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Neurons can detect weak target signals from complex background signals through stochastic resonance (SR) and vibrational resonance (VR) mechanisms. However, random phase variation of rapidly fluctuating background signals is generally ignored in classical VR or SR studies. Here, the rapidly fluctuating background signals are modeled by bounded noise with random rapidly fluctuating phase derived from Wiener process. Then, the influences of bounded noise on the weak signal detection are discussed in the FitzHugh–Nagumo (FHN) neuron. Numerical results reveal the occurrence of bounded noise-induced single- and biresonance as well as a transition between them. Randomness in phase can enhance the adaptability of neurons, but at the cost of signal detection performance so that neurons can work in more complex environments with a wider frequency range. More interestingly, bounded noise with appropriate parameters can not only optimize information transmission but also simultaneously reduce energy consumption. Finally, the potential mechanism of bounded noise is explained.

Neurons are the basic information processing devices of the nervous system, and they work in very noisy and complex environments [

Nonlinear systems are driven by two periodic forces: a low-frequency (LF) one (considered as a signal) and an HF one (considered as a carrier) [

However, most previous studies mainly paid much attention to individual or combined effects of amplitude and frequency of HF force and did not consider the effect of random phase variation on the response of excitability systems [

It is very important to find a model that appropriately describes the effects of noise when investigating a dynamical system under random perturbation. Gaussian noise is usually adopted in many cases for convenience of analysis. It is worth mentioning that the widely used Gaussian noise has the probability of taking very large values, which violates the notion that real physical quantities in real systems always take values in bounded intervals [

Many bounded noise-induced phenomena, such as resonance [

As a simplified variant of the famous Hodgkin–Huxley model, the FHN model is simple but can represent the major characteristics of the electrophysiological activity of neurons. The FHN model is described by the following coupled equations [

In (

Time series of HF bounded noise

In this study, the Fourier coefficient (also called response measure and synchronization factor)

We firstly discuss how the intensity

Time series of

To further confirm the above results quantitatively and investigate the effect of angular frequency ratio

Dependencies of response measure

Next, we investigate how the amplitude

Dependencies of response measure ^{0.56}. (b) Landscape and contour plot of ^{0.56}.

In what follows, we investigate how angular frequency ratio

Contour plots of

These phenomena raise a question: what is the mechanism of HF bounded noise? To answer this problem, we plot the time series of the FHN model during a period of LF signal in Figure

The potential mechanism analyses of HF bounded noise.

Real neurons are often surrounded in very noisy environments. Neurons are requested to detect generally feeble target signals (or stimuli) from noisy background signals. It is very interesting and significant to elucidate the mechanisms of weak signal detection in an excitable nervous system. A number of efforts have been contributed to this area. But the understanding of these mechanisms for a nervous system subjected to HF bounded noise has remained incomplete. Bounded noise with constant amplitude and time-varying random phase not only is a versatile model for simulating irregularity included in real-world external signals, but also meets the fact that the real physical quantities in a real system are always bounded.

In this study, we study in detail the effect of HF bounded noise on the detection of a weak LF periodic signal in the FHN neuron. It is found that HF bounded noise has a facilitation or repression effect on synchronization between the input signal and the output of the system. Moreover, the numerical results also demonstrate the occurrence of HF bounded noise-induced single- and biresonance as well as a transition between them. Through this resonance mechanism, the response of neurons in random rapidly fluctuating background signals to the external subthreshold target signal can be significantly enhanced. The numerical results also reveal that randomness in phase (i.e.,

The authors declare that they have no conflicts of interest.

Yuangen Yao implemented the project, analyzed data, and wrote the manuscript. Lijian Yang, Canjun Wang, Quan Liu, Rong Gui, Juan Xiong, and Ming Yi extensively shared and discussed results. All authors read and approved the final manuscript.

This work was supported by the National Natural Science Foundation of China (Grants nos. 31601071, 11675060, 91730301, and 11205006), the Huazhong Agricultural University Scientific and Technological Self-Innovation Foundation Program (Grant no. 2015RC021), and the Natural Science New Star of Science and Technologies Research Plan in Shaanxi Province of China (Grant no. 2014KJXX-77).