Dynamic hebbian learning in adaptive frequency oscillators

TLDR: A learning rule for oscillators which adapts their frequency to the frequency of any periodic or pseudo-periodic input signal, which is easily generalizable to a large class of oscillators, from phase oscillators to relaxation oscillators and strange attractors with a generic learning rule.

About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 2006-04-15 and is currently open access. It has received 344 citations till now. The article focuses on the topics: Synchronization networks & Learning rule.

Figures

Fig. 2. Plot of the evolution of ω for four different values of . Here we set µ = 1, x(0) = 1 and y(0) = 0, the perturbing force is F = cos(30t). For every value of , we see that ω converges to 30, which is the frequency of the input signal. Therefore, the system is able to learn the frequency of the input signal. We also notice that controls the convergence rate, the higher it is, the faster the system learns.

Fig. 9. We show the adaptation of the Van der Pol oscillator to the frequencies of various input signals: (a) a simple sinusoidal input (F = sin(40t)), (b) a sinusoidal input with uniformly distributed noise (F = sin(40t) + uniform noise in [−0.5, 0.5]), (c) a square input (F = square(40t)) and (d) a sawtooth input (F = sawtooth(40t)). For each experiment, we set = 0.7 and α = 100 and we show three plots. The right one shows the evolution of ω(t). The upper left graph is a plot of the oscillations, x , of the system, at the beginning of the learning. The lower graph shows the oscillations at the end of learning. In both graphs, we also plotted the input signal (dashed). In each experiment, ω converges to ω ' 49.4, which corresponds to oscillations with a frequency of 40 rad s−1 like the input and thus the oscillator correctly adapts its frequency to the frequency of the input.

Table 1 This table summarizes the maximum errors of the simulation for the first and second order approximations discussed from Fig. 3

Fig. 3. Results of the simulation of the first and second order approximations. For a simple input, here F = sin(40t), = 0.9, initial conditions are t0 = 0, w0 = 30. The upper figure shows the evolution of the ω variable for the initial dynamical system (Eq. (38)), the first order approximation ω (t) and the second order approximation ω 2 (t). The lower figure shows quadratic errors between the initial system and the two approximations, for the evolution of ω.

Fig. 8. This figure shows the convergence of ω for several initial frequencies. The Van der Pol oscillator is perturbed by F = sin(30t), with coupling = 0.7, α = 50. We clearly see that the convergence directly depends on the initial conditions and as expected the different kinds of convergence correspond to the several entrainment basins of Fig. 7.

Fig. 1. We illustrate the coordinate system in which synchronization is most naturally discussed. The figure shows an arbitrary limit cycle. The system is strongly damped in the direction perpendicular to the limit cycle Eer and marginally stable in the direction tangential to the limit cycle Eeφ . This is the reason for the structurally stable limit cycle in the first place and allows for a resetting of the phase on the other hand. Note that the two-dimensional representation is always valid for discussing a limit cycle since there exists always a two-dimensional manifold which contains the limit cycle. Refer to the text for a discussion of the perturbation EP .

Frequently asked questions

Q1. Why can a perturbation affect the phase of the oscillator?

Due to the stability properties of a limit cycle system a perturbation can in the long term only affect the phase of the oscillator. 

Q2. What are the contributions mentioned in the paper "Dynamic hebbian learning in adaptive frequency oscillators" ?

In this paper, the authors propose a learning rule for oscillators which adapts their frequency to the frequency of any periodic or pseudo-periodic input signal. 

Q3. What future works have the authors mentioned in the paper "Dynamic hebbian learning in adaptive frequency oscillators" ?

The possible relevance to biology has to be investigated in further research. 

Q4. Why does the frequency spectrum contain an infinite number of frequency components?

(50)Because of the relaxation property of the oscillator, the frequency spectrum contains, in addition to the frequency of the oscillations, an infinite number of frequency components. 

Q5. What is the effect of an increase in the frequency spectrum of an oscillator?

an increase in the complexity of the frequency spectrum of an oscillator also generates side effects, like adaptation toward synchronization of the higher frequency components of the oscillator and the frequency of an input signal. 

Q6. What is the main implication of adaptive oscillators?

By using adaptive oscillators, one could build CPGs that can dynamically adapt their frequencies and consequently, create a desired pattern of oscillations. 

Q7. How many frequency components of the oscillator are phase-locked?

The authors also explained that the oscillator may phase-lock its higher frequency components, as these frequencycomponents are equally spaced, one should expect phase-lock for fractions of the frequency of the perturbing force. 

Q8. What is the r variable required to keep the oscillator coupled with the input?

The authors must also note that it is required to keep the oscillator coupled with the input, because it is the evolution of φ(t), i.e. change of frequency correlated with ω̇, that enables adaptation in Eq. (7). 

Q9. What is the average contribution of the function E6?

The integration of E6 gives a function oscillating with some frequency but with its amplitude varying because of the t term, the average contribution of thisfunction is zero. 

Q10. What is the effect of a small perturbation on the oscillator?

Especially in a small neighborhood of the limit cycle a small perturbation can only affect the phase strongly if it perturbs the oscillator in the direction tangential to the limit cycle. 

Q11. What is the general learning rule for the hopf oscillator?

The authors have the general learning rule ω̇ = − F y√x2 + y2 . (55)Only the sign in front of F may change according to the orientation of the flow of the oscillator in the phase space. 

Q12. How many errors are there in the numerical integration of the dynamical system?

It must be noted that numerical integration of the dynamical system is done with an embedded Runge–Kutta–Fehlberg(4, 5) algorithm, with absolute and relative errors of 10−6. 

Q13. Why are there so few attempts to model nonlinear oscillators?

But these attempts are so far limited to very simple classes of oscillators, equivalent to phase oscillators, mainly because this is the only class of oscillators that can be analytically studied and for which convergence can be proved, when adding adaptivity to the system. 

Q14. Why do nonlinear oscillators fail to synchronize with external input?

This is mainly the case because oscillators lack plasticity, they have fixed intrinsic frequencies and cannot dynamically adapt their parameters.∗ 

Q15. Why do the authors call their adaptive mechanism dynamic Hebbian learning?

The authors call their adaptive mechanism1 dynamic Hebbian learning because it shares similarities with correlation-based learning observed in neural networks [11]. 

Q16. What is the learning rule for the Van der Pol oscillator?

So the learning rule is ω̇ = F y√x2 + y2 . (51)The authors do not give an analytical proof of convergence for the Van der Pol oscillator because to use perturbation methods, as the authors did for the Hopf oscillator, the authors need to know the solution for the unperturbed Van der Pol oscillator, but to the best of their knowledge, only implicit solutions are known [7] and thus such a proof is beyond the scope of this article. 

Q17. What is the evolution of the parameter controlling the frequency of the adaptive oscillators?

The evolution of the parameter controlling the frequency of the adaptive oscillators that the authors discussed can be viewed as the correlation between the phase of the oscillator and the input signal. 

Q18. What is the proof for the adaptive hopf oscillator?

the authors only give a proof for the adaptive Hopf oscillator and even if the authors numerically show that more complex adaptive oscillators can be designed, a general rigorous proof for a larger class of oscillators is still missing.