Directional Hearing by the Mauthner 
System 
Audrey L. Gusik 
Department of Psychology 
University of Colorado 
Boulder, Co. 80309 
Robert C. Eaton 
E. P.O. Biology 
University of Colorado 
Boulder, Co. 80309 
Abstract 
We provide a computational description of the function of the Mau- 
thner system. This is the bralnstem circuit which initiates fast- 
start escapes in telcost fish in response to sounds. Our simula- 
tions, using backpropagation in a realistically constrained feedfor- 
ward network, have generated hypotheses which are directly inter- 
pretable in terms of the activity of the auditory nerve fibers, the 
principle cells of the system and their associated inhibitory neu- 
rons. 
1 INTRODUCTION 
1.1 THE MAUTHNER SYSTEM 
Much is known about the brainstem system that controls fast-start escapes in teleost 
fish. The most prominent feature of this network is the pair of large Mauthner 
cells whose axons cross the midline and descend down the spinal cord to synapse 
on primary motoneurons. The Mauthner system also includes inhibitory neurons, 
the PHP cells, which have a unique and intense field effect inhibition at the spike- 
initiating zone of the Mauthner cells (Faber and Korn, 1978). The Mauthner system 
is part of the full brainstem escape network which also includes two pairs of cells 
homologous to the Mauthner cell and other populations of reticulospinal neurons. 
With this network fish initiate escapes only from appropriate stimuli, turn away 
from the offending stimulus, and do so very rapidly with a latency around 15 msec 
in goldfish. The Mauthner cells play an important role in these functions. Only one 
574 
Directional Hearing by the Mauthner System 575 
fires thus controlling the direction of the initial turn, and it fires very quickly (4-5 
msec). They also have high thresholds due to instrinsic membrane properties and 
the inhibitory influence of the PHP cells. (For reviews, see Eaton, et al, 1991 and 
Faber and Korn, 1978.) 
Acoustic stimuli are thought to be sufficient to trigger the response (Blaxter, 1981), 
both Manthner cells and PHP cells receive innerration from primary auditory fibers 
(Faber and Korn, 1978). In addition, the Manthner cells have been shown physio- 
logically to be very sensitive to acoustic pressure (Canfield and Eaton, 1990). 
1.2 LOCALIZIN( SOUNDS UNDERWATER 
In contrast to terrestrial vertebrates, there are several reasons for supposing that 
fish do not use time of arrival or intensity differences between the two ears to localize 
sounds: underwater sound travels over four times as fast as in air; the fish body 
provides no acoustic shadow; and fish use a single transducer to sense pressure which 
is conveyed equally to the two ears. Sound pressure is transduced into vibrations 
by the swim bladder which, in goldfish, is mechanically linked to the inner ear. 
Fish are sensitive to an additional component of the acoustic wave, the particle 
motion. Any particle of the medium taking part in the propagation of a longitudehal 
wave will oscillate about an equilibrium point along the axis of propagation. Fish 
have roughly the same density as water, and will experience these oscillations. The 
motion is detected by the bending of sensory hairs on auditory receptor cells by 
the otolith, an inertial mass suspended above the hair cells. This component of the 
sound will provide the axis of propagation, but there is a 180 degree ambiguity. 
Both pressure and particle motion are sensed by hair cells of the inner ear. In 
goldfish these signals may be nearly segregated. The linkage with the swim bladder 
impinges primarily on a honey chamber containing two of the endorgans of the inner 
ear: the saccule and the lagena. The utricle is a third endorgan also thought to 
mediate some acoustic function, without such direct input from the swimbladder. 
Using both of these components fish can localize sounds. According to the phase 
model (Schuijf, 1981) fish analyze the phase difference between the pressure com- 
ponent of the sound and the particle displacement component to calculate distance 
and direction. When pressure is increasing, particles will be pushed in the direc- 
tion of sound propagation, and when pressure is decreasing particles will be pulled 
back. There will be a phase lag between pressure and particle motion which varies 
with frequency and distance from the sound source. This, and the separation of the 
pressure from the displacement signals in the ear of some species pose the greatest 
problems for theories of sound localization in fish. 
The acoustically triggered escape in goldfish is a uniquely tractable problem in 
underwater sound localization. First, there is the fairly good segregation of pressure 
from particle motion at the sensory level. Second, the escape is very rapid. The 
decision to turn left or right is equivalent to the firing of one or the other Manthner 
cell, and this happens within about 4 msec. With transmission delay, this decision 
relies only on the initial 2 msec or so of the stimulus. For most salient frequencies, 
the phase lag will not introduce uncertainty: both the first and second derivatives 
of particle position and acoustic pressure will be either positive or negative. 
576 Guzik and Eaton 
1.3 THE XNOR MODEL 
A 
Active 
pressure 
inpui 
Active 
displacemeni 
input 
Leh 
Mauthner 
outpu 
Right 
Mauthner 
output 
B 
P+ 
P+ 
p- 
p- 
DL 
DR 
DL 
DR 
On 
off 
off 
On 
Off 
On 
On 
Off 
Left sound 
SOLlICe 
DR 
P+ 
p- 
DL 
Mauthner  Mauthncr 
cell X cell 
 0 = excitntory 
No response 
DL 
P+ 
P- 
DR 
Figure 1 Truth table and minimal network for the XNOR model. 
Given the above simplification of the problem, we can see that each Mauthner 
cell must perform a logical operation (Guzik and Eaton, 1993; Eaton et al, 1994). 
The left Mauthner cell should fire when sounds are located on the left, and this 
occurs when either pressure is increasing and particle motion is from the left or 
when pressure is decreasing and particle motion is from the right. We can call 
displacement from the left positive for the left Mauthner cell, and immediately we 
Directional Hearing by the Mauthner System 577 
have the logical operator exclusive-nor (or XNOR). The right Mauthner cell must 
solve the same problem with a redefinition of right displacement as positive. The 
conditions for this logic gate are shown in figure 1A for both Mauthner cells. This 
analysis simplifies our task of understanding the computational role of individual 
elements in the system. For example, a minimal network could appear as in figure 
lB. 
In this model PHP units perform a logical sub-task of the XNOR as AND gates. 
This model requires at least two functional classes of PHP units on each side of 
the brain. These PHP units will be activated for the combinations of pressure 
and displacement that indicate a sound coming from the wrong direction for the 
Mauthner cell on that side. Both Mauthner cells are activated by sufficient changes 
in pressure in either direction, high or low, and will be gated by the PHP cells. This 
minimal model emerged from explorations of the system using the connectionist 
paradigm, and inspired us to extend our efforts to a more realistic context. 
2 THE NETWORK 
We used a connectionist model to explore candidate solutions to the left/right dis- 
crimination problem that include the populations of neurons known to exist and 
include a distributed input resembling the sort available from the hair cells of the 
inner ear. We were interested in generating a number of alternative solutions to be 
better prepared to interpret physiological recordings from live goldfish, and to look 
for variations of, or alternatives to, the XNOR model. 
2.1 THE ARCHITECTURE 
As shown in figure 2, there are four layers in the connectionist model. The input 
layer consists of four pools of hair cell units. These represent the sensory neurons 
of the inner ear. There are two pools on each side: the saccule and the utricle. 
Treating only the hori.ontal plane, we have ignored the lagena in this model. The 
saccule is the organ of pressure sensation and the utricle is treated as the organ 
of particle motion. Each pool contains 16 hair cell units maximally responsive for 
displacements of their sensory hairs in one particular direction. They are activated 
as the cosine of the difference between their preferred direction and the stimulus 
deflection. All other units use sigmoidal activation functions. 
The next layer consists of units representing the auditory fibers of the VIIIth nerve. 
Each pool receives inputs from only one pool of hair cell units, as nerve fibers have 
not been seen to innervate more than one endorgan. There are 10 units per fiber 
pool. 
The fiber units provide input to both the inhibitory PHP units, and to the Mauthner 
units. There are four pools of PHP units, two on each side of the fish. One set 
on each side represents the collateral PHP cells, and the other set represents the 
commissural PHP cells (Faber and Korn, 1978). Both types receive inputs from the 
auditory fibers. The collaterals project only to the Mauthner cell on the same side. 
The commissurals project to both Mauthner cells. There are five units per PHP 
pool. 
578 Guzik and Eaton 
The Mauthner cell units receive inputs from saccular and utricular fibers on their 
same side only, as well as inputs from a single collateral PHP population and both 
commissural PHP populations. 
Left Saccule Left Utricle 
Hair Cells I ] I I 
! l 
Auditory Nerve i 
Fiber Pools 
Right Utricle Right Saccule 
PHPs c-- 
Left Mauthner Right Mauthner 
Figure 2 The architecture. 
Weights from the PHP units are all constrained to be negative, while all others are 
constrained to be positive. The weights are implemented using the function below, 
positive or negative depending on the polarity of the weight. 
f(w) = 1/2 (w + In cosh(w) + In 2) 
The function asymptotes to zero for negative values, and to the identity function for 
values above 2. This function vastly improved learning compared with the simpler, 
but highly nonlinear exponential function used in earlier versions of the model. 
2.2 TRAINING 
We used a total of 240 trnlnlng examples. We began with a set of 24 directions for 
particle motion, evenly distributed around 360 degrees. These each appeared twice, 
once with increasing pressure and once with decreasing pressure, making a base set 
of 48 examples. Pressure was introduced as a deflection across saccular hair ceils of 
either 0 degrees for low pressure, or 180 degrees for high pressure. These should be 
thought of as reflecting the expansion or compression of the swim bladder. Targets 
for the Mauthner cells were either 0 or I depending upon the conditions as described 
in the XNOR model, in figure 1A. 
Directional Hearing by the Mauthner System 579 
Next by randomly perturbing the activations of the hair cells for these 48 patterns, 
we generated 144 noisy examples. These were randomly increased or decreased up 
to 10%. An additional 48 examples were generated by dividing the hair cell activity 
by two to represent sub-threshold stimuli. These last 48 targets were set to zero. 
The network was trained in batch mode with backpropagation to minimize a cross- 
entropy error measure, using conjugate gradient search. Unassisted backpropaga- 
tion was unsuccessful at finding solutions. 
For the eight solutions discussed here, two parameters were varied at the inputs. In 
some solutions the utricle was stimulated with a vector sum of the displacement and 
the pressure components, or a "mixed" input. In some solutions the hair cells in the 
utricle are not distributed uniformly, but in a gaussian manner with the mean tuning 
of 45 degrees to the right or left, in the two ears respectively. This approximates 
the actual distribution of hair cells in the goldfish utricle (Platt, 1977). 
RESULTS 
Analyzing the activation of the hidden units as a function of input pattern we found 
activity consistent with known physiology, nothing inconsistent with our knowledge 
of the system, and some predictions to be evaluated during intracellular recordings 
from PHP cells and auditory afferents. 
First, many PHP cells were found exhibiting a logical function, which is consistent 
with our minimal model described above. These tended to project only to one 
Mauthner cell unit, which suggests that primarily the collateral PHP cells will 
demonstrate logical properties. Most logical PHP units were NAND gates with 
very large weights to one Mauthner cell. An example is a unit which is on for all 
stimuli except those having displacements anywhere on the left when pressure is 
high. 
Second, saccular fibers tended to be either sensitive to high or low pressure, consis- 
tent with recordings of Furukawa and Ishii (1967). In addition there were a class 
which looked like threshold fibers, highly active for all supra-threshold stimuli, and 
inactive for all sub-threshold stimuli. There were some fibers with no obvious se- 
lectivity, as well. 
Third, utricular fibers often demonstrate sensitivity for displacements exclusively 
from one side of the fish, consistent with our minimal model. Right and left utricular 
fibers have not yet been demonstrated in the real system. 
Utricular fibers also demonstrated more coarsely tuned, less interpretable receptive 
fields. All solutions that included a mixed input to the utricle, for example, pro- 
duced fibers that seemed to be "not 180 degree",or "not 0 degree", countering the 
pressure vectors. We interpret these fibers as doing clean-up given the absence of 
negative weights at that layer. 
Fourth, sub-threshold behavior of units is not always consistent with their supra- 
threshold behavior. At sub-threshold levels of stimulation the activity of units may 
not reflect their computational role in the behavior. Thus, intracellular recordings 
should explore stimulus ranges known to elicit the behavior. 
580 Guzik and Eaton 
Fifth, Mauthner units usually receive very strong inputs from pressure fibers. This 
is consistent with physiological recordings which suggest that the Mauthner cells 
in goldfish are more sensitive to sound pressure than displacement (Canfield and 
Eaton, 1990). 
Sixth, Mauthner cells always acquired relatively equal high negative biases. This is 
consistent with the known low input resistance of the real Mauthner cells, giving 
them a high threshold (Faber and Korn, 1978). 
Seventh, PHP cells that maintain substantial bilateral connections tend to be ton- 
ically active. These contribute additional negative bias to the Manthner cells. The 
relative sies of the connections are often assymetric. This suggests that the commis- 
sural PHP cells serve primarily to regulate Mauthner threshold, ensure behavioral 
response only to intense stimuli, consistent with Faber and Korn (1978). These cells 
could only contribute to a partial solution of the XNOR problem. 
Eighth, all solutions consistently used logic gate PHP units for only 50% to 75% 
of the training examples. Probably distributed solutions relying on the direct con- 
nections of auditory nerve fibers to Mauthner cells were more easily learned, and 
logic gate units only developed to handle the unsolved cases. Cases solved without 
logic gate units were solved by assymetric projections to the Mauthner cells of one 
polarity of pressure and one class of direction fibers, left or right. 
Curiously, most of these cases involved a preferential projection from high pressure 
fibers to the Mauthner units, along with directional fibers encoding displacements 
from each Mauthner unit's positive direction. This means the logic gate units 
tended to handle the low pressure cases. This may be a result of the presence of 
the assymetric distributions of utricular hair cells in 6 out of the 8 solutions. 
4 CONCLUSIONS 
We have generated predictions for the behavior of neurons in the Mauthner system 
under different conditions of acoustic stimulation. The predictions generated with 
our connectionist model are consistent with our interpretation of the phase model 
for underwater sound localization in fishes as a logical operator. The results are also 
consistent with previously described properties of the Mauthner system. Though 
perhaps based on the characteristics more of the training procedure, our solutions 
suggest that we may find a mixed solution in the fish. Direct projections to the 
Manthner cells from the auditory nerve perhaps handle many of the commonly 
encountered acoustic threats. The results of Blaxter (1981) support the idea that 
fish do escape from stimuli regardless of the polarity of the initial pressure change. 
Without significant nonlinear processing at the Mauthner cell itself, or more com- 
plex processing in the auditory fibers, direct connections could not handle all of 
these cases. These possibilities deserve exploration. 
We propose different computational roles for the two classes of inhibitory PHP 
neurons. We expect only unilaterally-projecting PHP cells to demonstrate some 
logical function of pressure and particle motion. We believe that some elements of 
the Mauthner system must be found to demonstrate such minimal logical functions 
if the phase model is an explanation for left-right discrimination by the Mauthner 
system. 
Directional Hearing by the Mauthner System 581 
We are currently preparing to deliver controlled acoustic stimuli to goldfish during 
acute intracellular recording procedures from the PHP neurons, the afferent fibers 
and the Manthner cells. Our insights from this model will greatly assist us in 
designing the stimulus regimen, and in interpreting our experimental results. Plans 
for future computational work are of a dynamic model that will include the results of 
these physiological investigations, as well as a more realistic version of the Manthner 
cell. 
Acknowledgements 
We are grateful for the technical assistance of members of the Boulder Connectionist 
Research Group, especially Don Mathis for help in debugging and optilniing the 
original code. We thank P.L. Edds-Walton for crucial discussions. This work was 
supported by a grant to RCE from the National Institutes of Health (RO 1 N S22621). 
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