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A Computational Model of Auditory Processing for Sound Localization
Diana Stan, Michael Reed
Department of Mathematics, Duke University
Abstract
We have formulated and worked with a model for azimuthal sound
localization, as done by the Lateral Superior Olive (LSO), a nucleus in
the brain. The inputs to our model are the intensity in decibels at each
ear, as the LSO does sound localization based upon Interaural Intensity
Difference (IID). The output is independent of absolute intensity, and is
instead dependent upon IID. The output is the number of LSO cells (out
of a total of 60) that fire, which can be translated into the degree on the
horizon from which the sound originated, up to 3 degrees for high
frequency sounds. If IID is 0, this means that 30 of 60 LSO cells are
firing, then this indicates that the sound is coming from straight ahead,
at 0 degrees. When 0 LSO cells fire, and this indicates that the sound is
coming from -90 degrees, on the ipsilateral side, since the LSO receives
inhibitory input from the medial nucleus of the trapezoid body (MNTB)
on the ipsilateral side, whereas it receives excitatory input from the
anterior ventral cochlear nucleus (AVCN) on the contralateral side.
Similarly, when all 60 LSO cells fire, and this indicates that the sound is
coming from +90 degrees, on the contralateral side. Our model is
shown to adapt to synapse death and randomness in synaptic
connections between the AVCN, MNTB, and LSO. The LSO cell number
at which cell firing in spikes per second increases linearly with an
increase in IID, since the cells in the AVCN and MNTB are arranged in
columns so that the firing thresholds of the cells increase monotonically.
This model can be adapted as a teaching tool.
The auditory system is such that certain topological regions respond
to a certain frequency much better than all others; this is shown
through tuning curves. Thus, when thinking of our model, we can
study only one such “frequency slab”, as we assume that all distinct
tonotopic regions make computations in the same way. We can view
the neurons of the three participatory brain parts as being columns for
ease, and then assign cell numbers to neurons going up the columns.
We assume that both the AVCN and MNTB have 40 cells project onto
the LSO per tonotopic region, and that the LSO has 60 cells with
which to make connections. We then assume that even numbered
cells from the AVCN and MNTB make 8 synapses with target cells in
the LSO, and that odd numbered cells make 7 synapses. Thus, each
LSO cell receives 37 or 38 synapses as inputs. We have created the
model so that all parameters could be changed if desired. Of course,
each of these neurons must have a threshold at which to fire: we
have made the assumption for our model the thresholds of AVCN
neurons increase monotonically as cell number increases, but that the
thresholds of MNTB neurons decrease monotonically as cell number
increases, as shown below, to account for this clear, everday
phenomenon: if a sound is a louder in, say the right ear, it is coming
from the right side of the head.
AVCN and MNTB Output Curves; LSO Inputs
This curve shows the
firing rate of the MNTB
as a function of cell
number. The curve for
the AVCN at the same
intensity level is
precisely the mirror
image.
The figure on the left shows how the model responds to
extreme synaptic death. Even though the summed excitatory
and inhibitory curves are rather distorted, LSO cell number
at which firing rate becomes 0 spikes/sec is the same, and
thus the LSO can still do azimuthal sound localization even
in cases of synaptic death. The figure on the right shows the
case of uneven synaptic death, here more AVCN synapses
are removed, and thus the model responds a bit poorly, as
LSO cell number at which firing ceases decreases, since the
model assumes IID has decreased.
This figure shows the case
of cell death, when 3 cells
are removed from the
AVCN and MNTB. The
model responds similarly to
above, as LSO cell number
has decreased. Yet in both
cases, the model is still
fairly accurate, and the
LSO can still encode
azimuthal degree fairly
well.
The input to the model is the intensity level in decibels at each ear.
The LSO computes based upon Interaural Intensity Difference
(IID), the difference of the two.
LSO Output
Introduction
The LSO works in conjunction with the Medial Superior Olive (MSO); the
LSO doing calculations based upon intensity differences in sound
between the two ears, and the MSO doing calculations based upon time
arrival differences of sound to each ear. Thus, the LSO is able to
compute easier with high frequency sounds, whereas the MSO does
better with lower frequency sounds. Both can handle all frequencies,
however.
Model Basics
Shown above is the circuitry of the LSO. Blue connections are
excitatory, whereas pink are inhibitory. Connections are made
“tonotopically”, that is areas that respond well to a certain frequency
connect to areas that respond well to the same frequency.
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Shown on the left is the output of the model set up in a deterministic
way: each AVCN and MNTB neuron project onto specific LSO
neurons. On the right, the set up is more random, and the LSO
neurons pick with which AVCN and MNTB neurons to make synapses
using a modified normal distribution. The two curves are very similar.
The equation for the firing rate of the ith AVCN or MNTB neuron is:
S = B + V * [(I - Ti) / Ki + (I - Ti)],
where I - Ti = I - Ti, I ≥ Ti; I - Ti = 0, I < Ti
Similarly, the equation for the firing rate of the jth LSO neuron is:
S’ = 0, if Ej – Ij < 0
S’ = B + V * [((Ej – Ij) – Tj) / Kj + ((Ej – Ij) - Tj)],
Here, S and S’ are respective firing rates in spikes per second; B is
the firing rate of a neuron when the inputs are below its threshold, set
to 10 spikes/sec in the model; V is the maximum “plateau” firing rate,
here set to be 300 spikes/sec; Ki and Kj are half-saturation constants,
set at 10 decibels; Ej is the summed excitatory input to the LSO in
spikes/sec; Ij is the summed inhibitory input to the LSO in spikes/sec;
Ti and Tj are the thresholds of the ith and jth AVCN, MNTB, and LSO
neurons, Ti range from 0.5 to 20, stepping up by 0.5, Tj are all 0.
This figure shows the models
dependence on difference in
intensities. Even though the
absolute intensities are
different, the IID is the same,
and thus the same number of
LSO cells fire as above.
LSO output curve: example of
behavior when intensities at ears
are not equal. Note how LSO
cell number at which firing rate
becomes 0 has increased, as IID
has increased.
Summary
1.
2.
3.
4.
5.
We have developed a model for azimuthal sound localization as
computed by the LSO.
We have shown the model to be robust in terms of synapse
death and randomness in the connections, and responds
reasonably well to cell death.
We have shown the model to be independent of actual
intensities, and to instead be dependent upon interaural intensity
differences.
A teaching tool has been designed and can be viewed at:
www.people.duke.edu/~dms60.
Further work: Examining projections from the LSO onto the
Inferior Colliculus.
References
1.
Michael Reed, Jacob Blum. J. Acoust. Soc. Am. Volume 88, Issue 3, pp.
1442-1453 (1990)