8.2 The “ Minimal Model ” of Therapeutic Interventions
_
xth¼ kthxthþgthkthþaxclxth ð8:8Þ _
xcl ¼ kclxclþgclkclþbx2th ð8:9Þ In contrast to Sect.8.1, the relative values ofkth/kclshall not befixed a priori here.
Thus, our discussion is now departing from the slaving principle (cf. Info-Box8.1) that we assumed to hold in the previous section. The stationary solutions (i.e., where no longer behavior changes occur:x_ ¼0) of (8.8) and (8.9) shall be written as
xth¼Xth, xcl¼Xcl ð8:10Þ In principle, stationary states need not signify attractors but may be merely locally stable, like a ball resting on the top of a hill. Thus, in order to analyze the (asymptotic) stability behavior of the system in the neighborhood of stationary states, we apply linear stability analysis: we add small values ξ, ςto the client’s and therapist’s state variables to test the stability of the stationary states:
xth¼Xthþξ, xcl¼Xclþς ð8:11Þ Inserting into (8.8) and (8.9), we then obtain a modified formulation of the minimal system:
ξ_ ¼ kthðXthþξÞ þgthkthþa Xð clþξÞðXthþςÞ ð8:12Þ _
ς ¼ kclðXclþςÞ þgclkclþb Xð thþξÞ2 ð8:13Þ The solutions of (8.8) and (8.9) describe the changes of therapist’s and client’s behavior, which due to (8.11) can be expressed by the sum of the stationary terms containingXth,Xcland the varying terms containingξ,ς. Only the latter are relevant for changes. This yields
ξ_ ¼ kthξþaXclξþaXthςþðaξςÞ ð8:14Þ _
ς ¼ kclςþbξ2
þ2bξXth ð8:15Þ Under the assumption thatξ,ςare small, we may also disregard the nonlinear terms printed in brackets in (8.14) and (8.15). Wefinally obtain a system of linear homogeneous equations, and we may explore the solutions of this system. To solve the linearized equations, we make use of a connection between complex numbers and trigonometric functions (i.e., functions that describe oscillatory behavior). This step is reasonable because our goal is to explore possible oscillatory behavior in the therapist-client system. For mathematical details on the solution of this equation system, see Info-Box8.2.
8.2 The“Minimal Model”of Therapeutic Interventions with. . . 101
Info-Box 8.2: Solution of the Linear Equation System (8.14) and (8.15) We set
ξ¼ξ0eðΛkÞt,ς¼ς0eðΛkÞt ð8:16Þ where Λ is a complex number Λ ¼ Λr + iΛi. This provides us with the opportunity to make use of a well-known connection between complex num- bers and trigonometric functions:eiα¼ cosα+isinα!
To illustrate the further steps, we discuss as an example that the relaxation times of therapist and client are identical (which is counter to the assumptions of the slaving principle):
kcl¼kth¼k ð8:17Þ
We obtain after dividing Eqs. (8.14) and (8.15) byeΛt:
Λξ0¼aXclξ0þaXthς0 ð8:18Þ Λς0¼2bξ0Xth ð8:19Þ In algebra, the coefficients determine which solutions exist for a linear equation system; there may be no solution at all. According to the theorems of algebra, in order to possess solutions, the determinant of the system must be zero; thus in case of (8.18) and (8.19),
ΛaXcl aXth
2bXth Λ
¼0 ð8:20Þ Then to satisfy (8.20):
ΛaXcl
ð ÞΛ2abX2th¼0, i:e::
Λ2þaXclΛ2ab¼0 ð8:21Þ The solution is
Λ¼1
2aXcl 1
4a2X2clþ2abX2th 12
ð8:22Þ
If1
4a2X2clþ2abX2th<0 ð8:23Þ thenΛis a complex number that we write in the form
(continued)
Info-Box 8.2 (continued)
Λ¼γþiω where the frequencyωis given by
ω¼ 1
4a2X2clþ2abX2th
1=2
:
Consequently, the exponential functions in (8.16) can be written as eiωt e(γ k)t. Becauseeiωtcan be expressed by cos (ωt) and sin (ωt) (see above!), the solutions ξ, ς (8.16) become periodic functions of time representing oscillations.
As the result of the argumentation in Info-Box 8.2, we find that the minimal model of client-therapist interaction can show oscillations expressed by the periodic functions cos (ωt) and sin (ωt). Because of (8.16), which depends on the relaxation behavior (throughk), these oscillations are damped (since the exponential function e(Λk)tin (8.16) containsekt, which approximates zero with increasing timet).
IfaXcl0 and we start counting timetsuch that fort¼0 the client’s variable is zero,ξ(0)¼0, then the solution to (8.14) and (8.15) reads
ξð Þ ¼t Acos ð Þeωt kt ð8:24Þ ζð Þ ¼t Bsin ð Þeωt kt ð8:25Þ The amplitudeAisfixed by the requirement that att¼0,Aequals the therapist’s variable,A ¼ξ(0)¼Xth, i.e., the steady-state solution. The amplitudeBmust be chosen such that both Eqs. (8.24) and (8.25) are satisfied. By insertingξandζin the linearized Eqs. (8.14) and (8.15) withaXcl¼0, we readily obtain
B¼ ð1=aÞð2jabjÞ1=2, where |. . .| means: take the positive value.
Our solutionξ(t),ζ(t) shows that both variables oscillate at thesamefrequencyω. Yet the curves of cos (ωt) and sin (ωt) are shifted against each other. Figure 8.2 shows the curves for ξ(t), ζ(t) under the assumption that A > 0, B >0 (which requiresa<0), i.e., the therapist’s behavior is impeded by the client. For the purpose of illustration, we putA¼B¼1.
The sizeϑof this shift in Fig.8.2is called phase shift. In the present example, we may use the formula sin ð Þ ¼ωt cos ωtπ2
or, usingϑ, sin (ωt)¼ cos (ωtϑ), whereϑ¼π2.
Thus, the phase shift is positive. This means that the client’s ζ reaches its maximum at a later time than the therapist’sξ(t). The therapist is “leading,”and the client is lagging behind and following the therapist. Leading and following are common empirical findings in the literature on interpersonal synchrony (e.g., Karvonen, Kykyri, Kaartinen, Penttonen, & Seikkula, 2016; Kupper, Ramseyer, 8.2 The“Minimal Model”of Therapeutic Interventions with. . . 103
Hoffmann, & Tschacher, 2015). Such empirical studies will be addressed more closely in Sect.8.3.
So far, we have derived the phase shiftϑunder the assumptionsaXcl0, and a<0. The treatment of the casesaXcl6¼0, anda>0 ora<0 is somewhat involved.
To summarize the result of an elaboration, the phase shiftϑis determined by the equation
tan ϑ¼ω
γ ð8:26Þ
Whenγ<0 and we letγ!0, the right-hand side of (8.26) goes to minus infinity:
ωγ ! 1. The corresponding solution of the tangent function thus is ϑ¼π2, in accordance with our previous result.
The extent of oscillatory processes depends on the parameters. The prerequisites for the presence of oscillations areab<0 andbsufficiently large. In other words, amust be negative, i.e., the client tends to resist the therapist’s actions.bis positive, i.e., the therapist has an impact on the client’s states. With (8.20), i.e., withkclkth, we have left the scope of the slaving principle (Info-Box8.1). The mathematical treatment of the Fokker-Planck equation is then becoming very tedious. Our approach (8.8) and (8.9) nevertheless contains both options, the assumption of fixed-point dynamics following the slaving principle and the assumption of oscilla- tory dynamics of therapist and patient, hence of limit cycle attractors, depending on kcl,kth.
As a further step in discussing oscillatory behavior, we may also address the question ofsynchronization between the therapist and the client from the mathe- maticalansatzof the minimal evolution equations (see Info-Box8.3). To this end, we calculate how the phase shift ϑ between the therapist’s and client’s actions (cf. Fig.8.2) depends on the parameters of the minimal model (Eqs. (8.8) and (8.9)).
Fig. 8.2 Oscillations of therapist’s and client’s state variables as a solution of the minimal model.
ϑ, phase shift;ωt, period of the oscillation
Info-Box 8.3
As our vantage point, we go back to Eqs. (8.18) and (8.19).
But we now, corresponding to (8.16), formulateΛdifferently:
Λ¼γiω, whereγ¼1=2aXcl,ω¼ 1=4a2X2clþ2abX2th1=2
>0 ð8:27Þ Using the abbreviation
c¼ 1 2bXth
ð8:28Þ we can rewrite (8.19):
ξ0 ¼Λcζ0 ð8:29Þ and because of (8.26)
ξ0 ¼ðγiωÞcζ0 ð8:30Þ Because of (8.16) and (8.17), the general solution of (8.18) is
ξ¼ξþ0 eiωtþξ0eiωt
eKt ð8:31Þ
with the abbreviation
K¼k1=2aXcl ð8:32Þ Because of (8.30) we rewrite (8.31) as
ξ¼ðγþiωÞcζþ0eiωtþðγiωÞcζ0eiωt
eKt ð8:33Þ The general solution to (8.15) is
ζ¼ζþ0eiωtþζ0eiωt
eKt ð8:34Þ
We are looking for real solutions, and for (8.33) and (8.34) to be real, the following must be given:
(continued) 8.2 The“Minimal Model”of Therapeutic Interventions with. . . 105
Info-Box 8.3 (continued)
ζþ0 ¼ ζ0 , ð8:35Þ where ζ0 is the complex conjugate ofζþ0. We now expressζ+by
ζþ¼ζ0eiϑ ð8:36Þ where ζ0 is a real positive amplitude and ϑ denotes a phase. Thus (8.34) becomes
ζ¼ζ0ðeiωtiϑþeiωtþiϑÞeKt ð8:37Þ For the sake of simplicity, we abbreviateωtϑ¼Δ. Thus, written with real expressions, (8.37) becomes
ζ¼2ζ0cos ð ÞeΔ Kt ð8:38Þ In precisely the same way, we may cast (8.33) into the form
ξ¼2cζ0ðγcos ð Þ Δ ωsin ð ÞΔ ÞeKt ð8:39Þ To introduce the phase shiftϑ as defined in and after Fig. 8.2, we write (8.39) as
ξ¼2cζ0Dcos ðΔþϑÞeKt ð8:40Þ By applying the trigonometric law cos (Δ+ϑ)¼ cos (Δ) cosϑ sin (Δ) sinϑto (8.40), and comparing the result with (8.39), we obtain an equation for Dandϑ:
Dcos ϑ¼γ, Dsin ϑ¼ω ð8:41Þ Because of sin2ϑ+ cos2ϑ¼1, we must require
D2 ¼ω2þγ2 ð8:42Þ
whereD>0 because of the definition of the phase shift. Using (8.42) we form by divisionDcosDsin ϑϑ¼ tan ϑso that again due to (8.41),
(continued)
Info-Box 8.3 (continued)
tan ϑ¼ω
γ ð8:43Þ
Finally, we useωtϑ¼Δin (8.38) and (8.40) so that
ζ¼2ζ0cos ðωtϑÞeKt ð8:44Þ ξ¼2cξ0Dcos ð Þeωt Kt ð8:45Þ
The central result of Info-Box8.3is a formula for the phase shift, supporting what we already claimed above in (8.26). Thus, the oscillatory properties of the therapist- client system can in principle be linked with the parameters of the minimal modelk, a, andb.
The discussion in this section has shown that the minimal model of psychother- apy, in the shape of two-dimensional evolution equations, allows for both oscilla- tions and synchronized phase-shifted coupling of therapist and client behavior. This theoretical model is therefore consistent with empirical findings of interactional synchrony, where following and leading synchronies are commonly found.