Can You Hear Me Now? - University of Pennsylvaniated/210F10/References/...Can You Hear Me Now? An...

IntroductionHow We Hear

Cochlear MechanicsCochlear Implants

Conclusion

Can You Hear Me Now?An Introduction to the Mathematics of Hearing

Joshua Goldwyn

Department of Applied MathematicsUniversity of Washington

April 26, 2007

Joshua Goldwyn Can You Hear Me Now?



Conclusion

Some Questions

I How does hearing work?

I What are the important structures and mechanisms of theauditory system?

I How can mathematics improve our understanding of hearing?

I What is a cochlear implant and what role can mathematicalmodeling play in improving implant performance?




Conclusion

Physical Basis of Sound

Any vibrating object produces sound. The sound we hear comesfrom pressure waves propagating through the air.




Conclusion

Perceptual Basis of Sound

Our perception of sound is due to neural response in the auditorycortex of the brain.




Conclusion

Anatomy of the Ear




Conclusion

The Middle Ear




Conclusion

The Inner Ear




Conclusion

The Cochlea




Conclusion

Cross-Section of the Cochlea




Conclusion

Detail of Cochlear Partition




Conclusion

Hair Cells and Transduction




Conclusion

Cochlear Mechanics: Place Theory of Hearing

I The 1961 Nobel Prize in Medicine was awarded to Georg vonBekesy for his lifetime of experimenatl research into cochlearmechanics.

I Two of his key findings were the traveling wave and tonotopicresponse of the basilar membrane.

I The location of maximum BM vibration is determined by thefrequency of the stimulus.

I Modern techniques have enabled researchers to observe theresponse of cochleae in live subjects, leading to the discoveryof the active mechanism.




Conclusion

Cochlear Mechanics: What to Model?

I Traveling Wave

I Frequency Dependence of Traveling Wave Envelope

I Spiral Geometry in Three Dimensions

I Micromechanics of the Organ of Corti

I Outer Hair Cell Motility and Active Mechanism




Conclusion

Cochlear Mechanics: Fluid Dynamics

Since the cochlear fluid is irrotational the velocity field is:

v = OΦ.

Since the fluid is incompressible, linear, and inviscid, the fluiddynamics reduce from Navier Stokes to the potential equation:

O2Φ = 0.

Conservation of momentum requires:

P1 + ρ∂Φ

∂t= 0

where P1 is the pressure in the upper chamber, ρ is fluid density.




Conclusion

Cochlear Mechanics: Boundary Conditions

Consider the interface at z = 0 and let n be the displacement ofthe basilar membrane normal to the interface.Continuity at the interface requires:

∂n

∂t=

∂Φ

∂z.

Assume the mechanics of the membrane are governed by:

m∂2n

∂t2+ β

∂n

∂t+ κn = P2 − P1 = −2P1.

This is a commonly used model that assumes the membranevibrates as a series of uncoupled springs of mass m, damping β andstiffness κ. β and κ vary exponentially with distance from the baseof the cochlea.




Conclusion

Cochlear Mechanics: Simplifying the BVP

We will restrict ourselves to the steady state response of thecochlea to pure tone of frequency ω. Assuming linearity of thesystem we can transform the BVP:

Define Φ = e iωtφ and n = e iωtη, and P1 = e iωtp1

Then the BVP for the upper chamber becomes:

O2φ = 0∂φ

∂z=

2ρiωφ

Zat z = 0 (Basilar Membrane)

∂φ

∂x= ω at x = 0 (Oval Window)

∂φ

∂n̂= 0 at rigid walls

where Z = iωm + β + κiω .




Conclusion

Finite Differences

One method for numerically solving diffential equations is to usefinite difference operators. Recall the definition of a derivative:

df

dx= lim

h→0

f (x + h)− f (x)

hif the limit exists.

On a computer, we approximate derivatives by computing thisfraction for small (but finite) h. When solving differentialequations, this results in the need to solve systems of equations.




Conclusion

Traveling Waves in a Straight Rectangular Cochlea

Basilar Membrane Response to 1000Hz and 200Hz Stimuli




Conclusion

Physical Domains using Logically Rectangular Grids




Conclusion

Cochlear Implants

I As we have seen, healthy hearing relies on complicated systemof finely tuned structures in the ear to convert sound (pressurewaves) into a neural response that is sent to the brain.

I For many individuals with hearing loss, some part of thistransduction process does not work.

I Cochlear Implants work to restore hearing by stimulatingauditory nerves directly with electical pulses.




Conclusion

Cochlear Implants




Conclusion

Cochlear Implants

I According to the National Institute on Deafness and OtherCommunication Disorders, the FDA reported in 2005 thatnearly 100,000 people worldwide have received cochlearimplants. In the United States, roughly 22,000 adults andnearly 15,000 children have received implants.

I Successful cochlear implants can allow otherwise deaf patientsto communicate without lip-reading or other visual clues, talkon the telephone, and help deaf children develop speech andlanguage skills.




Conclusion

Mathematical Model

Mathematical models of cochlear implants require twocomponents:

I Electrical Field Model

I Neural Excitation Model




Conclusion

Electrical Field

Electrostatic field generated by electrodes can be modeled byPoisson’s Equation with point sources representing the electrodes:

O2φ = δ(x − x0, y − y0, z − z0).

In simplified geometries, exact solutions are possible. To capturefull biological complexity (spiral shape, characteristics of tissues,bones, etc.) numerical methods must be employed.




Conclusion

Neural Excitation

In response to an electric field, neurons respond by producingaction potentials that are transmitted from neuron to neuron untilthey reach the brain.




Conclusion

Modeling Action Potentials

The 1963 Nobel Prize in Medicine was awarded to Hodgkin andHuxley for their research into the dynamics of action potentials.They developed the following set of Ordinary DifferentialEquations:

dV

dt=

1

Cm(I − (INa + IK + IL))

dm

dt= αm(1−m)− βmm

dh

dt= αh(1− h)− βhh

dn

dt= αn(1− n)− βnn.




Conclusion

Spatial Selectivity

I Perception of frequency is related to location of stimulatedneurons (tonotopy)

I Original cochlear implants were monopolar, thus a broadcluster of neurons is stimulated in response to sound.

I More restricted neural excitation can be achieved throughtripolar cochlear implants.




Conclusion

Monopolar vs Tripolar Electric Field




Conclusion

Monopolar vs Tripolar Neural Excitation




Conclusion

Conclusion

I Our ability to hear relies on a complicated system of biologicalprocesses that converts sound waves to vibrations in themiddle and inner ear which trigger neural responses that aresent to the brain for processing.

I The auditory system is not fully understood and there arenumerous opportunities for mathematical modeling tocontribute to our understanding of hearing.

I The full complexity of biological systems can never becaptured by (tractable) mathematical equations, but wellformulated models can still improve our understanding ofphysiological and other biological systems.


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