This is a somewhat sprawling paper, which is probably because it's hard to talk about dynamical systems or computational modeling to psychologists without a lot of introduction. For example, there is a whole section describing the concept of an attractor. This may not be a bad reference for future pedagogy...
The paper is also somewhat confusing because all the models have been described elsewhere (apparently), so there isn't always a lot of details, and some of the methods are only fully described in book chapters. Ugh.
They present two main models. The first one, presented in a fair amount of detail, is a model of individual behavior based on a logistic model: x_n+1 = rx_n(1 - x_n). Why they use that model is not explained, other than it's a simple model of the growth of a system. They use x_n as an individual's current state, and r_i as a measure of the individual's "personality" or something. Then they couple this and show how coupled individuals can synch up their behaviors and/or more deep lying personalities.
The idea for this analysis apparently draws heavily on a paper in BBS by M.D. Lewis (2005) titled "Bridging emotion theory and neurobiology through dynamic systems modeling."
An important point: "The causal loops in this process [of emotion interpretation] not only span different levels of emotional-cognitive linkage, but also different time-scales. At the lower level are emotional reactions, which last seconds or minutes. On the intermediate level are moods. On the highest level are personalities, which are relatively stable." (p. 359)
Coupled logistic maps are used to model the synchronization of people in social interaction:
Modeling the synchronization of internal states
Assume also that on each time step, the value of each person's control parameter r drifts somewhat in the direction of the value of the other person's control parameter. The rate of this drift and the size of the initial discrepancy between values determine how quickly the control parameters begin to match.
Behavioral coordination occurs when each person modifies his control parameter in an attempt to match the other person's pattern of behavior. The exact value of the partner's r may be invisible, but he remembers the partner's most recent set of behavior (i.e., the most recent values of x) and his own. The person compares the two sets of behaviors, and adjusts his control parameter until their is a match. Basically, if the partner's behavior is more complex than one's own, he increases his r, and vice versa.
The mathematics of coupling:
Two kinds of coupling: fast and slow.
Fast coupling:
This is the coupling between behaviors, already described above.
Slow coupling:
This is the coupling between control parameters.
Each player estimates the Liapunov exponent of his behavior and his partner's behavior as a running average. The two resulting Liapunov exponents are then compared via a function
where sgn(x) is the sign function that simply returns +1 if x > 0, -1 if x < 0, and 0 if x = 0. The control parameter is then updated as a function of itself, the difference in Liapunov exponents, and a small contribution of the local shape parameter surface. This last thing basically an inertial term which means that a slow-changing thing continues to change slowly, and vice versa.
They get an interesting result. With weak coupling of behaviors, convergence of behavior takes a long time, but also involves the convergence of the control parameter. With strong coupling, they get a very fast convergence of behavior, but much much slower convergence of the control parameter.
They interpret this in terms of influence and control.
"When influence is too weak, synchronization may fail to develop. Very strong influence, on the other hand, can prevent the development of a relationship based on mutual understanding and empathy. Highly controlled partners may fully synchronize their behavior, but they are correspondingly less likely to internalize the values of control parameters necessary to maintain such behavior in the absence of interpersonal influence. For such internalization to occur, intermediate levels of mutual influence would seem to be most effective." (pp. 365-366)
Modeling personality development
The authors also interpret their model in terms of parental vs. peer influence on personality (the control factor. This work is (allegedly) described in detail in Nowak, Vallacher, & Zochowski (2002) The emergence of personality: Personality stability through interpersonal synchronization, In: Advances in personality science.
They assume a coupled system in which individual A's control parameter is fixed, and individual B adapts. They consider the following scenarios in which they varied the attractor for B's control parameter:
- No attractor. Here, the control parameter of B matched that of A over a moderate amount of time.
- B has an attractor whose position matched the control parameter of A. Here synchronization occurred very rapidly.
- B has a weak attractor at a value that did not match the value of A's control parameter. Here, B eventually matched its control parameter to that of A, but it took a relatively long time. Matching was eventually achieved because the tendency to synchronize was stronger than the pull of its own relatively weak attractor.
- B has a strong attractor at a non-matching value. Here, B was unable to match the value of A's control parameter.
They interpret these results and the previous ones in terms of the development of personality during childhood:
"Despite the disproportionate amount of time that children spend with their parents as opposed to any single peer—particularly when they are very young and thus most impressionable—there is reason to think that children develop personality traits that are more similar to those of their peers than to those of their parents. This conclusion is controversial, but it is not particularly surprising in light of the simulation results. Because of the strong coupling characterizing parent–child relationships, children have little need to internalize the values of their parents' control parameters to develop and maintain behavioral coordination with them. In monitoring a child's behavior, praising it when it is deemed appropriate and attempting to suppress or redirect it when it is less so, parents exert fairly constant and strong influence over what the child does. Because parents' behavior control is obvious to the child, he or she learns to act in accordance with the underlying reinforcement contingencies, rules, and expectations." (p. 368)
The Society of Self
They also analyze a simple cellular automata model which is suppose to show how an individual's personality can congeal and be relatively stable. Imagine each cell represents some item of information relevant to the self. It's color represents a positive or negative evaluation, represented by +/- 1. It's height represents it's importance (from 2 to 10). There is local correlation between the importance of "related" information. Each cell is influenced by its neighbors, representing related concepts. At each time step, a random cell is chosen. It then sums up the weighted (by importance) sum of its (nearest 8) neighbors' elements, and switches positions if that value is the opposite sign as itself and the total importance is higher than it's own. They show that this can lead to stable zones of opinion, and that it is relatively robust to perturbation.
There is also some interesting discussion about social influences on personality and interaction at the end of the paper.
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