Coherence and stochastic resonance in threshold crossing detectors with delayed feedback
Introduction
Feedback is an important concept in many areas of science, particularly in physics [1]. The temporal evolution of a system with delayed feedback is dependent not only on the current value of state variables but also on their past values. Delayed feedback occurs in many nonlinear systems such as optical resonators [2], [3], [4], chemical reactions [5], biological [6], [7] and artificial [8] neural nets, and genetic and other physiological control systems [9], [10]. The importance of delayed feedback has further been highlighted in chaos control [4], [11], chaos communication [3], and anticipatory chaos control [12]. Memory effects due to feedback strongly affect the dynamics when the associated delays are commensurate with or longer than the time scales of the system without feedback.
Many such feedback systems, including excitable biological or optical systems, further involve sharp activation functions, or “thresholds”, and are driven by noise [13] and one or more periodic inputs. The effect of noise on the switching properties of threshold systems has received much interest in the context of coherence resonance (CR) [15] and stochastic resonance (SR) [14], [16], [17], [18], [19]. In particular, there have been very recent studies of the enhancement by noise of the oscillations in a network of delayed-coupled oscillators [21] and in a bistable discrete-time stochastic map [22], of stochastic resonance in a non-Markovian system [23] and of control of noise-induced motion in relaxation oscillators using delayed feedback [24]. Recent studies have also highlighted the potential usefulness of instantaneous feedback on SR [25], [26], [27].
The novel synergistic effects of noise and delayed feedback on the synchronization of threshold elements to external input are the subject of our Letter. We focus on single and parallel arrays of threshold-crossing detectors (TCD). Each such thresholding element produces a stereotyped brief “spike” at the instants when a threshold level is exceeded, thus capturing in a simple manner the key nonlinear features of excitable systems, biological, optical or other. Our results apply however to a large class of noisy systems with delayed feedback and external forcing. We find that the interaction of an intrinsic mode due to delayed feedback with external forcing and noise leads to a strong amplification of subthreshold input when the input period matches the delay. Another way of describing this behavior is to say that the TCD acts as a strong frequency selector, i.e., it will tend to resonate with and thus amplify a narrow range of frequencies from a broadband input.
Our Letter is organized as follows. In Section 2, we present the model used to study threshold elements with noise, deterministic input and delayed feedback. Section 3 shows numerical results on the behavior of a single threshold system without feedback (“open-loop”) and with feedback (“closed loop”), and in particular, on the behavior of the signal-to-noise ratio (SNR) as the input noise intensity is varied. This section includes behavior of the system before and after sinusoidal forcing, which reveals the presence of reverberating oscillations. Section 4 then focuses on eliminating these reverberatory oscillations which are enhanced by a harmonic input of a certain frequency and outlast its presence. This is done by considering arrays of TCD's, each with their individual feedback. An approximate analytical characterization of the reverberations and dependence of the SNR on input frequency and feedback delay follows in Section 5.
Section snippets
Model TCD with feedback
The TCD is a nonlinear system that generates a brief spike—which for our purposes will be a Dirac delta function—whenever its input crosses a fixed threshold. In our study, we will confine the spike generation to crossings in the positive direction only. Also, we will consider a TCD driven by lowpass-filtered zero-mean Gaussian white noise, a sinusoidal input, and pulses (“spikes”) from the output of this same TCD at a previous time (Fig. 1(a)). This latter driving force is the delayed
Resonances for a single TCD
We first consider the case where a single TCD is driven by noise and periodic input (subthreshold throughout our study) as well as feedback spikes. The ratio of the spectral power at the forcing frequency to the noise floor at was computed from the averaged spectrum. The resulting signal-to-noise ratio or SNR is plotted as a function of the noise intensity D in Fig. 3 for open-loop and closed loop conditions. For open-loop (), the SNR exhibits a unimodal curve, i.e., stochastic
Parallel network of TCDs with feedback
This ongoing reverberation seen in Fig. 3, Fig. 4, Fig. 5 before and after the signal can interfere with signal detection. Reverberation can be eliminated, all the while significantly increasing the SNR, by using a summing network of parallel feedback loops (Fig. 6) driven by the same periodic signal. In our model, the noise in one loop is uncorrelated with that in other loops, and a given TCD receives feedback only from itself. By analyzing the superposed spike trains from all TCDs, we observe
Analysis of the resonances
A full theoretical analysis of these SNR data is not yet possible, due to the nonlinear and non-Markovian nature of these stochastic delayed dynamics, and to the multistability it may exhibit with respect to initial conditions on [20]. It is possible however to analyze certain novel dynamical features of this system. To understand the resonant nature of the system in Fig. 1 and its implications for CR and SR, we formulate a dynamical equation for , the voltage input right at the
Acknowledgements
This work was funded by Defeating Deafness (R.M.) and NSERC Canada (A.L.).
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