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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The “Sonar Hopf” cochlea is a recently much advertised engineering design of an auditory sensor. We analyze this approach based on a recent description by its inventors Hamilton, Tapson, Rapson, Jin, and van Schaik, in which they exhibit the “Sonar Hopf” model, its analysis and the corresponding hardware in detail. We identify problems in the theoretical formulation of the model and critically examine the claimed coherence between the described model, the measurements from the implemented hardware, and biological data.

The description of biological processes in terms of mathematics and physics is a challenge of great importance as an advanced step in explaining natural phenomena. Well worked-out examples connecting biology, nonlinear physics or mathematical bifurcation theory are relatively rare, implying that each new example is of great interest. The mammalian hearing system provides in this context an important example. The recent article “

In many applications, a differential equation

This system admits a fixed point at (^{iωt}

Rewriting the forced variant of this equation [

From the cubic term, the authors conclude “^{2} ∈ , the promised Hopf bifurcation will not occur. Explicitly, the bifurcation analysis of the system in the absence of drive yields

This system has a fixed point

The first fixed point at the origin has the eigenvalues _{a1,2} = _{0}. Increasing _{0} ≥|

For observing the Hopf bifurcation, the term | ^{2} | ^{3}, is essential. This problem propagates through several works of the authors, leading to the conclusion that its origin cannot be accidental.

The aim in the design of natural and artificial sensors is mainly the implementation of signal sensitivity,

In addition to this local notion of bifurcation used, also a global notion exists. In the Hopf case, a bifurcation from a stable fixed point to a stable limit cycle of initially zero amplitude and fixed frequency is called “supercritical”. If we observe the collapse of an unstable limit cycle with a stable fixed point at the origin, by which the system is expelled to second, larger and stable limit cycle, this would be called “subcritical”. The second limit cycle also depends on the bifurcation parameter. Reducing the bifurcation parameter, the limit cycle generally becomes smaller, until it breaks down and the system falls on the stable solution at the origin, introducing hysteresis in this way. The problem in the terminology as used is that local and system level (

“_{0}_{0} + Δ

When the “Sonar Hopf” system is said to be “

Even more importantly, a genuine Hopf bifurcation might not necessarily be involved in the described “Sonar Hopf” cochlea framework. In the approach, a feedback system is used to vary the quality factor of the basilar membrane at each the intensity (and frequency) of the input signal [Note: In contrast to how the authors of [

From a general discussion point of view, it is true that by means of an appropriately operated feedback loop, a large class of nonlinear dynamical systems can be brought from fixed-point into oscillatory behavior—some of them by means of a Hopf bifurcation, but not necessarily—as well as from chaos into regular oscillations, and so on [

More specifically, the “Sonar Hopf cochlea” extends a resonator model by positive feedback, with the objective to use a parametrized positive feedback function so that a limit cycle emerges by means of the Andronov–Hopf mechanism. Unfortunately the discussion of the Andronov–Hopf bifurcation in feedback systems as provided is fragmentary, partly wrong and historically inadequate. The key point made in [

The analysis provided up to this point does not reach beyond a steady state, pure tone analysis. If a feedback mechanism is needed to keep it at the desired bifurcation location, even if the amplification had the correct form, this would not be instantaneous. Different unwanted effects all known from delay control mechanisms must be expected. The dynamic response, a very essential feature of a cochlear device, cannot safely be retrieved from this oversimplified picture. Whereas the ultimate mathematically precise description would probably have to be given in terms of spectra of resonances [

From an engineering point of view, ignoring relevant mathematic details in this description, the “Sonar Hopf” construct is an example of well-known feedback systems with limit cycles, known under the name of Lure systems. For a systematic engineering approach to Lure type systems with nonlinear feedback and limit cycles, the theory of Sepulchre and Stan [

The feedback system of _{k*}_{k}

We feel that for a “scholarly” approach as taken in the paper, these relations fundamental for the understanding of the “Sonar Hopf” concept should not be ignored. We also note that the electronic literature notion of an “

The main claim then is that the response of the constructed electronic device is faithfully represented by _{eff}

The concluding statement “

For the reader it would also have been interesting to be able to compare the “Sonar Hopf” implementation against a direct implementation of active Hopf hair cells amplifiers, such as the earlier electronic Hopf cochlea [

Recently, this device was used to gain insight into the perception of mammalian hearing [

Our analysis shows that the close correspondence between the “Sonar Hopf” device and behavior of a genuine system at a Hopf bifurcation point claimed in [

We warmly acknowledge the contributions provided by R. Mathis for this manuscript.

Vector field plot with isoclines (red lines) and fixed points (filled circles). Two solutions emanating from different initial conditions (green, blue) illustrate that a bifurcation from a central previously stable, fixed point to two stable fixed points has occurred (_{0} = 1,

Equivalent representations of the Lure SISO nonlinear system described.

Forcing the Hopf bifurcation with an integrator in the feedback loop and

Blue: Envelope of the exponential decay from larger towards smaller stimulation according to

(a) Original measurements (2004) from a cochlea with directly implemented Hopf amplifier outer hair cells, reproduced in [