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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 [

The above-mentioned techniques can stabilize unstable nodes (UFP with even number of real positive eigenvalues

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.

To be specific we consider an array of FitzHugh–Nagumo (FHN) oscillators [

An array of mean-field coupled (star coupling) oscillators is sketched in Figure

Network of mean-field coupled oscillators,

The array in Figure

When an RC tracking filter is applied to the Ctrl node (Figure

For

System (

The corresponding characteristic equation, obtained from differential equation (

The fixed point of the mean field is stable, if the real parts of all three eigenvalues

Real parts of the eigenvalues,

In addition, the necessary and sufficient conditions of stability can be estimated analytically from the Hurwitz matrix

We start the analysis with

Analytical solution of (

System (

Waveforms from (

Fixed point spectra

It is worth noting that stabilization of the UFP can be achieved in the unsynchronized (uncoupled) array (Figure

In Figure

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 [

An individual FHN type oscillator is presented in Figure

Circuit diagrams. (a) FHN asymmetric electronic oscillator. OA is an operational amplifier, for example, NE5534,

The negative resistor “

Negative impedance converter: active circuit implementation of the negative resistor “

Experimental waveforms, the mean-field voltage

Experimental waveforms have been taken by means of a digital camera from the screen of a multichannel analog oscilloscope and are shown in Figure

Similarly to the numerical results (Figure

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) [

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 [

Specifically, in [

In addition, a recently found phenomenon of oscillation quenching in the systems of coupled nonlinear oscillators is worth mentioning [

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 [

Our future work will focus on the investigation of an array of weakly coupled FHN oscillators (

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 (