In the descriptive approach initiated in this chapter, we have distinguished three different types of therapeutic interventions that were instrumental in lifting a para- digmatic depressed client out of her psychopathology. We will now need to model these three steps more precisely in the context of the Fokker-Planck model of change that was introduced in Chap.4.
Thefirst stepof a destabilization of an attractor by an increase of stochastic inputs is directly addressed in the Fokker-Planck equation:
dP xð Þ;t dt ¼ d
dxðkxP xð Þ;t Þ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{D
þQd2P xð Þ;t dx2 zfflfflfflfflfflffl}|fflfflfflfflfflffl{S
ð7:1Þ
The change of the probabilitydP of a state x(such as the client’s depression score) is a process, i.e., depending on time t. With deterministic inputs (term D) constant or zero, theSterm of the Fokker-Planck equation (7.1) entails an expansion of the distribution of depression values, as we have observed in the changes between Figs.7.4and7.5. Thus, the stochastic step is directly represented by theSterm where Qis increased. Increased diffusion tends to destroy the structure of psychopatho- logical states, and this is frequently an important step in psychotherapeutic treatment.
Thesecond stepwas a deterministic intervention by which the state of the system was “pushed” out of a local attractor. We assumed then that the input affects behaviorxat a specific valuex0, such as in our example at a depression score of 32 (see Fig.7.3). The mathematical idea that implements a momentary deterministic input within the Fokker-Planck framework is presented in Info-Box7.1.
7.3 Modeling Interventions of Psychotherapeutic Change 89
Info-Box 7.1: Modeling the Effect of a Momentary Deterministic Input Let us consider deterministic interventions in the framework of the Fokker- Planck equation. First, we abbreviate the right-hand side of the equation in the following way:
dP xð Þ;t dt ¼ d
dxðkxP xð Þ;t Þ þQd2P xð Þ;t dx2 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{L xð ÞP xð Þ;t
ð7:2Þ
Next, we regard the direct deterministic input as a momentary change of the distribution functionP(x) that is added to the time-dependent Fokker-Planck equation as in the following:
dP xð Þ;t
dt ¼L xð ÞP xð Þ þ;t δðtt0Þδðxx0Þg xð ÞP xð Þ;t ð7:3Þ
The term added to the right-hand side of (7.3) stands for the intervention. It makes use of Dirac’s delta function (δ), which is a mathematical instrument to represent a sharply peaked input at a momentt¼t0and a locationx¼x0. The exact form of this input is modeled byg(x). In order to observe how the input affects the change ofdP(x;t), we may integrate the extended Fokker-Planck equation (7.3) over time fromt0θtot0+θwithθvery small.
The result of integration is not given in elaborate form here, as it can be illustrated in graphical form by Fig. 7.7, where we start from a simple Gaussian distribution in (a).
The intervention changes the Gaussian probability distribution (Fig.7.7a) to the graph at Fig.7.7b, which has a markedly decreased probability at locationx0. This means that the system state is likely to change to one of the two peaks in the
Fig. 7.7 Result of a momentary deterministic intervention into the state atx0. The probability distribution of behaviorx(e.g., depression) becomes“indented”as a result. (a) before intervention;
(b) immediately after intervention
neighborhood ofx0. In the context of Fig.7.3, where the client showed a stable state with a depression score of 32, this opens up a possibility of her moving toward more attenuated depression, i.e., depression scores of 20 as in Fig.7.4.
The modified probability distribution of Fig.7.7b, however, is only a momentary transformation of the probability distribution. Because of the general action of the Fokker-Planck dynamics, it will soon after the intervention again relax toward the previous distribution of Fig.7.7a. Therefore, a sustained effect on the probability function is only to be expected when interventions are repeated or enduring.
Repeating the intervention may then pave the way to reach the left (healthy) attractor of Fig.7.4.
One must be aware, however, of one limitation of our argument in this chapter:
in descriptive Figs. 7.2,7.3, 7.4, 7.5, and 7.6, the attractor landscapes had three minima—the underlying distribution functions were thus more complicated than the single-peak Gaussians that can be derived from the Fokker-Planck framework. This means we have to model the data region by region, as, in certain regions of state space x, the distributions are approximately Gaussian. In compliance with this region-wise procedure, the landscapes in thefigures were approximated by three additive Gaussians. When we assume that some effective intervention can be installed permanently, this will amount to an enduring and sustainable intervention, because it will modify the attractor landscape. In other words, we then have arrived at thethird step, where a contextual intervention altered the probability distribution permanently and qualitatively changed the attractor landscape, so that the risk of relapse is largely reduced or even absent.
In summary, in this chapter we started from the insight that all real psychother- apies rest on a mixture of deterministic, stochastic, and contextual interventions. In psychotherapy practice, these different interventions cannot be disentangled in pure forms, in the same way as the specific and the nonspecific (“common”) factors of therapy effects cannot be easily separated from each other. We have nevertheless shown in a simulation example, the treatment of depression, how the different interventions may act on a client’s symptoms. In their idealized forms, the three types of interventions, deterministic, stochastic, and contextual, possess distinguish- able profiles of action. Stochastic interventions have a diffusive and destabilizing effect on preexisting attractors as they deconstruct the attractor landscape. Deter- ministic interventions change the state of the system by a deterministic“push”in an unaltered attractor landscape, or they shift and modify an attractor. Contextual interventions come about by changing the control parameters that shaped this landscape in thefirst place, and thus such interventions modify the landscape. All three types of intervention may be manifest in a single therapy course but at different stages of treatment. There is probably no single best approach to treat symptoms.
Several roads can lead to Rome.
Finally, we have discussed the different intervention types in the context of our main model, the Fokker-Planck equation. All types can be located within this modeling approach. Stochastic interventions simply correspond to the stochastic term of the equation, when the deterministic term is downregulated or zero. Then all states of symptom x receive random inputs. Deterministic interventions can be
7.3 Modeling Interventions of Psychotherapeutic Change 91
modeled by the local impact of delta functions, which lead to a momentary change of distributions that destabilize distinct states of symptom x and allow to shift the attractor. Andfinally, contextual interventions generalize on the deterministic inter- ventions by introducing enduringlocal interventions, which results in sustainable changes of distributions and thus of the underlying attractor landscape.
References
Beck, A. T., Freeman, A., & Davis, D. D. (2004).Cognitive therapy and the emotional disorders.
New York, NY: Guilford Press.
Beck, A. T., Steer, R. A., & Brown, G. K. (1996).Manual for the Beck depression inventory–II.
San Antonio, TX: Psychological Corporation.
Greeno, J. G. (1994). Gibson’s affordances.Psychological Review, 101, 236–342.
Haken, H. (1977). Synergetics—An introduction. Nonequilibrium phase-transitions and self- organization in physics, chemistry and biology. Berlin, Germany: Springer.
Michalak, J., Troje, N. F., Fischer, J., Vollmar, P., Heidenreich, T., & Schulte, D. (2009). Embodi- ment of sadness and depression–Gait patterns associated with dysphoric mood.Psychosomatic Medicine, 71, 580–587.
Tschacher, W. (1997).Prozessgestalten—Die Anwendung der Selbstorganisationstheorie und der Theorie dynamischer Systeme auf Probleme der Psychologie. Göttingen, Germany: Hogrefe.
Tschacher, W., & Dauwalder, J.-P. (1999). Situated cognition, ecological perception, and syner- getics: A novel perspective for cognitive psychology? In W. Tschacher & J.-P. Dauwalder (Eds.),Dynamics, synergetics, autonomous agents(pp. 83–104). Singapore: World Scientific.
Tschacher, W., & Haken, H. (2007). Intentionality in non-equilibrium systems? The functional aspects of self-organized pattern formation.New Ideas in Psychology, 25, 1–15.