Biol Cybern DOI 10.1007/s00422-014-0607-5
ORIGINAL PAPER
Critical slowing down and noise-induced intermittency in bistable perception: bifurcation analysis Alexander N. Pisarchik · Rider Jaimes-Reátegui · C. D. Alejandro Magallón-García · C. Obed Castillo-Morales
Received: 23 October 2013 / Accepted: 21 April 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Stochastic dynamics and critical slowing down were studied experimentally and numerically near the onset of dynamical bistability in visual perception under the influence of noise. Exploring the Necker cube as the essential example of an ambiguous figure, and using its wire contrast as a control parameter, we measured dynamical hysteresis in two coexisting percepts as a function of both the velocity of the parameter change and the background luminance. The bifurcation analysis allowed us to estimate the level of cognitive noise inherent to brain neural cells activity, which induced intermittent switches between different perception states. The results of numerical simulations with a simple energy model are in good qualitative agreement with psychological experiments. Keywords Dynamics
Perception · Bifurcation · Noise · Brain ·
This work has been supported by the COECYTJAL-UdeG through project 05-2010-1-783. A. N. Pisarchik Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico A. N. Pisarchik (B) Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain e-mail: [email protected] R. Jaimes-Reátegui · C. D. A. Magallón-García · C. O. Castillo-Morales Centro Universitario de Los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Paseo de la Montaña, Lagos de Moreno, Jalisco, Mexico
1 Introduction The perception of visual signals in the brain was among the first issues discussed in terms of multistability which has been introduced to provide mechanisms for information processing in biological neural systems (Atteneave 1971). Multistability reveals itself when a single physical stimulus or noise induces alternations between coexisting subjective percepts (Schwartz et al. 2012). Multistable perception characterizes the wavering percepts that can be brought about by certain visually ambiguous patterns, such as the Necker cube shown in the upper part of Fig. 1. When this figure with all frames is viewed for an extended time, the two percepts alternate spontaneously, changing as often as every few seconds. This alternation has been attributed to neural adaptation or satiation (Hebb 1949; Köhler and Wallach 1944). The mechanisms underlying the switches between different visual percepts are not well understood. Most current models imply that the biological origin of the perceptual switches lies in noise, inherent to brain neural cells activity, which results from relatively random spiking times of individual neurons (Deco et al. 2009). Furthermore, external noise can also induce perception alternations (Kanai et al. 2005). Contributed by the probabilistic spiking times of neurons (Tolhurst et al. 1983), the brain noise plays an important and advantageous role in signal detection and decisionmaking by preventing deadlocks and allowing switches between different perception states (Moreno-Bote et al. 2007; Gigante et al. 2009). Decisions may be difficult without noise which produces probabilistic choice. The switches are stochastic processes according to a Markov chain (Meyn and Tweedie 2008) and are measured by recording either phase durations or percept frequencies (Leopold et al. 2002). In a bistable image, such as the Necker cube, two different percepts are associated with two stable steady states (attrac-
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Fig. 1 Hysteresis in the Necker cube when wireframe contrast is varied. The isometric perspective of a wireframe drawing makes the spot perceived plan ambiguous. (Up) Images of the Necker cube for different wires’ contrasts. (Down) Bifurcation diagram of bistable model Eq. (4) when control parameter c is varied with velocity v = 0.01. The arrows indicate directions of the parameter change. c f and cb are, respectively, forward and backward saddle-node bifurcations
tors). A change in a control parameter leads to the deformation of their basins of attraction and finally to a change in their stability, that occurs at a critical point. The system behavior near the onset of a dynamical bifurcation has gained considerable attention in recent decades because bifurcations play a crucial role in characterization of system dynamics. While the statistical properties of bistable visual perception have been extensively studied and reported in many papers [see, for instance, (Merk and Schnakenberg 2002; Ta’edd et al. 1988; Aks and Sprott 2003)], the bifurcation analysis still remains an interesting research task which can provide additional information about brain functionality. In this paper, we use the bifurcation analysis for studying visual perception of the Necker cube which wireframe contrast c is taken as a time-variable control parameter. It should be noted that the parameter variation is practically necessary while dealing with a huge amount of data, in particular, for biological systems. This allows one to collect as many data as possible to perform statistical or bifurcation analyses. By increasing c, one can find a forward saddle-node bifurcation c f where one of the percepts changes its stability, and by decreasing c, one comes to a backward saddle-node bifurcation cb where another percept appears. While one percept is stable for c < c f and another for c > cb , bistability exists for cb < c < c f . When c is fixed, the brain has enough time for image recognition. Instead, if c is continuously varied in time, the bifurcation points are shifted from their statical values, no matter how slowly c is varied. The postponement of a bifurcation in a system with a time-dependent parameter known as critical slowing down
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is a general phenomenon. First discovered in laser equations (Mandel and Erneux 1984), it was then found numerically and experimentally in many dynamical systems (Kapral and Mandel 1985; Pisarchik et al. 1997; Pisarchik 1998; Tredicce et al. 2004). Since the shift of bifurcation depends on the velocity of the parameter change, this effect reveals itself as rate-dependent hysteresis. In addition, this phenomenon is intrinsically accompanied by an enhanced sensitivity to noise (Broggi et al. 1986; Huerta-Cuellar et al. 2009; Pisarchik et al. 2012). Relatively strong noise destabilizes coexisting attractors so that the system intermittently escapes from one state to another, the process related to the so-called Kramers’ escape problem (Kramers 1940). Since noise is ubiquitous in the brain at multiple scales, from vesicular release and spiking variability to fluctuations in global neurotransmitter levels, deterministic brain dynamics can be studied only theoretically using noiseless models. To numerically simulate our experiment, in this work, we use a noise-driven attractor model (Moreno-Bote et al. 2007), although there also exist oscillator models (Kalarickal and Marshall 2000; Lago-Fernandez and Deco 2002) where noise is assumed to be an inessential component. The paper is structured as follows. Section 2 is devoted to psychological experiments with the Necker cube, demonstrating the phenomena of critical slowing down and noiseinduced intermittency between two coexisting percepts. In Sect. 3, we present the results of numerical simulations with a simple energy model and compare with our experimental results. Finally, main conclusions are given in Sect. 4.
2 Experiment 2.1 Experimental methodology The majority of psychological experiments with the Necker cube were focused on the statistical measurements of switching times between two different percepts [see, e.g., (Merk and Schnakenberg 2002)]. In this work, we search the critical points for onset of intermittency using the wires’ contrast and background luminance as control parameters. In our experiments, the Necker cube was placed at the middle of a computer screen as black lines on a white background with a small spot in the left-middle corner which served as the fixation point in order to concentrate attention. The contrast of the middle lines was used as a time-dependent parameter. Initially, the contrast of three middle lines centered in the leftmiddle corner (marked by the spot) was set to white (255), so that these lines were indistinguishable on the white background, while the contrast of three middle lines centered in the right-middle corner was set to black (0). First, we changed the contrast in the forward direction, i.e., the contrast of the lines centered in the left-middle corner was increasing from
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0 to 255, while the contrast of the lines centered in the rightmiddle corner was simultaneously decreasing from 255 to 0, so that the image was smoothly changing from the cube on the left side of Fig. 1 to the cube on the right side during time T . Then, the same procedure was repeated in the backward direction, i.e., the contrast of the left-middle lines was increasing, while the contrast of the right-middle lines was simultaneously decreasing. To observe the critical slowing down phenomenon, we varied the wires’ contrast with different velocities and detected the instant of time when a subject recognized a first switch in the percept. In a bistable system, such switches occur in forward and backward saddle-node bifurcation points related to the direction of the parameter change. Generally, the distance between these two bifurcation points yields dynamical hysteresis determined by both the velocity of the parameter variation and noise. This behavior was observed in many bistable systems (Kapral and Mandel 1985; Pisarchik et al. 1997; Pisarchik 1998; Tredicce et al. 2004). In addition, critical slowing down is intrinsically accompanied by an enhanced sensitivity to noise (Broggi et al. 1986; Huerta-Cuellar et al. 2009; Pisarchik et al. 2012). Strong noise can destabilize coexisting attractors resulting in a metastable attractor when the system jumps intermittently from one state to another (Pisarchik and Feudel 2014). This phenomenon is referred to as noise-induced intermittency. There are two types of hysteresis; one (positive) is in bistability and another one (negative) is in noise-induced intermittency. The former type of hysteresis shown in Fig. 1 is observed in a noiseless bistable system or in a system with relatively low noise. The assumption that noise is responsible for switches between different percepts predicts smaller hysteresis in bistability and larger in intermittency. The effect of critical slowing down due to parameter variation is opposite: The faster the parameter change, the larger the hysteresis in bistability and the shorter in intermittency. We also assume that brain noise of each person is constant, or at least, that it is not gradually increasing or decreasing during the experiment. Of course, we cannot exclude brain adaptation (Shpiro et al. 2009) or that Bayesian inference dynamics (MorenoBote et al. 2011) might lead to a change in noise; however, it is unlikely that adaptation acts only at high parameter change rates and has no effect at slow velocities. In all our experiments, we varied the wires’ contrast linearly in time t, first in the forward and then in the backward direction. Starting from the left-side or from the right-side image in Fig. 1, we recorded, respectively, the times t f and tb when a subject sighted the first switch in percept. Raw data typically look like those displayed in Fig. 2, where we plot t f and tb as a function of the run time T . Usually, t f and tb are very close to each other. The first switch in percept occurs for the critical value of the control parameter corresponding to the onset of noise-induced intermittency.
tb
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Fig. 2 Typical dependences of forward and backward switch times on run time T (subject 12)
Since in all our experiments, the contrast is always a linear function of time, i.e., c ∼ vt, and the velocity v of the contrast change is proportional to the inverse run time T −1 , the normalized time t/T can also be used as a control parameter because this value is always associated with the corresponding contrast c. Though there is no principal difference which of these parameters is used as a control parameter, the time is more convenient, because it can be recorded directly in experiments. In total, twenty student volunteers took part in our experiments. All subjects gave written informed consent to participate and were naive as to the specific experimental question. The studies were performed in accordance with the ethical standards laid down in the Declaration of Helsinki (World Medical Association 2000). All subjects were well aware about bistability of the Necker cube and were able to distinguish the percept switches. They were instructed to fixate their attention on the small spot and press a key on the computer keyboard in front of them only once at the moment when they fixed a first change in the spot position. In each direction, only the time of the first switch in percept was recorded. After the first switch which indicated the onset of intermittency, all subjects observed other switches between two different percepts; however, these later switches were ignored. We carried out two series of experiments. The first series were performed with white background, and the run time T was used as a control parameter. For every fixed T , we measured the times of the first switches in the forward and backward directions (t f and tb ) as shown in Fig. 2. In the second series, we fixed the run time to T = 30 s and used the background luminance as a control parameter. From twenty subjects involved in our experiments, fifteen subjects (1–15) participated in the first series and eight sub-
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2.2 Experiments with white background From fifteen students participated in this experimental series, only the data of eight subjects were selected for further analysis because of several reasons. The possible reasons were the following: (i) a person did not well understand the task, for instance, he/she pressed the key not at the moment of the first switch in the percept, but when the intermittent switches disappeared, and (ii) the loss of attention that could be caused by personal problems, for example, the person illness, tiredness, worrying or thinking about something else, etc. All these factors could result in a very high deviation from the average and after revealing the subjective reasons the data of these subjects were excluded from the analysis. The onset of intermittency was measured for 14 different velocities of the contrast variation in each direction, starting from the longest run time T = 30 s (slow parameter variation) to the shortest run time of T = 4 s (fast parameter variation) with the interval of 2 s. Figure 3 shows the normalized hysteresis h = (t f + tb − T )/T
(1)
as a function of T . For large run times, all subjects detected the first switch in percept at times less than one half of the run time (t f ≈ tb ≤ T /2), that resulted in negative hysteresis in the intermittent regime. As we expected, h saturates as the run time T increases, i.e., when the velocity of the contrast variation decreases; the faster the parameter change, the shorter the hysteresis range. Such a behavior is a sequence of critical slowing down near the saddle-node bifurcations. Although this phenomenon was previously observed in many other systems with time-dependent parameters, this is the first, to the best of our knowledge, experimental demonstration of critical slowing down in visual perception.
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subject 2 subject 4 subject 5 subject 6 subject 8 subject 11 subject 12 subject 15
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jects (3, 4, 5, 8, 13, 16, 18, 19) in the second series, due to personal reasons. Only five subjects (3, 4, 5, 8, 13) took part of both series of the experiments. The data of eight subjects (2, 4, 5, 6, 8, 11, 12, 15) participated in the first series and eight subjects (3, 4, 5, 8, 13, 16, 18, 19) participated in the second series were included in the analyses. The data of three subjects (4, 5, 8) were used in the analyses of both series. Since the analysis of each series was carried out independently, the participation of the same subject in both experiments was not necessary. To exclude a possible brain adaptation to the cognitive task, the same test was not repeated for the same subject. In the following, we present the experimental results of the two experimental series.
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Fig. 3 Experimental data for eight subjects viewing the Necker cube with white background. The dashed lines are the exponential fits by Eq. (2) demonstrating critical slowing down. The saturation at large run times is determined by internal brain noise, and the slope is related to the reaction time
The dependences in Fig. 3 are well approximated by an exponential decay h = k exp(−T /γ ) + h s ,
(2)
where k is the scaling coefficient, γ is the scaling exponent, and h s is the saturation value. Since we assumed that noise induces intermittent switches between the coexisting states, the existence of the saturation in the rate-dependent hysteresis confirms our hypothesis that the hysteresis at saturation h s is determined by brain noise. In other words, the higher the brain noise, the larger the absolute value of hysteresis. This simple experiment allowed not only measuring brain noise, but also estimating the reaction time which is determined by the scaling exponent γ . Specifically, when the contrast was changed too fast, the subjects did not have enough time to react adequately to this change that resulted in decreasing the hysteresis region. The dynamical characteristics of eight subjects extracted from the experimental dependences in Fig. 3 using the exponential fitting Eq. (2) are summarized in Table 1. One can see that h s varied from −0.57 to −0.92 with the mean value −0.76 and the standard errors for all subjects are less than 15 %. The comparison of h s for different subjects provides us with information about the difference in the relative levels of personal brain noise. For example, we can conclude that subject 6 had minimum and subject 11 had maximum brain noise among all subjects in this group. The value γ allows us to compare the reaction time of different participants. The very big SD for mean γ (44 %) means that the reaction time significantly varied across the individuals. Indeed, γ of subject 2 was approximately four times larger than γ of subject 12 (6.35 against 1.52), although in our experiments the error in γ for subject 12 was very large (41 %).
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Table 1 Main characteristics evaluated from experimental data with white background Saturation h s
SE
γ
SE
2
−0.75
0.03
6.35
0.74
4
−0.80
0.02
5.66
1.39
5
−0.70
0.05
7.64
2.00
6
−0.57
0.03
5.54
0.71
8
−0.90
0.01
4.14
0.08
11
−0.92
0.01
5.35
0.86 0.63
12
−0.72
0.02
1.52
15
−0.74
0.09
10.44
6.73
Mean
−0.76
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5.83
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Negative hysteresis observed at saturation in all subjects means that their brain noise was so strong that there was not bistability in their perception. In other words, noise completely destabilized the coexisting attractors. In this content, the interesting question arises: How does the hysteresis depend on noise? Since we do not know how to change internal brain noise, a simple solution could be to add external noise and use it as a control parameter. Another way to infer the perception dynamics is to change the perception sensitivity to brain noise. In the second series of experiments, we chose the latter way. Namely, we varied the intensity of background pixels and used their luminance as a control parameter. Although the luminance is not a stochastic signal, its effect is rather similar to that of external noise. In fact, it acts as a bias which increases the energy in the double-well potential of the bistable system, thus making perception more sensitive to brain noise. Eight subjects participated in this series of experiments. Similarly to the previous case, we varied the wires’ contrast and measured hysteresis. However, now the run time was fixed to a relatively large value T = 30 s, and the background luminance was varied. Although for dark background, the data differed across individuals much more than for white background; nevertheless, for four subjects, the dependences were well approximated by a linear fit with less than 12 % standard error, and for four other subjects exhibited a resonance-like behavior. Figure 4 shows the dependences of the normalized hysteresis on the background darkness N = 1 − L (L ∈ [0, 1] being the pixels luminance). The linear dependences in Fig. 4a exhibit a linear growth of the intermittency range h = h 0 + δ N . Since for N = 0 all background pixels are white, the hysteresis h 0 at the intersection of the extrapolated
subject 5 subject 16 subject 18 subject 19
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Fig. 4 Experimental normalized hysteresis as a function of N = 1− L (L being the background pixels luminance) for T = 30 s. a Monotonous and b nonmonotonous dependences. The dashed lines are linear fits
fitting lines with the ordinate axis N = 0 corresponds to the case studied in the first series of experiments, whereas the slope δ characterizes the perception sensitivity to the background luminance. The results of the analysis of the data in Fig. 4a are summarized in Table 2. The coincidence of the mean value of the intercept h 0 from Table 1 with the mean value of saturation h s from Table 2 indicates on a good accuracy of the obtained experimental results, that confirms our hypothesis that both values are associated with brain noise. Our method allowed the comparison of brain noise and perception sensitivity for different persons, for example, we can conclude that subject 3 had much smaller brain noise that subject 4. On the other hand, subject 4 had a very low sensitivity to the background luminance. The nonmonotonous behavior of other four subjects shown in Fig. 4b is probably related to the phenomenon of stochastic resonance. The external bias signal added to
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Biol Cybern Table 2 Main characteristics evaluated from experimental data in Fig. 4a Subject
Intercept h 0
SE
Slope δ
SE
3
−0.41
0.05
−0.27
0.08
4
−0.75
0.06
−0.12
0.10
8
−0.62
0.06
−0.29
0.10
13
−0.61
0.06
−0.24
0.10
Mean
−0.60
0.14a
−0.23
0.08a
SE means a standard error a Standard deviation from the mean
internal brain noise got in resonance with the Kramers time in the bistable system that facilitated intermittent switches between the coexisting states in the Necker cube, thus enlarging the intermittency area. Such a behavior is not surprising. Early, stochastic resonance was observed in psychological experiments (Chialvo and Apkarian 1993) with bistable Haken’s images (Haken 1983). Furthermore, another kind of stochastic resonance, subthreshold stochastic resonance, was detected in visual perception (Simonotto et al. 1997). The latter kind of stochastic resonance occurs in monostable excitable systems near the threshold. Such a resonance enhances recognition of obscured images when small noise neurons are below the threshold of action activity.
3 Theory Having the results as described above, we will now proceed to construct a simple one-parameter theoretical model that is nonlinear and stochastic. We consider the simplest doublewell potential model which exhibits the coexistence of two fixed points, that is the double-well potential model. The model is based on the assumption that each of two neuronal populations (say A and B) represent a different interpretation of the stimulus (c A and c B ) (Moreno-Bote et al. 2007) x˙ = −4x(x 2 − 1) − 2c A (x − 1) − 2c B (x + 1) + αξ(t), (3) where x is the state variable proportional to the difference between the dimensionless firing rates of the two competing populations, ξ(t) is zero mean Gaussian white noise, and α is the noise intensity. Equation (3) is derived from the energy function d E/d x = −τ d x/dt describing perceptual alternation dynamics, where the minima are located close to x = ±1. In our simulations, for simplicity, the time scale τ is set to 1. In our case, c A and c B are associated with the wire contrasts responsible for different image interpretations. Since in our experiments, we changed the contrasts of two wireframes simultaneously in opposite directions (while one was increasing the other was decreasing), we can use only one-parameter c = c A = −c B that makes Eq. (3) more
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simple: x˙ = −4x(x 2 − 1) + 4c + αξ(t),
(4)
The important advantage of the theoretical consideration over the experimental one is that in the theory, we can study the behavior of the noiseless bistable system (α = 0), while in experiments, it is impossible since internal brain noise is not only inevitable but also very high because, as our experiments show, the hysteresis in visual perception is always negative (see Fig. 4). Without noise (α = 0), the system Eq. (4) flows into one of the stable states (depending on the initial condition) and stays there forever. If the control parameter c is varied as c = c0 ± vt
(5)
with velocity v, the system passes through forward and backward bifurcation points c f and cb , where the system changes its attractor, as shown in the bifurcation diagram in Fig. 1. The system state depends on both the initial condition c0 and the direction of the parameter change determined by the sign in Eq. (5). The bistability is accompanied by hysteresis h = c f − cb . Due to critical slowing down, the position of bifurcation points depends on the velocity of the parameter change, i.e., h enlarges when v increases. Such a behavior is illustrated in Fig. 5a, b. Since the perception of one stimulus occurs when the firing rate of its population is higher than that for another population, the position of the bifurcation point is detected at the moment when the variable x crosses zero. While the increasing velocity enlarges h, the increasing noise produces an opposite effect, i.e., hysteresis decreases when noise increases. This effect is clearly seen in Fig. 5c. For sufficiently strong noise, instead of bistability, two-state intermittency takes place when the system intermittently switches between two coexisting states. In this case, hysteresis is negative, as seen in Fig. 5d. Figure 6 shows how the hysteresis range depends on the noise intensity at different velocities of the parameter change. Positive hysteresis (h > 0) indicates bistability and negative hysteresis (h < 0) means two-state intermittency. Since in our experiments hysteresis was negative, we concluded that the internal brain noise was strong enough to induce intermittent switches between coexisting percepts. The theoretical noise dependences in a large range of the noise intensity are well approximated by a sigmoidal fit. Nevertheless, within a small range, a linear fit yields a rather good approximation that is in agreement with our experimental results. To compare the results of the numerical simulations with our experiments, in Fig. 7, we plot the hysteresis value versus inverse velocity v −1 for different noise intensities α. These dependences are well approximated by exponential decay Eq. (2), in good agreement with experimental results (see Fig. 3).
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A good qualitative agreement between theoretical and experimental results indicates that even such a simple model allows simulation of brain cognitive dynamics. However, our model cannot describe the nonmonotonous noise dependences observed in the experiments. We expect that more realistic models, such as, e.g., the Hodgkin–Huxley neuron model with a stochastic term (Borisyuk et al. 2009) will allow a more detailed description of the experimentally observed features.
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Fig. 5 Bifurcation diagrams demonstrating a shift of bifurcations due to critical slowing down when control parameter c is varied for a v = 0.18 and b v = 0.45 without noise and for c α = 18 and d α = 60 at fixed velocity v = 0.01. While for weak noise the hysteresis is positive (c), for strong noise it is negative (d)
4 Conclusions The phenomena of critical slowing down and noise-induced two-state intermittency were studied in the bistable visual perception of the Necker cube in order to estimate important
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cognitive characteristics of the brain, such as brain noise and reaction-cognition time. The results obtained with the developed energy-based model are in good qualitative agreement with the experiments. Both the experimental and numerical results confirmed the hypothesis that the alternation between competing percepts associated with activation of different states of neural activity is driven by noise. It is important to stress that the noise originated from random spiking fluctuations may be behaviorally adaptive, and therefore should not be considered only as a problem in terms of how brain works. We believe that the testing methodology proposed in this work can find applications in understanding pathological brain stability states, such as schizophrenia and obsessivecompulsive disorder. A state with a weak stability at one end of the distribution contributing to the schizophrenia symptoms (Rolls 2008) may result from very strong brain noise. Instead, a too stable state at the other end of the distribution may contribute to the symptoms of obsessive-compulsive disorder (Rolls 2008) due to very little brain noise. Large deviations of the cognition reaction time from its mean value can indicate on serious brain diseases, such as delayed response syndrome (Figley and Sprenkle 1978) or reactive attachment disorder (Hanson and Spratt 2000). Of potentially great importance is that the measures of the noise level and reaction-cognition time can make the diagnosis of these diseases easier. Acknowledgments We thank K. M. Prado-Tabares for her help in the data collection and all voluntaries for their participation in the experiments. A. N. P. acknowledges support from CONACYT (Mexico).
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Kalarickal GJ, Marshall JA (2000) Neural model of temporal and stochastic properties of binocular rivalry. Neurocomputing 32–33:843– 853 Kanai R, Moradi F, Shimojo S, Verstraten FAJ (2005) Perceptual alternation induced by visual transients. Perception 34:803–822 Kapral R, Mandel P (1985) Bifurcation structure of the non-autonomous quadratic map. Phys Rev A 32:1076–1081 Köhler W, Wallach H (1944) Figural after-effects, an investigation of visual processes. Proc Am Philos Soc 88:269–357 Kramers HA (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7:284–360 Lago-Fernandez LF, Deco G (2002) A model of binocular rivalry based on competition in IT. Neurocomputing 44:503–507 Leopold DA, Wilke M, Maier A, Logothetis NK (2002) Stable perception of visually ambiguous patterns. Nat Neurosci 5:605609 Mandel P, Erneux T (1984) Laser Lorenz equations with a timedependent parameter. Phys Rev Lett 53:1818–1820 Merk I, Schnakenberg J (2002) A stochastic model of multistable visual perception. Biol Cybern 86:111–116 Meyn SP, Tweedie RL (eds) (2008) Markov Chains and stochastic stability. Cambridge University Press, Cambridge Moreno-Bote R, Rinzel J, Rubin N (2007) Noise-induced alternations in an attractor network model of perceptual bistability. J Neurophysiol 98:11251139 Moreno-Bote R, Knill DC, Pouget A (2011) Bayesian sampling in visual perception. PNAS. doi:10.1073/pnas.1101430108 Pisarchik AN, Feudel U (2014) Control of multistability. Physics Reports (in press). doi:10.1016/j.bbr.2011.03.031 Pisarchik AN, Chizhevsky VN, Corbalán R, Vilaseca R (1997) Experimental control of nonlinear dynamics by slow parametric modulation. Phys Rev E 55(3):2455–2461 Pisarchik AN (1998) Dynamical tracking of periodic orbits. Phys Lett A 242(3):152–162 Pisarchik AN, Pochepen ON, Pisarchyk LA (2012) Increasing blood glucose variability is a precursor of sepsis and mortality in burned patients. PLoS One 7(10):e46582 Rolls ET (2008) Emotion, higher order syntactic thoughts, and consciousness. In: Weiskrantz I, Davies MK (eds) Frontiers of consciousness. Oxford University Press, Oxford, pp 131–167 Rolls ET (2008) Memory, attention, and decision-making: a unifying computational neuroscience approach. Oxford University Press, Oxford Schwartz J-L, Grimault N, Hupé J-M, Moore CJBCJ, Pressnitzer D (2012) Multistability in perception: sensory modalities, an overview. Phil Trans R Soc B 367:896–905 Shpiro A, Moreno-Bote R, Rubin N, Rinzel J (2009) Balance between noise and adaptation in competition models of perceptual bistability. J Comput Neurosci. doi:10.1007/s10827-008-0125-3 Simonotto E, Riani M, Seife C, Roberts M, Twitty J, Moss F (1997) Visual perception of stochastic resonance. Phys Rev Lett 78(6):1186–1189 Ta’edd LK, Ta’eed O, Wright JE (1988) Determinants involved in the perception of the necker cube: an application of catastrophe theory. Behav Sci 33(2):97–115 Tolhurst DJ, Movshon JA, Dean AF (1983) The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vis Res 23:775–785 Tredicce JR, Lippi GL, Mandel P, Charasse B, Chevalier A, Picqué B (2004) Critical slowing down at a bifurcation. Am J Phys 72:799–809 World Medical Association (2000) Declaration of Helsinki: ethical principles for medical research involving human subjects. JAMA 284(23):3043–3045
ORIGINAL PAPER
Critical slowing down and noise-induced intermittency in bistable perception: bifurcation analysis Alexander N. Pisarchik · Rider Jaimes-Reátegui · C. D. Alejandro Magallón-García · C. Obed Castillo-Morales
Received: 23 October 2013 / Accepted: 21 April 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Stochastic dynamics and critical slowing down were studied experimentally and numerically near the onset of dynamical bistability in visual perception under the influence of noise. Exploring the Necker cube as the essential example of an ambiguous figure, and using its wire contrast as a control parameter, we measured dynamical hysteresis in two coexisting percepts as a function of both the velocity of the parameter change and the background luminance. The bifurcation analysis allowed us to estimate the level of cognitive noise inherent to brain neural cells activity, which induced intermittent switches between different perception states. The results of numerical simulations with a simple energy model are in good qualitative agreement with psychological experiments. Keywords Dynamics
Perception · Bifurcation · Noise · Brain ·
This work has been supported by the COECYTJAL-UdeG through project 05-2010-1-783. A. N. Pisarchik Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, 37150 Leon, Guanajuato, Mexico A. N. Pisarchik (B) Center for Biomedical Technology, Technical University of Madrid, Campus Montegancedo, 28223 Pozuelo de Alarcon, Madrid, Spain e-mail: [email protected] R. Jaimes-Reátegui · C. D. A. Magallón-García · C. O. Castillo-Morales Centro Universitario de Los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Paseo de la Montaña, Lagos de Moreno, Jalisco, Mexico
1 Introduction The perception of visual signals in the brain was among the first issues discussed in terms of multistability which has been introduced to provide mechanisms for information processing in biological neural systems (Atteneave 1971). Multistability reveals itself when a single physical stimulus or noise induces alternations between coexisting subjective percepts (Schwartz et al. 2012). Multistable perception characterizes the wavering percepts that can be brought about by certain visually ambiguous patterns, such as the Necker cube shown in the upper part of Fig. 1. When this figure with all frames is viewed for an extended time, the two percepts alternate spontaneously, changing as often as every few seconds. This alternation has been attributed to neural adaptation or satiation (Hebb 1949; Köhler and Wallach 1944). The mechanisms underlying the switches between different visual percepts are not well understood. Most current models imply that the biological origin of the perceptual switches lies in noise, inherent to brain neural cells activity, which results from relatively random spiking times of individual neurons (Deco et al. 2009). Furthermore, external noise can also induce perception alternations (Kanai et al. 2005). Contributed by the probabilistic spiking times of neurons (Tolhurst et al. 1983), the brain noise plays an important and advantageous role in signal detection and decisionmaking by preventing deadlocks and allowing switches between different perception states (Moreno-Bote et al. 2007; Gigante et al. 2009). Decisions may be difficult without noise which produces probabilistic choice. The switches are stochastic processes according to a Markov chain (Meyn and Tweedie 2008) and are measured by recording either phase durations or percept frequencies (Leopold et al. 2002). In a bistable image, such as the Necker cube, two different percepts are associated with two stable steady states (attrac-
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Fig. 1 Hysteresis in the Necker cube when wireframe contrast is varied. The isometric perspective of a wireframe drawing makes the spot perceived plan ambiguous. (Up) Images of the Necker cube for different wires’ contrasts. (Down) Bifurcation diagram of bistable model Eq. (4) when control parameter c is varied with velocity v = 0.01. The arrows indicate directions of the parameter change. c f and cb are, respectively, forward and backward saddle-node bifurcations
tors). A change in a control parameter leads to the deformation of their basins of attraction and finally to a change in their stability, that occurs at a critical point. The system behavior near the onset of a dynamical bifurcation has gained considerable attention in recent decades because bifurcations play a crucial role in characterization of system dynamics. While the statistical properties of bistable visual perception have been extensively studied and reported in many papers [see, for instance, (Merk and Schnakenberg 2002; Ta’edd et al. 1988; Aks and Sprott 2003)], the bifurcation analysis still remains an interesting research task which can provide additional information about brain functionality. In this paper, we use the bifurcation analysis for studying visual perception of the Necker cube which wireframe contrast c is taken as a time-variable control parameter. It should be noted that the parameter variation is practically necessary while dealing with a huge amount of data, in particular, for biological systems. This allows one to collect as many data as possible to perform statistical or bifurcation analyses. By increasing c, one can find a forward saddle-node bifurcation c f where one of the percepts changes its stability, and by decreasing c, one comes to a backward saddle-node bifurcation cb where another percept appears. While one percept is stable for c < c f and another for c > cb , bistability exists for cb < c < c f . When c is fixed, the brain has enough time for image recognition. Instead, if c is continuously varied in time, the bifurcation points are shifted from their statical values, no matter how slowly c is varied. The postponement of a bifurcation in a system with a time-dependent parameter known as critical slowing down
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is a general phenomenon. First discovered in laser equations (Mandel and Erneux 1984), it was then found numerically and experimentally in many dynamical systems (Kapral and Mandel 1985; Pisarchik et al. 1997; Pisarchik 1998; Tredicce et al. 2004). Since the shift of bifurcation depends on the velocity of the parameter change, this effect reveals itself as rate-dependent hysteresis. In addition, this phenomenon is intrinsically accompanied by an enhanced sensitivity to noise (Broggi et al. 1986; Huerta-Cuellar et al. 2009; Pisarchik et al. 2012). Relatively strong noise destabilizes coexisting attractors so that the system intermittently escapes from one state to another, the process related to the so-called Kramers’ escape problem (Kramers 1940). Since noise is ubiquitous in the brain at multiple scales, from vesicular release and spiking variability to fluctuations in global neurotransmitter levels, deterministic brain dynamics can be studied only theoretically using noiseless models. To numerically simulate our experiment, in this work, we use a noise-driven attractor model (Moreno-Bote et al. 2007), although there also exist oscillator models (Kalarickal and Marshall 2000; Lago-Fernandez and Deco 2002) where noise is assumed to be an inessential component. The paper is structured as follows. Section 2 is devoted to psychological experiments with the Necker cube, demonstrating the phenomena of critical slowing down and noiseinduced intermittency between two coexisting percepts. In Sect. 3, we present the results of numerical simulations with a simple energy model and compare with our experimental results. Finally, main conclusions are given in Sect. 4.
2 Experiment 2.1 Experimental methodology The majority of psychological experiments with the Necker cube were focused on the statistical measurements of switching times between two different percepts [see, e.g., (Merk and Schnakenberg 2002)]. In this work, we search the critical points for onset of intermittency using the wires’ contrast and background luminance as control parameters. In our experiments, the Necker cube was placed at the middle of a computer screen as black lines on a white background with a small spot in the left-middle corner which served as the fixation point in order to concentrate attention. The contrast of the middle lines was used as a time-dependent parameter. Initially, the contrast of three middle lines centered in the leftmiddle corner (marked by the spot) was set to white (255), so that these lines were indistinguishable on the white background, while the contrast of three middle lines centered in the right-middle corner was set to black (0). First, we changed the contrast in the forward direction, i.e., the contrast of the lines centered in the left-middle corner was increasing from
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0 to 255, while the contrast of the lines centered in the rightmiddle corner was simultaneously decreasing from 255 to 0, so that the image was smoothly changing from the cube on the left side of Fig. 1 to the cube on the right side during time T . Then, the same procedure was repeated in the backward direction, i.e., the contrast of the left-middle lines was increasing, while the contrast of the right-middle lines was simultaneously decreasing. To observe the critical slowing down phenomenon, we varied the wires’ contrast with different velocities and detected the instant of time when a subject recognized a first switch in the percept. In a bistable system, such switches occur in forward and backward saddle-node bifurcation points related to the direction of the parameter change. Generally, the distance between these two bifurcation points yields dynamical hysteresis determined by both the velocity of the parameter variation and noise. This behavior was observed in many bistable systems (Kapral and Mandel 1985; Pisarchik et al. 1997; Pisarchik 1998; Tredicce et al. 2004). In addition, critical slowing down is intrinsically accompanied by an enhanced sensitivity to noise (Broggi et al. 1986; Huerta-Cuellar et al. 2009; Pisarchik et al. 2012). Strong noise can destabilize coexisting attractors resulting in a metastable attractor when the system jumps intermittently from one state to another (Pisarchik and Feudel 2014). This phenomenon is referred to as noise-induced intermittency. There are two types of hysteresis; one (positive) is in bistability and another one (negative) is in noise-induced intermittency. The former type of hysteresis shown in Fig. 1 is observed in a noiseless bistable system or in a system with relatively low noise. The assumption that noise is responsible for switches between different percepts predicts smaller hysteresis in bistability and larger in intermittency. The effect of critical slowing down due to parameter variation is opposite: The faster the parameter change, the larger the hysteresis in bistability and the shorter in intermittency. We also assume that brain noise of each person is constant, or at least, that it is not gradually increasing or decreasing during the experiment. Of course, we cannot exclude brain adaptation (Shpiro et al. 2009) or that Bayesian inference dynamics (MorenoBote et al. 2011) might lead to a change in noise; however, it is unlikely that adaptation acts only at high parameter change rates and has no effect at slow velocities. In all our experiments, we varied the wires’ contrast linearly in time t, first in the forward and then in the backward direction. Starting from the left-side or from the right-side image in Fig. 1, we recorded, respectively, the times t f and tb when a subject sighted the first switch in percept. Raw data typically look like those displayed in Fig. 2, where we plot t f and tb as a function of the run time T . Usually, t f and tb are very close to each other. The first switch in percept occurs for the critical value of the control parameter corresponding to the onset of noise-induced intermittency.
tb
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Fig. 2 Typical dependences of forward and backward switch times on run time T (subject 12)
Since in all our experiments, the contrast is always a linear function of time, i.e., c ∼ vt, and the velocity v of the contrast change is proportional to the inverse run time T −1 , the normalized time t/T can also be used as a control parameter because this value is always associated with the corresponding contrast c. Though there is no principal difference which of these parameters is used as a control parameter, the time is more convenient, because it can be recorded directly in experiments. In total, twenty student volunteers took part in our experiments. All subjects gave written informed consent to participate and were naive as to the specific experimental question. The studies were performed in accordance with the ethical standards laid down in the Declaration of Helsinki (World Medical Association 2000). All subjects were well aware about bistability of the Necker cube and were able to distinguish the percept switches. They were instructed to fixate their attention on the small spot and press a key on the computer keyboard in front of them only once at the moment when they fixed a first change in the spot position. In each direction, only the time of the first switch in percept was recorded. After the first switch which indicated the onset of intermittency, all subjects observed other switches between two different percepts; however, these later switches were ignored. We carried out two series of experiments. The first series were performed with white background, and the run time T was used as a control parameter. For every fixed T , we measured the times of the first switches in the forward and backward directions (t f and tb ) as shown in Fig. 2. In the second series, we fixed the run time to T = 30 s and used the background luminance as a control parameter. From twenty subjects involved in our experiments, fifteen subjects (1–15) participated in the first series and eight sub-
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2.2 Experiments with white background From fifteen students participated in this experimental series, only the data of eight subjects were selected for further analysis because of several reasons. The possible reasons were the following: (i) a person did not well understand the task, for instance, he/she pressed the key not at the moment of the first switch in the percept, but when the intermittent switches disappeared, and (ii) the loss of attention that could be caused by personal problems, for example, the person illness, tiredness, worrying or thinking about something else, etc. All these factors could result in a very high deviation from the average and after revealing the subjective reasons the data of these subjects were excluded from the analysis. The onset of intermittency was measured for 14 different velocities of the contrast variation in each direction, starting from the longest run time T = 30 s (slow parameter variation) to the shortest run time of T = 4 s (fast parameter variation) with the interval of 2 s. Figure 3 shows the normalized hysteresis h = (t f + tb − T )/T
(1)
as a function of T . For large run times, all subjects detected the first switch in percept at times less than one half of the run time (t f ≈ tb ≤ T /2), that resulted in negative hysteresis in the intermittent regime. As we expected, h saturates as the run time T increases, i.e., when the velocity of the contrast variation decreases; the faster the parameter change, the shorter the hysteresis range. Such a behavior is a sequence of critical slowing down near the saddle-node bifurcations. Although this phenomenon was previously observed in many other systems with time-dependent parameters, this is the first, to the best of our knowledge, experimental demonstration of critical slowing down in visual perception.
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subject 2 subject 4 subject 5 subject 6 subject 8 subject 11 subject 12 subject 15
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jects (3, 4, 5, 8, 13, 16, 18, 19) in the second series, due to personal reasons. Only five subjects (3, 4, 5, 8, 13) took part of both series of the experiments. The data of eight subjects (2, 4, 5, 6, 8, 11, 12, 15) participated in the first series and eight subjects (3, 4, 5, 8, 13, 16, 18, 19) participated in the second series were included in the analyses. The data of three subjects (4, 5, 8) were used in the analyses of both series. Since the analysis of each series was carried out independently, the participation of the same subject in both experiments was not necessary. To exclude a possible brain adaptation to the cognitive task, the same test was not repeated for the same subject. In the following, we present the experimental results of the two experimental series.
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Fig. 3 Experimental data for eight subjects viewing the Necker cube with white background. The dashed lines are the exponential fits by Eq. (2) demonstrating critical slowing down. The saturation at large run times is determined by internal brain noise, and the slope is related to the reaction time
The dependences in Fig. 3 are well approximated by an exponential decay h = k exp(−T /γ ) + h s ,
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where k is the scaling coefficient, γ is the scaling exponent, and h s is the saturation value. Since we assumed that noise induces intermittent switches between the coexisting states, the existence of the saturation in the rate-dependent hysteresis confirms our hypothesis that the hysteresis at saturation h s is determined by brain noise. In other words, the higher the brain noise, the larger the absolute value of hysteresis. This simple experiment allowed not only measuring brain noise, but also estimating the reaction time which is determined by the scaling exponent γ . Specifically, when the contrast was changed too fast, the subjects did not have enough time to react adequately to this change that resulted in decreasing the hysteresis region. The dynamical characteristics of eight subjects extracted from the experimental dependences in Fig. 3 using the exponential fitting Eq. (2) are summarized in Table 1. One can see that h s varied from −0.57 to −0.92 with the mean value −0.76 and the standard errors for all subjects are less than 15 %. The comparison of h s for different subjects provides us with information about the difference in the relative levels of personal brain noise. For example, we can conclude that subject 6 had minimum and subject 11 had maximum brain noise among all subjects in this group. The value γ allows us to compare the reaction time of different participants. The very big SD for mean γ (44 %) means that the reaction time significantly varied across the individuals. Indeed, γ of subject 2 was approximately four times larger than γ of subject 12 (6.35 against 1.52), although in our experiments the error in γ for subject 12 was very large (41 %).
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Table 1 Main characteristics evaluated from experimental data with white background Saturation h s
SE
γ
SE
2
−0.75
0.03
6.35
0.74
4
−0.80
0.02
5.66
1.39
5
−0.70
0.05
7.64
2.00
6
−0.57
0.03
5.54
0.71
8
−0.90
0.01
4.14
0.08
11
−0.92
0.01
5.35
0.86 0.63
12
−0.72
0.02
1.52
15
−0.74
0.09
10.44
6.73
Mean
−0.76
0.11a
5.83
2.58a
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Negative hysteresis observed at saturation in all subjects means that their brain noise was so strong that there was not bistability in their perception. In other words, noise completely destabilized the coexisting attractors. In this content, the interesting question arises: How does the hysteresis depend on noise? Since we do not know how to change internal brain noise, a simple solution could be to add external noise and use it as a control parameter. Another way to infer the perception dynamics is to change the perception sensitivity to brain noise. In the second series of experiments, we chose the latter way. Namely, we varied the intensity of background pixels and used their luminance as a control parameter. Although the luminance is not a stochastic signal, its effect is rather similar to that of external noise. In fact, it acts as a bias which increases the energy in the double-well potential of the bistable system, thus making perception more sensitive to brain noise. Eight subjects participated in this series of experiments. Similarly to the previous case, we varied the wires’ contrast and measured hysteresis. However, now the run time was fixed to a relatively large value T = 30 s, and the background luminance was varied. Although for dark background, the data differed across individuals much more than for white background; nevertheless, for four subjects, the dependences were well approximated by a linear fit with less than 12 % standard error, and for four other subjects exhibited a resonance-like behavior. Figure 4 shows the dependences of the normalized hysteresis on the background darkness N = 1 − L (L ∈ [0, 1] being the pixels luminance). The linear dependences in Fig. 4a exhibit a linear growth of the intermittency range h = h 0 + δ N . Since for N = 0 all background pixels are white, the hysteresis h 0 at the intersection of the extrapolated
subject 5 subject 16 subject 18 subject 19
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Fig. 4 Experimental normalized hysteresis as a function of N = 1− L (L being the background pixels luminance) for T = 30 s. a Monotonous and b nonmonotonous dependences. The dashed lines are linear fits
fitting lines with the ordinate axis N = 0 corresponds to the case studied in the first series of experiments, whereas the slope δ characterizes the perception sensitivity to the background luminance. The results of the analysis of the data in Fig. 4a are summarized in Table 2. The coincidence of the mean value of the intercept h 0 from Table 1 with the mean value of saturation h s from Table 2 indicates on a good accuracy of the obtained experimental results, that confirms our hypothesis that both values are associated with brain noise. Our method allowed the comparison of brain noise and perception sensitivity for different persons, for example, we can conclude that subject 3 had much smaller brain noise that subject 4. On the other hand, subject 4 had a very low sensitivity to the background luminance. The nonmonotonous behavior of other four subjects shown in Fig. 4b is probably related to the phenomenon of stochastic resonance. The external bias signal added to
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Biol Cybern Table 2 Main characteristics evaluated from experimental data in Fig. 4a Subject
Intercept h 0
SE
Slope δ
SE
3
−0.41
0.05
−0.27
0.08
4
−0.75
0.06
−0.12
0.10
8
−0.62
0.06
−0.29
0.10
13
−0.61
0.06
−0.24
0.10
Mean
−0.60
0.14a
−0.23
0.08a
SE means a standard error a Standard deviation from the mean
internal brain noise got in resonance with the Kramers time in the bistable system that facilitated intermittent switches between the coexisting states in the Necker cube, thus enlarging the intermittency area. Such a behavior is not surprising. Early, stochastic resonance was observed in psychological experiments (Chialvo and Apkarian 1993) with bistable Haken’s images (Haken 1983). Furthermore, another kind of stochastic resonance, subthreshold stochastic resonance, was detected in visual perception (Simonotto et al. 1997). The latter kind of stochastic resonance occurs in monostable excitable systems near the threshold. Such a resonance enhances recognition of obscured images when small noise neurons are below the threshold of action activity.
3 Theory Having the results as described above, we will now proceed to construct a simple one-parameter theoretical model that is nonlinear and stochastic. We consider the simplest doublewell potential model which exhibits the coexistence of two fixed points, that is the double-well potential model. The model is based on the assumption that each of two neuronal populations (say A and B) represent a different interpretation of the stimulus (c A and c B ) (Moreno-Bote et al. 2007) x˙ = −4x(x 2 − 1) − 2c A (x − 1) − 2c B (x + 1) + αξ(t), (3) where x is the state variable proportional to the difference between the dimensionless firing rates of the two competing populations, ξ(t) is zero mean Gaussian white noise, and α is the noise intensity. Equation (3) is derived from the energy function d E/d x = −τ d x/dt describing perceptual alternation dynamics, where the minima are located close to x = ±1. In our simulations, for simplicity, the time scale τ is set to 1. In our case, c A and c B are associated with the wire contrasts responsible for different image interpretations. Since in our experiments, we changed the contrasts of two wireframes simultaneously in opposite directions (while one was increasing the other was decreasing), we can use only one-parameter c = c A = −c B that makes Eq. (3) more
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simple: x˙ = −4x(x 2 − 1) + 4c + αξ(t),
(4)
The important advantage of the theoretical consideration over the experimental one is that in the theory, we can study the behavior of the noiseless bistable system (α = 0), while in experiments, it is impossible since internal brain noise is not only inevitable but also very high because, as our experiments show, the hysteresis in visual perception is always negative (see Fig. 4). Without noise (α = 0), the system Eq. (4) flows into one of the stable states (depending on the initial condition) and stays there forever. If the control parameter c is varied as c = c0 ± vt
(5)
with velocity v, the system passes through forward and backward bifurcation points c f and cb , where the system changes its attractor, as shown in the bifurcation diagram in Fig. 1. The system state depends on both the initial condition c0 and the direction of the parameter change determined by the sign in Eq. (5). The bistability is accompanied by hysteresis h = c f − cb . Due to critical slowing down, the position of bifurcation points depends on the velocity of the parameter change, i.e., h enlarges when v increases. Such a behavior is illustrated in Fig. 5a, b. Since the perception of one stimulus occurs when the firing rate of its population is higher than that for another population, the position of the bifurcation point is detected at the moment when the variable x crosses zero. While the increasing velocity enlarges h, the increasing noise produces an opposite effect, i.e., hysteresis decreases when noise increases. This effect is clearly seen in Fig. 5c. For sufficiently strong noise, instead of bistability, two-state intermittency takes place when the system intermittently switches between two coexisting states. In this case, hysteresis is negative, as seen in Fig. 5d. Figure 6 shows how the hysteresis range depends on the noise intensity at different velocities of the parameter change. Positive hysteresis (h > 0) indicates bistability and negative hysteresis (h < 0) means two-state intermittency. Since in our experiments hysteresis was negative, we concluded that the internal brain noise was strong enough to induce intermittent switches between coexisting percepts. The theoretical noise dependences in a large range of the noise intensity are well approximated by a sigmoidal fit. Nevertheless, within a small range, a linear fit yields a rather good approximation that is in agreement with our experimental results. To compare the results of the numerical simulations with our experiments, in Fig. 7, we plot the hysteresis value versus inverse velocity v −1 for different noise intensities α. These dependences are well approximated by exponential decay Eq. (2), in good agreement with experimental results (see Fig. 3).
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Fig. 7 Theoretical dependences of hysteresis on the inverse velocity of parameter change for different noise intensities. The dashed lines are exponential fits
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A good qualitative agreement between theoretical and experimental results indicates that even such a simple model allows simulation of brain cognitive dynamics. However, our model cannot describe the nonmonotonous noise dependences observed in the experiments. We expect that more realistic models, such as, e.g., the Hodgkin–Huxley neuron model with a stochastic term (Borisyuk et al. 2009) will allow a more detailed description of the experimentally observed features.
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Fig. 5 Bifurcation diagrams demonstrating a shift of bifurcations due to critical slowing down when control parameter c is varied for a v = 0.18 and b v = 0.45 without noise and for c α = 18 and d α = 60 at fixed velocity v = 0.01. While for weak noise the hysteresis is positive (c), for strong noise it is negative (d)
4 Conclusions The phenomena of critical slowing down and noise-induced two-state intermittency were studied in the bistable visual perception of the Necker cube in order to estimate important
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cognitive characteristics of the brain, such as brain noise and reaction-cognition time. The results obtained with the developed energy-based model are in good qualitative agreement with the experiments. Both the experimental and numerical results confirmed the hypothesis that the alternation between competing percepts associated with activation of different states of neural activity is driven by noise. It is important to stress that the noise originated from random spiking fluctuations may be behaviorally adaptive, and therefore should not be considered only as a problem in terms of how brain works. We believe that the testing methodology proposed in this work can find applications in understanding pathological brain stability states, such as schizophrenia and obsessivecompulsive disorder. A state with a weak stability at one end of the distribution contributing to the schizophrenia symptoms (Rolls 2008) may result from very strong brain noise. Instead, a too stable state at the other end of the distribution may contribute to the symptoms of obsessive-compulsive disorder (Rolls 2008) due to very little brain noise. Large deviations of the cognition reaction time from its mean value can indicate on serious brain diseases, such as delayed response syndrome (Figley and Sprenkle 1978) or reactive attachment disorder (Hanson and Spratt 2000). Of potentially great importance is that the measures of the noise level and reaction-cognition time can make the diagnosis of these diseases easier. Acknowledgments We thank K. M. Prado-Tabares for her help in the data collection and all voluntaries for their participation in the experiments. A. N. P. acknowledges support from CONACYT (Mexico).
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