Control of stochastic and inverse stochastic resonances in a liquid-crystal electroconvection system using amplitude and phase noises

Stochastic and inverse stochastic resonances are counterintuitive phenomena, where noise plays a pivotal role in the dynamics of various biological and engineering systems. Even though these resonances have been identified in various systems, a transition between them has never been observed before. The present study demonstrates the presence of both resonances in a liquid crystal electroconvection system using combined amplitude and phase noises, which correspond to colored noises with appropriate cutoff frequencies (i.e., finite correlation times). We established the emergence of both resonances and their transition through systematic control of the electroconvection threshold voltage using these two noise sources. Our numerical simulations were experimentally confirmed and revealed how the output performance of the system could be controlled by combining the intensity and cutoff frequency of the two noises. Furthermore, we suggested the crucial contribution of a usually overlooked additional phase noise to the advancements in various noise-related fields.


Control of stochastic and inverse stochastic resonances in a liquid-crystal electroconvection system using amplitude and phase noises
Jong-Hoon Huh * , Masato Shiomi & Naoto Miyagawa Stochastic and inverse stochastic resonances are counterintuitive phenomena, where noise plays a pivotal role in the dynamics of various biological and engineering systems.Even though these resonances have been identified in various systems, a transition between them has never been observed before.The present study demonstrates the presence of both resonances in a liquid crystal electroconvection system using combined amplitude and phase noises, which correspond to colored noises with appropriate cutoff frequencies (i.e., finite correlation times).We established the emergence of both resonances and their transition through systematic control of the electroconvection threshold voltage using these two noise sources.Our numerical simulations were experimentally confirmed and revealed how the output performance of the system could be controlled by combining the intensity and cutoff frequency of the two noises.Furthermore, we suggested the crucial contribution of a usually overlooked additional phase noise to the advancements in various noise-related fields.
Stochastic resonance (SR) is an attractive counterintuitive phenomenon induced by noise combined with a deterministic signal 1 , which enhances the output performance of unknown weak signals below the threshold of detection tools.Usually, SR shows a maximal output performance peak corresponding to the signal-to-noise ratio at a moderate optimal noise level.Since Benzi et al. first suggested this phenomenon and its underlying mechanism in a study on ice-age cycles 2,3 , SR has been extensively investigated in various fields, including physics 4,5 , chemistry 6,7 , biology 8,9 , information technology 10,11 , and brain science 12,13 .By contrast, an opposing phenomenon known as the inverse SR (ISR) was initially discovered in a neural system [14][15][16] , showing a minimal output performance peak at a moderate optimal noise intensity.Later, ISR was also reported in other systems such as ecological systems and during electroconvection (EC) 17,18 .
The mechanisms of both SR 1 and ISR 19 are generally described using an expanded Langevin equation as follows: In standard SR, the presence of a weak deterministic signal (i.e., A 0 ≠ 0) and noise ζ(t) dictates that the potential function ϕ(x) for the system of interest must be bistable (i.e., two stable minima) 1 .Conversely, ISR can be obtained in the absence of a weak deterministic signal (i.e., A 0 = 0) if the two minima have different depths and widths.Consequently, the reflection symmetry (x → − x) in the quartic double-well potential is broken for ISR 19 , whereas the deterministic signal (A 0 ≠ 0) breaks the symmetry in SR.In other words, the symmetry-broken potential is periodic (Ω ≠ 0) for SR and stationary (Ω = 0) for ISR 19 .In contrast to the standard SR introduced by Benzi et al., Sutera suggested a pure noise-induced transition (A 0 = 0) to explain the ice-age cycles 20 .Interestingly, nonstandard SR was also observed 19 in the absence of a signal (A 0 = 0), which is often referred to as coherent resonance 1 .Breaking the reflection symmetry of the two-minimum potential is essential to generate standard (1) SR and ISR.Moreover, the noise ζ(t) in Eq. ( 1) is additive, i.e., independent of the variable x 21,22 .Similarly, multiplicative noise ξ(t) [i.e., xξ(t)] can provide SR [22][23][24] and ISR 17 .
To the best of our knowledge, SR and ISR have never been observed concurrently in any actual system.To expand the use of SR and ISR into advanced applications, it is necessary to control both resonances and provide a transition between them, enabling the subsequent control of both desired and undesired system performances according to actual needs.In our previous studies, we confirmed the presence of both SR 25 and ISR 17 independently in a nonequilibrium system using different noises, i.e., phase noise for SR and amplitude noise for ISR.In the present study, we appropriately combined these two types of noise in order to control both resonances and the transition between them.In addition to the commonly used amplitude noise, our method utilized the usually overlooked phase noise 25,26 .Moreover, we used a colored noise with a finite autocorrelation time (τ c ≠ 0) 17,21,22,[27][28][29][30][31] , instead of the conventional quasi-white noise (τ c ≈ 0) [2][3][4][5][6][7][8][9][10][11][12][13]27 .
In this report we demonstrate the transition between SR and ISR via a smooth variation of the output performance using ac-driven electroconvection in a nematic liquid crystal (NLC) [32][33][34][35][36] .Our findings show how to control monotonic and nonmonotonic variations of the EC threshold using colored amplitude and phase noises.The unusual nonmonotonic behavior of the threshold confirms the presence of both SR and ISR, providing maximal and minimal peaks of the EC pattern performance by efficiently facilitating and suppressing EC around a moderate optimal level of the amplitude and phase noises.
In this numerical study, the threshold voltage V c of the EC was calculated using the one-dimensional Carr − Helfrich equations as follows [32][33][34][35] : Here, q(t) and ψ(t) represent the space-charge density and curvature (ψ = ∂φ/∂x) of the director for the deviation angle φ from the initial director n 0 (// x ) at V = 0, respectively (Fig. 1a), and n corresponds to a unit vector indicating a locally averaged direction of the rod-like molecules of NLCs.The values of τ, σ H , λ, E 0 2 , and η are determined by material parameters such as the electric and viscoelastic properties of the NLC with a thickness

Experimental results for SR and ISR
In an electro-optical system for EC 35,36 , we observed the emergence of SR and ISR by controlling both noises (i.e., ϕ N and V N ) with appropriate f cp and f ca values.For reference, a conventional pattern evolution with increasing ϕ N (at as shown in Fig. 5a [see the corresponding V c (ϕ N , V N = 0) in Fig. 2a].Since α decreased with increasing ϕ N , the performance of EC patterns (or optical (4) intensity I ∝ φ 2 ) 40 decreased with increasing ϕ N .Then, EC disappeared at ϕ N = 40° for α < 0 (i.e., V 0 < V c ).Such a pattern evolution is trivial and intuitive in the presence of conventional noise.Conversely, at V 0 = 16.5 V [< V c (V N = 5 V) = 17.1 V], SR was found, as shown in Fig. 5b [see the corresponding V c (ϕ N , V N = 12 V) in Fig. 2a].By increasing ϕ N , α < 0 changed to α > 0 and then again to α < 0. Thus, EC smoothly appeared and disappeared with increasing ϕ N .The EC performance (i.e., φ) showed a typical bell-shaped curve (i.e., SR), which is similar to the SR obtained from a single phase noise 25 .
Figure 5c shows a unique pattern evolution with increasing V N at ϕ N = 90° and V 0 = 18.3 V [> V c (V N = 0) = 17.8 V].A typical reversed bell-shaped curve of the EC performance indicating ISR was observed, which is similar to that of the ISR obtained from a single amplitude noise 17 .In addition, the present patterns at high V N (= 6 − 8 V) showed localized ECs, which were not observed in a previous study based on a single amplitude ).A nonmonotonic V c (ϕ N ) [e.g., at V N = 12 V in (a,c)] indicates SR, as shown in Fig. 1c.When V N is increased, a monotonic increase in V c (ϕ N ) in (a) smoothly changes into a monotonic decrease through a nonmonotonic behavior.In contrast, the nonmonotonic behavior of V c (ϕ N ) in (c) changes into a monotonic decrease for high V N (≥ 19 V). noise 17 .Such localized patterns are attributed to the combined effect of both noises, indicating that comparatively high intensities of both noises may play a role in EC pattern structures as well as EC thresholds.In localized ECs that are stationary (not transient), a noise-induced abnormal distribution of electric charges for ECs may occur 41 , which can be distinguished from the normal distribution of conventional ECs 34 .Such localized ECs are beyond the scope of our analysis using Eqs.( 2) and (3).Evidently, our experimental observations revealed the crucial effects of colored amplitude and phase noises on the generation of both resonances through the smooth variation of the EC threshold.Unfortunately, the transition between SR and ISR (Fig. 3) was not observed due to experimental limitations, such as material parameters [e.g., τ, σ H , λ, and η in Eqs. ( 2) and ( 3)] that are highly sensitive to V c but difficult to tune to the values in the numerical study.

Discussion and conclusion
During the last four decades, SR has been intensively investigated in useful concepts of randomness 42 and extensively applied for noise benefits [42][43][44] .In this study, we presented an SR transitioned from ISR which has been less addressed so far in the literature.By examining the threshold of ac-driven EC, we showed that a suitable combination of colored amplitude and phase noises could induce the emergence of both resonances and the transition between them.Therefore, we demonstrated that SR and ISR, which have been independently reported so far, could be handled in a single framework.A recent numerical study on the co-occurrence of SR and ISR 16 reported in a neural system should be distinguished from this study.Such co-occurrence implies that an ISR exhibiting the minimal output for one performance measure (mean firing rate) can induce an SR exhibiting the maximal output for another performance measure (mutual information).Note that colored noise can vary the probability density of the state of systems, which is governed by the Fokker-Planck equation 45 .According to the correlation time of colored noise, the probability density distribution can also provide two peaks for two possible states (φ = 0 and φ ≠ 0 in this study) 45 .In the Carr-Helfrich mechanism of ac-driven EC [32][33][34][35][36] , the combined electric noises can play critical roles in the occurrence of EC through their effects on the motion of electric charges (by Coulomb force against the electro-elastic restoring force of the NLC) and vary the EC threshold.In particular, in the appropriate conditions of correlation times and intensities of the two noises, their roles can contradict each other, i.e., one may suppress EC and the other may promote EC.Consequently, this competition between the noise-induced stabilization and destabilization effects on EC is the underlying reason for the emergence of both resonances and their transition.If there exists a nonequilibrium, free energy-like potential 46 with its minima at φ = 0 and φ ≠ 0 (i.e., a rest state and a convection state, respectively), which correspond to an ice state and a warm state, respectively, in the study of ice-age cycles 2,3 , such a potential should be investigated along with its symmetry breaking 1,19 to understand the detail of the competition mechanism; and this is an important open question.Our numerical and experimental findings suggest that the control of SR and ISR by combining amplitude and phase noises can be very useful for electrical applications, such as sensing technologies 36,47,48 and brain science 12,13 , for which additional phase noise can be readily introduced.Furthermore, the transition between SR and ISR may provide effective controls for desired and undesired performances in various related fields 49,50 .

Figure 1 .
Figure 1.Colored noises-induced threshold variation of ac-driven electroconvection (EC).(a) A schematic representation of EC driven by Coulomb forces on electric charges (blue circle with plus sign, red circle with minus sign) in a nematic liquid crystal (NLC).The rods in the vortices of EC indicate the director n of the NLC.Above a threshold V c , EC is observed as a roll pattern (i.e., Williams domain) in the xy plane, which results from the periodic director angle φ(x).(b) Original sinusoidal voltage V(t) (i.e., ϕ N = V N = 0) (top) and voltages superposed by amplitude noise (i.e., ϕ N = 0, V N ≠ 0) (middle) and phase noise (i.e., ϕ N ≠ 0, V N = 0) (bottom).(c) Threshold voltage V c as a function of phase-noise intensity ϕ N .(d) Threshold voltage V c as a function of amplitude-noise intensity V N .(e) Experimental system for EC under two noise sources (NG-1 noise generator for amplitude noise, NG-2 noise generator for phase noise, SWG sinusoidal wave generator, A amplifier, C combiner).The functions of V c (ϕ N ) and V c (V N ) highly depend on the cutoff frequency f c of the colored noise.In (c), nonmonotonic V c (ϕ N ) is observed for f cp ~ f cp * of phase noise, indicating stochastic resonance (SR).In (d), reversed nonmonotonic V c (V N ) is observed for f ca ~ f ca * of amplitude noise, indicating inverse stochastic resonance (ISR).See Refs 17,25 .for our previous results [V c (V N ) and V c (ϕ N )] and corresponding EC pattern changes.

Figure 3 .
Figure 3. Behavior of V c (V N ) for various phase-noise intensities ϕ N .In the case of f 0 = 2.5 kHz and f ca = 1 kHz (≈ f ca * ≈ 1 kHz), V c was determined using the phase noise with (a) f cp = 100 Hz (≈ f cp * ) and (b) f cp = 50 Hz (< f cp * ).The nonmonotonic V c (V N ) [e.g., at ϕ N = 140 deg in (a,b)] indicates SR.Furthermore, the reversed nonmonotonic V c (V N ) [e.g., at ϕ N = 0 in (a,b)] indicates ISR, as shown in Fig. 1d.With an increase in the ϕ N , the nonmonotonic behavior of V c (V N ) smoothly changes to reversed nonmonotonic behavior by exhibiting a nearly constant behavior [at ϕ N = 60 deg in (a) and ϕ N = 80 deg in (b)]; thus, a transition from ISR to SR is found by increasing ϕ N .

Figure 4 .
Figure 4. Phase diagrams of V c in the f ca and f cp planes.(a) V c (ϕ N ) for various V N values.Notably, the symbols indicating V c (ϕ N ) are arranged in the order in which V N increases from top to bottom.Case11 (C11), C21 and C31 correspond to Fig. 2a, c, b, respectively.(b) V c (V N ) for various ϕ N values.The symbols indicating V c (V N )are arranged in the order in which ϕ Ν increases from top to bottom.C12 and C32 correspond to Fig.3a, b, respectively.The symbols filled triangle, downward filled triangle, and filled rectangle indicate monotonic increase, monotonic decrease, and constant behavior of V c , respectively; blue circle filled at bottom and red circle filled at top indicate nonmonotonic and reversed nonmonotonic behavior of V c , providing SR and ISR, respectively.In this numerical study, f 0 = 2.5 kHz, f ca * ≈ 1 kHz, f cp * ≈ 100 Hz, Δf ca ≈ 500 Hz, and Δf cp ≈ 50 Hz.See Supplementary Information for details.

Figure 5 .
Figure 5. EC pattern evolutions with increasing ϕ N or V N .ECs were observed at f 0 = 1.5 kHz in a cell (MBBA, d = 25 μm).Phase noise ϕ N with f cp = 2 kHz smoothly increased when (a) V 0 = 18.8 V and V N = 0 and (b) V 0 = 16.5 V and V N = 5 V with f ca = 1 kHz.(c) Amplitude noise V N with f ca = 4.5 kHz smoothly increased when V 0 = 18.3 V and ϕ N = 90 deg with f cp = 1.6 kHz.Notably, (a,b) show pattern evolutions for V c (ϕ N ) at V N = 0 and V N = 12 V in Fig. 2a, respectively, and (c) indicates that for V c (V N ) at ϕ N = 20 deg in Fig. 3b.SR and ISR are observed in (b,c), respectively.See Supplementary Information for details.