Complexity

Volume 2018 (2018), Article ID 5632650, 10 pages

https://doi.org/10.1155/2018/5632650

## Subthreshold Periodic Signal Detection by Bounded Noise-Induced Resonance in the FitzHugh–Nagumo Neuron

Correspondence should be addressed to Ming Yi

Received 27 October 2017; Revised 12 January 2018; Accepted 24 January 2018; Published 20 February 2018

Academic Editor: Dimitri Volchenkov

Copyright © 2018 Yuangen Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Neurons are the basic information processing devices of the nervous system, and they work in very noisy and complex environments [1, 2]. The roles and environmental features of neurons demand them to reliably detect extremely weak extrinsic target stimuli (or signals) from noisy background signals with certain specialized mechanisms [1]. Therefore, a pivotal and interesting problem in neuroscience is to explore the relation between noise and signal detection and to look for signal detection mechanisms in the nervous system [2]. In the last century, many efforts had revealed the constructive role of noise. While investigating the periodically recurrent ice ages, Benzi and collaborators firstly discovered noise-induced stochastic resonance (SR) where an external weak signal can be enhanced and optimized by noise [3]. Later, it was discovered that noise can also induce coherent oscillations in a simple two-dimensional system even without an external signal [4]. This noise-induced coherent oscillation phenomenon (called coherence resonance) also appears in the excitable FitzHugh–Nagumo (FHN) system subjected only to noise [5]. In fact, noise-induced resonance can be widely observed in various neuronal models, such as Hodgkin–Huxley (HH) [6], Hindermarsh–Rose (HR) [7], and Morris–Lecar (ML) [8]. In particular, in the last two decades, many efforts revealed that sensory systems can use noises to enhance their sensitivity for detecting weak periodic signals [9–11]. Therefore, noise-induced enhancement of response is considered to be a possible mechanism for signal detection.

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) [12]. The HF force with an appropriate strength and frequency may augment the processing of a weak LF signal [13]. This phenomenon is called vibrational resonance (VR) [14]. It should be mentioned that two-frequency driving systems can be found in many different fields, such as the communication field where HF carrier waves are usually used to enhance the target signal [13], neuroscience [15], acoustics [16], and laser physics [17]. The influences of VR on signal detection and amplification have been widely investigated in these systems subjected to LF and HF driving forces. For example, Ullner et al. studied the influence of additive noise and the interplay between SR and VR and found that the response of an excitable system to a low-frequency signal can be enhanced by high-frequency driving [12]. Deng et al. investigated the effect of HF force on the detection of a subthreshold low-frequency signal in neuron populations with different topology structures and hybrid synapses [18, 19]. Yao and Zhan found that signal detection and transmission can be effectively enhanced by VR in one-way coupled bistable systems [20] or a spatially periodic force in the reaction-diffusion system [21]. In our previous study, the occurrence and mechanism of VR were discussed in detail within the whole parameter plane of amplitude and frequency of HF force based on an excitable FHN neuron model driven by two regular periodic signals [13]. We found that VR can be induced by the transition between different phase-locking modes since each maximum of response measure appears exactly at the transition boundary of phase patterns of the LF-signal-free FHN model [13]. Therefore, VR is also regarded as a possible mechanism for understanding how neurons extract target signals from a noisy environment. As a matter of fact, besides noise and HF force, there are many other factors that influence signal detection, such as network topology [22], autapse [23–25], time delay [26, 27], and electromagnetic induction [28–30].

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 [13]. In fact, besides amplitude and frequency, phase is also an important factor that influences the response of excitability systems to external signals [1, 31, 32]. Furthermore, real-world external signals are always irregular and the phase of signals may vary randomly with time. For example, a time-varying phase occurs when a periodic wave travels through a random medium or interface [33].

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 [34–36]. In fact, unbounded Gaussian noise is not always suitable for modeling all realistic random processes. Many stochastic processes in real-life systems do not follow either white or colored Gaussian laws [37]. Moreover, most parameters in real-life systems are required to be strictly positive quantities, such as the time-varying fluctuation of reaction rates [37]. If unbounded Gaussian noise is used rather than bounded noise, unrealistic model-based inferences may occur due to the infinite domain of Gaussian noise [37]. For example, biologically paradoxical results occur in the pharmacokinetics model of antitumor chemotherapy: antitumor drugs have the probability of increasing the number of the tumor cells [37, 38]. Moreover, experimental studies in sensory and other biological systems support the necessity of using non-Gaussian noises [39–41]. More importantly, Gaussian noise is inappropriate to directly mimic the signals with a rapidly fluctuating phase. The rapidly fluctuating background signals around neurons can be modeled by bounded noise with a random rapidly fluctuating phase derived from the Wiener process. In addition, bounded noise is a simple non-Gaussian noise in mathematical presentation, and its spectrum may be either narrowband or broadband by appropriately choosing relevant parameters [42]. Therefore, as a reasonable and versatile random model, it has been long used in electrical engineering and has been recently used in mechanical and structural engineering and biological systems [43–45].

Many bounded noise-induced phenomena, such as resonance [46, 47], transitions [39, 48], and spatial synchronization [42], have been reported. We have also investigated the effect of bounded noise on the formation and instability of spiral waves [35, 36]. In particular, the combined effects of correlated bounded noises and weak periodic signal were investigated in a recent study, and related results suggest that bounded noise can enhance the detection of external signals [49, 50]. Although the effects of the time-varying initial phase of the received signal on signal detection have been reported in excitable systems using phase noise to mimic the time-varying initial phase of the target signal [1, 51–53], the related studies mainly focus on target signals and take no account of rapidly fluctuating background signals received by neurons. To the best of our knowledge, the resonance behavior and signal detection and amplification in rapidly fluctuating background signals have not previously been studied. In this study, the rapidly fluctuating background signals are modeled by bounded noise with constant amplitude and random rapidly fluctuating phase derived from the Wiener process. Then, the effects of bounded noise on signal detection and amplification are mainly investigated based on the Fourier coefficient for measuring the system response to input signals.

#### 2. Model and Simulation

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 [13, 15]. where fast variable stands for the membrane voltage of the neuron, whereas slow variable is linked to the conductivity of the potassium channels embedded in the neuron membrane. A small parameter of time scale results in the separation of all motions into fast and slow ones. The characteristics of solutions of the system are controlled by a time independent external signal . Only one stable excitable steady state occurs and the system is excitable when is less than 0.898. Small but finite deviations from steady state may give rise to a transition to the unstable point only after a large excursion. Here, is set to 0 throughout this paper. denotes an external LF periodic signal. We set and throughout this paper so that this signal is too weak to evoke firing by itself. Namely, this signal is subthreshold.

In (1), represents the external bounded noise with a random rapidly fluctuating phase. For convenience, we call it simply HF bounded noise or irregular HF signal in the rest of this article. Notice that HF bounded noise is not an exclusive name for noise. The expression of is described as follows [44]: where denotes the amplitude of the external HF bounded noise and the angular frequency of HF bounded noise is (>1) times the angular frequency of the LF signal; is the unit Wiener process, and *σ* denotes the intensity of the unit Wiener process . For , the mean, autocorrelation function, and power spectral density of bounded noise are, respectively, represented as follows [44]: Clearly, (6) shows two symmetrical peaks at . When , the two peaks merge into one. The position of the peak of power spectral density is mainly dominated by , and the bandwidth of bounded noise mainly depends on the value of . Thus, the spectrum of bounded noise may be either narrowband or broadband by appropriately choosing relevant parameters. It turns into a narrowband process when is small enough, whereas it approaches white noise when . The random phase fluctuation of bounded noise can be tuned by changing , and larger results in faster fluctuation (Figure 1). When is sufficiently large, the phase of is mainly dominated by Wiener process and the behavior of is completely like bounded noise rather than a regular HF periodic signal. Conversely, the decreasing makes the behavior of more like a regular periodic signal. In particular, bounded noise turns into a cosine periodic signal when is equal to 0 (Figure 1(a)). Therefore, in this study, the intensity of the unit Wiener process, angular frequency ratio , and amplitude of are important parameters that influence the weak signal detection. In the computation, bounded noise can be obtained by directly simulating the unit Wiener process [54]. As reported previously, the time evolution of a unit Wiener process is determined in our numerical simulations by the following formula [35, 48]:Here, and are two independent random numbers which are uniformly distributed on the unit interval. We set a time step time units through this paper. In addition, according to Euler’s method, (1) and (2) are discretized in our numerical simulations as follows: