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.

Appendices

Appendix: A: Derivation of (17b) and (25b)

Recall that α _{r e l,c,2} is the critical control parameter value at which the transition from mode 1 to mode 2 occurs when α is gradually increased (ascending condition). From (16) it follows that an increase in α implies that the ratio λ ₂/λ ₁ increases. The critical condition for the ratio λ ₂/λ ₁ is defined by (8). Substituting (16) into (8) and replacing α _{r e l} by α _{r e l,c,2}, we obtain

$$ \frac{L_{0} + \beta \, \alpha_{rel,c,2}}{g} = L_{0} - \beta \, \alpha_{rel,c,2} \ . $$

(44)

Solving for α _{r e l,c,2} yields (17a). By analogy, the critical value α _{r e l,c,1} can be determined. Recall that α _{r e l,c,1} denotes the critical control parameter value at which the transition from mode 2 to mode 1 occurs when α is decreases gradually (descending condition). The decrease of α implies an increase of the ratio λ ₁/λ ₂, see (16) again. The critical condition for the ratio λ ₁/λ ₂ is given by (9) Substitute (16) into (9) and replacing α _{r e l} by α _{r e l,c,2}, we obtain

$$ \frac{L_{0} - \beta \, \alpha_{rel,c,1}}{g} = L_{0} + \beta \, \alpha_{rel,c,1} \ . $$

(45)

Solving for α _{r e l,c,1} yields (17b). Note that (17a) and (17b) exemplify the two control parameter principles given by (14) and (15) that hold for any dynamical system that exhibits hysteresis. In particular, using (17a) and (17b) we see explicitly that

$$ {\Delta} \alpha=\alpha_{c,2}-\alpha_{c,1}= \alpha_{rel,c,2}-\alpha_{rel,c,1}=2\alpha_{rel,c,2}=-2\alpha_{rel,c,1} $$

(46)

holds, which is (15). Consequently, we have α _{r e l,c,2}=Δα/2 and α _{r e l,c,1}=−Δα/2, as stated see (17b) again.

The critical control parameter values in the case of negative hysteresis can be determined by a similar argumentation. To this end, however, we need to assume that a time scale separation holds for the variables involved in the order parameter and system parameter model. The characteristic time scales are summarized in Table 5.

Table 5 Key variables used in the model sketched in Fig. 1 and their characteristic time scales

First of all, it is assumed that order parameters ξ _k evolve quickly every time a perceptual process or behavioral response is required. The L _k (n) system parameters evolve slowly relative to ξ _k (t). However, they evolve quickly on the time scale of the repetitively performed judgment tasks or executed behavioral responses. That is, they are considered as fast evolving variables relative to changes of the control parameter α. Consequently, at the bifurcation point as defined by the critical control parameter values α _c,1 and α _c,2 the system parameters L _k are in good approximation at their respective stationary values.

Let us determined the critical control parameter value α _{r e l,c,2} for transitions from mode 1 to mode 2 when α is increased (ascending condition) provided that mode “1” has been “on” or active in several previous behavioral responses. Increasing α implies that the ratio λ ₂/λ ₁ increases. In (16) we replace α _{r e l} by α _{r e l,c,2} such that λ _k =L _k −β α _{r e l,c,2} For L _k we use the corresponding stationary values s _k : L ₁=s ₁ and L ₂=s ₂. That is, as mentioned above, we assume that at the transition point the system parameter dynamics is close to its fixed point. From (19) we obtain L ₁=s ₁=L ₀−h and L ₂=s ₂=L ₀. Substituting these results into (8), we obtain

$$ \frac{L_{0} + \beta \, \alpha_{rel,c,2}}{g} = L_{0} -h \, \beta \, \alpha_{rel,c,2} \ . $$

(47)

Solving for α _{r e l,c,2} yields (25a). By analogy, (25b) can be derived.

Appendix : B: More than two alternatives for the special case of a set of almost indistinguishable stimuli

In the conclusions (Section 4), a generalized order parameter model involving more than two alternatives has been discussed and a heuristic argument has been developed that suggests that increasing the number of alternatives or choices will increase the response times predicted by that generalized model. This argument can be worked out in detail for a decision-making process in the special case when all order parameters evolve in exactly the same way for most of the time of the process. That is, we consider M alternatives or choices, where each alternative is described by an order parameter ξ _k . Based on certain stimuli decisions have to be made (e.g., in an experiment on associative memory involving stimulus-response pairs the learned associations are tested by presenting the stimuli in an random order). The stimuli are assumed to be almost indistinguishable such that all ξ _k assume that same values for most of the decision-making processes: ξ _k (t)=z(t) for k=1,…,M. Note that in this context we also assume that λ _k =λ holds for k=1,…,M. That is, all growth parameters are identical. Eventually, the order parameters ξ _k will diverge (i.e., ξ _k ≠ξ _j for all k≠j) due to the impact of fluctuations and only one order parameter will ”survive” (winner-takes-all dynamics, see Section 2.1). In the following calculations, we do not account for this final phase and assume that the final phase does not qualitatively affect the relationship between the number of choices M and the response times.

For ξ _k (t)=z(t)>0 we need to take the interaction terms $g \xi _{k} {\xi _{j}^{2}}$ into consideration. That is, (34) has to be supplemented with M−1 interaction terms describing the inhibitory impacts of the other patterns or choices on the pattern (choice) of consideration. Since we have ξ _k (t)=z(t) for all k it follows that all interaction terms have the same form and are equal to g z ³(t). Adding M−1 terms g z ³(t) to the right-hand side of (34) and using ξ _k rather than z as variable (for sake of consistency with our presentation in Section 3), we obtain

$$ \frac{\mathrm{d}}{\mathrm{d}\tau} \xi_{k} = \lambda \, \xi_{k} - g(M-1){\xi^{3}_{k}}- {\xi^{3}_{k}} = \lambda \, \xi_{k} - G {\xi^{3}_{k}} $$

(48)

with

$$ G=g(M-1)+1 >1\ . $$

(49)

Note that as long as ξ _k (t)=z(t) for all k holds, the trajectories converge to an unstable saddle point with $\xi _{k}=\sqrt {\lambda /G} \ \forall \ k$. Therefore, we assume that the trajectories ξ _k converge from ξ _k =ξ ₀ at t=0 towards a value ξ _k =ξ _T at time T with ξ _T >ξ ₀ and $\xi _{T}<\sqrt {\lambda /G}$. After that point the trajectories start to diverge and one of the trajectories will approach the finite value $\xi _{k}=\sqrt {\lambda }>\sqrt {\lambda /G}$, while all others will converge to zero. Consequently, the response time RT exceeds the time point T: R T>T. The time T can be calculated just as in Section 2.1. We obtain

$$ T = \frac{1}{2\lambda} \left\{ \ln\left(\frac{{\xi^{2}_{T}}}{{\xi_{0}^{2}}}\right) + \ln\left( \frac{\lambda-G{\xi_{0}^{2}}}{\lambda-G{\xi_{T}^{2}}}\right)\right\} = \frac{1}{\lambda} \left\{ \ln\left(\frac{\xi_{T}}{\xi_{0}}\right) - \frac{1}{2} \ln\left( \frac{\lambda-G{\xi_{T}^{2}}}{\lambda-G{\xi_{0}^{2}}}\right)\right\} \ . $$

(50)

Note that for G=1 and $\xi _{0}=D\sqrt {\lambda }$, $\xi _{T}=\theta \sqrt {\lambda }$ we re-obtain (35). Importantly, the auxiliary parameter G increases with the number of choices M, see (49). Therefore, the question arises how does the time T depend on the factor G. Differentiating T with respect to G yields

$$ \frac{\mathrm{d}T}{\mathrm{d}G}=\frac{1}{2W} \left({\xi_{T}^{2}}-{\xi_{0}^{2}}\right)>0 $$

(51)

with $W=(\lambda -G{\xi _{0}^{2}})(\lambda -G{\xi _{T}^{2}})>0$. We see that the duration T contributing to the response time RT increases when G increases. This implies that T increases when the number M of choices increases. As mentioned above, this calculation is a crude estimate for the response time RT. The response time includes a final phase in which ξ _j ≠ξ _k holds. In fact, the assumption ξ _k (t)=z(t) for t<T and ξ _j ≠ξ _k for t>T should be considered as a simplified two-phases scenario that is used to derive by analytical methods an estimate for the lower bound T of the response time RT. Nevertheless, this estimate suggests that the response time RT increases as a function of M. More detailed calculations (involving simulation studies) are needed to obtain an advanced understanding of this issue but are left to the future.