Resonance phenomena controlled by external feedback signals and additive noise in neural systems

Chaotic resonance is a phenomenon that can replace the fluctuation source in stochastic resonance from additive noise to chaos. We previously developed a method to control the chaotic state for suitably generating chaotic resonance by external feedback even when the external adjustment of chaos is difficult, establishing a method named reduced region of orbit (RRO) feedback. However, a feedback signal was utilized only for dividing the merged attractor. In addition, the signal sensitivity in chaotic resonance induced by feedback signals and that of stochastic resonance by additive noise have not been compared. To merge the separated attractor, we propose a negative strength of the RRO feedback signal in a discrete neural system which is composed of excitatory and inhibitory neurons. We evaluate the features of chaotic resonance and compare it to stochastic resonance. The RRO feedback signal with negative strength can merge the separated attractor and induce chaotic resonance. We also confirm that additive noise induces stochastic resonance through attractor merging. The comparison of these resonance modalities verifies that chaotic resonance provides more applicability than stochastic resonance given its capability to handle attractor separation and merging.

of stochastic resonance to implement surgical grasping forceps, enhancing human tactile sensory performance through vibration. Enders et al. 27 and Seo et al. 28 proposed a method to improve touch sensation in paralyzed patients and stroke survivors. Moreover, at cognitive levels of brain function, Van der Groen et al. 30 developed a method to enhance perceptual decision-making by exploiting stochastic resonance. Specifically, the optimal amount of noise applied by transcranial random noise stimulation to the visual cortex conforms a non-invasive brain stimulation technique that enhances the accuracy of perceptual decisions.
A chaotic system exhibits various kinds of dynamical characteristics, such as intermittency chaos, hyperchaos, and bubbling transition 31,32 (reviewed in 33,34 ). Among them, chaos induces phenomena such as chaos synchronization and chaotic resonance (reviewed in 1,2,35 ). In this study, we focused on chaotic resonance, which can be interpreted as the phenomenon replacing the fluctuation source in stochastic resonance by chaos instead of using additive noise 35 . Chaotic resonance can be applied in two forms. First, the signal response can be enhanced by applying an external deterministic chaotic signal instead of external stochastic noise [36][37][38] . Specifically, the signal generated by external chaotic systems is applied to a dynamical system with bi-stable states by inputting weak signals [36][37][38] . Second, the signal response can be enhanced by dynamics with intrinsic chaotic behavior instead of applying external chaotic signals produced in other systems (we considered the second form in this study) 37,[39][40][41][42] . Chaotic resonance is fed to the system with chaos-chaos intermittency, where the chaotic orbit goes back and forth among separate regions 36,37,[39][40][41]43 . In chaotic resonance, synchronization of chaos-chaos intermittency against small external signals can be induced, and its degree can be maximized close to the condition for attractor merging bifurcation (reviewed in 35 ). This chaotic resonance has been found in many chaotic systems including one-dimensional cubic maps, the Chua's circuit, Lorenz system, and Duffing oscillator 36,37,[39][40][41] . In neural systems, the study of chaotic resonance has led to findings such as associative chaotic neural network models and a discrete neural system which is composed of excitatory and inhibitory neurons [43][44][45][46] . Moreover, recent studies of chaotic resonance have been focused on various types of neural systems, such as cerebellar learning systems and spiking neuron models with various types of spiking patterns [47][48][49][50][51][52] .
Compared to stochastic resonance, several studies have reported that the sensitivity of chaotic resonance is higher 45,46 . Still, few studies have addressed applications of chaotic resonance, possibly because in chaotic resonance, the chaotic state must be properly adjusted for obtaining resonance by the parameters of the internal system. In a large proportion of cases, particularly in biological system, adjusting internal parameters from outside cannot be realized. To solve this problem, we previously developed a method to control the chaotic state for generating chaotic resonance through an external feedback signal 53 . The feedback signal reduces the local maximum and minimum of a map function inducing chaos-chaos intermittency and separating the merged attractor. Consequently, chaotic resonance is achieved without rectifying internal parameters, in a method denominated reduced region of orbit (RRO) feedback. Although other conventional methods to control chaotic states by external signals exist, such as the Ott-Grebogi-Yorke method 54 , delayed feedback 55,56 , and H ∞ control 57 , they eliminate the chaotic dynamics by stabilizing equilibrium and transitioning to a stable periodic state by applying external perturbations. In contrast, RRO feedback adjusts the chaotic state without eliminating it to generate chaotic resonance 53 .
RRO feedback has been used in a discrete cubic map, coupled cubic maps, and a discrete neural system composed of excitatory/inhibitory neurons 44 , successfully inducing chaotic resonance 53,58,59 . However, the feedback signal has been only utilized for separating the merged attractor in these studies. Therefore, if the system behaviour exhibits the condition for separating the attractor, chaotic resonance cannot be controlled because attractor merging is needed instead of separation. Further, the performance of chaotic resonance induced by feedback and that of stochastic resonance by additive noise remain to be evaluated and compared.
We hypothesized that the separated attractor can be merged by negative feedback in RRO feedback. Therefore, chaotic resonance can be induced even under a separated attractor. To prove this hypothesis, we applied negative feedback to a discrete neural system 44 in this study. The induced chaotic resonance was evaluated regarding feedback strength, internal parameters of the neural system, and input signal amplitude/frequency. Finally, we compared the signal sensitivity of induced chaotic resonance with that of stochastic resonance by additive noise. Methods neural system model. Figure 1 illustrates the discrete neural system composed of excitatory/inhibitory neurons developed by Sinha 44 and considered in this study. The dynamics of the states for excitatory neuron x(t) and inhibitory neuron y(t) is expressed as Here, w EE and w EI represent excitatory synaptic weights between excitatory neurons and from excitatory to inhibitory neurons, respectively, w II and w IE represent inhibitory synaptic weights between inhibitory neurons and from inhibitory to excitatory neurons, respectively. Activation functions F a and F b are given by F a (X) = −1 for X < −1/a, F a (X) = aX for −1/a ≤ X ≤ 1/a, F a (X) = 1 for . Parameters 1/a and 1/b correspond to the activation threshold for the states of excitatory and inhibitory neurons, respectively. Under constraint w EI /w EE = w II /w IE = k, the two-dimensional dynamics of x(t) and y(t) reduces to effective neural potential z(t) = x(t)−ky(t), whereas the one-dimensional dynamics is given by www.nature.com/scientificreports www.nature.com/scientificreports/ To control the signal response in the neural system, we use chaotic resonance or stochastic resonance (Fig. 1). In both control approaches, the signal response is controlled by attractor merging (chaos-chaos intermittency). In chaotic resonance, attractor merging is controlled by an external feedback signal, whereas in stochastic resonance, attractor merging is controlled by additive noise. Specifically, to induce chaotic resonance by controlling the chaos-chaos intermittency of effective neural potential z(t), we applied an RRO feedback term u(z) 53 as follows: Here, K, z d , and σ are the RRO feedback strength, merging point of each chaotic attractors, and a parameter to determine the region for RRO feedback effect, respectively. In this simulation, we used set parameters to a = 5.95, 5.96, 5.97, b = 3.42, and k = 1.3811 44 . Consequently, the orbit of z(t) is trapped to its positive or negative regions. Based on our previous study 53 , z d and σ were set to z d = 0 for the divided points of each chaotic region z = 0, and the distance from this divided point z(0) to the local maximum/minimum σ = a 1 . In addition, the positive region of K was investigated, where RRO feedback signal Ku(z) suppresses chaos-chaos intermittency and separates the merged chaotic attractor 59 . In contrast, we focused on the negative region in the present study. In the negative region, we assumed that RRO feedback signal Ku(z) enhances chaos-chaos intermittency and merges the separated chaotic attractor.
To explain the effect of RRO feedback signal Ku(z), Figs 2 and 3 show map function of F(z) + K(u(z)) according to external feedback signals. Attractor merging (chaos-chaos intermittency) occurs if F(f max ) + K(u(f max )) < 0 and F(f min ) + K(u(f min )) > 0, where f max and f min are the local maximum and minimum of the map function. For internal neural parameter a = 6.03 and feedback strength K = 0, the attractor merging conditions are satisfied (left graph in Fig. 2). Applying positive feedback (K = 0.1, Fig. 4), the absolute values of f max and f min are reduced, and the attractor merging conditions are not satisfied, as shown in the right graph of Fig. 2. In our previous work 59 , we controlled chaotic resonance using this suppressing effect. For a = 5.96 and K = 0, the attractor merging conditions are not satisfied (left graph in Fig. 3). However, applying negative feedback signal Ku(z) (K = −0.1, Fig. 4), the absolute values of f max and f min increase, and the attractor merging conditions are satisfied, as shown in the right graph of Fig. 3.
To evaluate the signal response during chaotic resonance, an external sinusoidal signal S(t) = Asin2πΩt can be applied: For stochastic resonance, additive white Gaussian noise ξ(t) with zero mean and unit variance can be applied to Eq. (6) as follows:   www.nature.com/scientificreports www.nature.com/scientificreports/ Here, D represents the noise strength.
indices for signal response, chaotic state, and controlling attractor merging. We consider the synchronization between sign change of effective neural potential z(t) and input signal S(t) as signal response by utilizing the correlation coefficient between binarized time series z(t) being Z(t) (Z(t) = 1 if z(t) ≥ 0; Z(t) = −1 otherwise) and the time series of input signal S(t): where 〈⋅〉 denotes averaging over t and τ is the time delay.
To evaluate the instability of the z(t) orbit as chaos index, the Lyapunov exponent is employed 60 : , with z′(t) denoting a perturbed orbit. Finally, to confirm the frequency control effect of chaos-chaos intermittency by external RRO feedback signal, we use the occurrence probability of chaos-chaos intermittency: www.nature.com/scientificreports www.nature.com/scientificreports/ t cc where T and f cc denote the number of iterations and frequency of chaos-chaos intermittency, respectively.

Results
Attractor merging induced by internal neural parameters. We demonstrate the dependence of dynamics on internal neural parameter a. Figure 5 shows the effective neural potential z(t) bifurcation diagram, occurrence probability of chaos-chaos intermittency P t , Lyapunov exponent λ and F(f max ), F(f min ) according to internal neural parameter a. In the bifurcation diagram, two kinds of initial values for z(0) (i.e. negative and positive values) are used. The chaotic attractor (λ > 0), which is divided into positive and negative regions in   .
. control of attractor merging. Consider the effect of RRO feedback signal for separating and merging the attractor. Figure 6 shows the z(t) map function and its orbit for a = 6.03 without RRO feedback signal (K = 0) and with positive RRO feedback signal (K = 0.1). Without RRO feedback, attractor merging condition F(f max ) < 0,  www.nature.com/scientificreports www.nature.com/scientificreports/ F(f min ) > 0 is satisfied, and therefore z(t) exhibits chaos-chaos intermittency between the negative and positive z(t) regions. With RRO feedback signal, this condition is not satisfied, and the merged attractor is separated, while z(t) is trapped in the negative or positive region, depending on the z(t) initial value. Figure 6 also shows the z(t) map function and its orbit for a = 5.96 without RRO feedback signal (K = 0) and with negative RRO feedback signal (K = −0.1). Without feedback, the condition for attractor merging, F(f max ) < 0, F(f min ) > 0, is not satisfied, and therefore z(t) is trapped in the negative or positive region. Applying negative feedback (K = −0.1), the merging condition is satisfied, and thus z(t) exhibits chaos-chaos intermittency. To evaluate the dependence of the system behaviour on negative RRO feedback strength K in more detail, Fig. 7 depicts the z(t) bifurcation diagram, P t , λ, F(f max ) + K(u(f max )), and F(f min ) + K(u(f min )) as function of K for a = 5.95, 5.96, 5.97. The separated chaotic attractor (λ > 0) merges as it satisfies the condition for attractor merging, 0 068, 0 051, 0 035 for a = 5.95, 5.96, 5.97, respectively. Decreasing internal neural parameter a from the attractor merging, a ≈ 5.99, demands a larger absolute value of negative feedback strength K.
Next, we evaluate the effect of additive noise for merging the attractor. Figure 8 shows the map function of z(t) and its orbit for a = 5.96 without noise (D = 0). The attractor is separating (F(f max ) > 0, F(f min ) < 0). Adding noise (D = 0.01), the attractor is merged, and z(t) exhibits chaos-chaos intermittency. The dependence of z(t) behaviour on noise strength D is investigated in detail. Figure 9 shows the bifurcation diagram of z(t) and P t according to D for a = 5.95, 5.96, 5.97. If  . × − D 2 5 10 3 , 2.0 × 10 −3 , 1.5 × 10 −3 in a = 5.95, 5.96, 5.97, respectively, the attractor is merged for P t > 0. Decreasing internal neural parameter a from the attractor merging, a ≈ 5.99, shown in Fig. 5, demands a larger noise strength D.
control signal response in chaotic resonance. We first evaluate the signal response according to feedback strength K under chaotic resonance. Figure 10 shows correlation coefficient max τ C(τ) between input sinusoidal signal S(t) and binarized z(t) according to RRO feedback strength K, F(f max ) + K(u(f max )), and F(f min ) + K(u(f min )). Correlation coefficient max τ C(τ) exhibits a unimodal peak at the slightly merged settings for feedback strength. To observe the system behaviour in more detail, Fig. 11 shows time series z(t) for a = 5.96. Without RRO feedback signal (K = 0), attractor switching occurs around the peaks of input signal S(t). However, switching is not maintained for more than one iteration. Stronger feedback (K = −0.035) increases the switching www.nature.com/scientificreports www.nature.com/scientificreports/ frequency, but again, switching is not maintained for more than one iteration. In contrast, at appropriate feedback strength, where K corresponds to the peak of max τ C(τ) (K = −0.075), the attractor switches at the period of S(t), and thus chaos-chaos intermittency synchronizes with the input signal. At even stronger negative K (K = −0.15), the very high chaos-chaos intermittency frequency does not allow to confirm its synchronization with input signal S(t).
Then, we evaluate the sensitivity of the signal response in chaotic resonance. Figure 12 shows correlation coefficient max τ C(τ) between input sinusoidal signal S(t) and binarized z(t) according to feedback strength K and signal amplitude A under chaotic resonance. In amplitude region To investigate the relationship between input signal amplitude and signal response frequency, Fig. 13 shows correlation max τ C(τ) according to the amplitude and frequency of S(t) for various values of K and a, where K corresponds to peaks in correlation max τ C(τ) (Fig. 10). High values of max τ C(τ)  0. control of signal response in stochastic resonance. By controlling attractor merging using additive noise, stochastic resonance can be evaluated without feedback strength (K = 0). Figure 14 shows correlation coefficient max τ C(τ) between input sinusoidal signal S(t) and binarized z(t) according to noise strength D. A single www.nature.com/scientificreports www.nature.com/scientificreports/ Figure 11. Time series of z(t) (black line) and input sinusoidal signal S(t) (red line) for internal neural parameter a = 5.96 (corresponding to Fig. 10). Without RRO feedback signal (K = 0) and with weak RRO feedback strength (K = −0.035), attractor switching occurs around the peaks of input signal S(t). However, switching is not maintained for more than one iteration. In contrast, at appropriate feedback strength (K = −0.075), where K corresponds to the peak of max τ C(τ) (K = −0.075), the attractor switches at the period of S(t), and thus chaos-chaos intermittency synchronizes with the input signal. At even stronger negative K (K = −0.15), the very high chaos-chaos intermittency frequency does not allow to confirm its synchronization with input signal S(t). www.nature.com/scientificreports www.nature.com/scientificreports/ peak is confirmed around the slightly merged settings for noise strength D ≈ 5.0 × 10 −3 , 4.0 × 10 −3 , 3.0 × 10 −3 at a = 5.95, 5.96, 5.97 respectively, given in Fig. 9. Therefore, stochastic resonance is induced. Figure 15 shows time series z(t) for a = 5.96 (corresponding to Fig. 14). Without additive noise (D = 0) and with weak noise strength (D = 1.5 × 10 −3 ), attractor switching occurs around the peaks of input signal S(t). However, switching is almost lost in more than one iteration. For stronger noise corresponding to D where the peak of max τ C(τ) occurs (D = 4.0 × 10 −3 ), the attractor switches at the period of S(t), and chaos-chaos intermittency synchronizes with the input signal. At even stronger D (D = 2.0 × 10 −2 ), the very high frequency of chaos-chaos intermittency does not allow to confirm its synchronization with input signal S(t).
Then, to evaluate the sensitivity of the signal response under stochastic resonance, Fig. 16 shows correlation coefficient max τ C(τ) according to noise strength D and signal amplitude A. In amplitude region  Correlation coefficient max τ C(τ) between input sinusoidal signal S(t) and binarized z(t) according to noise strength D under stochastic resonance. Solid lines and shaded areas indicate the mean and standard deviation of max τ C(τ) among 10 trials, respectively. Correlation coefficient max τ C(τ) exhibits a unimodal peak around the slightly merged settings for noise strength given in Fig. 9. Therefore, stochastic resonance is induced.   Next, we investigate the relationship between input signal amplitude and signal response frequency. Figure 17 shows correlation max τ C(τ) according to the amplitude and frequency of S(t) for various values of D and a, where D corresponds to peaks in correlation max τ C(τ) (Fig. 14). High values of max τ C(τ)  0.

Discussion
To control chaotic resonance under a separating attractor into different regions, we propose the application of negative RRO feedback. This feedback was applied to a discrete neural system which is composed of excitatory and inhibitory neurons. As a result, it has been confirmed that negative RRO feedback can merge the separated attractor and induce chaotic resonance. For comparison on attractor merging, white Gaussian noise was also applied to the neural system, inducing stochastic resonance through attractor merging.
Regarding the characteristics of chaotic resonance induced by negative RRO feedback, around the attractor merging strength, the correlation between the effective neural potential and input signal as index of signal response exhibits a unimodal peak. This signal response is maximized at appropriate input signal amplitude and frequency. These characteristics are similar to those induced by positive RRO feedback 59 and agree with those of chaotic resonance in other systems 36,37,[39][40][41]43 .
Compared to stochastic resonance, additive noise and negative RRO feedback have the same effect for attractor merging and enhancing the signal response. However, the sensitivity against weak input signal in chaotic resonance is higher than that in stochastic resonance (see Figs 12,13,16 and 17). Besides high sensitivity, the degree of signal response in chaotic resonance is higher compared to stochastic resonance (see the peak values in the resonance regions of Figs 12, 13, 16 and 17, max τ C(τ) ≈ 0.7 for chaotic resonance and max τ C(τ) ≈ 0.4 for stochastic resonance). This higher ability of chaotic resonance than stochastic resonance agrees with our findings in the case where chaotic resonance is controlled by an internal system parameter 45,46 . Therefore, in chaotic resonance induced by negative RRO feedback, this higher sensitivity is maintained. Moreover, additive noise only merges the separated attractor, whereas RRO feedback can either merge or separate the attractor by applying negative or positive feedback, respectively (chaotic resonance under positive RRO feedback is detailed in our previous study 59 ). Therefore, RRO feedback can be adopted for more varied attractor conditions compared to additive noise.
Some limitations of this study should be considered. The neural system proposed by Sinha is the simplest neuron model for eliciting chaos-chaos intermittency. Therefore, for applying the RRO feedback method to actual neural systems, chaotic resonance induced by RRO feedback must be evaluated in more realistic neural systems www.nature.com/scientificreports www.nature.com/scientificreports/ (i.e. complex, continuous, and high-dimensional neural networks). We are currently developing a method to apply RRO feedback in continuous chaotic systems utilizing the dynamics on Poincaré sections. This approach might be the suitable for applying RRO feedback to continuous and high-dimensional systems.
In this paper, we have reported that chaotic resonance can be induced by an external RRO feedback signal in neural systems without constraints on whether the attractor is merging or separating. Moreover, we confirm that chaotic resonance controlled by RRO feedback signal provides wider applicability than stochastic resonance. The outcomes of this study might promote the development of devices to strengthen signal responses by the effect of chaotic resonance in actual neural systems, where internal parameters cannot be controlled from outside the system.