An Inner Ear Hair Cell Model and Visualization of its Deflection

Final Project

ESM 4984

Principle Investigator:

John Cotton


May 8, 1996


This project discusses the development of a vestibular hair cell model and its visualization tools. It begins with an introduction to the anatomy and physiology of the vestibular system. It next discusses the development of the hair cell model. Finally, the creation of a set of graphics tools to help interpret the model results is covered.

Anatomy and Physiology

Vestibular Labyrinth

All vertebrates have the ability to sense changes in the positioning of the skull. This ability, called the vestibular sense, is needed for activities from walking to looking around.

The vestibular senses are made possible by the vestibular labyrinth, found within the inner ear. The entire labyrinth is about the size of a marble and is located next to the cochlea, used in hearing.

Figure 1. The Vestibular Labyrinth, showing the semicircular canals and the otolith organs. (Freidmann and Ballantyne, 1984)

The vestibular labyrinth consists of five parts: three semicircular canals, and the two otolith organs, known as the utricle and the saccule. All of these sections are connected by bony passageways filled with a endolymphatic fluid. The labyrinth also connects to the cochlea, which is responsible for hearing.

Semicircular Canals

There are three semicircular canals in each ear. They are laid out in orthogonal planes. An individual canal senses rotations in one of three orthogonal directions. Thus the three canals, by combining their results, give the ability to sense rotations in any direction in three-dimensional space.

The canal consists of a circular tube (much like the inner-tube of a bicycle) filled with viscous endolymphatic fluid. At one end of the tube is a small triangular shaped mass of gelatinous material called the cupula, which partially blocks the canal. The cupula sits on a bulged area covered with hair cells, known as the crista.

Figure 2. A cross-section of a semicircular canal. (Grant, 1995)

When the head is rotated, the fluid lags behind because of inertia. The fluid pushes against the cupula, deflecting it. This deflection is measured by the hair cells in the crista.

Otolith Organs

There are two otolith organs in each ear. They are the saccule, which senses motion in the vertical plane; and the utricle, which senses motion in the horizontal plane. By combining the results of these two organs, motion or acceleration can be deduced in any arbitrary direction in three dimensional space.

Each otolith organ consists of a cavity in the bony labyrinth. Within this cavity lies a flat mass of small dense bodies called otoconia or otoliths (literally: ear stones). These are held together by an extracellular matrix. This mass, called the otoconial layer, is connected to the wall of the labyrinth on one side by a viscoelastic gel, conveniently called the gel layer. The labyrinth wall where the gel layer attaches is full of hair cells. The entire cavity is filled with a viscous fluid.

When the head is moved, the dense mass initially lags behind due to inertia. This causes the gel layer to shear, which is sensed by the hair cells beneath. Under the influence of gravity, the denser otoconial layer will deflect towards the earth, causing shear also.

Figure 3. A schematic of the cross section of the otolith organ.

Hair Cells

The hair cells transduce, or change, deflection into nerve impulses which are then carried to the brain. The upper part of the hair cell projects small hairs, or cilia, into the cupula of the semicircular canal, or the gel layer of the otolith organ. When these cilia are deflected, the information is sent to the brain.

Figure 4. A diagram of a hair cell, showing the apical surface and cilia extending from it. (adapted from Freidmann and Ballantyne, 1984)

The top of the hair cell consists of an epithelial surface, a number of stereocilia, and a single kinocilium. There may be anywhere from ten to one hundred stereocila in a bundle. The kinocilium is thicker and, usually, taller than the stereocilia. Hair cells are packed in arrays where each cilia has six neighbors an equal distance apart (hexagonal packing).

The stereocilia are interconnected by small fibers called subapical bands. These bands extend down the length of the cilia and connect the cilia to each of its neighbors. Each stereocilia also connects to its next tallest neighbor by a single tip link. It is thought that tension in these tip links caused by the bundle being deflected opens small channels in the cilia. As these channels open, ions from the surrounding fluid rush in due to an electrochemical gradient, causing the cell to fire.

Figure 5. Schematic of hair cell showing interconnections. (adapted from Goodyear and Richardson, 1994)

Hair cells vary greatly from species to species, organ to organ, and even the location within the same organ. These variations lead us to ask many questions such as:

How do variations in bundle dimensions effect hair bundle stiffness?

What effect does stiffness have on the filtering ability of a hair cell?

Why are certain types of hair cells found in different areas of the vestibule?

In a more general sense, we have other questions dealing with the actual hair cell mechanism. These questions include:

Does tip link tension account for the observed cell output?

What is the structure and material that makes up the subapical bands?

Hair Cell Model

In order to answer some of the questions we have about hair cell mechanics, we have created a mathematical model to test some of the various configurations of hair cell bundles. Our primary goal is to be able to be able to predict bundle stiffness given any geometry. We also hope to use the model to investigate theories regarding tip links, subapical bands, and the cell's firing mechanisms.

Model Challenges

There are four main challenges that make this problem unique from an engineering point of view. First off, the individual stereocilia taper as they insert into the epithelial surface. Any model of bundle behavior must not assume stereocilia have a constant cross section.

For most tall, slender structures under transverse loading, the deflection due to shear is negligible when compared to the deflection due to bending. However, stereocilia have been shown to exhibit significant shearing. This indicates we must use the more complex Timoshenko, or Shear Deformable Theory in our calculations, rather than the more common, and easier to use, Euler Theory.

Cilia interconnections are assumed to be thin strands of proteins. It is widely believed that these interconnections will only carry a tensile force and will give no resistance to compression. Hence the relationship between deflection and force is nonlinear for the interconnections. This results in the first of our nonlinearities.

The second of our nonlinearities is due to geometry. As the bundle deflects, the interconnections also move and become reoriented. This means, the line of action of their forces varies with deflection.

Finite Element Analysis

To solve the problem of what happens to the bundle when a force is applied to it, we utilize a Finite Element Analysis (FEA). In FEA a boundary value problem made up of differential equations is transformed into a large number of linear algebraic equations, each valid over a small subset of the original problems space. The linear algebraic equations, stored as a matrix are solved via digital computer.

Because our problem is nonlinear, we must modify our analysis. Finite Element Analysis is, by itself, a linear technique. We solve this problem by breaking our problem up into many small linear steps. We apply our force to the stereocilia and determine the deflection of the stereocilia. We then use that deflection to determine the forces generated by the interconnections and apply those forces to the stereocilia as well.


The ability to visualize the deflected bundle configuration is essential to the model. It will enable us to check the validity of the model, as well as provide a quick overview of new results. An additional bonus is the ability to convince non-engineers of the validity of the model.

Cilia positions and deflections are read from the output file and converted into Inventor format. The resulting file can be displayed in an Inventor viewer, which provides full rotation, zoom, and animation capabilities. Unfortunately, its exporting capabilities and portability are limited.

Figure 6. Output of Inventor format on SGI. A utricular hair cell is depicted.

To get around this, a number of tools were created using PV-Wave. All tools are run from a main driver routine. Different options are selected from a pull down menu. The tools enable the bundle to be viewed as a shaded surface or a simple line drawing. Both styles allow rotations, zooming, and scaling of deflections. The user can also animate the above. Below we have the line drawing of a bundle using the "line draw" feature. Deflections are scaled from zero to one, which gives the impression the bundle is being deflected. Further below is the full shaded bundle.

Figure 7. Line drawn output from PV-Wave

Figure 8. PV-Wave depiction of utricular hair cell.


For General Anatomy

Friedmann, I., and J. Ballantyne, 1984. Ultrastructural Atlas of the Inner Ear, Butterworths and Co., London.

Gaudin, A. J., and K. C. Jones, 1989. Human Anatomy and Physiology, Harcourt Brace and Jovanovich, Washington, D.C.

Grant, J.W., 1995. Chapter 39: Vestibular Mechanics, in The Bioengineering Handbook, CRC Press, Inc., Boca Raton, FL.

For Hair cells

Bagger-Sjoback, D. and M. Takumida, 1988. Geometrical Array of the Vestibular Sensory Hair Bundle, Acta Otolaryngologica, 106:393-403.

Furness, D. N. and C. N. Hackney, 1985. Crosslinks Between Stereocilia in the Guinea Pig Cochlea, Hearing Research, 18:177-188.

Goodyear, R. and G. Richardson, 1994. Differential Glycosylation of Auditory and Vestibular Hair Bundle Proteins Revealed by Peanut Agglutinin, The Journal of Comparative Neurology, 345:267-278.

Tilney, L. G., et. al., 1983. Actin Filaments, Stereocilia, and Hair Cells of the Bird Cochlea. II. Packing of Actin Filaments in the Stereocilia and in the Cuticular Plate and What Happens to the Organization When the Stereocilia Are Bent., Journal of Cell Biology, 96:822-834.

For Finite Element Analysis

Duncan, K., 1993. Finite Element Analysis of Inner Ear Hair Bundles: A Parameter Study of Bundle Mechanics, Masters Thesis, Virginia Polytechnic Institute and State University.

Duncan, K. and J. W. Grant, 1996. A finite Element Model of Inner Ear Hair Bundle Micromechanics, in press.

Reddy, J. N., 1993. An Introduction to the Finite Element Method, McGraw Hill, New York.

Appendix: Source Files

Fortran files

viz_lin.f, viz_shade.f

PV-Wave proceedure files,,,,,,,

Posted May 26, 1996