In conclusion, we may state that the phase shiftϑand the frequencyωcan be determined by a bestfit between cos (ω(t+ L)ϑ) and the empirically derived cross-correlation C L~ð Þ. The coupling constantsa,bcan be calculated from (8.57) and (8.46).
8.4 Formulation of the Two-Dimensional Fokker-Planck
Qth ∂2
∂x2thf xð th;xcl;tÞ þQcl ∂2
∂x2clf xð th;xcl;tÞ ð8:71Þ In the framework of our model, we will assume that the therapist is well
“shielded”against chance events, so that we may putQth0. Psychologically this means that the therapist, at least in the therapy sessions, is not prone to environmen- tal random “kicks,” endogenous affective instability, or like influences. We see resilience, stamina, and perseverance as therapists’virtues, which is supported by findings on “therapist effects”(Pereira et al.,2017). Especially therapist mindful- ness, which contains the aspect of non-reactivity (Bergomi et al., 2013), is likely associated with low values of therapist’s Q. Therefore, the complete two-dimensional model reads
df xð th;xcl;tÞ
dt ¼ ∂
∂xth fkthxthþgthkthþaxtclxthgf
xth;xcl;t
∂
∂xcl
kclxclþgclkclþbx2th
f
xth;xcl;t
þQcl
∂2
∂x2clf xð th;xcl;tÞ
ð8:72Þ
The crucial question is now, of course, how we may solve Eq. (8.72)? Whereas in the one-dimensional case, the Fokker-Planck equation could be solved in a simple closed form at least for the steady state, in the present two-dimensional case, this is only possible under the specific conditions of the slaving principle of synergetics (see Info-Box8.1). These are however not fulfilled for Eq. (8.72)—we sacrificed the assumptions of the slaving principle in order to obtain the oscillations that are occasionally found empirically.
The therapist acts as the zeitgeber for the synchronization with the client.
This assumption can be put under empirical scrutiny, for example, by exploring asymmetries in the observed synchronization behavior of therapeutic dyads, where therapist as zeitgeber amounts to so-called therapist leading. Ramseyer and Tschacher (2011) found that therapist leading was in fact more pronounced at least in initial stages of psychotherapy.
The slaving principle suggests a specificansatzfor the total distribution function f(xth,xcl;t) as a product of two factors that refer to the therapist,fth, and the client,fcl. Thus we put
f xð th;xcl;tÞ ¼fth fcl ð8:73Þ The crucial question is how to determine these factors. According to the slaving principle, the behavior of the fast variable, in the present casexcl, is“enslaved”by the slow variable, i.e.,xclis entrained byxth. This means that the distribution function fcldoes not only depend onxclbut also on a“prescribed”interventional variablexth. Therefore we may writefclin a special form as a conditional probability:
fcl¼fclðxcljxth;tÞ ð8:74Þ In other words, we view xcl “under condition of” xth. Additionally, we have claimed above that the therapist’s variablexthis only weakly or not at all influenced byxcl. This entails thatfthdepends only onxth, i.e.:
fth¼fthðxth;tÞ ð8:75Þ Thus ouransatzfor the solution of (8.72) reads
f xð th;xcl;tÞ ¼fthðxth;tÞfclðxcljxth;tÞ ð8:76Þ Our further mathematical procedure is purely technical (cf. Haken’s book on synergetics (Haken,1977)) and boring for the reader, so that we may skip it here. Yet the result is highly relevant and sheds light on the processes in the present context.
It turns out that the client’s distribution function approximately obeys a one-dimensional Fokker-Planck equation, namely:
dfcl dt ¼ ∂
∂xcl
kclxclþgclkclþbx2th
f
þQcl∂2f
∂x2cl ð8:77Þ wherexthcan be treated as a time-independent parameter that is prescribed by the therapist. Psychologically, we assume high self-efficacy and self-management abil- ities on the therapist’s side, so that the therapist decides on his or her own state. Since the client adapts quickly, we need only consider the steady-state solution to (8.77), which reads
fclðxcljxthÞ ¼Nexp 1
2Q1cl kclxclgclbx2th=kcl2
ð8:78Þ
whereNis a normalization factor. Eq. (8.78) is a Gaussian of width 1
2Q1cl kcl ð8:79Þ
and a maximum at
xcl¼gclþk1cl bx2th ð8:80Þ As outlined above in Sect. 8.1, the results (8.79) and (8.80) imply that the therapist’s interventions may shiftxclbut do not influence the client’s response to chance events. As we have formulated there, the client’s stochastic behavior depends solely on the client’s filtering of environmental events. This stochastic filtering process cannot be taken over by the therapist directly—the therapist has no input 8.4 Formulation of the Two-Dimensional Fokker-Planck Equation 117
into Qcl and thus on the diffusion in (8.79). Yet indirect work in the sense of stochastic interventions, which can shape therapeutic boundary regulations, is fea- sible (cf. Chap.6).
How isxthto befixed? Atfirst sight,xthmay be arbitrarily chosen by the therapist by an act of“free will.”But the answer given by our model is somewhat different.
We have initially assumed that the therapist is well“shielded”againstfluctuations so that we can put Qth ¼0. In this case, the therapist’sxth obeys the deterministic Eq. (8.73). Whenxth, as a function of time, starts from a freely chosenxth(initial), it comes to a rest atxth¼gth! From this follows thatgthis the maximum value ofxth that can be used to shiftxclaccording to Eq. (8.80).
This means, the therapist intervenes through his or her personality. Therapist personality is the “Archimedean function”that can effect therapeutic change in a client. However, becausekthis small (i.e., the therapist’s relaxation time is long), this process may go on for a long time, and the therapist must choose his or her
“influence parameter”blarge enough.
With this, we have concluded the structural-mathematical modeling of the pro- cess of psychotherapy. Psychotherapy researchers will now wish to apply the ideas expressed in the structural-mathematical equations to reality, i.e., to their measure- ments and observations. This is the topic of Chap.9.
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