Mathematical modelling of the active hearing process in the mamalian inner ear

The human inner ear (or cochlea) is a remarkable device that out-performs any human-made system. For example, it is sensitive to displacements at sub-atomic length scales, smaller than background noise, can distinguish signals separated by microseconds, can process sounds over a million-fold intensity range (from 0dB to 120dB SPL), operates over a frequency range of over ten octaves (from 20Hz to 20kHz), can discriminate frequencies only 0.2% apart, and intensity changes of 1dB. Largely speaking, all mamals use a similar system to hear, and their ears have similarly remarkable performance. While physiologists can describe many of the processes that underlie this performance, there is a lack of agreement among them about what are the key ingredients that make it all work. Using data from rats rather than humans, we will seek to understand this process. The detailed structure of the cochlea is complicated and involves a fluid filled tube that is wrapped up into a spiral. The tube is divided in two by the so-called 'basilar membrane' that vibrates up and down like a drum. This is a very strange drum though. Sitting on top of the membrane is a device known as the organ of Corti that acts like a very special microphone. Not only does the microphone pick up the signal from the drum and relay it via nerve cells to the brain, but it also acts like an amplifier that actually makes the drum beat up and down more vigorously. However, each of the array of amplifiers at different distances along the spiral tube responds to a different frequency. This grant aims to understand how the cochlear amplifier works. It is widely believed that the key parts of the organ of Corti responsible for the amplification are the so-called outer hair cells. These have small hairs on them which can open and close tiny gates that allow calcium to flow into the cell. It is thought that the flow of calcium is the trigger that causes the cell to rapidly pull and push on the basilar membrane drum to make it beat with larger amplitude. We will use a mixture of experimental measurements (at Bristol and Keele) together with mathematical modelling and simulation. In the Bristol experiments, we will determine how the opening and closing of the gates on the outer hair cells can change the flow of calcium, how they lead to the pulling and pushing of the hair cell itself, and also how the hairs on neighbouring outer hair cells influence each other. The Keele experiments will look at detailed images of the motion of the basilar membrane as one changes the input amplitude of single-frequency sounds. This way we can look at a specific microphone/amplifier and see the dynamic response of its active process. These two sets of experiments will be used to inform a set of mathematical equations that capture the physics of the situation and enable accurate computer similation and ultimately an answer to the question of how hearing works. Firstly we shall write down equations governing the relation between the concentration of calcium, the opening of the gates on the hairs, and the pulling and pushing of the hair cell. Second we shall explore a so-called feedforward mechanism where the output of one hair cell causes amplification slightly further along the spiralling drumhead. Finally we shall look at the dynamics of how the hairs themselves couple together to cause a large response in the hair-cell microphone. Ultimately we shall use the mathematical models to decide which of a number of competing explanations is the most plausible for explaining how the active process occurs. We expect that this will make it easier for doctors to diagnose hearing probelms more accurately, and will alIow them to propose better remedies when a person's hearing does fail.

The aim of this project is to apply a predictive biology approach to understand mechanisms behind the active process of hearing in the mammalian cochlea. In particular we wish to combine state-of-the-art in vivo and in vitro measurements on the function of outer hair cells (OHCs) and on the dynamic response of the basilar membrane (BM), with models that accurately capture electro-mechanical feedback, geometric feedforward, temporal delay, and coupling between OHCs and the BM. We will collect in vitro time series data of the force exerted by the OHCs at the 4, 14 and 20kHz region of the rat cochlea due to stimulation of the hair bundles, as well as their transducer currents. We will apply both step stimuli, to observe the time-delayed response of the OHCs, as well as continuously oscillatory stimulus at realistic auditory frequencies. In vivo measurements will then be made to discover how the partition vibrates; it is vital to make the observations in healthy, living ears, where the feedback processes of the cochlear amplifier are functioning normally. From these physiological measurements we will formulate a hierarchy of biologically-derived, nonlinear, dynamical models of the cochlea. Throughout, mathematical modelling will go hand in hand with the experiments. We will include the two possible sources of active response in the OHC models: movement of tip links caused by closure of the ion channel, and mechanical contraction/expansion of prestin due to changes in trans-membrane potential. We will also investigate geometric feedforward on the BM, by including spatial and temporal delay in the BM models. Finally, we will analyse a compound model for frequency tuning and amplification using phase-space techniques, asymptotics, direct numerical simulation and parameter continuation. Ultimately we aim to answer the question of which effects in the OHCs and BM explains the mechanism by which the active process of cochlear amplification occurs.