Stochastic catastrophe analysis of switches in the perception of apparent motion

Abstract

An important objective when the dynamics of visual perception are studied is to discover the nonlinear basis for cooperative phenomena such as bistability and hysteresis, phenomenathat are not readily explainedby a linear model. Bistability means that more than one percept is possible for a single stimulus. Hysteresis means that what is perceived at any moment depends on the preceding state of the visual system. As a result, perception depends on the direction of parameter change. When a parameter is gradually increased, changes in percept will occur at a different point from changes in percept that occur when the parameter is gradually decreased. That is, the initial percept is likely to persist, although the parameter has changed to a value that favors the alternative percept.It is clear that a linear model ( y = ax + b) cannot explain bistability and hysteresis. According to a linear model, one stimulus should provoke one and only one percept (i.e., in the formula of the linear model, every value of x gives a single value of y, not two or more different values). A nonlinear model is needed to deal with these dynamical phenomena. A logistic model is not sufficient; it can explain rapid changes in responses, but not the occurrence of two different percepts for the same stimulus. We will propose a subset of dynamical system models, so-called stochastic catastrophe models, to explain phenomena such as bistability and hysteresis. A new prediction, that hysteresis must be accompanied by another dynamical phenomenon called divergence, is derived. The derivation of this new prediction is one of the main advantages of using catastrophe theory to explain nonlinear phenomena. Other theories do not have this mathematical link between hysteresis and divergence.We will show on a conceptual level why hysteresis must go together with divergence, and we will present the first empirical evidence of the existence of divergence in perception. We also show that stochastic catastrophe theory offers indices for model selection, which is seen as a major problem in psychology, and we show that we can test the fit of different catastrophe models efficiently.We use two existing paradigms based on apparent motion (Hock, Kelso, & Schöner, 1993;Hock, Kogan, & Espinoza, 1997) that clearly show that hysteresis and bistability do exist in visual perception. Apparent motion is an important subject in the study of the nonlinear dynamics of perception because, contrary to static stimuli, it has an extended time scale and parameters, such as speed and direction, that are relatively easy to manipulate. A lot of studies on dynamics have been done with static stimuli like Fisher's man's-face/woman's-figure-picture (Fisher, 1967). This drawing can be interpreted either as a man's face or as a woman's figure (i.e., the percept is bistable). These studies show that the dynamics of visual perception can be studied with static stimuli. However, by changing qualitative parts of pictures, it is not exactly clear which parameter is changed. The changing pa...

Estimation and interpretation of 1/fα noise in human cognition

“…We will argue here that the presence of 1/ f a noise does not necessarily indicate the presence of nonlinearity or deterministic chaos: In fact, 1/ f a noise and nonlinearity are to a large extent orthogonal concepts. This is an important point, not only because the necessary relation between 1/ f a noise and nonlinearity sounds (falsely) plausible given the relationship elsewhere between chaotic functions and selfsimilarity (e.g., the Mandelbrot set; Mandelbrot, 1982), but also because the presence of nonlinear structure in the dynamics of human cognition has received some attention recently (see, e.g., Busemeyer, Weg, Barkan, Li, & Ma, 2000;Elman, 1998;Heath, 2000;Heath, Kelly, & Longstaff, 2000;Kelly, Heathcote, Heath, & Longstaff, 2001;Kelso et al, 1998;Ploeger, van der Maas, & Hartelman, 2002;Port & van Gelder, 1995;Pressing, 1999a;Roe, Busemeyer, & Townsend, 2001;Thelen & Smith, 1994;Usher & McClelland, 2001;Ward, 2002).…”

Nonnormality and divergence in posttreatment alcohol use: Reexamining the Project MATCH data "another way."

“…where z indicates the behavior variable (dependent variable), β and α represent two control parameters (called the splitting and normal parameters, respectively), and C (α, β) is an integration constant (Ploeger et al, 2002 One of the more difficult aspects of stochastic catastrophe modeling is the identification of proper control parameters for the model. In previous studies (Hufford et al, 2003), we have experienced great success with proximal and distal risk as the normal and splitting control parameters, respectively.…”

Linking dynamical perceptual decisions at different levels of description in motion pattern formation: Computational simulations

“…Dynamical modeling for the motion quartet has also been based on catastrophe analysis (Ploeger, van der Maas, & Hartelman, 2002). …”