## Making Wavelets: A Profile of Ingrid Daubechies

Skip to main content Skip to table of contents. Advertisement Hide. Wavelets in Neuroscience. Authors view affiliations Alexander E. Hramov Alexey A. Koronovskii Valeri A. Makarov Alexey N. Front Matter Pages i-xvi. Mathematical Methods of Signal Processing in Neuroscience. Alexander E. Hramov, Alexey A. They are the extremal functions for the uncertainty inequality.

Extremals for the uncertainty principle satisfy differential equations. Since the membrane transfer function is described by an extremal function and its transforms under the symmetry group and since the extremal functions are preserved under this action, it is possible to derive differential equations for the output of the signal.

The resulting equations are called the structure equations. In this situation differentiation of the wavelet transform W f a , t with respect to the parameters of the symmetry group directly leads to a differential equation. A linearization process for the kernel brings it back to a differential equation that is then satisfied approximately.

The quantities in the equation at first are derivatives of the output function W f a , t and its Hilbert transform. A further calculation then shows that the result can be formulated as an inhomogeneous system of linear partial differential equations for the phase and for the logarithm of the amplitude of the output signal. This is particularly satisfying because these are exactly the physiologically relevant quantities.

The wavelet transform is then. The normalized extremal function h satisfies the differential equation.

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This gives the basic equation. This transform is a unitary operator on L 2 R , C. It extends to a bigger class of functions to all temperate distributions. On the basic trigonometric functions it operates very simply:. It immediately follows that. The linearity assumption then implies that this holds for arbitrary input signals f.

The Hilbert transform thus appears naturally in this setting.

Taking the factor a into account this is. Notice the shift by 1 2 that has its origin in the factor 1 a. Only the constants are slightly different. The structure equations can be written in x , t -coordinates:. Signal processing in the cochlea is non-linear. The main - but certainly not the only - source of non-linearity is the compressive nature inherent in the hearing process. In the abstract model pursued here this is taken care of with a single parameter that represents the level of sound intensity. The model then describes the linear approximations at these levels. The structure equations are at the core of this abstract model, in fact they comprise all the essential features.

First of all, they are linear as would be expected from a linear approximation. From a mathematical point of view, the equations therefore are very simple. On top, the system is quite special. Its solutions can be realized in complex form as products of two factors, the first of which is entirely determined by the system and the second is a holomorphic function that can be calculated from the signal. At every level c it is thus possible to associate to an input signal in a unique way a holomorphic function that describes the output signal in terms of the physiological parameters. The phase and the logarithm of the amplitude are used in the description of the experiments and they are omnipresent in all the representations of the auditory pathway.

In themselves they are of limited significance, because they are not coded as such. What really is essential in any cochlear or in any neural model are the changes of these quantities, both with respect to time and with respect to the place. The structure equations precisely relate the local and temporal derivatives of phase and logarithm of amplitude. The geometry of the cochlea implicitly is inherent in the extremality property of the basilar membrane filter. But in the structure equations this only shows in terms of the constants.

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## Wavelets in Neuroscience

The implicit appearance of the tonotopic axis is an expression of the basic invariance principle that stands at the outset of all considerations. The structure equations clearly exhibit the dichotomy in cochlear signal processing. The signals can either be analyzed in terms of their phase or in terms of their amplitudes.

Then the second equation. Inserted in the first equation. Conversely, the complete knowledge of amplitude information determines the phase information. From an abstract point of view, phase information and amplitude information each individually contain the full information of the signal. In the auditory pathway both phase and amplitude information is being processed. It is commonly assumed that phase information dominates in the low frequency range and amplitude information in the regions that process high frequencies.

The equations tell us that phase processing and amplitude processing are equally significant. Complete information on derivatives with respect to the position gives complete information on time derivatives - and vice versa. The structure equations are so simple that they can be solved in explicit mathematical terms. In its complex form the structure equation is the linear inhomogeneous equation. The general solution of an inhomogeneous linear differential equation can be presented as the linear combination of a particular solution any chosen solution of the equation and the general solution of the associated homogeneous differential equation.

A particular solution log Y p of the above complex equation is the function. Its distinguished feature is the time independence.

It follows that the general solution log Y is of the form. As a side remark, observe that the complex structure equation is obtained from the basic equation. With the variable change.

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The function G z is uniquely defined up to a constant. The situation can now be summarized as follows: An incoming signal f t gives rise to a family of analytic wavelet transforms. The functions Zf approximately satisfy the complex structure equation. The solutions. They can in fact be determined directly from the Fourier transform of the incoming signal f t. Since the system is linear, the superposition principle holds:. In the following section these are identified as the functions.

This is in fact the correct linear approximation.

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The approximate value for log Z f therefore is an over estimation. Together with. The outcome depends on the ratio between the amplitudes of the coefficients. But it should be warned that the approximation is valid only in the frequency interval specified above. Both the amplitude and the phase derivatives show this oscillation.

No doubt, the distinguished feature of a violin sound is the extraordinary big number of harmonics in the frequency spectrum. It is not uncommon to observe around twenty harmonics at an intensity level at which the sounds can still be detected. Except possibly for the first few, the harmonics show a gradual decrease in amplitude with some oscillation.