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

Lead Research Organisation: University of Bristol
Department Name: Engineering Mathematics

Abstract

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.

Technical Summary

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.

Publications

10 25 50
 
Description The most significant achievements in this project were:

1. Derivation of a mathematical model of the cochlea, based on sound biological principles, incorporating both longitudinal coupling and local nonlinearity. The model reproduces experimental data very well, and also predicts that:
(a) temporal and spatial delay are not directly equivalent,
(b) models that do not include longitudinal coupling may need un-physiological local nonlinearities to have good agreement with data,
(c) spontaneous emissions arise from Hopf bifurcations and not from localized instabilities, and that reflections from the oval window are not necessary for their generation.
2. Development of numerical tools for efficient exploration of dynamics and stability of nonlinear spatially-coupled PDEs, and their application to cochlear models.
3. Discovery that neither the 'Hopf oscillator', nor tuning in the neighbourhood of a bifurcation, are necessary to generate realistic excitation patterns; rather it is a combination of local nonlinearity and spatial coupling that appears to be key.


Initial models considered the nonlinear dynamics of outer hair cells, and their resulting amplification/compression of basilar membrane vibration. Using our methodology, we were able to compare and contrast popular existing models, with a range of new models we developed during the project including different biological mechanisms in the outer hair cell. The results showed that neither the popular 'Hopf oscillator', nor tuning in the neighbourhood of a bifurcation, are necessary to generate realistic excitation patterns, in contrast to some claims in the literature.

Subsequent models included the effects of time-delay and longitudinal coupling in the cochlea. There is evidence that longitudinal coupling is well supported by the organ of Corti structural properties; it has been argued in the literature that it can be replaced by an equivalent time-delay term (which presents fewer computational difficulties). Using the mathematical methodology developed in the project, we were able to successfully compute existence and stability of solutions of both time time-delay and longitudinally coupled models, and further to relate them using new analytical results. We were thus able to conclude that time-delay and longitudinal coupling are not equivalent, and yield very different dynamics and stability. Furthermore, by comparing to our own experimental data, as well as that in the literature and from collaborators, we were able to conclude that longitudinal coupling appears to offer the most promising agreement with experimental data with physiologically reasonable parameter values.

Finally, we combined the two approaches, and developed whole-cochlear models of basilar membrane motion and outer hair cell charge dynamics, including both spatial coupling (representing organ of Corti structural properties and/or longitudinal energy transfer in, e.g., the tunnel of Corti or tectorial membrane) and local nonlinearity (representing the integration of ionic currents flowing through outer hair cell ion channels). We investigated these models using our mathematical methodologies and, as a result, were able to draw several important conclusions:

a) Our models with both spatial coupling and local nonlinearity accurately reproduce experimentally observable features of mammalian hearing: sharp tuning, compressive amplification, spontaneous and stimulus frequency otoacoustic emissions. We were unable to find such good agreement by neglecting either mechanism; we believe that it does not suffice to study the behaviour of outer hair cells alone to understand the active hearing process in the mammalian inner ear
(b) By calibrating the model to only a single animal, we were able to achieve good agreement with experimental data obtained for a range of different animals, increasing confidence in our results.
(c) The mathematical methodology developed in the project allowed us to calculate stability of the cochlea; for realistic cochlear models this is highly nontrivial, and often neglected.
(d) Spontaneous emissions arise from supercritical Hopf bifurcations and not from localized instabilities. Reflections from the oval window are not necessary for their generation, contrary to existing theories in the literature.
(e) Our full model exhibits complicated dynamical phenomena, including coexisting attractors - that can explain why perturbations can make the cochlea switch between different emissions - and chaotic vibrations - that could explain experimental observation of temporal variations of spontaneous emissions.

The results have been reported in refereed journals, at key meetings in the field (ARO Midwinter meeting & Mechanics of Hearing), as well as at numerous seminars during the lifetime of the project.
Exploitation Route There is potential for future commercial exploitation of the results of the project. Companies such as google are developing techniques for efficient machine implementation of auditory filtering; our results from this project on longitudinal coupling and local nonlinearity could be useful in such technological advances, as could the mathematical methodologies for analysing their dynamics.

There is also potential to build on the work in this project by developing our understanding of inner hair cell dynamics, and the resultant neural encoding, which could be important in the development of cochlear implants and other related hearing aid devices.
Sectors Digital/Communication/Information Technologies (including Software),Healthcare,Pharmaceuticals and Medical Biotechnology

 
Title Local and global cochlear model 
Description Whole-cochlear models of basilar membrane motion and outer hair cell charge dynamics, including both spatial coupling (representing organ of Corti structural properties and/or longitudinal energy transfer in, e.g., the tunnel of Corti or tectorial membrane) and local nonlinearity (representing the integration of ionic currents flowing through outer hair cell ion channels). 
Type Of Material Model of mechanisms or symptoms - mammalian in vivo 
Provided To Others? No  
Impact a) Our models with both spatial coupling and local nonlinearity accurately reproduce experimentally observable features of mammalian hearing: sharp tuning, compressive amplification, spontaneous and stimulus frequency otoacoustic emissions. We were unable to find such good agreement by neglecting either mechanism; we believe that it does not suffice to study the behaviour of outer hair cells alone to understand the active hearing process in the mammalian inner ear (b) By calibrating the model to only a single animal, we were able to achieve good agreement with experimental data obtained for a range of different animals, increasing confidence in our results. (c) The mathematical methodology developed in the project allowed us to calculate stability of the cochlea; for realistic cochlear models this is highly nontrivial, and often neglected. (d) Spontaneous emissions arise from supercritical Hopf bifurcations and not from localized instabilities. Reflections from the oval window are not necessary for their generation, contrary to existing theories in the literature. (e) Our full model exhibits complicated dynamical phenomena, including coexisting attractors - that can explain why perturbations can make the cochlea switch between different emissions - and chaotic vibrations - that could explain experimental observation of temporal variations of spontaneous emissions. 
 
Title Local cochlear nonlinearity 
Description Initial models considered the nonlinear dynamics of outer hair cells, and their resulting amplification/compression of basilar membrane vibration. 
Type Of Material Model of mechanisms or symptoms - in vitro 
Year Produced 2011 
Provided To Others? Yes  
Impact Using our methodology, we were able to compare and contrast popular existing models, with a range of new models we developed during the project including different biological mechanisms in the outer hair cell. The results showed that neither the popular 'Hopf oscillator', nor tuning in the neighbourhood of a bifurcation, are necessary to generate realistic excitation patterns, in contrast to some claims in the literature. 
 
Title Time-delayed and feed-forward model 
Description Time-delayed and feed-forward model of the mammalian cochlea fluid-structure interaction. 
Type Of Material Model of mechanisms or symptoms - mammalian in vivo 
Year Produced 2011 
Provided To Others? Yes  
Impact Demonstration that temporal and spatial delay in models of the mammalian cochlea are not directly equivalent, as previously claimed. Enables the further investigation of hair cell nonlinearities in whole-cochlear models. 
 
Title Computational tools 
Description Development of a methodology, based on numerical bifurcation theory, to fully explore the dynamics of local nonlinear models of outer hair cell dynamics. Enabled by a significant suite of computational tools for efficient exploration of the dynamics and stability of PDE models with nonlinearity, spatial coupling and time-delay. 
Type Of Material Data analysis technique 
Provided To Others? No  
Impact One key aim of this project was to use mathematical models to make predictions about the behaviour of mammalian hearing systems, and hence to work to reduce and replace animal experiments. Our modelling approach has been to focus on relatively simple models, which capture the essential features of the physiology of the cochlea, and have comparatively few parameters. By calibrating the model with data obtained from only a single animal, we were able to achieve good agreement with existing experimental data obtained for a range of different animals. As well as being an important test for our model, this also suggests that mathematical modelling is an approach that can have significant benefits is the quest to reduce and replace animal experiments. 
 
Title Local & global cochlear model 
Description Whole-cochlear models of basilar membrane motion and outer hair cell charge dynamics, including both spatial coupling (representing organ of Corti structural properties and/or longitudinal energy transfer in, e.g., the tunnel of Corti or tectorial membrane) and local nonlinearity (representing the integration of ionic currents flowing through outer hair cell ion channels). 
Type Of Material Computer model/algorithm 
Provided To Others? No  
Impact a) Our models with both spatial coupling and local nonlinearity accurately reproduce experimentally observable features of mammalian hearing: sharp tuning, compressive amplification, spontaneous and stimulus frequency otoacoustic emissions. We were unable to find such good agreement by neglecting either mechanism; we believe that it does not suffice to study the behaviour of outer hair cells alone to understand the active hearing process in the mammalian inner ear (b) By calibrating the model to only a single animal, we were able to achieve good agreement with experimental data obtained for a range of different animals, increasing confidence in our results. (c) The mathematical methodology developed in the project allowed us to calculate stability of the cochlea; for realistic cochlear models this is highly nontrivial, and often neglected. (d) Spontaneous emissions arise from supercritical Hopf bifurcations and not from localized instabilities. Reflections from the oval window are not necessary for their generation, contrary to existing theories in the literature. (e) Our full model exhibits complicated dynamical phenomena, including coexisting attractors - that can explain why perturbations can make the cochlea switch between different emissions - and chaotic vibrations - that could explain experimental observation of temporal variations of spontaneous emissions. 
 
Title Local cochlear nonlinearity 
Description Initial models considered the nonlinear dynamics of outer hair cells, and their resulting amplification/compression of basilar membrane vibration. 
Type Of Material Computer model/algorithm 
Year Produced 2011 
Provided To Others? Yes  
Impact Using our methodology, we were able to compare and contrast popular existing models, with a range of new models we developed during the project including different biological mechanisms in the outer hair cell. The results showed that neither the popular 'Hopf oscillator', nor tuning in the neighbourhood of a bifurcation, are necessary to generate realistic excitation patterns, in contrast to some claims in the literature. 
 
Title Time-delayed and feed-forward model 
Description Time-delayed and feed-forward model of the mammalian cochlea fluid-structure interaction. 
Type Of Material Computer model/algorithm 
Year Produced 2011 
Provided To Others? Yes  
Impact Demonstration that temporal and spatial delay in models of the mammalian cochlea are not directly equivalent, as previously claimed. Enables the further investigation of hair cell nonlinearities in whole-cochlear models. 
 
Description Bastian Epp 
Organisation Carl von Ossietzky University of Oldenburg
Country Germany 
Sector Academic/University 
PI Contribution Development of mathematical modelling and analysis techniques.
Collaborator Contribution Initial model and data.
Impact Mathematical model and analysis. Two publications (one refereed journal, one refereed conference proceedings)
Start Year 2010
 
Description Dáibhid O Maoiléidigh 
Organisation Rockefeller University
Country United States 
Sector Academic/University 
PI Contribution Development and analysis of mathematical model.
Collaborator Contribution Mathematical model and biophysical parameters.
Impact Development and analysis of mathematical model. One refereed journal publication.
Start Year 2008