Eur. Phys. J. Special Topics 224, 1477–1484 (2015) © EDP Sciences, Springer-Verlag 2015 DOI: 10.1140/epjst/e2015-02473-0

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Controlling bistability in a stochastic perception model A.N. Pisarchik12,a , I.A. Bashkirtseva3 , and L.B. Ryashko3 1 2 3

Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico Computational Systems Biology, Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain Institute of Mathematics and Computer Sciences, Ural Federal University, Ekaterinburg, Russia Received 12 March 2015 / Received in final form 20 May 2015 Published online 27 July 2015 Abstract. Using a simple bistable perception model, we demonstrate how coexisting states can be controlled by periodic modulation applied to a control parameter responsible for the interpretation of ambiguous images. Because of stochastic processes in the brain, any percept is statistically recognized and multistability in perception never occurs. A stable periodic orbit created by the control modulation splits in two limit cycles in an inverse gluing bifurcation, which occurs when the modulation frequency increases. The statistical analysis of transitions between the coexisting states in the presence of noise reveals conditions under which an ambiguous image can be interpreted in a desired way determined by the control.

1 Introduction For a long time, manipulation of human consciousness has attracted great attention of politicians, militaries, and scientists around the world. In particular, if an ambiguous image can be interpreted by the same or different person in distinct ways, it would be extremely attractive to cause people perceive the image in a desired manner. If we consider the brain as a multistable dynamical system, control of perception is closely related to the problem of controlling multistability (see [1–3] and references therein). Signal processing in the brain is known to have probabilistic character, since cognitive brain noise provides a decision-making function to access different possible states [4]. Visual perception of ambiguous images, such as the Necker cube [5], Rubin’s face-vase [6], rotating sphere [7], tristable plaids [8], etc. are affected by stochastic brain activity which plays an important role in changing image interpretation [8, 9]. Although neural noise bases of perception for ambiguous stimuli are still controversial [10], diverse psychophysical and physiological experiments inspired formal models of multistable perception, which try to explain as much as possible the variability of a

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perceptual dynamics with simple mechanisms that could be implemented in relatively low-level sensory systems (like the visual cortex for visual multistability) [11]. Control of perception is an important problem in psychology, engineering, and robotics. Perceptual control theory [12] based on negative feedback control explains how an organism reacts on external stimuli to make predictions. Experimental results demonstrated that an organism controls neither its own behavior, nor external environmental variables, but rather its own perceptions of those variables. Although it is possible to control perception of ambiguous images by selective attention [13], actions are not controlled, they are varied so as to cancel the effects that unpredictable environmental disturbances would otherwise have on controlled perceptions [11]. According to the biased competition model of selective attention, attention is able to enhance the attended percept and suppress the unattended percept. The purpose of the present work is to develop a control method which would allow to decrease the frequency of switches between coexisting percepts so that the probability of these switches becomes so small that the perception is practically considered unambiguous, i.e. a person will see only one image determined by the initial condition of the control. In this paper we propose such a method based on periodic modulation of a parameter responsible for different interpretations of an ambiguous image. As distinct from traditional control, this control does not require a feedback. However, the control is not small, in the sense that the parameter change is relatively large. The limit cycle created by external periodic modulation is more stable than the original fixed points [14]. Noise-induced transitions between coexisting cycles in threedimensional flows have been studied in [15, 16]. Here, we also study the robustness of our control method to noise and its efficiency in controlling bistability. The rest of the paper is organized as follows. In Sect. 2 we introduce a stochastic perception model with bistability control in the form of harmonic modulation. Then, using the modulation frequency as a control parameter we demonstrate the emergence of bistability in an inverse gluing bifurcation and study noise-induced transitions between two coexisting limit cycles. The main characteristics of the control method and its efficiency are analyzed in Sect. 3. Finally, the main conclusions are given in Sect. 4.

2 Bistable perception model with periodic modulation 2.1 Model A simple stochastic model to describe bistable perception can be written as follows [17]: x˙ = −4x(x2 − 1) + 4c + εξ(t), (1) where x is the the state variable proportional to the difference between the dimensionless firing rates of the two competing populations, c is the parameter responsible for different interpretations of an ambiguous image, ξ(t) is zero-mean Gaussian white noise, and ε is the noise intensity. Equation (1) is derived from the potential energy function dE/dx = τ dx/dt describing perceptual alternation dynamics (Fig. 1), where the two minima are located close to x = ±1 [18]. In our simulations, for simplicity, the time scale τ is set to 1. √ √ In the deterministic bifurcation points c1 = −2 3/9 ≈ −0.385, c2 = 2 3/9 ≈ 0.385. For c < c1 and c > c2 , the system exhibits either single stable equilibrium ¯3 (c), respectively. In zone c1 < c < c2 , the sysx ¯1 (c) or single stable equilibrium x ¯3 (c), separated tem exhibits the coexistence of two stable fixed points, x ¯1 (c) and x by unstable point x ¯2 (c). In this bistability zone, the system state can be selected

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E(x)

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c

1

0

t

-1 Fig. 1. Periodic modulation (right) of the internal frame contrast c of the Necker cube (left) changes the potential shape (center).

by changing the initial condition x(0). As the parameter c changes, hysteresis-type ¯3 (c) occur. These transitions take transitions between the stable equilibria x ¯1 (c) and x place for c = c1,2 , and the distance between c1 and c2 define the hysteresis range. In the stochastic system (ε > 0), for weak noise, transitions between basins of attraction of equilibria are localized near c1 and c2 . Increasing noise shrinks the distance between these points leading to their convergence as soon as noise exceeds some threshold value. Our psychological experiments with the Necker cube [17] have shown that brain noise is so strong that bistability never occurs in visual perception, so that a person always detects intermittent switches between coexisting percepts. Here, we propose a simple method which provides bistability even in the presence of strong brain noise. The method implies periodic parameter modulation (c → c(t) = A cos wt). Now, the system under consideration becomes x˙ = −4x(x2 − 1) + 4A cos wt + εξ(t).

(2)

Note that considered system exhibits the phenomenon of stochastic resonance [20, 21]. For simplicity, the modulation amplitude is fixed to A = 1, and the modulation frequency w is used as a control parameter. Practically, this type of perception control can be realized in the Necker cube, where the contrast of the internal frame is periodically modulated. The effect of the periodic modulation in Eq. (2) is illustrated in Fig. 1. The modulation changes periodically the shape of the potential E(x). 2.2 Gluing bifurcation Without modulation red (w = 0), the deterministic (ε = 0) system Eq. (2) exhibits hysteresis-type transitions at fixed points c1,2 . When c varies, the dynamic hysteresis range increases due to critical slowing down [14, 17]. Since in the present paper, we focus on the case when the parameter c is periodically modulated (w = 0), the coexisting fixed points are replaced by a new attractor (stable limit cycle), as shown in Fig. 2(a). For small modulation frequencies (w < w∗ ), the deterministic system Eq. (2) has a single stable limit cycle (Fig. 2(a)), which intersects with x = 0 at two points c1,2 (w). As w increases, the attractor enlarges in size, so that the distance between c1 (w) and c2 (w) increases. When the modulation frequency approaches critical value w∗ ≈ 4.4, the single limit cycle splits in two coexisting stable limit cycles (blue curves in Fig. 2(b)) in the inverse gluing bifurcation [19]. A further growth in w separates the coexisting cycles so that the gap between them arises (green curves in Fig. 2(b))

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Fig. 2. Limit cycles of deterministic (ε = 0) system Eq. (2) with A = 1 and different modulation frequencies w.

Fig. 3. Random states of stochastic system Eq. (2) with periodic modulation. The deterministic attractors are shown by red lines and the random states by dots for noise intensities ε = 1 (light grey), ε = 0.5 (dark grey), ε = 0.1 (black) and modulation frequencies (a) w = 0.001, (b) w = 0.1, (c) w = 1, and (d) w = 20.

and increases so that the trajectory never crosses x = 0 anymore. The last situation occurs for very high w.

2.3 Noise-induced transitions The additive noise term εξ(t) in the model Eq. (2) makes the system solutions probabilistic. Figure 3 shows the random states of the system Eq. (2) under modulation with A = 1 for different w and ε. Figure 3 demonstrates noise-induced transitions through x = 0, whereas such transitions never occur in the deterministic system.

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3 Characterization of the control 3.1 Probability density functions Random transitions through x = 0 in the stochastic system Eq. (2) can be characterized by the probability density function (PDF) of the c-coordinates of intersection points of random trajectories (c(t), x(t)) with x = 0. Figure 4 shows PDFs for different noise intensities ε and modulation frequencies w. For weak noise, there are narrow PDF peaks which occur in the vicinity of c1,2 (w). When the modulation is relatively slow (w < 1) (Fig. 4(a,b)), the PDF peaks approach each other and merge at c = 0 as the noise intensity increases. This means that the transitions through x = 0 are concentrated near c = 0. As ε further increases, the distribution first becomes close to uniform, and then begins to grow near c = ±1. For faster modulation (1 < w < w∗ ), the transitions through x = 0 are mainly concentrated near c = ±1 (Fig. 4(c–e)). Below the gluing bifurcation, where the cycles are separated, the deterministic trajectories never intersect x = 0, and therefore, for weak noise the probability of intersections is very small, practically zero (Fig. 4(f–h)); for instance, our calculations during time T = 2 × 105 do not display any intersection for w = 20 and ε = 0.1 (blue horizontal line in Fig. 4(h)). 3.2 Efficiency of the control The efficiency of our control method can be quantitatively characterized by noisedependent factor k (3) ν(ε) = , T where k is the number of intersections of a stochastic trajectory with x = 0 during the time interval [0, T ]. The factor ν coincides with the Kramers frequency. Figure 5 shows this function for T = 2 × 105 and different modulation frequencies w. For low frequencies (w < w∗ ), above the gluing bifurcation the behavior of the functions ν(ε) is very similar for different w; they monotonically increase and are mainly distinguished by their deterministic values ν(0). However, below the gluing bifurcation (w > w∗ ), the behavior of ν(ε) crucially changes. It is worth noting that for w > w∗ we do not detect intersections with x = 0, i.e. ν(0) = 0. This means that for fast modulation the coexisting cycles are so separated that the PDF is practically zero (blue horizontal line in Fig. 4(h)). 3.3 Quality of the control Due to critical slowing down, the increasing w postpones noise-induced hopping between the two coexisting limit cycles. There is a noise-dependent threshold frequency for which this transition occurs during time T ; the stronger noise, the higher the threshold is. Figure 6 shows how this threshold frequency depends on noise for two different values of ν (ν1 = 0.1 and ν2 = 0.01) for T = 2 × 105 . The curves represent probabilistic boundaries between statistically monostable and statistically bistable regimes corresponding to high-quality control (ν2 = 0.01) and low-quality control (ν1 = 0.1) of bistability. Above the corresponding curve we deal with almost bistability, whereas below the curve the intermittent switches between the two cycles take place with higher probability. For weak noise, these dependences are well approximated by an exponential growth (note the lineal slope in the semilog scale). The notable saturation of these curves (when w → ∞) at strong noise εs means that

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Fig. 4. Probability density functions of intersection points with x = 0 for (a) w = 0.001, (b) w = 0.1, (c) w = 1, (d) w = 3, (e) w = 4, (f) w = 5, (g) w = 6, and (h) w = 20 and ε = 0.1 (blue), ε = 0.5 (green), ε = 1 (red), and ε = 2 (black).

the control is limited by a certain noise level (εs ≈ 0.55 and εs ≈ 0.70), above which the control is inefficient for practical purposes, because the probability of switches is so high that two coexisting percepts will be always visible, even for very fast modulation.

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Fig. 5. Normalized number of intersections with x = 0 as a function of noise intensity ε for different frequencies w during T = 2 × 105 .

Fig. 6. Threshold frequency versus noise intensity for ν1 = 0.1 (lower red curve) and ν2 = 0.01 (upper blue curve). The curves separate statistically bistable and statistically monostable regimes. Above the curves, two stable orbits coexist, whereas below the curves, intermittent random transitions between the two states occur with lower (between the curves) or higher (below the red curve) probability.

4 Conclusion We have introduced a nonlinear non-feedback control of visual perception in the form of periodic modulation applied to the parameter responsible for recognition of different coexisting percepts. The control aim is to cause a person interpret an ambiguous image in a desired way, determined by initial conditions of the control. Using a simple stochastic model of bistable perception, we have found critical parameters for a gluing bifurcation and boundary conditions between statistically bistable and statistically monostable (intermittent) regimes. Similar control can be applied to other dynamical systems with coexisting attractors. This work has been supported by the BBVA-UPM Isaac Peral BioTech Program.

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