in the z-direction. In this
manner, the model is `thick' enough to neglect variations in the
z-direction. Second-order Lagrange elements (triangles) were used in
the finite element mesh. We chose a pre-defined mesh mode (`normal' mode),
which automatically selected element sizes based on the size of the objects
located within the geometry. This mode typically produced meshes with 
90
000 elements. In order to calibrate the software, we studied a simple problem
whose analytical solution is known: a line charge in the middle of a grounded
tube. We found an RMS error of 0.27%
(Babineau, 2006).
Our studies involve three different fish geometries: one morphologically
accurate model (referred to as the `fish' model) and two greatly simplified
models (referred to as the `taper' and `box' models; see later for
descriptions). Each model is enclosed in a 70x70 cm2
aquarium, which also holds grounding and reference electrodes
(Fig. 1A). The size, location
and conductivity of these components correspond to those found in Assad's
experimental setup (Assad,
1997), with whose experimental data we calibrated our fish
model.
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) by
approximating the actual shape with an appropriately sized rectangle. The
100-µm thick skin layer was selected as an acceptable upper bound, based on
previous measurements (Bennett,
1971; Zakon,
1986). The fish model therefore has three compartments: an EO, a
thin skin layer, and a body component located between the EO and the skin. The
reference electrode is placed laterally to the fish, while the grounding
electrode is placed in the corner of the tank
(Assad, 1997). Water
conductivity was set to 0.023 S m-1
(Assad, 1997), in order to
mimic experimental conditions. The potential values were obtained by moving an
array of electrodes around an immobilized fish
(Assad, 1997). These potentials
were differentiated with respect to a fixed electrode located lateral to the
fish, near the fish's zero-potential line (see
Fig. 1A).
Electric field model parameters
Four of the fish model's parameters were varied systematically in order to
minimize the error between the simulated and measured potential fields: skin,
body and EO conductivity, as well as EO current density profile. EO
conductivity was varied from 0.01 to 100 S m-1, body conductivity
from 0.01 to 10 S m-1, and uniform skin conductivity from 0.00001
to 10 S m-1. A non-uniform skin conductivity profile, as predicted
by several studies (Heiligenberg,
1975; Hoshimiya et al.,
1980; Assad, 1997),
was also tested. This profile had three parts: a head conductivity of 0.00025
S m-1 (first 60% of body length, BL), a tail conductivity
of 0.0025 S m-1 (last 10% of BL) and a mid-body
conductivity which varied linearly from 0.00025 to 0.0025 S m-1 for
the middle 30% of the fish's BL (see
Fig. 8A). These bulk
conductivity values, which depend on skin thickness, were re-calculated from
Assad (Assad, 1997) for our
chosen skin thickness of 100 µm (versus 200 µm). Assad had
chosen 200 µm as a conservative assumption, but we have reduced the skin
thickness in order to more accurately mimic values found in the literature, as
noted previously.
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For the head-positive phase of the EOD cycle studied in the present work,
the potential changes polarity along the EO from positive at the fish's head
to negative at the tail. In addition, the field is positive for most of the
body length and is much stronger (negative) in the tail (see
Fig. 3A). Thus, several
zero-mean, bimodal EO current density profiles (linear profile, two component
point-line EO, etc.) were tested in order to mimic this field. One profile,
however, allowed for a more thorough investigation due to its general nature
and amenability to mimic various EOD phases. This current density profile,
referred to as `skewed' in this paper, is composed of two Gaussian curves: a
rostral positive one and a caudal negative one (see
Fig. 2A). Such a current
density was chosen for its generality; knowledge of the exterior field cannot
uniquely determine the distribution of sinks and sources inside of the fish
(Rasnow and Bower, 1996), and
experimental data is currently unavailable.
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;
Assad, 1997). The mean and
standard deviation of the two Gaussian curves that make up the skewed profile
were also initially adjusted in order to mimic the approximately dipolar
field. Afterwards, these four current density parameters were altered
sequentially in order to minimize the model's error (see next section for
error measure). For every set of current density parameters varied, the
amplitudes of the two Gaussian curves were adjusted in order to make the mean
of the skewed profile as close to zero as possible. This was accomplished by
optimizing the rostral curve's amplitude while fixing the caudal curve's
amplitude (hence the rostral curve's amplitude was a fifth free current
density parameter). Once each skewed current density profile parameter had
been optimized, a standard `nonlinear grid search' method
(Bevington and Robinson, 2003)
was applied in order to obtain a more precise value of the optimal parameter.
In short, this method entailed finding the minimum of the error parabola
generated by the `optimal' parameter value and the pair of parameter values
(less than and greater than the optimal value) for which the model's error
increased. Once the EO current density parameters were established, the EO,
body and skin conductivities were varied within the aforementioned ranges and
were similarly optimized using the nonlinear grid search method. A second
series of optimizations were abandoned since accuracy did not improve
greatly.
Model calibration
We calibrated our fish model to experimental data
(Assad, 1997) using a weighted
RMS error. The weighting was done to put more emphasis on the near field, in
the range of active electrolocation
(MacIver et al., 2001), and
because potential falls off to zero in the far field. The overall,
n-node weighted RMS error is given by:
 | (2) |
where 
and
are the simulated and
experimental potentials at a given node. The weighting function
wi was a factor proportional to the average rostro-caudal
field strength: this factor was set to a maximal value of 1 within 2 cm of the
fish's skin and had a minimal value of 0.138 near the lateral tank wall (the
wi were set to one when the non-weighted, standard RMS
error was calculated). The nodes used to calculate this weighted error were
also re-sampled from 361 to 325 (36 nodes withdrawn) in order to remove any
bias caused by the original, uneven data sampling
(Assad, 1997). Furthermore, the
field images shown in Fig. 3
were generated using these re-sampled nodes.
Geometrically simple electric field models
In order to study independently the effect of fish geometry and electrical
conductivities of the different body compartments on the electric field and
images, two geometrically simple electric field models were created, differing
only in body contour from the fish model. The first geometrically simple
model, referred to as the `taper' model
(Fig. 1C), has a triangular
body contour that touches the EO at both ends. The value of the taper is given
by the ratio of the lateral and horizontal extents of the skin contour (on one
of the fish's sides). In order to study a given taper (found along the fish
model's outer contour), the rostral lateral segment of the taper model was
adjusted appropriately. The second geometrically simple model, referred to as
the `box' model (Fig. 1D), has
a rectangular body contour that touches the EO at both ends. The effects of
the various body conductivities and of EO-to-skin distance were studied using
this model, independent of the effects of taper. For experiments in which the
EO-to-skin distance was not varied, a width of 2.42 cm was used by default
(see Fig. 1D). Results obtained
with various widths were qualitatively the same; therefore, the exact choice
of the box's width was not important for the comparative studies conducted in
this paper. Uniform conductivity parameters were used in these models (optimal
values from the fish model; see Results), unless otherwise stated.
In addition to the optimal skewed EO current density (see Results), a sinusoidal current density, which held an integer number of wavelengths along the EO, was also tested in these models (profiles with, e.g. 1 or 5 wavelengths, are respectively referred to as `1-cycle' or `5-cycle' sinusoidal waves in this paper). Sinusoidal profiles, which are standard periodic functions, were chosen in order to better compare the results obtained with the more realistic, but skewed current density profile.
Electric field characterization tools
In order to characterize the degree of uniformity or `smoothness' of the
electric field around the fish, a quantitative measure was needed. To this
end, we used a measure of `energy' (proportional to electric energy) to
quantify the degree of smoothness of the potential along the EO as well
as along the interior and exterior skin boundaries. It is defined as:
 | (3) |
This measure is analogous to the (potential) energy held within a string
stretched along a dimension `x' that is perturbed in the `y'
dimension. When no energy is applied to the string, there is no variation in
the vertical dimension (y; analogous to ) as a function of the
horizontal dimension (x). However, if the string is suddenly moved up
and down repeatedly so that, for example, a sinusoidal pattern results,
y will change as a function of x (hence
dy/dx=0), and therefore Eqn 3 yields a non-zero value for
energy. In this sense, the `rougher' the potential variation along a given
line is, the higher its associated energy value will be.
To study the electrical filtering effects of the fish body, i.e. how the
uniformity of the electric potential profile changes from inside to outside
the fish, the energy at the EO level (EnergyEO) was compared with
the energy at the exterior (Energyext) skin level. This filtering
effect was quantified as the percentage of energy lost across the body:
 | (4) |
With higher filtering values, more energy is being lost (filtered) from the EO to the skin exterior. A 50-cycle sinusoidal current density (see previous section) was used in order to calculate filtering. The EO segment was divided into 50 separate sections and one filtering value was calculated per section (i.e. one filtering value per wavelength). This spatial `frequency', although higher than that of the skewed current density, was selected because it made it possible to visualize the body's filtering effects along the body with good spatial resolution (see e.g. Fig. 4B). Further, we found that the filtering curve's main feature, namely a head-to-tail decrease in filtering, remained the same for all frequencies; only the absolute values differed.
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 | (5) |
where Vext and VEO are the
potentials at the exterior skin level and at the EO level, respectively,
skin and body are the conductivities of
the skin and body, respectively, and where tskin and
tbody are the thicknesses of the skin and body,
respectively.
Electric image calculations
Electric images were calculated as the difference between transdermal
potentials in the presence (OB) and in the absence (NO) of an object, all
along the rostro-caudal axis, as in previous studies
(Hoshimiya et al., 1980;
Chen et al., 2005;
Migliaro et al., 2005):
 | (6) |
where `ext' and `int' denote the exterior and interior boundaries of the skin layer, respectively.
Some of the electric images produced by the skewed current density profile used in this paper are generally bimodal in nature; rostral and caudal peaks of differing polarities are present. Throughout our analysis, the potential difference between the two largest peaks will be referred to as the `peak-to-peak potential', while the horizontal distance between them will be called the `peak-to-peak distance' or `delta'. Please note also that the words `images' and `electric images' will be used interchangeably throughout the text.
All electric images in this paper were produced by a 1.1 cm-radius metal (brass; conductivity=2.13x107 S m-1) disc, unless otherwise stated.
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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Fig. 2B shows a sensitivity analysis relative to the optimal (uniform) conductivity values of the different fish compartments. The vertical axis shows error above the minimal weighted RMS error, found with optimal parameters (28.6%); the horizontal axis shows normalized parameter values (optimal conductivity values each set to one). The optimal parameter values are: EO conductivity, 0.927 S m-1; uniform skin conductivity, 0.0017 S m-1; body conductivity, 0.356 S m-1. Skin conductivity is the least sensitive of the three conductivity parameters. A non-uniform skin conductivity profile was tested as well (see Fig. 8A), giving slightly better results (0.5% less error). Contrary to previous studies, however, we did not find that non-uniform skin conductivity was necessary to reproduce the rostrally leaning zero-potential line (not shown). Nevertheless, since the error associated with this type of profile was better, it was used by default throughout this paper, except for when the effects of other parameters were studied independently and hence the optimal uniform skin conductivity was used (see later section).
Model calibration
Fig. 3A shows the measured
(top) and simulated (bottom) field potentials (simulated field shown for
optimal set of parameters, including non-uniform skin conductivity). Most
qualitative aspects of the measured field, such as the uniform potential in
the head region, the elongated dipole field shape, and the rostrally leaning
zero-potential line, are reproduced. Aspects which are not reproduced as
effectively are the high potential and rate of field decay in the tail region.
This can further be seen in Fig.
3C, which shows the absolute potential difference between
simulated and experimental data; notice that only the tail region features a
discrepancy of several mV (differences over 5 mV are mapped to dark red;
maximal difference is 22 mV). Fig.
3B shows the un-weighted RMS error. The error close to the fish is
relatively low (10% in the rostral near field and similar to that of a
recent analytical model) (Chen et al.,
2005). The error increases further away; this is especially
apparent near the zero-potential line, where the measured absolute potentials
are very small (µV magnitude; see Fig.
3C), hence creating large relative discrepancies with the
simulated field. Fig. 3D,E show
the fall-off of potential with lateral distance for two different
rostro-caudal locations, one at the head
(Fig. 3D) and one at the tail
(Fig. 3E). These figures show
that even though the error increases further away (see
Fig. 3B), the fall-off is
qualitatively similar between our model and the data. The average weighted RMS
error is 28%; the average un-weighted RMS error is 43%; the average potential
difference over the entire field is approximately 1 mV.
Electric field characterization
Filtering
Previous studies have noted that the electric field in the head region of
A. leptorhynchus is relatively uniform along the body
(Rasnow et al., 1993;
Nelson, 2005), but have not
investigated its possible origin. In this section, we quantitatively analyze
the degree of electric field uniformity and its possible origin for the first
time. In Fig. 4A, the potential
at the EO level and at the inner and outer skin boundaries of the fish model
are shown as a function of normalized EO length (zero corresponding to the
rostral end, one to the caudal end) for a 5-cycle sinusoidal current density.
The potential at the fish's exterior is much smoother than at the EO level in
the head region, and becomes more spatially heterogeneous, or rougher, towards
the tail. This was also apparent for sinusoidal current densities of higher
and lower frequencies as well as for other current density profiles (data not
shown). The EO potential also increases caudally, due to the `narrowing' taper
of the fish (this is not due to the EO current density, since in this example
its amplitude was the same all along the EO).
Fig. 4B further characterizes the smoothness of the exterior potential. The green trace shows that the normalized energy at the exterior skin level approaches a value of zero towards the head, implying that the potential is almost perfectly `flat' in this region; the energy is minimal in the head region situated to the left of the EO (left of red line). The blue trace shows that filtering is maximal rostrally and decreases caudally, with a mean filtering value of 91.5% (shown for a 50-cycle sinusoidal current density). By contrast, the mean filtering for a 1-cycle sinusoidal current density is 47.3%. Therefore, the mean filtering value obtained with the skewed current density will be in between these two values (recall that a higher frequency was used in order to better visualize filtering along the body axis). While the body and skin tend to filter out more of the potential at higher frequencies, the qualitative shape of the filtering curve obtained with lower frequencies remains the same as in Fig. 4B (not shown).
Several tests with the two geometrically simple models were carried out in order to understand how different parameters affect the filtering due to the fish's body. By comparing the exterior (red trace) and interior (green trace) potentials in Fig. 4A, it is clear that the skin acts as a filter. However, the skin alone cannot account for the head-to-tail drop-off in filtering of nearly 35% in the fish model (see Fig. 4C, green trace). Nor can this effect be explained by the different head and tail conductivities found in the non-uniform skin conductivity profile. Tests conducted with body conductivity only predicted differences of a few percent in filtering when varied systematically. Fig. 4C shows the filtering effect for two versions of the taper model. The shape of the fish model's filtering curve (green trace) resembles the ones found with the taper models, either rostral to (taper of 0.05, blue trace) or caudal to (taper of 0.0178, black trace) the point of taper change (red broken line) in the fish model (see Fig. 1B for fish model geometry). This can also be understood as differences in effective distance, as a bigger taper value implies that the EO is effectively farther from the exterior skin. The slight caudal discrepancy (green versus black traces) is likely due to edge effects: the fish model has body tissue located between the end of the EO and the end of the tail that is not present in the taper model.
While a smooth external field may be of functional importance for the fish,
as noted previously, it remains that the relevant stimuli for the skin
electroreceptors is transdermal potential
(Migliaro et al., 2005). It is
thus reasonable to assume that a uniform transdermal potential would be of
greater significance to the fish. We calculated the energy of the transdermal
potential and found that it was equally `smoothest' in the head region (data
not shown). Its shape was similar to the external potential energy curve shown
in Fig. 4B (green trace),
although its peak was shifted rostrally. This can be understood by comparing
the external and internal potentials in
Fig. 4A. At the tail end, both
the internal and external potentials are `rough', but the difference between
them is relatively uniform.
Voltage divider
Previous studies have likened the fish to a voltage divider
(Rasnow, 1996;
McAnelly et al., 2003), and
Fig. 4D characterizes the
accuracy of such an assumption for the first time. Transdermal potentials
calculated for the taper model (simulated, blue trace) and for the ideal
voltage divider (theoretical, green trace), as well as the difference between
these two quantities (red trace), are shown as a function of normalized EO
length for a 5-cycle sinusoidal current density (taper model; see Materials
and methods for details). A taper of 0.05 was chosen, as this is the taper
that makes up the longest segment, approximately 25%, of the fish model's
body. Both the simulated and theoretical curves have the same quasi-sinusoidal
shape, although their amplitudes differ. The match between these two curves is
especially good towards the tail. On the whole, this analysis validates the
assumption that the fish body functions as an ideal voltage divider.
Electric image characterization
Effect of object location on electric images
As in several previous studies
(Hoshimiya et al., 1980;
Rasnow, 1996;
Rother et al., 2003;
Chen et al., 2005;
Migliaro et al., 2005), we
have characterized the electric images produced by simple (circular
cross-sections) conductive objects. The effects of different lateral and
rostro-caudal object locations on electric image shape are displayed in
Fig. 5. The electric image gets
smaller (in amplitude) and wider, as the object is moved away laterally from
the fish (Fig. 5A). The image
amplitude increases (in absolute terms) and then decreases, as the object
moves caudally (Fig. 5B). Most
of the electric images shown in Fig.
5B are bimodal in nature: they have a negative rostral peak and a
positive caudal one. These images become increasingly more bimodal towards the
tail. Also, the distance between successive negative peaks diminishes
caudally, even though the different traces are for regularly spaced object
locations (as shown in the inset). This signifies that the offset between the
object's location and the location of the electric image's dominant peak on
the skin varies with rostro-caudal location.
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;
Chen et al., 2005) and
numerically using different modeling techniques
(Hoshimiya et al., 1980;
Caputi et al., 1998;
Migliaro et al., 2005).
However, some qualitative differences exist. In order to investigate these
differences, normalized electric images
(Fig. 6B), which were produced
by a rostro-caudally centered metal sphere, were calculated for three separate
current densities (Fig. 6A): a
1-cycle sinusoidal current density, the optimal skewed current density, as
well as a unimodal, pulse-like current density similar to that found in
pulse-type electric fish (e.g. Migliaro et
al., 2005). The main differences in image shape are a variation in
the caudal trough size and discrepancies in the location of the rostral and
caudal peaks.
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The offset between the object and image locations, as observed in Fig. 5B, was characterized in Fig. 7. Tests were carried out using the box model with 1-cycle sinusoidal (Fig. 7A,B) and skewed (Fig. 7C,D) current densities (for reference, see also Fig. 6A). These figures show that the locations of the electric image's characteristic points (xR, x0 and xC) change as a function of rostro-caudal object location. The solid black line (identity line) indicates where the location of the image at the skin corresponds exactly to the location of the object. For the sinusoidal current density (Fig. 7A), xR (blue trace) is closest to the object's location for the first rostral third of the fish body, x0 (red trace) is closest to it for the middle third, and xC (green trace) is closest to it for the last, caudal third of the body (the black broken lines delimit these zones). This indicates that either the rostral or caudal peak of the bimodal electric image is located closest to the object's position, depending on whether the object is located close to the head or near the tail. A similar phenomenon occurs for the skewed current density (Fig. 7C), although in this case the bimodal electric image's rostral peak is closest to the object's actual location over a wider range of the fish's body. It should be noted that for a specific rostro-caudal object location, the electric image's amplitude, at this point, is zero (intersection of solid black and red curves).
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The absolute amplitudes of the rostral and caudal peaks of the bimodal electric image as a function of rostro-caudal object location are shown for sinusoidal (Fig. 7B) and skewed (Fig. 7D) current densities. Here it can be seen that image amplitude is roughly proportional to the EO potential; hence, the locations of the two amplitude peaks are approximately the same as the locations of the two peaks in a given current density profile (for both profiles). It should be noted that the potentials would have been relatively smaller in the tail region, had the non-uniform skin conductivity profile been used instead of the uniform one.
The effect of skin conductivity profile on electric image shape is studied with the box model in Fig. 8. The image due to the `real', non-uniform skin conductivity profile (green trace; Fig. 8A) either resembles the one obtained with the uniform head conductivity (0.00025 S m-1; red trace) or the one obtained with the uniform tail conductivity (0.0025 S m-1; blue trace), depending on rostro-caudal location. The electric image in the middle region shows a transition from the image obtained with one of the uniform conductivities to the other. From this figure, it is also apparent that, as expected, the electric image's amplitude decreases as skin conductivity increases.
In Fig. 9 we show the
bimodal electric images produced by different-sized objects as a function of
lateral distance (of the object centers), using the box model (with skewed
current density and uniform skin conductivity).
Fig. 9A,C show un-normalized
(actual amplitudes) and normalized (with respect to the caudal peak's
amplitude) bimodal electric images, respectively, produced by three disc-like
objects of different diameters. The amplitude (or peak-to-peak potential) of
the electric image produced by the 2 cm object
(Fig. 9A, red trace) is the
largest since all the objects were centered at the same lateral distance of 4
cm; therefore, this object's edge was closest to the skin layer and thus
affected the image more. The normalized bimodal electric images, however, are
all very similar (Fig. 9C).
Fig. 9B shows how the
peak-to-peak amplitudes of the electric images change as a function of lateral
object distance for the three objects: the curves are separated one from
another by approximately an order of magnitude (e.g. note peak-to-peak
differences at black dotted line). This is in agreement with previous studies
that have reported a similar correspondence between un-normalized image
amplitude and lateral object distance
(Heiligenberg, 1975;
Bastian, 1981;
Rasnow, 1996). Thus, the fish
cannot unambiguously determine lateral object distance using this measure.
This is because small objects placed near the fish's body may create electric
images that have the same amplitude as large objects placed farther away from
the fish. On the other hand, our analysis shows that the difference between
the rostral and caudal peak locations (defined here as `delta') varies
consistently with lateral object distance for all three objects
(Fig. 9D) and could therefore
be used by the fish to unambiguously measure lateral object distance. The
black broken line in Fig. 9B,D
shows the lateral distance (4 cm) for which the electric images in
Fig. 9A,C were calculated. It
should be noted that the results shown for the box model also hold for the
fish model (not shown). However, the caudal peaks in the fish model are
smaller because of the increased conductivity in the tail section (due to the
nature of the non-uniform skin conductivity profile).
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Fig. 10 shows the same set of panels as in Fig. 9, except that lateral distance is now calculated as a function of the object's edge, instead of the object's center. The 0.5 cm radius (blue) and 1.1 cm radius (green) objects were moved closer to the skin layer in Fig. 10A,C so that their edges were at 2 cm from the midline (same distance as the 2 cm-radius object). The amplitudes have therefore increased for these two objects since they are closer to the skin surface (Fig. 10C). There are, however, bigger differences between the normalized electric images of the differently sized objects, compared to when lateral distance is measured from the object's center. This is reflected in Fig. 10D, where one can see that the delta values no longer vary consistently with lateral distance for the three distinct objects. Indeed, this measure seems no more accurate in determining lateral distance of an object's edge than the peak-to-peak potentials shown in Fig. 10B. These results suggest specific predictions for the localization of objects of different sizes (see Discussion).
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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;
Caputi et al., 1998;
Migliaro et al., 2005), there
are some potentially important differences.
Electric field modeling
Our two-dimensional, realistic electric field model (referred to as the
`fish' model; see Fig. 1B)
reproduces many spatial aspects of the fish's self-generated EOD potential
(Fig. 3). For instance, the
model duplicated the electric field's dipolar nature and rostrally leaning
zero-potential line. The model was also very accurate near the fish. In
particular, the rostral part of the field, where most electroreceptors are
located and hence where active electrolocation is thought to be mediated
(Knudsen, 1975;
Carr et al., 1982), has a very
low error based on comparisons with experimental data (10%, which is
comparable to that found in a recent study)
(Chen et al., 2005). The high
potentials and field decay in the tail region, however, were not reproduced
with such a low error. This is possibly due to the two-dimensional nature of
the model in which the third spatial dimension does not contribute to the
mid-planar field (see also Assad,
1997). Also, the error between our model and the experimental data
does increase further away lateral to the fish. This is because potential
values become very small in the far field and hence percent differences can
become very large, even though the absolute errors are not large (see e.g.
Fig. 3C).
Fig. 3D,E show, however, that
the field's fall-off is qualitatively similar between the model and the data.
Therefore, we consider that results that were shown for lateral distances at
which the error is larger are still valid, at least qualitatively.
The effects of EO, body and skin conductivity on external potential were
studied directly and independently using the two geometrically simple models
(`taper' and `box' models; see Fig.
1C,D). The taper model also allows for an independent study of the
fish's geometry. Each model was simple to implement, using the finite element
software COMSOL Multiphysics. The models are also computationally fast,
solving a mesh containing 89 817 nodes in 7.6 s (on an IBM computer with a 3.2
GHz Intel Xeon processor). With its realistic electric fish geometry and
parameters, the fish model presented in this paper is also an improvement over
previous finite element models
(Heiligenberg, 1975;
Hoshimiya et al., 1980). In
particular, compared with the FEM model done
(Hoshimiya et al., 1980), our
model is more realistic morphologically (rather than ellipse-like) and has a
skin thickness approximately 30x smaller (and in the range of the
measured thickness). Our EO current density was also distributed
rostro-caudally along an EO, rather than a two-point dipole used in the
previous model. These improvements were enabled in part by our access to
increased computational power. The model's main shortcomings are that it is
two-dimensional and that it does not reproduce the potential in the far-field
and in the tail region. Also, the model currently only simulates the EOD
potential for a single phase of the EOD cycle [as in other recent models (see,
for example, Chen et al.,
2005)]. However, the model could easily be extended to other
phases by modifying the shape of the EO current density appropriately.
The parameters that provide the best fit to Assad's experimental data are
in the range of those measured or predicted in the literature. An average body
conductivity of 1 S m-1 was reported
(Scheich and Bullock, 1974),
whereas our optimal value is 0.356 S m-1. The optimal EO
conductivity, 0.927 S m-1, is also very similar. Skin conductivity
measurements have shown rostro-caudal variations along the fish's body
(Scheich and Bullock, 1974;
Assad, 1997), and other
numerical modeling studies have used a three-component skin conductivity
profile in order to model this inhomogeneity
(Heiligenberg, 1975;
Hoshimiya et al., 1980;
Assad, 1997). We have also
concluded that this profile (Fig.
8A) results in more accuracy, in comparison with uniform skin
conductivity profiles. However, we found that in contrast to these previous
studies, such a profile was not necessary to reproduce the rostrally leaning
zero-potential line (Fig. 3).
Thus, for some modeling studies, a uniform conductivity could be used to
simplify the models without loss of generality. Here, a uniform skin
conductivity of 0.0017 S m-1 was used for studying the effects of
other parameters independently (such as body geometry, for instance). The
skewed current density profile (Fig.
2A) can be easily adapted to model the time-varying EOD: one only
needs to change the mean and standard deviation of the two Gaussian functions
in order to mimic different phases. It would be interesting to compare our
`simplistic' skewed current density profile with the experimentally measured
one. However, these experiments have only been carried out, thus far, in
pulse-type electric fish (Caputi,
1999).
Electric field characterization
It has been suggested that the uniform field near the head of weakly
electric fish could improve their ability to resolve objects, because
electroreceptors respond to current flowing perpendicular to the skin surface
(Knudsen, 1975;
Rasnow and Bower, 1996). We
carried out several tests in order to better understand which attributes of
the fish body are responsible for this uniform field
(Fig. 4). We have shown that
this spatial filtering is dictated mainly by the tapered shape of the body,
and not only, for instance, because of a rostral current density of low
amplitude [experiments in pulse-type fish have shown that the electromotive
force varies along the fish's body length (see, for example,
Caputi et al., 1989)]. The
field is less smooth caudally because the skin is effectively closer to the
EO, therefore reducing the amount of body and skin that can filter out the
electric organ's potential. The smooth field in the head region is also aided
by the fact that the head region of the fish lacks an EO
(Caputi et al., 2002). In
fact, we have seen that this region had the lowest exterior energy value (see
Fig. 4A, green curve; energy
rostral of vertical red line is minimal). The importance of skin and body
tissues for filtering purposes had been suggested in the past; for example,
Rasnow et al. hypothesized that this was the reason why the fish's EOD
propagates less in the trunk (Rasnow et
al., 1993). Our geometrically simple models have allowed us to
study conductivity and body shape independently and to verify, for the first
time, such hypotheses.
We also found that the relevant stimulus for skin electroreceptors, i.e. transdermal potential, was most uniform in the head region. It is possible that electric fish geometry has evolved, in part, to its current shape to increase the field uniformity near the fish's head. The fish's tapered body shape also contributes to the large potentials in the tail region, again due to the effective closeness of the EO and skin layer.
We have also shown that the weakly electric fish acts approximately as an
ideal voltage divider (Fig.
4D). This is important since certain analytical
(Chen et al., 2005) and
semianalytical models (Rasnow,
1996; Nelson et al.,
2002), which treat electric images as perturbations of the
simulated or measured field at the fish's exterior, are based on such an
assumption. Our results indicate that this assumption is good, at least to a
first order approximation (especially in the tail region).
Electric image characterization
By positioning an object at various lateral and rostro-caudal locations
(Fig. 5), we were able to
simulate electric images that resemble some of those found experimentally
(von der Emde et al., 1998;
Chen et al., 2005) and
obtained with other numerical models
(Hoshimiya et al., 1980;
Migliaro et al., 2005). In
particular, we found that electric images diminished in amplitude and widened
with increasing lateral object distance
(Fig. 5A), as previously
reported (see Rasnow, 1996;
Caputi et al., 1998). We also
found that the electric image's peak-to-peak amplitude decreases with
increasing skin conductivity (Fig.
8B), which is in agreement with Migliaro et al.
(Migliaro et al., 2005). The
changes in image shape that occur as the object is `moved' rostro-caudally
(Fig. 5B) can be explained by
using the insights obtained with the geometrically simple models, some of
which confirm previous findings. The image amplitude increases at first, due
to the EO potential which increases caudally, but then decreases, even though
the potential is still increasing towards its (absolute) maximal value. This
happens because the conductivity of the skin is increasing, hence reducing the
potential drop across the skin and, consequently, the image's amplitude. The
electric image also widens because the object was moved on a straight line,
and therefore the object-to-skin distance increases as the object moves
towards the tail. The differences in electric image shape, such as trough
sizes, produced by wave-type (skewed current density) and pulse-type (impulse
current density) fish can also be explained, at least in part, by the distinct
shapes of the EO current densities (Fig.
6).
Electric image analysis has previously been used to predict a set of
electrolocation rules (Rasnow,
1996; Caputi et al.,
1998; von der Emde et al.,
1998). It was postulated that the fish could, in principle,
compute an object's rostro-caudal location by simply using the location of the
electric image's peak (Rasnow,
1996; Caputi et al.,
1998) and an object's lateral distance, regardless of its size, by
computing the ratio of maximal electric image slope and maximal electric image
amplitude (von der Embe et al.,
1998). These rules, however, were proposed in the context of
either Gaussian-like (Rasnow,
1996) or `Mexican-hat'-like
(Caputi et al., 1998;
von der Emde et al., 1998)
images. While some of the images found here have a `Mexican-hat'-like shape,
with a dominant peak surrounded by troughs of opposite polarity on either
side, we have also found that in some cases only one of these troughs was
significant, resulting in two dominant peaks
(Fig. 5B). We have
characterized these bimodal electric images in order to see how
electrolocation rules might differ for such image shapes.
In Fig. 7 we studied the
bimodal electric images for different rostro-caudal object locations and found
that there was an offset between object and electric image locations. Offsets
were found in Rasnow's study (Rasnow,
1996), but he concluded that these were minimal with respect to
the width of the electric image. We have shown here that different components
of the bimodal electric image, not a single one, are closest to the object's
location, as it moves rostro-caudally. When the object is near the head of the
fish, the rostral peak of the electric image is closest to the object's
location, while the caudal peak is closest when the object is near the
tail.
One way that the fish could unambiguously determine which peak is closest
to the object's location is by simply comparing the absolute amplitudes of the
two peaks: the one with the biggest amplitude indicating that it is closest to
the object. The offset between the image peak location and object location
could be of functional importance, possibly serving as a prediction of future
object location (for an object moving rostro-caudally beside the fish). Nelson
and MacIver postulated this was happening at the electrosensory afferent
level, as the peak in afferent activity was located ahead of the transdermal
potential peak (Nelson and MacIver,
1999). The offset in the bimodal electric image could provide
another cue for future object location prediction, but would require the use
of different peaks, depending on the object's direction of travel. Bacher has
also suggested an algorithm capable of extracting object location using
multimodal image shapes (Bacher,
1983); however, his method required the fish to know the object's
shape beforehand (and was only valid for spheres). The cues suggested here are
independent of object shape: the rostro-caudal object location is given by the
location of the larger peak of the bimodal electric image (obtained by
comparing the amplitudes of the two peaks), with the offset between peak and
object location possibly serving as a future object location predictor. In
fact, we have also performed the analysis shown in
Fig. 9 using cube-like objects
and have obtained similar results, i.e. that the fish could use delta as an
unambiguous measure of lateral distance (not shown). One example clearly
illustrates how bimodal electric images differ from unimodal ones; for certain
object locations, the image has zero amplitude at the corresponding lateral
skin position.
In Fig. 9 we studied the
bimodal electric images produced by different-sized objects as a function of
lateral distance. We have shown that the fish could, in principle, use the
distance between the rostral and caudal peaks of the electric image (delta) in
order to unambiguously determine lateral object distance, regardless of object
size. The lateral distance measure presented here is different and
computationally simpler than the one advanced earlier
(von der Emde et al., 1998):
only the locations of the two peaks need to be determined by the fish
(normalization is not required). The electric fish could subsequently
determine the object's size using the bimodal electric image's peak-to-peak
amplitude (with the reasonable assumption that the object's conductivity does
not change during the electrolocation task). Furthermore, we have seen that
measuring lateral object distance from the center of objects
(Fig. 9) seemed much more
fruitful than measuring it with respect to the object edges
(Fig. 10). This corroborates a
previous study, which noted that measuring lateral object distance with
respect to object center had simpler functional forms
(Rasnow, 1996). It seems
likely, however, that determining the distance of an object's edge would be
more relevant for the fish. The fish could then, in principle, extract the
edge's distance by using the delta-lateral distance curve associated with the
object's center and then subtract its radius (obtained via the
peak-to-peak potential value).
Conclusions
In order to fully understand electrosensory processing, one needs to
understand the mechanisms that weakly electric fish use in order to generate
electric signals, and how information is extracted from the electric images
produced by surrounding objects. Many previous studies have focused on the
posteffector mechanisms (Caputi and
Budelli, 1995; Rother et al.,
2003; Migliaro et al.,
2005) and electrolocation principles
(Rasnow, 1996;
Caputi et al., 1998;
von der Emde, 1999;
Lewis and Maler, 2002;
Chen et al., 2005) employed by
weakly electric fish. We have gained further insight into the electric sense
using a realistic model of the electric fish A. leptorhynchus, as
well as two geometrically simple models that have allowed us to study the
fish's conductivity and geometry in an independent way. We have also
characterized bimodal electric images and seen that electrolocation rules
obtained with such images differ from the ones found with Gaussian-like and
`Mexican-hat'-like images. Electrolocation rules found using bimodal electric
images suggest that weakly electric fish could determine an object's
rostro-caudal position by using the location of the peak whose amplitude is
greatest and determine its lateral distance by using the distance between the
rostral and caudal peaks. More detailed behavioral studies are required to
determine which rules are actually used for electrolocation. Finally, our
modeling approach also sets the stage for future studies on two poorly
understood aspects of the electrolocation behavior: the influence of the
time-varying EOD and the nature of complex electrosensory environments.
, V
|
Acknowledgments |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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