1. Aims To use the linear fractional transformations (LFT) modelling framework to reconstruct an existing state-space model of the cochlea in terms of a stable linear system with interconnections to an operator containing any desired parametric uncertainties (or perturbations such as inhomogeneities) and any non-linearities. To use this framework to answer the question: How large can the inhomogeneities be before the model becomes unstable, thus generating SOAEs? This question is non-trivial because the inhomogeneities take the form of different perturbations in the BM parameters at each point along the BM, all of which interact with each other. Investigating cochlear model stability using linear fractional transformations Ben Lineton 1 and Emmanuel Prempain 2 1 ISVR, University of Southampton, UK; 2 Department of Engineering, University of Leicester, UK LFT modelling: Block diagram of single element To apply LFT methods to the cochlear model, first the second-order finite-difference equations representing the fluid-BM coupling, and BM micromechanical dynamics are implemented as a block diagram for a single element, as in Fig. 2. (Prempain E & Lineton B. 2009) + + _ + + + active outer hair cell pressure state-vector of i th element transformed stereocilia deflection Fig 2: Block diagram of the i th element, with inputs p i-1, p i+1 and p a,i, and outputs, p i, u i, and z i. The matrices A i, B i, C i, and F i contain the structural parameters defining the micromechanics, and  is a constant arising in the finite difference approximation to the governing wave-equation. The relationship of p a,i to u i remains unspecified at this “open-loop” stage in the formulation. Motivation for using LFT LFT is an analytical method used in the field of robust control system design, which has many uses, such as investigating the effect of parametric uncertainty on a system’s stability and performance, or the effect of introducing non-linearities (Zhou et al., 1995). It is also particularly easy to implement such models using packages such as SIMULINK. LFT modelling: Interconnecting elements The block diagram in Fig. 2 can be written as a single block, T i. The blocks for all the elements can then be connected together as shown in Fig. 3, for the first three blocks. Continuing the interconnection of all N elements, and applying the boundary conditions at the stapes and helicotrema, leads to the full cochlear model in LFT form. vector of transformed stereocilia deflections p0p0 p4p4 p1p1 p2p2 p2p2 p1p1 p2p2 p3p3 p3p3 p2p2 p1p1 p3p3 u1u1 z1z1 u2u2 z2z2 u3u3 z3z3 p a,1 p a,2 p a,3 T1T1 T2T2 T3T3 p0p0 p4p4 p3p3 G3G3 vector of active pressures Fig 3: Left: Interconnection of the first three elemental blocks. T i is a 6 x 3 matrix representing Fig 2 in the Laplace domain. Right: The three interconnected elements are combined to form, G 3 (expressed as a 16 x 5 matrix), together with the feedback block,  3. The transformation labelled “LFT” is the so-called Redheffer star-product (Zhou et al., 1995), which allows the calculation of the G i s from the T i s recursively. G 0 is a 2 x 2 matrix depending on the basal boundary condition. p0p0 GNGN Fig 4: Full model in LFT form. Thus the LFT formulation expresses the model as a stable linear system, G N, with a feedback connection to the system,  N, which contains all the parametric uncertainty (and any non-linearity, if desired). In the following analysis, this formulation has been used to find the bounds on the size of the inhomogeneities within which the linear model is guaranteed to remain stable. Fig. 5: Bounds on 1/  MAX. Results: Instability due to parametric perturbations Inhomogeneities along the BM can be represented as a vector of perturbations,  = [  1  2...  N ]. The block in Fig. 4 is then the matrix,  N =  [ I + diag(  ) ] where  is a constant determining the overall gain. The question is, if we allow all the  i ’s to vary arbitrarily, within bounds of ±  MAX then what is the value of  MAX that ensures stability? The exact solution to this problem is not trivial (it is NP-hard, which is practice means that computational time becomes prohibitive for large values of N). However, a partial solution can be obtained using by a technique called  -analysis. This is a structured singular value test (Packard and Doyle, 1993; Zhou et al., 1995), which can be applied to a matrix that is readily determined from the formulation in Fig. 4, and which yields bounds on the value of  MAX. The result for the model of Elliott et al. (2007) is shown in Fig. 5, with  =0.8. Thus, from the peak in the curve of 12.5%, it reveals that a variation of up to 8.5% in feedback gain at each element can be tolerated, with guaranteed system stability. Summary and conclusions LFT methods can be applied to existing discrete cochlear models, giving a model formulation in terms of a stable linear system connected via a closed-loop feedback path to a block containing any desired non-linearities and parametric uncertainties. The LFT formulation admits various analyses, such as a test of the degree of parametric uncertainty that may lead to system instability. An example described here yields the bounds on the variation in the BM gain that guarantee model stability. This can be interpreted as the bounds on the size of BM inhomogeneities below which SOAEs will not be generated (irrespective of the actual pattern of inhomogeneities). % References Elliott SJ, Ku EM, & Lineton B. 2007. A state space model for cochlear mechanics. J Acoust Soc Am. 122, 2759-71. Packard A & Doyle JC. 1993. The complex structured singular value. Automatica, 29, 71-109. Prempain E & Lineton B. 2009. Cochlear Modelling using a Linear Fractional Transformation Approach, ICCAS-SICE, Aug. 18-21, Fukuoka, Japan, August 2009. Zhou K, Doyle J, & Glover K. 1995. Robust and Optimal Control. New Jersey, Prentice-Hall, pp. 247-69. Cochlear Model The 1-D longwave model was assumed, with micromechanics shown in Fig 1. Fig 1: The discrete model of the cochlea including the details of the micromechanical model shown for element number 3. The micro-mechanical model is based on a Neely and Kim con- figuration, with details given in Elliott et al. (2007). For element i, p i (or p ) is the BM pressure difference; w i, (or w BM ) is the BM displacement; p a,i (or p a ) is the active pressure; and w TM is a transformed tectorial membrane displacement. p1p1 p2p2 p3p3 p N-1 pN=0pN=0 sound source, middle ear, and stapes helico- trema s2s2 r2r2 s3s3 r3r3 s1s1 r1r1 pdpd p papa Acknowledgements The study is part of the project entitled “Modelling and Measurements of Cochlear Gain Control” funded by the EPSRC within the Collaborating for Success in System Biology Discipline Hopping scheme.