Advanced Topics in Modeling, Bifurcation Analysis, and Control Theory of Complex SystemsView this Special Issue
Stabilization of a Network of the FitzHugh–Nagumo Oscillators by Means of a Single Capacitor Based RC Filter Feedback Technique
We suggest employing the first-order stable RC filters, based on a single capacitor, for control of unstable fixed points in an array of oscillators. A single capacitor is sufficient to stabilize an entire array, if the oscillators are coupled strongly enough. An array, composed of 24 to 30 mean-field coupled FitzHugh–Nagumo (FHN) type asymmetric oscillators, is considered as a case study. The investigation has been performed using analytical, numerical, and experimental methods. The analytical study is based on the mean-field approach, characteristic equation for finding the eigenvalue spectrum, and the Routh–Hurwitz stability criteria using low-rank Hurwitz matrix to calculate the threshold value of the coupling coefficient. Experiments have been performed with a hardware electronic analog, imitating dynamical behavior of an array of the FHN oscillators.
A large number of adaptive control techniques have been developed so far to stabilize unstable fixed points (UFP) of dynamical systems. These include derivative control [1–3], tracking filter technique, based on either low-pass or high-pass first-order RC filters [4–9], and notch filter technique that employs two second-order Wien-bridge filters with the incommensurate resonance frequencies . The delayed feedback technique, though originally designed to control chaos, that is, to stabilize unstable periodic orbits [11, 12], under appropriate setting of parameters can stabilize the UFP as well [5, 6, 13–19].
The above-mentioned techniques can stabilize unstable nodes (UFP with even number of real positive eigenvalues , e.g., and no imaginary parts of the eigenvalues, i.e., ) and unstable spirals (UFP with even number of complex eigenvalues with positive real parts, e.g., ). However, the methods fail to stabilize saddle-type UFP, more specifically, UFP with an odd number of real positive eigenvalues, for example, , . To solve the problem of the odd number limitation, Pyragas and coauthors proposed to use an unstable first-order filter [20, 21]. It was an elegant idea to fight one instability by means of another instability. The method was demonstrated for a variety of mathematical models and experimental systems [20–23]. Later an unstable filter control was developed to stabilize saddle-type UFP in conservative and weakly damped systems [24–26] also under the influence of delay (inertia) in the feedback loop of the controller.
The first-order RC filters, based on a single capacitor, as well as other methods developed for stabilizing the UFP have been applied to single oscillators only. The question thus arises: can a single capacitor stabilize a network of oscillators? The answer depends on the properties of the network. Evidently, if the oscillators in the array are uncoupled or weakly coupled, a single capacitor is insufficient to control the entire network. Each individual oscillator should be provided with a separate controller. Such solution is impractical for applications. However, when the oscillators are coupled strongly enough, one could expect that it is possible to stabilize the entire network using a single controller.
In this paper, we demonstrate the possibility of stabilizing the network analytically, numerically, and experimentally.
2. Mathematical Model and Its Analysis
To be specific we consider an array of FitzHugh–Nagumo (FHN) oscillators , also known in literature as Bonhoeffer–van der Pol (BVP) oscillators [28, 29]. The FHN (or BVP) oscillator actually is a simplified version of the Hodgkin–Huxley (HH) oscillator, imitating the dynamics of spiking neurons . A set of the FHN oscillators is described byHere is a nonlinear function approximated by a three-segment piecewise linear function  In (2) . Therefore, is an essentially asymmetric function, in contrast to common FHN or BVP cubic function [27–29]. The bias parameters in (1) are intentionally set to be different for each individual oscillator thus making them nonidentical units.
An array of mean-field coupled (star coupling) oscillators is sketched in Figure 1.
The array in Figure 1 is given by the 2N-dimensional systemHere is the mean value of the variables
When an RC tracking filter is applied to the Ctrl node (Figure 1) of the network the overall system becomes ()-dimensional systemThe cut-off frequency of the filter should be low () to ensure tracking the state of the system under control. Note that, in comparison with (3), here in the first equation the mean is replaced with its filtered variable . The case of a single oscillator (, yielding the 3-dimensional system) has been investigated in  Analysis of 2N-dimensional system (3) and ()-dimensional systems (5) is very complicated. Therefore, we consider a mean-field approach. The mean-field variables are obtained by directly averaging the variables and and the parameters over all oscillators in (3) and (5), respectively: Note that (6a) lacks the term , since .
For and the steady-state solutions of (1), (6a), and (7a) are presented by the following fixed points:Note that the RC tracking filter in (7a) does not change the position of the fixed point: compare (10) with (9). The values given by (10) are independent on , since and the coupling term () in (7a) vanishes for the fixed point. The fixed points (9) and (10) are similar in form to fixed points of the individual uncoupled oscillators (8), except the fact that the fixed points (9) and (10) are single points, whereas (8) yields points in the phase space.
The corresponding characteristic equation, obtained from differential equation (11) using a standard procedure, has the following algebraic form:where , , .
The fixed point of the mean field is stable, if the real parts of all three eigenvalues are negative. Equation (12) has been solved numerically for different values of coupling coefficient k (Figure 2) and the threshold value , for which the largest , is found.
In addition, the necessary and sufficient conditions of stability can be estimated analytically from the Hurwitz matrixAccording to the Routh–Hurwitz stability criterion if all diagonal minors of the -matrix are positive
We start the analysis with . Since should be positive according to the second inequality, the third inequality for can be simplified to . This can be further simplified to , because by definition. The inequality is always satisfied, since it was used to derive the fixed points (8)–(10). Consequently, we are left with the first and the second inequalities in (14). We define the threshold requiring that for the both minors, and are positive. The first minor is rather simple and is readily obtained:For the parameter values given in Figure 2, . The second inequality in (14) is more cumbersome and yields quadratic equation with respect to . where , , .
Analytical solution of (16) gives two different values: and . Finally, we obtain = 3.23, which is in an excellent agreement with the numerical result, derived from Figure 2(b), where cross the abscissa axis at .
3. Numerical Results
System (5) have been solved numerically using MATHEMATICA, version 9.0 software package. The results are presented in Figure 3. The waveforms of the mean variable (Figure 3(a)) and of the individual oscillators, say (Figure 3(b)), look nearly the same, since the array is synchronized. Other variables , not shown in Figure 3, have similar form with only small phase shifts, as expected for nonidentical elements. The main difference is in the UFP values due to different bias parameters . The differences between the fixed points of the individual oscillators () are brought out by the fixed point spectra, presented in Figure 4.
It is worth noting that stabilization of the UFP can be achieved in the unsynchronized (uncoupled) array (Figure 3(c)) applying the whole control term in (5) at with a sufficiently large coefficient . However, this feature is important from a theoretical point of view only. The term in (5) means that it controls every individual oscillator “.” It is easy to do in a mathematical model, but in practical (experimental) situations, it requires direct access to every neuronal oscillator. Moreover, from a practical point of view, especially for a possible application to neuronal systems there is no need to stabilize the UFP, if the oscillators are not synchronized. Unsynchronized oscillators yield low mean field, as evidenced by the left hand part () of the plot in Figure 3(c). For larger arrays, say , the mean field becomes even lower. We recall that in this paper we consider the case of coupled and synchronized oscillators.
In Figure 4(a), is in a good agreement with the value calculated from (9). In the case of coupled and stabilized array (Figure 3(b)) the spectrum is narrower in comparison with the case of uncoupled oscillators due to strong interaction between oscillators. In Figures 3(a), 3(c), and 4(b). It well coincides with the value of in Figure 4(a).
4. Experimental Setup
The experiments have been carried out using an electronic analog array, composed of 30 mean-field coupled FHN type oscillators and described in detail elsewhere . This electronic array has been employed earlier to implement experimentally both the repulsive synchronization [33, 34] and the mean-field nullifying techniques [34, 35].
An individual FHN type oscillator is presented in Figure 5(a). Dimensionless variables , , , and t, as well as the parameters , b, c, d, , and introduced in (5), are related to the electrical values of the analog circuits (Figure 5) in the following way: Here is the breakpoint voltage of the forward current-voltage characteristic of the diodes.
The negative resistor “” in Figure 5(b) has been implemented by means of the negative impedance converter (Figure 6) . The input resistance of the NIC . The capacitor with negative resistor “” in series should not be confused with an unstable RC filter, employed to stabilize saddle-type UFP [25, 26]. Here “” simply compensates the positive buffer resistor of the network. The RC tracking filters are actually composed of the coupling resistors (Figure 1) and the capacitor ; see also definition of the cut-off frequency in (18).
5. Experimental Results
Experimental waveforms have been taken by means of a digital camera from the screen of a multichannel analog oscilloscope and are shown in Figure 7.
Similarly to the numerical results (Figure 3) the experimental waveform in Figure 7 exhibits negative stabilized state (the nonzero value is due to the dc bias in Figure 5(a)). Whereas the control current in Figure 7 becomes vanishingly small after a relatively short transient process.
The investigation performed here is not an end in itself. The purpose of the study is the search of practical techniques inhibiting activity of neuronal arrays. It is widely believed that strong synchrony of spiking neurons in the brain causes the symptoms of Parkinson’s disease .
One of the simplest methods to damp spiking neurons is the external stimulation of certain brain areas with strong high frequency (about 100 to 150 Hz) periodic pulses. It is a conventional clinically approved therapy for patients with the Parkinson’s symptoms, so-called deep brain stimulation (DBS) [38–40]. Unfortunately, the DBS treatment is often accompanied with undesirable side effects. In recent papers [41–43], it has been demonstrated that the high frequency forcing eventually stabilizes the UFP of the neuronal oscillators in case of HH, FHN, and other neuronal models. Two shortcomings of the DBS have been emphasized . Firstly, though the spiking neurons are suppressed, relatively high amplitude (10 to 20%) high frequency artifact oscillations are observed. Secondly, the fixed points of the membrane voltages are essentially moved from their natural values because of the rectifying effect in the cells . This can be a reason of the side effects.
A number of more sophisticated methods to avoid synchronization of interacting oscillators in general and more specifically with the possible application to neuronal arrays have been described in literature, for example [33, 37, 44–54].
Specifically, in  suppression of synchrony of coupled oscillators by means of a passive oscillator is described. The controller is a four-terminal third-order device with separate recording and field application electrodes. The feedback loop contains a second-order damped oscillator, an integrator, an adder, and two amplifiers. Our controller is much simpler. It is a two-terminal first-order device with the same voltage sensing and current application electrode. The controller contains a single capacitor and a negative impedance converter (NIC) based on a single operational amplifier. The NIC is not necessary, if the buffer resistance is small.
In addition, a recently found phenomenon of oscillation quenching in the systems of coupled nonlinear oscillators is worth mentioning [55–58]. It can manifest via two different mechanisms, the so-called oscillation death and amplitude death. The effect, in particular the oscillation death, can be perspective for oscillation suppression in neuronal disorders, such as the Parkinson’s disease and essential tremor. This type of oscillation quenching depends on the intrinsic parameters of the individual oscillators, but even more on the way and the strength of coupling. The parameters of the oscillators and the parameters of coupling can be easily controlled in the artificially made physical, chemical, electronic, and so on systems. However, these parameters are difficult to tune in natural, for example, biological, systems. Therefore, the techniques employing external feedback loops seem to be advantageous solutions.
An array of coupled neuronal type oscillators, specifically the FitzHugh–Nagumo cells, can be stabilized by means of a single capacitor based RC filter feedback technique. The feedback signals become vanishingly small, when the UFP is stabilized, similarly to the feedback suppression of synchrony described in . This can be an advantage over the nonfeedback techniques, for example, the DBS employing external high frequency periodic forcing.
Our future work will focus on the investigation of an array of weakly coupled FHN oscillators (k < ), when stabilization of the UFP is impossible. We hope that a single capacitor based RC filter can desynchronize oscillators in the array, somewhat likewise to the repulsive coupling [33, 34] and the mean-field nullifying [34, 35] techniques.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors thank Dr. Nikolai Rulkov for critical discussion of the results and especially for the suggestion of emphasizing the effect of control at the level of individual elements and also for drawing their attention to the fact that there is no reason for the array, given by (5), to oscillate synchronously before the control term is applied. The authors are also grateful to Professor Michael Rosenblum for drawing their attention to the fact that, in paper , along with the suppression of synchrony in an ensemble of interacting units, stabilization of an active oscillator by means of a passive oscillator, included in the feedback loop, is also described.
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