Interplay between order-parameter and system parameter dynamics: considerations on perceptual-cognitive-behavioral mode-mode transitions exhibiting positive and negative hysteresis and on response times

Abstract

A mathematical model is presented for the emergence of perceptual-cognitive-behavioral modes in psychophysical experiments in which participants are confronted with two alternatives. The model is based on the theory of self-organization and, in particular, the order parameter concept such that the emergence of a mode is conceptualized as an instability leading to the emergence of an appropriately defined order parameter. The order parameter model is merged with a second model that describes adaptation in terms of a system parameter dynamics. It is shown that the two-component model predicts hysteretic mode-mode transitions when control parameters are increased or decreased beyond critical values. The two-component model can account for both positive and negative hysteresis effects due to the interaction between order parameter and system parameter dynamics. Moreover, the model-based analysis reveals that response time curves look rather flat when response times are relatively decoupled from the mode-mode transition phenomenon. In general, response time curves exhibit a peaked close to the mode-mode transition point. In this context, the possibility is discussed that such peaked response time curves belong to the class of critical phenomena of self-organizing systems. In order to illustrate the relevance of peaked response time curves for future research and research reported in the past, results from a perceptual judgment experiment are reported, in which participants judged their ability to stand on a tilted slope for various angles of inclination. Response time curves were found that exhibited a peak around the mode-mode-transition points between "yes" and "no" responses.

Fig. 1

Outline of a generalized order parameter model for switches between alternative modes of perceptual-cognitive-behavioral processes. The generalized model is based on Haken’s synergetic order parameter model for non-equilibrium pattern formation and pattern recognition systems [14]. The model takes the interplay between order parameter dynamics and system parameter dynamics into account. Variables: ξ _k order parameters, L _k system parameters, α (experimentally accessible) control parameter reflecting properties of the environment, s _k and λ _k coupling parameters. See text for details. Mathematical details can be found in Section 2

Fig. 2

Representative trajectories of order parameters ξ ₁(t) and ξ ₂(t) as obtained by solving (5) numerically (Euler forward with single time step 0.01). Parameters as in Fig. 3 for the first trial in the ascending sequence

Fig. 3

Simulation results for hysteretic mode-mode transitions in the case of positive hysteresis and a flat response time curve in the pre- and post-transition regions. Panel a λ ₁ (solid) and λ ₂ (dashed). Panel b L ₁ (solid) and L ₂ (dashed). Both lines are on top of each other. Panel c ξ _1,st (full circle) and ξ _2,st (open diamonds). Panel d response times in a logarithmic scale as obtained numerically (symbols) and as defined by (35) (solid line). Parameters: g=1.0168, h=0, α={5,7,...,33}, 𝜃=0.95, D=0.60, w=0.10, L ₀=0.7002, β=0.0023, T=5

Fig. 4

Simulation results for hysteretic mode-mode transitions in the case of negative hysteresis and a response time curve that is flat in the pre- and post-transition regions. Panels as in Fig. 3. Parameters: g=1.0, h=0.0219, α={11,13,...,41}, 𝜃=0.95, D=0.60, w=0.10, L ₀=0.7002, β=0.0022, T=5

Fig. 5

Illustration of the behavior of the auto-regulated pseudo control parameter ΔL involved in mode-mode transitions exhibiting negative hysteresis. ΔL was computed from L ₁ and L ₂, see (20), using the values shown in Fig. 4b

Fig. 6

Simulation results for hysteretic mode-mode transitions in the case of positive hysteresis and a peaked response time curve. Panels as in Fig. 3. Parameters: g=1.0168, h=0, α={5,7,...,33}, η=0.10, E=0.20, w=0.10, L ₀=0.7002, β=0.0023, T=5

Fig. 7

Simulation results for hysteretic mode-mode transitions in the case of negative hysteresis and a peaked response time curve. Panels as in Fig. 3. Parameters: g=1.0, h=0.0219, α={11,13,...,41}, η=0.10, E=0.20, w=0.10, L ₀=0.7002, β=0.0022, T=5

Fig. 8

Experimental results for visual (solid lines) and haptic (dashed lines) judgment tasks. Top panel percentage of “Yes” responses. Bottom panel response times (linear scale)