Chapter 8
The One- and Two-Dimensional Fokker-Planck Equation
8.1 A “ Minimal Model ” of Therapeutic Intervention,
Example 8.1 (continued)
“agreeableness,” which is the personality trait of a person commonly manifesting kindness and social positivity). In the client,xclmay denote the state variable“low anxiousness”of the client (i.e., low values of“worried, nervous, apprehensive momentary mood”) andgclthe client’s personality trait
“low generalized anxiety”(low values of long-standing behavioral properties of a person who is commonly anxious, self-conscious, and shy, such as the Big Five trait“neuroticism”). Note that we choose the polarity of variables in such a way that high values denote favorable states.
To obtain afirst intuition, we ignorefluctuations, i.e., the stochastic forces on the two actors of the therapeutic setting, therapist and client. We wish to model their interaction and propose the following Eqs. (8.1) and (8.2) forxth0,xcl0. The therapist’s behavior change is assumed to be
dxth
dt ¼ kthxthþgthkthþaxclxth ð8:1Þ In Eq. (8.1),kth¼1=τthis a force constant, which can be expressed by the inverse relaxation timeτthof the therapist. Thefirst two terms on the right-hand side of (8.1) can be written askth(xthgth) representing a deterministic force (or“tendency”) that restores the therapist’s state to the trait valuegthafter a perturbation ofxth. As mentioned in Chap.4, the relaxation time is the time needed for a system to return to steady state after some perturbation.gthis a personality constant of the therapist in steady state, and the termaxclxthrepresents the coupling between the therapist and the client.
Ouransatzfor the client’s behavior change is in Eq. (8.2):
dxcl
dt ¼ kclxclþgclkclþbx2th¼kclðgclxclÞ þbx2th ð8:2Þ kclstands for the inverse relaxation time of the client,gclis the client’s personality constant in steady state, and the coupling term bx2th represents the impact of the therapist on the client. As can be seen, (8.1) and (8.2) are analogous except for their coupling terms.
This choice of the model (8.1) and (8.2) needs some justification. It is our goal here to propose aminimal modelof therapeutic action. This minimal model is based on the expectation that the impact of the therapist on the client can be large, hence the quadratic coupling term bx2th in the differential equation for the client’s behavior change (8.2). Correspondingly, we also expect that vice versa the impact of the client on the therapist (in Freudian terms, the countertransference) should possibly be small, so that in a simplified case,a in (8.1) may even be zero. In this case, the therapist’s action would be completely unaffected by the client’s behavior; there is
no backward action from the client to the therapist. For valuesa,bsmaller than zero, the interactive influence of the client or therapist is negative in the sense that it contributes to a reduction of the respective counterpart’s behavior change. In Eq. (8.1),a>0 means that during a therapy process, some feedback is present, by which the client influences the therapist. Likewise in (8.2),b>0 indicates positive action of the therapist on the client.
In a therapeutic working alliance,b>0 is expected.amay be>0 or<0, although typically we may see a <0, i.e., the client delimits and weakens the therapist’s action.
Example 8.2
The client’s“low anxiousness”increases (i.e., anxiousness decreases) when there is a combination of circumstances, which each corresponds to the two terms of the right-hand side of (8.2):
(1) kcl(gclxcl): this term should be high, which is fulfilled by short client’s relaxation times.
(2) kcl(gcl xcl): the term in brackets should also be high, i.e., client’s generalized anxiety should be low (which is unlikely, asgcl is a long- standing trait) and state anxiety should be high.
(3) bx2th: since the coupling termbis assumed to be positive, positive affect of the therapist should be high.
We will now make further assumptions with the goal to allow the application of the synergetic slaving principle (see Info-Box8.1) to the present model. We there- fore assume thatkthkcl: in other words, the therapist’s relaxation times are much larger than the client’s. This means that the therapist reacts very slowly and carefully, whereas the patient is supposed to adapt quickly.
Interestingly, this assumption is supported empirically by research on those therapist variables that are associated with successful therapy outcomes. In large databases of therapeutic quality control, it is commonly found that some therapists are consistently more effective than others. There are however no simple answers to the question which therapist variables may account for such heterogeneity—thera- pist’s age, gender, gender-matching with the client, therapist’s preferred treatment modality, and years of therapist’s professional experience do not explain the over 5%
of outcome variance that can be attributed to therapist effects (Lutz & Barkham, 2015). Among the most reliablefindings to date is that it is the individual resilience and mindfulness of a therapist that can explain the heterogeneity among therapists.
This means a therapist’s personal and emotional stability and hardiness in the face of adversity, together with a stance of nonjudgmental awareness, constitute important personality traits and coping styles that make psychotherapists more successful (Pereira, Barkham, Kellett, & Saxon, 2017). These recent empirical findings are clearly compatible with our assumption that relaxation times are important features and that the slaving principle may be applied to psychotherapy processes.
8.1 A“Minimal Model”of Therapeutic Intervention, Assuming the Slaving Principle 95
Info-Box 8.1: Slaving Principle
The slaving principle of synergetics is the core of self-organization processes occurring in complex dynamical systems, i.e., systems composed of many variables. These“variables”describe the states (as functions of time) of all single elements of the system, which in a complex system (one may think of all single neurons in a brain) amounts to vast numbers of different variables.
When all these variables behave independently of one another, the whole system is in a state of maximal disorder. On the verge of self-organization, this disordered and high-dimensional regime changes in a short transition period because (few) slow variables entrain (i.e.,“enslave”) the behavior of the (many) quick variables.
Such disorder-order transitions occur only in open systems, i.e., systems that are“driven”by energy inputs (the “control parameters”in physical and biological systems, Haken,1977). Mental systems are driven by motivational and attentional parameters (often called “valence” or “affordance,” Lewin, 1936; Gibson, 1979). Self-organization occurs at a“critical point”of these driving parameters where highly ordered, self-organized behavior emerges from disordered behavior of the system. The slaving principle is thus a candidate mechanism by which we can model phenomena like (in psychology) gestalt perception, social synchrony, and entrainment. Gener- ally in complex dynamical systems, the slaving principle underlies the spon- taneous formation of temporal and spatial patterns.
Slow variables are those variables that have long relaxation times (i.e., smallk). These slow variables assume the role of“order parameters,”i.e., they begin to completely govern all other system variables. The order parameters produce the observable patterns of the self-organized state of the system. Since only few order parameters (often just a single one) prevail at a critical point, the system has transited from high disorder into a highly coherent and ordered state.
Importantly, the slaving principle also holds independent of the self- organizing processes of complex systems, i.e., when there is only one slow and one quick variable as in the two-dimensional“minimal model”presented in Chap.8. In physics, this process is called an adiabatic process. The adiabatic theorem was originally developed in quantum theory but also has applications in the macroscopic world and in thermodynamics.
Psychologically, a person who has long relaxation times may be character- ized as being resilient and mindful—resilience is the ability to deal with stress and to remain competent when encountering adverse situations (instead of responding to stimuli too quickly). Likewise, mindfulness is one’s ability to not automatically respond to adversity but maintain a state of nonjudgmental attention to the present moment.
The difference between the therapist’s and the client’s relaxation times, which was mentioned in Example8.2, requires a transition period that, because of the short client’s relaxation time, is short. During this period, the client adapts his/her state to that prescribed by the therapist. As this happens on a shorter time scale than that of the therapist, the therapist’s state appears as constant to the client. Mathematically, this amounts to setting in Eq. (8.2):dxdtcl¼0, since client behavior is now entrained by the (approximately constant) therapist behavior. Thus we obtain an (algebraic) equation that allows expressing the state xcl at time t by that of xth at the same time, i.e.:
xclð Þ ¼t gclþbxthð Þt 2
kcl ð8:3Þ
This expression forxclcan now be inserted into (8.1). Consequently, instead of the two differential Eqs. (8.1) and (8.2) that determine the time evolution (“trajec- tory”) in the two-dimensional state space (xcl,xth), we may deal with a one-dimensional state space. Since according to (8.3)xthfixes and entrainsxcl,xth
“enslaves”xcland hence acts, in the parlance of synergetics, as an order parameter.
According to dynamical systems theory, the only possible steady states in a one-dimensional state space arefixed points. After these insights, we now turn to the direct solution of (8.1) and (8.2).
The steady-state solution is given when the system comes to a rest, i.e.,
dxth
dt ¼dxdtcl ¼0. In the case of (8.1), this yields
0¼ kthxthþgthkthþaxclxth ð8:4aÞ or
xthðkthaxclÞ ¼gthkth ð8:4bÞ and thus
xth¼ gthkth
kthaxcl ð8:4cÞ
This means that ifkth¼axclthere is instability (a zero denominator is mathemat- ically not defined). Also, the therapist must choose a value ofa (i.e., the client’s reverse impact on the therapist) small enough in order not to minimize the denom- inator. Ifa>0 andxcl>0, the denominator in (8.3) may approach zero, and thus xth! 1: the therapist’s state becomes unstable. Therefore, it is essential that the
“countertransference”constantabe kept small, unless it is<0.
In the context of Sect.8.1, it suffices to put in (8.4c)a¼0, so that
8.1 A“Minimal Model”of Therapeutic Intervention, Assuming the Slaving Principle 97
xth¼gth ð8:5Þ Inserting (8.5) in (8.3) yields
xcl¼gclþ b kcl
g2th ð8:6Þ
Equation (8.6) says that the client’s state depends on the client’s long-standing personality state and the impact of the therapeutic intervention. This intervention term on the right-hand side of Eq. (8.5) describes the effective impact the therapist has on the client’s state.
Anticipating results of our solution of the two-dimensional Fokker-Planck equa- tion (see Sect.8.4below), we now include possible stochasticfluctuations in (8.6), assuming thatQth0, i.e., the therapist is not influenced by random events, hence is self-efficacious and largely under internal control. To present our result, we use a mathematical notation by which the functionexis written as exp(x). This convention is used when the exponentxis a complicated expression as in (8.7). The model of the client’s probability distributionfclin the steady-state limit, withtlarge, becomes
fclð Þ ¼xcl Nexp 1
2Q1cl kcl xclgcl b kclg2th
2!
ð8:7Þ
where Qcl is the effective noise, i.e., noise as sensed by the client. N is the normalization parameter, which takes care of the convention that all probabilities add up to 1.
Equation (8.7) describes a Gaussian distribution with a mean xcl¼gclþkbclg2th. The variance (or the“width”) of this distribution is proportional toQcl/kcl, i.e.,Qclτcl. This means that the therapist strongly influences the mean value of the client’s states (through his or her squared personality traitgth), but not the variance (Fig.8.1).
The variance in (8.7), i.e., the range of the client’s behaviors, can only be influenced by regulating the stochastic influences (Qcl) that act on the client (the result of boundary regulation, as we elaborated in Chap.6) and the client’s relaxation timeτcl. We must therefore focus on the client-specific sensitivity, which“filters”the objective environmental noiseQ(including the stochastic interventions triggered by the therapist). Thus,Qcl¼ϕQ.
There are a number of psychological constructs that describe aspects of such filtering processesϕ.“Openness to experience,”one of the factors in the Big Five model of personality (Costa & McCrae,1992), is an approximation ofϕbecause it describes the permeability and receptivity of the client’s personality. We have recently found empirically that “Openness to experience” in fact increases the duration of synchronization behavior in interacting dyads (Tschacher, Ramseyer,
& Koole, 2018). “Observe,” an aspect of mindfulness, likewise deals with the individual’s awareness of external events (Bergomi, Tschacher, & Kupper,2013) and is therefore also a psychological construct proportional toϕ.
In general, therapeutic interventions are applied on (8.7). Which targets are there of such interventions? As we have seen in Chap.6, interventions can be aimed atQcl, both by generally modulating environmental noise and the patient’sfilteringϕof the environment. Second, the original personality constant of the clientgclis still present in (8.7). It is a further goal of therapy to eventually alter long-standing personality traits by therapeutic interventions, by the therapist deterministically shiftinggclto a new value, using his or her own personality.
In conclusion, these general ideas mean that a minimal model of the therapist’s psychotherapeutic impact suggests the following guidelines: (1) the therapist should be resilient and“slow,”i.e., he/she should have long relaxation times (kthshould be small), (2) the therapist should not underlie strong countertransference or otherwise be strongly influenced by the client (a should be small), and (3) he/she must, however, strongly influence the client (bshould be large) and have a concise and stable personality (gth should be large). In other words, the exchanges between therapist and client are not on an equal footing, and the exchanges between therapist and client are clearly asymmetrical. Thus, good alliance and a working therapeutic relationship may mean a rather one-sided interaction. Atfirst sight, this claim may sound controversial to some psychotherapists, especially in the light of humanistic and systemic psychotherapeutic convictions. Behavior therapy and cognitive- behavioral therapies, on the other hand, would rather endorse this view of the minimal model. We will come back to a discussion of such questions in Chap.10.
Fig. 8.1 Distribution of client’s states, following the therapeutic interaction with the therapist 8.1 A“Minimal Model”of Therapeutic Intervention, Assuming the Slaving Principle 99