
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71897
10.1038/s41598-024-71897-z
Article
Manipulating conductivity and noise for transitioning between stochastic and inverse stochastic resonances in liquid–crystal electroconvection
Huh Jong-Hoon huh@phys.kyutech.ac.jp

Higashi Takumu
Sato Yuki
https://ror.org/02278tr80 grid.258806.1 0000 0001 2110 1386 Department of Physics and Information Technology, Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Fukuoka, 820–8502 Japan
18 9 2024
18 9 2024
2024
14 2182124 6 2024
2 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Noise can play a constructive role in nature and various engineering systems. Over the past four decades, noise-induced stochastic resonances (SRs) have been extensively documented, showing enhancement in system performance. Additionally, inverse SR has been observed in various systems. Typically, these resonances were studied independently. A transition between these resonances was recently observed in an alternating current-driven liquid–crystal electroconvection (EC) system using combined amplitude and phase noises. This study uses internal (material) and external (noise) parameters to demonstrate the control of this transition. Specifically, the nonmonotonic threshold voltage behavior of the EC system, indicative of the resonances, was numerically examined using additional parameters. Experimental tests were conducted to confirm the effects of these parameters. The findings reveal that the transition between these resonances can be systematically controlled to meet specific needs, whether desirable or undesirable system performances. Notably, this study illustrates how to modify the behavior of both resonances in colored noise by adjusting its cutoff frequency and steepness and phase noise, which is often overlooked. Moreover, this study provides valuable insights for various noise-related applications.

Subject terms

Phase transitions and critical phenomena
Statistical physics, thermodynamics and nonlinear dynamics
http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science 22K03470 Huh Jong-Hoon issue-copyright-statement© Springer Nature Limited 2024
==== Body
pmcIntroduction

Noise-induced stochastic resonance (SR) and inverse stochastic resonance (ISR) are intriguing counterintuitive phenomena1,2. SR is characterized by a bell-shaped signal-to-noise ratio (SNR) curve, indicating optimal output performance at a specific noise intensity. In contrast, ISR decreases performance, resulting in an inverted bell-shaped SNR curve. Benzi et al. first proposed SR and its underlying mechanism in the context of ice-age cycles1,3. Since then, SR has been extensively studied in various fields, including electronic devices4,5, nonlinear chemical reactions6,7, reaction mechanisms of living organisms8,9, image processing10,11, and neural networks12,13. Similarly, since the first identification of ISR in a neural network system2,14,15, it has been reported in other systems, such as ecological systems16 and electroconvection (EC) systems17. However, in most studies1–17, these resonances have been investigated separately. A recent study18 discovered a substantial transition between SR and ISR in a liquid–crystal EC system using specific combinations of amplitude and phase noises.

Generally, a weak deterministic signal and a double minimum potential function U(x) are crucial for the generation of SR and ISR19–21. Additionally, it is essential that the reflection symmetry (x →  − x) of the double minimum potential is broken for both resonances20. Previous studies19–21 found that the symmetry-broken potential is periodic and stationary for SR and ISR, respectively. For both resonances, additive noise and multiplicative noise have been employed21,22; additive noise is independent of the variable indicating output performance 21–23, while multiplicative noise is coupled to the variable (e.g., ψ in Eqs. (1) and (2))24,25. Moreover, most earlier studies1,3,6,9,10,12 used white noise to generate both resonances. However, colored noise was found to be essential for controlling their generation17,18,21–25. Notably, colored noise with a finite autocorrelation time (τc ≠ 0) can be distinguished from conventional white noise (τc = 0)18,21,26. Typically, amplitude noise has been used for both resonances; however, phase noise, which is often overlooked, was found to contribute to SR in a nanoelectromechanical membrane system27 and EC systems28. Additionally, Gaussian noise has usually been employed for both resonances1–28, but SR has also been observed in non-Gaussian noise29,30. Amplitude and phase noises that are multiplicative, colored, and of Gaussian-type were used in this study. The degree of colorization of both noises, which is controlled by the autocorrelation time and the steepness of their power spectra, is particularly important in this investigation18,26,31.

In a recent report18, SR and ISR in the EC system (Fig. 1a) were investigated using numerical calculations of the threshold voltage Vc of the alternating current (AC)-driven EC. The emergence of both resonances was also experimentally confirmed in the EC system17,18,28. It was discovered that amplitude noise-induced ISR17 and phase noise-induced SR28 in the EC system combined in the presence of both noises, revealing a transition between the competing resonances18. The noise-dependent curves of Vc exhibited bell-shaped and inverted bell-shaped behaviors corresponding to ISR and SR, respectively. Notably, a bell-shaped Vc curve indicates that the output performance (i.e., the growth of EC) shows an inverted bell-shaped SNR curve for ISR17 and vice versa for SR28. Thus, a transition between SR and ISR was first discovered in an actual system18.Fig. 1 AC-driven electroconvection (EC) with superposed external colored noise. (a) Schematic of EC driven by Coulomb forces on electric charges (+ ,−) in a nematic liquid crystal (NLC). The rods in the EC vortices indicate the director n(nx, 0, nz) of the NLC modulated from the initial director n0 = (1, 0, 0) (at V = 0). Above a threshold voltage Vc, EC is optically observed as a periodic roll pattern (i.e., the so-called Williams domains) in the xy plane by the lens effect of the periodic director angle φ(x). (b) Power spectra P(f) of colored noise characterized by the cutoff frequency fc of the pass band and the width W of the attenuation band. (c) Schematic of Gaussian amplitude noise (top) and phase noise (bottom) superposed on a sinusoidal AC signal. (d) Typical P(f) of noise in this study, characterized by the steepness s corresponding to W at a fixed fc; note that s = 1 indicates W = 0 for ideal noise filters. Each P(f) was obtained from the noises generated by a frequency filtering program integrated into the general-purpose software (MATLAB). Compare these P(f) from the filtering program with that (s = 0.80) obtained experimentally from a wave generator (HIOKI, 7075).

Two meaningful parameters, an internal (material) parameter and an external (noise) parameter, were introduced in this study to deepen the understanding of the transition in the EC system. The first parameter is the Helfrich conductivity σH, which is determined by adjusting the electric conductivity of nematic liquid crystals (NLCs)32. It can be experimentally realized by doping a compound, for example, tetrabutylammonium bromide (TBAB), into n-(4-methoxybenzylidene)-4-butylaniline (MBBA), as used in this study26 (see Eq. (1)). The second parameter is the steepness of the noise power spectra P(f), defined by the attenuation (frequency) band, as shown in Fig. 1b 31. A cutoff frequency fc of the noise in Fig. 1b is defined to characterize colored noise using a low-pass filter26. Subsequently, noise with pass-band frequency components (i.e., f < fc) is generated and used as a superposing noise to the EC system. In principle, the ideal low-pass filter has no attenuation band (i.e., W = 0), but real noises are experimentally and numerically generated with it (i.e., W ≠ 0)26,31. Generally, the steepness s of colored noise is defined as W = (1 − s)(fNy − fc), using a Nyquist frequency fNy33. Thus, s = 1 indicates the ideal cutoff frequency of an ideal low-pass filter (i.e., W = 0). The steepness s is first introduced to investigate the transition between both resonances. Specifically, two kinds of steepness, i.e., amplitude noise (i.e., sa) and phase noise (i.e., sp), are utilized in this study. The sinusoidal signals superposed by amplitude and phase noises must be noted (Fig. 1c). The power spectra P(f) of noise with different steepness in Fig. 1d are provided to confirm feature of the steepness. Evidently, the attenuation band changes with the steepness variation. Additionally, this study explored whether the attenuation frequencies varied by steepness can play a role in the occurrence of EC and the emergence of both resonances. Expanding SR and ISR into advanced applications necessitates controlling both resonances. Therefore, based on actual needs, one subsequently controls desired and undesired system performances from this idea. For instance, a desired effect (of SR) and an undesired side effect (of ISR) in actual applications may be controlled by employing such internal and external parameters of the system of interest.

In this numerical study, the threshold voltage Vc for the EC system was calculated using the one-dimensional Carr–Helfrich equations (Eqs. (1) and (2)) for an NLC, specifically MBBA, sandwiched between two parallel electric plates with a gap distance d32,34:1 q˙+qτ+σHV(t)dψ=0,

2 ψ˙+λE02+V(t)d2ψ+qηV(t)d=0,

where q(t) and ψ(t) represent the space-charge density and the variation of the director (ψ = ∂φ/∂x), respectively (Fig. 1a). Generally, the director n is defined by a unit vector indicating the locally averaged direction of the rodlike molecules in NLCs32. In principle, the Carr–Helfrich effect causes electrohydrodynamic instability for V > Vc, providing a periodic deviation angle φ(x) from the homogeneous initial director n0 [= (1, 0, 0) at V = 0]32. As a result, a well-ordered convection structure called Williams domains (Fig. 1a) can be observed through a lens effect due to the director modulation32,35. In other words, EC arises at the transition from φ = 0 (for V < Vc) to φ ≠ 0 (for V > Vc). Therefore, Vc can be determined as the lowest voltage in the numerical loop with an increment of ΔV, at which the director angle φ does not relax to zero36. Furthermore, the dielectric constant ε and electric conductivity σ define the Helfrich conductivity in Eq. (1) as σH =σ‖(ε⊥/ε‖-σ⊥/σ‖), where the subscripts ‖ and ⊥ indicate the directions parallel and perpendicular to the initial director n0, respectively32,34. Additionally, material parameters, such as the electric and viscoelastic properties of NLCs, determine the values of τ, λ, E02, and η32,36. In the absence of noise, the threshold voltage Vc of EC is analytically determined as follows32,36:3 Vc2(f)=V02(1+4π2f2τ2)δ2-(1+4π2f2τ2),

where V0 represents a specific voltage (V0 = E0d) at an AC frequency f and δ is a dimensionless coefficient influenced by the material properties of the NLC. Typically, V0 ranges from 10 to 25 V, and 1.5 < δ2 < 4 for MBBA, as used in this study36.

Additionally, in the presence of conventional white amplitude noise [i.e., ξa(t) in V(t)=2Vcos(2πft)+ANξa(t)], the threshold voltage Vc can be analytically determined as follows34:4 Vc2(f,VN)=Vc02+b(f)VN2,

5 b(f)=1+4π2f2τ2δ2-(1+4π2f2τ2),

where Vc0 represents the threshold voltage when the noise intensity VN=<(ANξa(t))2> = 0. Previous studies37,38 indicate that parameter b has a positive value for white noise but can be negative for properly adjusted colored noise. Thus, for colored noise, the noise correlation time τc and the charge relaxation time τ should modify b38. Technically, the cutoff frequency fc of low-pass filters (where fc = 1/(2πτc)) can control τc26,38. Under these noise conditions, the function Vc(VN) exhibits a monotonic behavior [Eq. (4)], showing no SR or ISR that appear in the nonmonotonic behavior of Vc(VN). Occasionally, experiments with uncontrolled material properties and noises have shown the nonmonotonic behavior of Vc(VN), although SR and ISR were not mentioned in those studies26.

Considering the Helfrich conductivity σH of EC and the steepness s of noise, the behavior of Vc in the presence of well-adjusted noises was examined. Numerical calculations were performed using an electric voltage V(t) with both amplitude and phase noises as follows18:6 V(t)=2Vcos2πft+ϕNξp(t)+ANξa(t),

Where ξa(t) and ξp(t) correspond to the amplitude and phase Gaussian-colored noises with steepness (s ≠ 1), respectively, and ϕN indicates the intensity of the phase noise (0 ≤ ϕN ≤ 180 degrees). Amplitude and phase noises were adjusted by setting specific values for s (0.9 < s < 1) and fc. The conditions of fca = 1 kHz and fcp = 50 Hz, representing the cutoff frequencies of the amplitude and phase noises, respectively, were fixed in this study for the numerical calculations. Indeed, s and fc can be determined using a built-in frequency filtering program integrated within the software (MATLAB solver) used in this study18. In a recent study18, nearly ideal colored noises with sa = sp = 0.999 were employed in the numerical analysis. While fc can be controlled experimentally through low-pass filters18,26, adjusting s experimentally poses challenges.

Results

Control of SR and ISR by the Helfrich conductivity

Based on a recent study utilizing colored amplitude and phase noises18, SR, ISR, and their transition were quantitatively investigated through numerical calculations of the threshold voltage Vc. A typical result of Vc from a recent report18 is presented in Fig. 2a to provide a clearer understanding of the current findings. In this result, obtained under fixed conditions (i.e., f = 2.5 kHz, fcp = 50 Hz, fca = 1 kHz, sa = sp = 0.999, and σH = 1.52 × 10−8Ω−1 m−1), Vc varies with the amplitude noise intensity VN in a smooth manner, correlating with the phase noise intensity ϕN. For instance, the bell-shaped curve of Vc(ϕN = 0), indicating ISR, transitions into a monotonic curve (ϕN = 80 degrees) with nearly the same value and further transitions into an inverted bell-shaped curve of Vc(ϕN > 80 degrees), indicating SR (e.g., ϕN = 140 degrees). Hereafter, the units of ϕN [degrees] and σH [10−8Ω−1 m−1] are omitted for convenience.Fig. 2 Behavior of the EC threshold voltage Vc in relation to amplitude noise intensity VN and phase noise intensity ϕN. (a) Behavior of Vc(VN) for different values of ϕN. Note that a bell-shaped curve of Vc (e.g., ϕN = 0) changes into an inverted bell-shaped curve (e.g., ϕN = 140 deg); the former and the latter indicate ISR and SR, respectively. (b) The height h, indicating the degree of the output performance of SR or ISR, was extracted from (a). Accordingly, h = 0 means that the performance of SR or ISR completely fades out. A characteristic phase noise intensity ϕN* (~ 70 deg) can be determined for h = 0. The data were obtained at the following fixed values: f = 2.5 kHz for EC, fca = 1 kHz, fcp = 50 Hz, and sa = sp = 0.999 for noise, and the Helfrich conductivity σH = 1.52 × 10−8Ω−1 m−. For convenience, the units of σH [10−8Ω−1 m−1] are omitted in the captions of Figs. 3, 4, and 9.

To elucidate the characteristics of SR and ISR, h is introduced as the height of the bell-shaped curve, as depicted in Fig. 2a. Here, h = Vc(VN = VN*) − Vc(VN = 0) at a specific intensity VN*, where Vc attains maximal (for ISR) or minimal (for SR) values. Accordingly, with h > 0 and h < 0 indicating ISR and SR, respectively, the transition from ISR to SR occurring at a characteristic phase noise intensity ϕN* (approximately 80 in Fig. 2a) can be assessed, where h approaches 0 (i.e., the disappearance of ISR and the onset of SR). Clearly, Fig. 2b illustrates a distinct transition between SR and ISR, as extracted from Fig. 2a. It is noteworthy that ISR (h > 0) transitions into SR (h < 0) at ϕN* when h = 0. The more precise value of ϕN* (approximately 70) can be observed in Fig. 2b. Moreover, apart from the transition (h = 0), the magnitude of |h|, which corresponds to maximal or minimal SNR, can be regulated. This highlights the influence of noise on the desirable or undesirable output performance for practical applications within the system of interest.

To comprehend the responses of both resonances to changes in the internal properties of EC, this study analyzed the threshold function Vc(VN) as a function of the Helfrich conductivity σH under fixed external conditions of amplitude and phase noises, with sa = sp = 0.999. An AC frequency f = 2.5 kHz was employed for EC, maintaining consistency with the conditions depicted in Fig. 2a, extracted from a recent report18. Following the derivation of Vc(VN) illustrated in Fig. 2a, the function h(ϕN) with different values of σH was determined, as depicted in Fig. 3. Notably, h increases with increasing ϕN for larger σH (> 2). In contrast, it decreases for smaller σH (< 2). The characteristic intensity ϕN*(h = 0) of the phase noise is identified for σH < 2, depending on each value of σH, signifying the transition from ISR (h > 0) to SR (h < 0). Particularly, ISR (h > 0) is solely observed for large values of σH (> 2). In other words, there is no ϕN*(h = 0); for example, refer to h(ϕN) for σH = 2.27. Consequently, as depicted in Fig. 4, three regions are delineated based on σH, including the regions of ISR (σH > σH** ~ 2.05) and transition from ISR to SR (σH* < σH < σH**) and an unknown region (σH < σH* ~ 1.15). Notably, in this unknown region, Vc seems to diverge, exhibiting values of h greater than 100 V (in numerical calculation). Importantly, in the region of the transition from ISR to SR, ϕN*(h = 0) increases with increasing σH. These findings indicate that the control of the transition from ISR to SR is feasible under limited conditions (σH* < σH < σH**). From an advanced application standpoint, an internal parameter (such as σH, which can be altered by electric conductivityσ) can regulate the magnitude of h, indicating the maximal or minimal output performance of application systems, and h = 0, indicating the transition between both resonances. Hence, these results offer a pivotal insight into various noise-related applications employing SR and ISR: a controllable (internal) parameter may furnish a more effective control mechanism for the output performance. Thus, it is crucial to identify a parameter that is easily controllable and sufficiently sensitive to the output performance of the system of interest. This study confirmed that the dielectric constants ε|| and ε⊥ are not easily controllable in experimentation and lack the requisite sensitivity for controlling both resonances.Fig. 3 Height h as a function of phase noise intensity ϕN for different values of σH. Note that the curve of h(ϕN) for σH = 1.52 is the same as in Fig. 2b. The regions where h > 0 and h < 0 indicate ISR and SR, respectively. Thus, the characteristic phase noise intensity ϕN* for h = 0 is determined for each value of σH. However, no ϕN* is found, because h > 0 for all values of ϕN. This means that the transition from ISR to SR does not occur. The data were obtained under the same conditions as in Fig. 2, except for σH.

Fig. 4 Three regions of SR and ISR depending on the Helfrich conductivity σH. The characteristic phase noise intensity ϕN*(h = 0) was extracted from h(ϕN) in Fig. 3. The region of ISR only is speculated for σH > σH** ~ 2.05, a higher characteristic value of σH. However, for σH < σH* ~ 1.15, a lower characteristic value of σH, h appears to diverge (larger than 100 V in the numerical calculation). Note that the region of the transition from ISR to SR is limited to σH* < σH < σH**.

Control of SR and ISR by the steepness of noise power spectra

Similarly, Vc(VN) was investigated while varying the steepness s of both noises under fixed conditions of fca = 1 kHz and fcp = 50 Hz. The parameters f = 2.5 kHz and σH = 1.52 were kept constant for the EC. Initially, the function h(ϕN) was established for different values of sa for the amplitude noise, with a fixed sp = 0.999 for the phase noise, as illustrated in Fig. 5. It is clear that h decreases almost linearly with increasing ϕN, regardless of sa. Furthermore, the h(ϕN) curve shifts into lower values with increasing sa. However, it is observed that for sa ≤ 0.995, h > 0, exclusively indicating ISR. Consequently, no transition from ISR to SR is discernible under these sa conditions. Thus, the ϕN* characteristic value for h = 0 was determined for sa > 0.995. Figure 6 illustrates the behavior of ϕN*(h = 0), demonstrating a smooth decrease with increasing sa. The result delineates two regions contingent on sa: the regions of ISR (sa < sa* ~ 0.995) and transition from ISR to SR (sa > sa*). Unlike the σH-dependent regions depicted in Fig. 4, no region of SR exclusively is identified due to s ≤ 1.Fig. 5 Height h as a function of phase noise intensity ϕN for different values of amplitude noise steepness sa and a fixed phase noise steepness sp = 0.999. The parameters f, fca, fcp, and σH were fixed at the same values as in Fig. 2b. Notably, h decreases with increasing ϕN. Furthermore, this decrease is almost linear and independent of sp. Only ISR (h > 0) is found for values of sa smaller than the characteristic value sa* ~ 0.995. In other words, the transition from ISR to SR does not occur. Compare this behavior to that exhibited in Fig. 7.

Fig. 6 Two regions of SR and ISR depending on amplitude noise steepness sa. The characteristic phase noise intensity ϕN*(h = 0) was extracted from h(ϕN) in Fig. 5. The region of ISR only is found for sa < sa* ~ 0.995, while the region of the transition from ISR to SR is found for sa > sa*. Note that no SR region is only found because the maximal value of sa is 1. Obviously, ϕN* smoothly decreases with increasing sa. Compare this to the behavior exhibited in Fig. 8.

Next, h(ϕN) was determined for different values of sp of the phase noise at a fixed sa = 0.999 of the amplitude noise, as depicted in Fig. 7. The conditions of f = 2.5 kHz and σH = 1.52 were also maintained. Similar to h(ϕN) obtained for different values of sa (Fig. 5), h(ϕN) also decreases smoothly with increasing ϕN, independent of sp. In contrast to the findings in Fig. 5, h(ϕN) shifts to higher values with increasing sp. Furthermore, h(ϕN) decreases nonlinearly with increasing ϕN for sp < 0.999 and decreases almost linearly with increasing ϕN for sp ≥ 0.999. Moreover, the transition from ISR (h > 0) to SR (h < 0) is observed for all values of sp. In other words, ϕN*(h = 0) is determined for the transition, independent of sp. Figure 8 illustrates the behavior of ϕN*(h = 0), demonstrating an almost exponential increase with increasing sp. In this scenario, the transition from ISR to SR always occurs for typical colored noises (sp > 0.8) used in experiment. This behavior of ϕN*(sp) is markedly different from that of ϕN*(sa) found in the case of h(ϕN, sa) (Fig. 6); the former exhibits a monotonic increase with the steepness sa, whereas the latter displays a monotonic decrease with sp. This indicates that the control of the transition from ISR to SR should involve the fine adjustment of the steepness of both noises. From a practical application standpoint, it is crucial to manipulate an external control parameter, such as the steepness of noise power spectra. In most previous studies17,18,22,23,28,37,38, although colored noise was utilized, only its cutoff frequency was adjusted without defining the steepness. The present results underscore the significance of steepness in controlling both resonances. The steepness function is typically unavailable in experimental setups, even in dedicated noise generators. Nonetheless, confirming the attenuation band is imperative to comprehend the edge effect of noise power spectra31. If feasible, the initialized noise should be appropriately tuned for noise-related applications. Computer-programmed noise with adjusted steepness may be a feasible alternative option for such applications.Fig. 7 Height h as a function of phase noise intensity ϕN for different values of phase noise steepness sp and a fixed amplitude noise steepness sa = 0.999. The parameters f, fca, fcp, and σH were fixed at the same values as in Fig. 2b. Similar to Fig. 5, h smoothly decreases with increasing ϕN. However, the behavior of h(ϕN) changes from a linear decrease (sp > 0.999) to a nonlinear one (sp < 0.999). Note that the value of h at ϕN = 0 is fixed (h ~ 1 V) because no phase noise was applied, and only amplitude noise was applied. Compare this to Fig. 5, where different amplitude noises with different sa were applied, resulting in different values of h at ϕN = 0 depending on sa.

Fig. 8 Transition from ISR to SR concerning phase noise steepness sp. The characteristic phase noise intensity ϕN*(h = 0) was extracted from h(ϕN) in Fig. 7. Considering h > 0 for ϕN = 0 (Fig. 7) and the maximal value sp = 1, the regions of SR only or ISR only are not found. Contrary to Fig. 6, ϕN* smoothly increases with increasing sp.

Experimental tests for the Helfrich conductivity and the steepness

To confirm the effects of the internal and external parameters (σH and s), experimental tests were conducted in a planarly-aligned cell [n0 = (1, 0, 0)] using an NLC (MBBA) with d = 50 µm. Initially, an EC (Williams domain) was observed at V = 5.9 V, f = 500 Hz, and T = 35 °C. Subsequent pattern changes were observed with increasing amplitude noise intensity VN (fc = 2 kHz and sa = 0.80), as shown in Fig. 9a. The optical intensity contrast of the pattern corresponding to the performance of EC slightly decreased at VN = 2.0 V and then increased with increasing VN, resulting in high intensity contrast at VN = 4.7 V. This indicates the presence of a weak ISR, suggesting a small SNR. The experiment was repeated at a different temperature (T = 30 °C) corresponding to a different value of the Helfrich conductivityσH, as shown in Fig. 9b. The EC nearly disappeared at VN = 2.0 V and then reappeared at VN = 3.4 V, indicating the presence of a clear ISR. Additionally, the pattern changes were quantified as the contrast intensity IC in Fig. 10. IC was determined by averaging the optical intensity difference between minimal and maximal peaks in each pattern. For instance, the pattern observed at 4.7 V in Fig. 9a has 15 maximal peaks, indicating upward and downward flows corresponding to the deviation angle φ of the director in the EC [Fig. 1 and Eqs. (1) and (2)]. The curves (a) and (b) in Fig. 10 correspond to the pattern changes in Fig. 9a and b, respectively. The inverted bell-shaped IC(VN) indicating ISR is found in both (a) and (b). Notably, the IC(VN) curve (b) for σH = 1.60 (T = 30 °C) is more pronounced than the curve (a) for σH = 1.88 (T = 35 °C), suggesting a clear ISR. This ISR completely disappeared at higher temperatures (T > 37 °C; σH = 2.02) (i.e., higher σH). These results are qualitatively consistent with the variation of h(ϕN = 0) shown in Fig. 3; h decreases with increasing σH (or T). The experimental results clearly demonstrate the effect of the internal parameter (σH).Fig. 9 Experimental tests for the Helfrich conductivityσH and noise steepness sa in an EC (V = 5.9 V and f = 500 Hz) conducted in MBBA. Pattern changes indicating the output performance with increasing amplitude noise intensity VN (fc = 2 kHz) were observed at: (a) T = 35 °C (σH = 1.88) and sa = 0.80, (b) T = 30 °C (σH = 1.60) and sa = 0.80, (c) T = 30 °C and sa = 0.96, and (d) T = 30 °C and sa = 0.98. The ISR in (b) was more clearly observed than in (a). Moreover, the ISR in (b) completely disappears at different values of sa in (c) and (d).

Fig. 10 Noise intensity VN-dependent contrast intensity IC of the pattern changes shown in Fig. 9. From the output performance (IC) corresponding to the signal-to-noise ratio (SNR), ISR is observed in (a) and (b), with (b) exhibiting a deeper inverted bell-shaped curve than (a). ISR is not observed in (c) and (d). IC was calculated using the minimal and maximal peaks indicating upward and downward flows in the EC patterns shown in Fig. 9.

Similarly, successive pattern changes were observed at T = 30 °C (σH = 1.60) under amplitude noises with different values of the steepness sa = 0.96 (Fig. 9c) and sa = 0.98 (Fig. 9d). Compared to the ISR observed in Fig. 9b (sa = 0.80), the ISR completely disappears in Figs. 9c and d. In these cases, the EC forms patterns with higher optical intensity with respect to VN. Consequently, the IC(VN) curves (c) and (d) do not exhibit the inverted bell-shaped feature seen in Fig. 10. These results are qualitatively consistent with the variation of h(ϕN = 0) shown in Fig. 5; h decreases with increasing sa, and then may reach to h = 0 (i.e., disappearance of ISR). Likewise, applying phase noises with different steepness sp may induce a similar effect on SR. This indicates that steepness plays a crucial role in the emergence of ISR and SR.

Discussion and conclusion

Over the past four decades, SR has undergone intensive investigation as a counterintuitive phenomenon and has found extensive application across various fields due to its utilization of noise21,39,40. Recent studies18,28 have proposed that phase noise, which is often overlooked, plays a substantial role in the emergence of SR and ISR when combined with conventional amplitude noise. Additionally, it was observed that the spectral colorization of noise is crucial for controlling SR and ISR17,18,28. Exploring these constructive aspects of noises, this study investigated the practical methods for controlling SR and ISR and their transition by manipulating the internal and external parameters. In the current experimental setup, the Helfrich conductivity (σH) and the noise power-spectrum steepness (sa and sp) were utilized as the internal and external parameters, respectively. By varying the internal (material) parameter, the performance levels of both resonances (denoted as h in Fig. 2a) can be modulated. Moreover, it was found that colored noises enabled the control not only of the typical SR effect (maximal performance) but also of the inverse ISR effect (minimal performance).

Conversely, the performance levels of both resonances can also be controlled when adjusting the external (noise) parameter. However, the impact of the steepness of the phase noise spectra differed markedly from that of the amplitude noise spectra. Specifically, the characteristic noise intensity (denoted as ϕN* in Fig. 2b) for the transition from ISR to SR exhibited a smooth decrease with the amplitude noise steepness (Fig. 6). In contrast, it displayed a smooth increase with the phase noise steepness (Fig. 8). These contrasting behaviors highlight the effectiveness of combining both types of noises for the precise control of performance levels and the transition (denoted as h = 0) between SR and ISR.

The current findings reveal a typical behavior of the EC threshold (Fig. 2a) observed under fixed colored noises at their respective cutoff frequencies. Internal and external parameters can influence the level of performance and the transition from ISR to SR depending on these cutoff frequencies (i.e., fca and fcp), which exhibit distinct behaviors as outlined in a recent study18. For instance, slight variations in these parameters may induce both resonances even under conditions where noise intuitively exhibits no SR and ISR. Therefore, one can more effectively control SR, ISR, and their transition to meet specific requirements by precisely adjusting the system parameters under investigation.

Finally, the underlying mechanism behind the current results, particularly the outcomes related to steepness, is worth mentioning. In the Carr–Helfrich mechanism of AC-driven EC32,34,36,41, the collective influence of electric noises on the AC field can substantially impact EC occurrence by affecting the motion of electric charges (specifically, the flow-induced torque to the director opposing the electro-elastic restoring torque to it), consequently altering the EC threshold Vc. Typically, white amplitude and phase noises decrease Vc due to random fluctuations contrasting with the sinusoidally alternating signal. However, appropriately colored amplitude and phase noises can modulate the AC signal to facilitate EC induction. Therefore, in the case of colored noises, the active attenuation (frequency) band (Fig. 1b) may play a crucial role in determining Vc alongside the conventional pass (frequency) band. In essence, the steepness, indicating the degree of the attenuation band, can influence the variation of Vc. However, the roles of amplitude and phase noises differ; one may suppress EC (i.e., stabilization effect), while the other may promote EC (i.e., destabilization effect), depending on the degree of the pass band (i.e., the cutoff frequency). Consequently, when both noises are combined, a certain interplay between noise-induced stabilization and destabilization effects on EC underlies the emergence of SR, ISR, and their transition. Phase noise (and its steepness) predominantly affects SR (Fig. 7). In contrast, amplitude noise (and its steepness) predominantly affects ISR (Fig. 5). More importantly, in amplitude noise, a smaller steepness (i.e., wider attenuation band) results in a greater height (|h|, i.e., better output performance from ISR). Similarly, in phase noise, a smaller steepness (i.e., wider attenuation band) leads to a greater height (|h|, i.e., better output performance from SR). Furthermore, the transition between SR and ISR can be controlled by varying the steepness of both noises. This implies that the attenuation band is susceptible to the appearance and disappearance of EC, consequently leading to the emergence of SR and ISR.

The current findings indicate that finely adjusting amplitude and phase noises can be highly beneficial for controlling SR and ISR in noise-related applications. Specifically, it is demonstrated that the steepness (i.e., the edge effect) of noise power spectra can influence the emergence and transition of both resonances31,42. Furthermore, varying appropriate internal parameters of the system may yield unexpected results for both resonances. The approach employed in this study holds promise for utilization in electrical applications where both amplitude and phase noises can be finely tuned. For instance, in sensing technology43,44, finely tuned noises could enhance the performance of sensors in detecting weak signals. In biotechnology39,45, the same principles could improve the accuracy of bio-detection systems. Moreover, our findings could be applied to neural networks for optimizing signal processing12,13. By tuning these noise parameters, our research can provide significant advancements across these fields, following the principles of SR and ISR demonstrated in previous studies. The ability to control the transition between both resonances may offer more effective methods to achieve the desired performance outcomes based on specific requirements.

Methods

Numerical calculation

The discrete fourth-order Runge–Kutta method within the general-purpose software MATLAB R2023b46 was employed to determine the threshold voltage Vc of EC based on the governing Eqs. (1) and (2) for the Carr–Helfrich mechanism32,36,41. Specifically, Vc was determined as the lowest value V in the numerical loop, with an increment of ΔV (= 0.1 V in this study), for which the director angle φ [see Fig. 1a] does not relax to zero due to instability36. Thus, through linear stability analysis, Vc was automatically established by confirming φ(t) → 0 or ∞ (for t → ∞) within a margin of error of ± ΔV [V]. In this study, the primary control parameters, represented by colored Gaussian noises with cutoff frequencies fca and fcp and steepness sa and sp, were provided by a frequency filtering program in the software. The material parameters for the NLC (i.e., MBBA) utilized in this study align with those detailed in our recent report18.

Tuning noise in numerical examination and for an experimental approach

In numerical studies, colored noises that are generated from computer-programmed noise filters must undergo tests to confirm their power spectra. For instance, the frequency filtering program integrated into general-purpose software like MATLAB, utilized in this study, provides two types of low-pass filters, i.e., finite impulse response and infinite impulse response, exhibiting distinct power spectra at the same cutoff frequency and steepness. This observation highlights the differences in noise characteristics between the two types of low-pass filters. Depending on the research subject, these differences may have noticeable effects. On the other hand, in experimental setup, noise generators or noise filters are typically employed for noise-related studies and applications. The details of colored noises can be adjusted through the built-in features of each generator and noise filter. For example, the programmable noise filters (NF, 3628) of this study offer maximum flat (s = 0.96) and phase linear (s = 0.98) low-pass filters. Although their cutoff frequency is set at the same values, their power spectra exhibit slight differences, indicating varying steepness.

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 22K03470. The authors would like to thank Enago (www.enago.jp) for the manuscript review and editing support.

Author contributions

J.-H.H. conceived the research and wrote the paper. T.H. and Y.S. performed the numerical analysis. J.-H.H. and Y.S. carried out the experimental examination. All authors contributed to all aspects of this work.

Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Benzi R Sutera A Vulpiani A The mechanism of stochastic resonance J. Phys. A 1981 14 L453 10.1088/0305-4470/14/11/006
Benzi, R., Sutera, A. & Vulpiani, A. The mechanism of stochastic resonance. J. Phys. A 14, L453 (1981).
2. Gutkin B Jost J Tuckwell HC Inhibition of rhythmic neural spiking by noise: the occurrence of a minimum in activity with increasing noise Naturwissenschaften 2009 96 1091 10.1007/s00114-009-0570-5 19513592
Gutkin, B., Jost, J. & Tuckwell, H. C. Inhibition of rhythmic neural spiking by noise: the occurrence of a minimum in activity with increasing noise. Naturwissenschaften 96, 1091 (2009).19513592
3. Benzi R Parisi G Sutera A Vulpiani A Stochastic resonance in climate change Tellus 1982 34 10 10.3402/tellusa.v34i1.10782
Benzi, R., Parisi, G., Sutera, A. & Vulpiani, A. Stochastic resonance in climate change. Tellus 34, 10 (1982).
4. Bhar B Khanna A Parihar A Datta S Raychowdhury A Stochastic resonance in insulator-metal-transition systems Sci. Rep. 2020 10 5549 10.1038/s41598-020-62537-3 32218495
Bhar, B., Khanna, A., Parihar, A., Datta, S. & Raychowdhury, A. Stochastic resonance in insulator-metal-transition systems. Sci. Rep. 10, 5549 (2020).32218495
5. Dodda A Stochastic resonance in MoS2 photodetector Nat. Commun. 2020 11 4406 10.1038/s41467-020-18195-0 32879305
Dodda, A. et al. Stochastic resonance in MoS2 photodetector. Nat. Commun. 11, 4406 (2020).32879305
6. Hohmann W Müller J Schneider FW Stochastic resonance in chemistry J. Phys. Chem. 1996 100 5388 10.1021/jp953269r
Hohmann, W., Müller, J. & Schneider, F. W. Stochastic resonance in chemistry. J. Phys. Chem. 100, 5388 (1996).
7. Suzuki Y Asakawa N Stochastic resonance in organic electronic devices Polymers 2022 14 747 10.3390/polym14040747 35215663
Suzuki, Y. & Asakawa, N. Stochastic resonance in organic electronic devices. Polymers 14, 747 (2022).35215663
8. Douglass JK Wilkens L Pantazelou E Moss F Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance Nature 1993 65 337 10.1038/365337a0
Douglass, J. K., Wilkens, L., Pantazelou, E. & Moss, F. Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 65, 337 (1993).
9. Russell DF Wilkens LA Moss F Use of behavioral stochastic resonance by paddle fish for feeding Nature 1999 402 291 10.1038/46279 10580499
Russell, D. F., Wilkens, L. A. & Moss, F. Use of behavioral stochastic resonance by paddle fish for feeding. Nature 402, 291 (1999).10580499
10. Simonotto E Riani M Seife C Roberts M Twitty J Moss F Visual perception of stochastic resonance Phys. Rev. Lett. 1997 78 1186 10.1103/PhysRevLett.78.1186
Simonotto, E. et al. Visual perception of stochastic resonance. Phys. Rev. Lett. 78, 1186 (1997).
11. Roy PK Rallabandi VPS Magnetic resonance imaging (MRI) enhancement using stochastic resonance Magn. Reson. Imaging 2010 28 1361 10.1016/j.mri.2010.06.014 20797832
Roy, P. K. & Rallabandi, V. P. S. Magnetic resonance imaging (MRI) enhancement using stochastic resonance. Magn. Reson. Imaging 28, 1361 (2010).20797832
12. Gluckman BJ Netoff TI Neel EJ Ditto WL Spano ML Schiff SJ Stochastic resonance in a neuronal network from the mammalian brain Phys. Rev. Lett. 1996 77 4098 10.1103/PhysRevLett.77.4098 10062387
Gluckman, B. J. et al. Stochastic resonance in a neuronal network from the mammalian brain. Phys. Rev. Lett. 77, 4098 (1996).10062387
13. Kai S Mori T Noise-induced entrainment and stochastic resonance in human brain waves Phys. Rev. Lett. 2002 88 218101 10.1103/PhysRevLett.88.218101 12059504
Kai, S. & Mori, T. Noise-induced entrainment and stochastic resonance in human brain waves. Phys. Rev. Lett. 88, 218101 (2002).12059504
14. Uzuntarla M Cressman JR Ozer M Barreto E Dynamical structure underlying inverse stochastic resonance and its implications Phys. Rev. E 2013 88 042712 10.1103/PhysRevE.88.042712
Uzuntarla, M., Cressman, J. R., Ozer, M. & Barreto, E. Dynamical structure underlying inverse stochastic resonance and its implications. Phys. Rev. E 88, 042712 (2013).
15. Zamani A Novikov N Gutkin B Concomitance of inverse stochastic resonance and stochastic resonance in a minimal bistable spiking neural circuit Commun. Nonlinear Sci. Numer. Simulat. 2020 82 105024 10.1016/j.cnsns.2019.105024
Zamani, A., Novikov, N. & Gutkin, B. Concomitance of inverse stochastic resonance and stochastic resonance in a minimal bistable spiking neural circuit. Commun. Nonlinear Sci. Numer. Simulat. 82, 105024 (2020).
16. Touboul JD Staver AC Levin SA On the complex dynamics of savanna landscapes Proc. Natl. Acad. Sci. U.S.A. 2018 115 E1336 10.1073/pnas.1712356115 29378933
Touboul, J. D., Staver, A. C. & Levin, S. A. On the complex dynamics of savanna landscapes. Proc. Natl. Acad. Sci. U.S.A. 115, E1336 (2018).29378933
17. Huh J-H Inverse stochastic resonance in electroconvection by multiplicative colored noise Phys. Rev. E 2016 94 052702 10.1103/PhysRevE.94.052702 27967018
Huh, J.-H. Inverse stochastic resonance in electroconvection by multiplicative colored noise. Phys. Rev. E 94, 052702 (2016).27967018
18. Huh J-H Shiomi M Miyagawa N Control of stochastic and inverse stochastic resonaces in a liquid-crystal electro convection system using amplitude and phase noises Sci. Rep. 2023 13 16883 10.1038/s41598-023-44043-4 37803168
Huh, J.-H., Shiomi, M. & Miyagawa, N. Control of stochastic and inverse stochastic resonaces in a liquid-crystal electro convection system using amplitude and phase noises. Sci. Rep. 13, 16883 (2023).37803168
19. Zhang X-J Qian H Qian M Stochastic theory of nonequilibrium steady states and its applications Part I. Phys. Rep. 2012 510 1 10.1016/j.physrep.2011.09.002
Zhang, X.-J., Qian, H. & Qian, M. Stochastic theory of nonequilibrium steady states and its applications. Part I. Phys. Rep. 510, 1 (2012).
20. Torres JJ Uzuntarla M Marro J A theoretical description of inverse stochastic resonance in nature Commun. Nonlinear Sci. 2020 80 104975 10.1016/j.cnsns.2019.104975
Torres, J. J., Uzuntarla, M. & Marro, J. A theoretical description of inverse stochastic resonance in nature. Commun. Nonlinear Sci. 80, 104975 (2020).
21. Gammaitoni L Hanggi P Jung P Marchesoni F Stochastic resonance Rev. Mod. Phys. 1998 70 223 10.1103/RevModPhys.70.223
Gammaitoni, L., Hanggi, P., Jung, P. & Marchesoni, F. Stochastic resonance. Rev. Mod. Phys. 70, 223 (1998).
22. Jia Y Zheng X Hu X Li J Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative and additive noise Phys. Rev. E 2001 63 031107 10.1103/PhysRevE.63.031107
Jia, Y., Zheng, X., Hu, X. & Li, J. Effects of colored noise on stochastic resonance in a bistable system subject to multiplicative and additive noise. Phys. Rev. E 63, 031107 (2001).
23. Nicolis G Altares V A new method of analysis of the effect of weak colored noise in nonlinear dynamical systems J. Stat. Phys. 1987 46 191 10.1007/BF01010340
Nicolis, G. & Altares, V. A new method of analysis of the effect of weak colored noise in nonlinear dynamical systems. J. Stat. Phys. 46, 191 (1987).
24. Gammaitoni L Marchesoni F Menichella-Saetta E Santucci S Multiplicative stochastic resonance Phys. Rev. E 1994 49 4878 10.1103/PhysRevE.49.4878
Gammaitoni, L., Marchesoni, F., Menichella-Saetta, E. & Santucci, S. Multiplicative stochastic resonance. Phys. Rev. E 49, 4878 (1994).
25. Seki K Barzykin AV Stochastic resonance driven by Gaussian multiplicative noise Europhys. Lett. 1997 40 117 10.1209/epl/i1997-00433-3
Seki, K. & Barzykin, A. V. Stochastic resonance driven by Gaussian multiplicative noise. Europhys. Lett. 40, 117 (1997).
26. Huh J-H Influence of external noise on various electrohydrodynamic instabilities in a nematic liquid crystal J. Phys. Soc. Jpn. 2012 81 104602 10.1143/JPSJ.81.104602
Huh, J.-H. Influence of external noise on various electrohydrodynamic instabilities in a nematic liquid crystal. J. Phys. Soc. Jpn. 81, 104602 (2012).
27. Chowdhury A Barbay S Clerc MG Robert-Philip I Braive R Phase stochastic resonance in a forced nanoelectromechanical membrane Phys. Rev. Lett. 2017 119 234101 10.1103/PhysRevLett.119.234101 29286702
Chowdhury, A., Barbay, S., Clerc, M. G., Robert-Philip, I. & Braive, R. Phase stochastic resonance in a forced nanoelectromechanical membrane. Phys. Rev. Lett. 119, 234101 (2017).29286702
28. Huh J-H Yano Y Miyagawa N Phase noise can induce stochastic resonance? J. Phys. Soc. Jpn. 2019 88 063001 10.7566/JPSJ.88.063001
Huh, J.-H., Yano, Y. & Miyagawa, N. Phase noise can induce stochastic resonance?. J. Phys. Soc. Jpn. 88, 063001 (2019).
29. Castro FJ Kuperman MN Fuentes M Wio HS Experimental evidence of stochastic resonance without tuning due to non-Gaussian noises Phys. Rev. E 2001 64 051105 10.1103/PhysRevE.64.051105
Castro, F. J., Kuperman, M. N., Fuentes, M. & Wio, H. S. Experimental evidence of stochastic resonance without tuning due to non-Gaussian noises. Phys. Rev. E 64, 051105 (2001).
30. Chen Y-F Wang K-K Ye H Wang Y-J Impact of non-gaussian noise and time delay on stability and stochastic resonance for a FitzHugh-Nagumo neural system subjected to a multiplicative periodic signal Fluct. Noise Lett. 2024 23 2450002 10.1142/S0219477524500020
Chen, Y.-F., Wang, K.-K., Ye, H. & Wang, Y.-J. Impact of non-gaussian noise and time delay on stability and stochastic resonance for a FitzHugh-Nagumo neural system subjected to a multiplicative periodic signal. Fluct. Noise Lett. 23, 2450002 (2024).
31. Fantini DA Emmerich DS Edge effects on frequency discrimination of tones presented in low- and high-pass noise backgrounds J. Acoust. Soc. Am. 1987 82 1593 10.1121/1.395148 3693698
Fantini, D. A. & Emmerich, D. S. Edge effects on frequency discrimination of tones presented in low- and high-pass noise backgrounds. J. Acoust. Soc. Am. 82, 1593 (1987).3693698
32. Prost J de Gennes PG The Physics of Liquid Crystals 1993 Claerendon
Prost, J. & de Gennes, P. G. The Physics of Liquid Crystals (Claerendon, 1993).
33. Mishali M Eldar YC From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals IEEE J. Sel. Top. Sig. Proc. 2010 4 375 10.1109/JSTSP.2010.2042414
Mishali, M. & Eldar, Y. C. From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals. IEEE J. Sel. Top. Sig. Proc. 4, 375 (2010).
34. Kawakubo T Yanagita A Kabashima S External noise effect on the onset of Williams domain in nematic liquid crystals J. Phys. Soc. Jpn. 1981 50 1451 10.1143/JPSJ.50.1451
Kawakubo, T., Yanagita, A. & Kabashima, S. External noise effect on the onset of Williams domain in nematic liquid crystals. J. Phys. Soc. Jpn. 50, 1451 (1981).
35. Williams R Domains in liquid crystals J. Chem. Phys. 1963 39 384 10.1063/1.1734257
Williams, R. Domains in liquid crystals. J. Chem. Phys. 39, 384 (1963).
36. Smith IW Galerne Y Lagerwall ST Dubois-Violette E Durand G Dynamics of electrohydrodynamic instabilities in nematic liquid crystals J. Phys. (Paris) 1975 36 C1 237 10.1051/jphyscol:1975142
Smith, I. W., Galerne, Y., Lagerwall, S. T., Dubois-Violette, E. & Durand, G. Dynamics of electrohydrodynamic instabilities in nematic liquid crystals. J. Phys. (Paris) 36, C1-237 (1975).
37. Huh J-H Noise-induced threshold shift and pattern formation in electroconvection by controlling characteristic time scales Phys. Rev. E 2011 84 025302 10.1103/PhysRevE.84.025302
Huh, J.-H. Noise-induced threshold shift and pattern formation in electroconvection by controlling characteristic time scales. Phys. Rev. E 84, 025302 (2011).
38. Huh J-H Kai S Colored noise-induced threshold shifts and phase diagrams in electroconvections J. Phys. Soc. Jpn. 2014 83 063601 10.7566/JPSJ.83.063601
Huh, J.-H. & Kai, S. Colored noise-induced threshold shifts and phase diagrams in electroconvections. J. Phys. Soc. Jpn. 83, 063601 (2014).
39. McDonnell MD Abbott D What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology PLoS Comput. Biol. 2009 5 e1000348 10.1371/journal.pcbi.1000348 19562010
McDonnell, M. D. & Abbott, D. What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology. PLoS Comput. Biol. 5, e1000348 (2009).19562010
40. Budrikis Z Forty years of stochastic resonance Nat. Rev. Phys. 2021 3 771 10.1038/s42254-021-00401-7
Budrikis, Z. Forty years of stochastic resonance. Nat. Rev. Phys. 3, 771 (2021).
41. Eber N Salamon P Buka A Electrically induced patterns in nematics and how to avoid them Liq. Cryst. Rev. 2016 4 101 10.1080/21680396.2016.1244020
Eber, N., Salamon, P. & Buka, A. Electrically induced patterns in nematics and how to avoid them. Liq. Cryst. Rev. 4, 101 (2016).
42. Jurado CA Pedersen CS Møller H Jurado CA Pedersen CS Møller H Auditory filters at low-frequencies: Filter shape in the range 50 Hz to 1000 Hz 8th European Cence on Noise Control 2009 (EURONOISE 2009) 2009 Institute of Acoustics
Jurado, C. A., Pedersen, C. S. & Møller, H. Auditory filters at low-frequencies: Filter shape in the range 50 Hz to 1000 Hz. In 8th European Cence on Noise Control 2009 (EURONOISE 2009) (eds Jurado, C. A. et al.) (Institute of Acoustics, 2009).
43. Saha AA Anand GV Design of detectors based on stochastic resonance Sig. Proc. 2003 83 1193 10.1016/S0165-1684(03)00039-2
Saha, A. A. & Anand, G. V. Design of detectors based on stochastic resonance. Sig. Proc. 83, 1193 (2003).
44. Li Q Li Z A novel sequential spectrum sensing method in cognitive radio using suprathreshold stochastic resonance IEEE Trans. on Vehicular Tech. 2014 63 1717 10.1109/TVT.2013.2287616
Li, Q. & Li, Z. A novel sequential spectrum sensing method in cognitive radio using suprathreshold stochastic resonance. IEEE Trans. on Vehicular Tech. 63, 1717 (2014).
45. Hänggi P Stochastic resonance in biology: How noise can enhance detection of weak signals and help improve biological information processing ChemPhysChem 2002 3 285 10.1002/1439-7641(20020315)3:3<285::AID-CPHC285>3.0.CO;2-A 12503175
Hänggi, P. Stochastic resonance in biology: How noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem 3, 285 (2002).12503175
46. Shampine LF Reichelt MW The MATLAB ode suite SIAM J. Sci. Comput. 1997 18 1 10.1137/S1064827594276424
Shampine, L. F. & Reichelt, M. W. The MATLAB ode suite. SIAM J. Sci. Comput. 18, 1 (1997).
