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.
this research in its overwhelming majority has been merely cross-sectional. Psycho- therapy research is for the most part still in a pre-dynamical stage, and even so-called process research in psychotherapy is seldom based on real process data, i.e., time series. We have criticized this state of affairs repeatedly (Salvatore & Tschacher, 2012; Tschacher,1997; Tschacher & Ramseyer,2009). Only very recently, things have started to change.
Concerning the aspect of coupling, with the advent of the research agenda of dyadic nonverbal synchrony, a rapidly growing interest in the dynamics of therapist- client relations has come to the fore (Koole & Tschacher,2016). Synchrony research explores social coupling via the entrainment of two or more individuals, which can be computed based on the individuals’ behavior over time. The synchronization phenomenon is sometimes also called resonance, mimicry, attunement, contagion, mirroring, etc. One of the novelties of this line of research in psychotherapy is that behavior is monitored with high frequency, so that the data permit to observe the
“here-and-now”of the therapeutic coupling. Previously, such high-resolution data were only monitored in phenomenological studies and video analyses, for instance, in supervision settings where single therapy cases are explored qualitatively. The novel development is that microscopic synchronization processes are now being studied using quantitative methods, so that quantification can complement the qualitative research of previous years. The majority of studies have used correla- tional statistics to detect synchrony. In short, the cross-correlation function (CCF) of the client’s and the therapist’s time series is computed, and the mean cross- correlations are regarded as an indicator of the degree of therapist-client synchrony (a precise account of the methodology of synchrony is provided in Sect.9.4).
The datasets of current synchrony research address different sources of nonverbal behavior. At present, most time series of this research approach represent body movement data, which can be conveniently generated by actigraphic sensors or other motion capture systems. Movement time series can alternatively be derived very economically using video analysis (through motion energy analysis, MEA, Grammer, Honda, Schmitt, & Jütte,1999). An example of MEA measures was given in Fig. 1.3. Some work has also been done studying conversations; then the assessment of synchrony can be based on prosodic variables, such as the pitch of interlocutors’voices (Reich, Berman, Dale, & Levitt,2014). A growing number of studies are focusing on physiological data, especially the electrodermal activity (skin conductance response) of participants in an interaction (Coutinho et al., 2018;
Karvonen et al.,2016). Some “hyperscanning” studies have begun to extend the range of recorded time series to central nervous physiology, i.e., to variables of brain activity. The acquisition of cortical activation using near-infrared spectroscopy (NIRS) may offer a viable method that monitors central nervous signals, which nevertheless can be applied even in more naturalistic settings (Koole & Tschacher, 2016; Zhang, Meng, Hou, Pan, & Hu,2018). There is also some research using hyperscanning on the basis of magnetic resonance tomography (MRT), but this setup does not allow to study natural social interaction as participants are severely restricted by a motionless supine position inside the noisy environment of an MRT scanner.
The current state of research has yielded clear proof of existence for nonverbal synchrony. It has been repeatedly shown that synchronies of motor behavior in natural psychotherapy settings have moderate to strong effect sizes, which may be assessed by testing against pseudo-synchrony generated by Monte Carlo bootstrapping, so-called surrogate tests (Paulick et al.,2018; Ramseyer & Tschacher, 2011,2016). This is also true for synchrony of physiological time series in couple therapy (Karvonen et al.,2016) and in interactions among spouses (Coutinho et al., 2018). Altogether, there are by now reliable indications that coupling in the sense of our minimal model of therapeutic relationship is a quite common phenomenon.
Importantly, this synchrony can be measured in behavioral variables that are not under conscious control of the participants—electrodermal activity is a signal of the involuntary sympathetic activation of the autonomous nervous system. In the case of nonverbal behavior, motor activity was usually monitored in naive participants—the participants were commonly not aware of the goals of analysis. In other words, nonverbal synchrony is a spontaneous phenomenon, not a willful act of imitating or mimicking one’s interaction partner. The spontaneous emergence of synchrony appears to be a phenomenon analogous to self-organized processes of pattern formation as described by synergetics (cf. Info-Box 5.1).
Concerning the aspect of the stability component of the minimal model, we must address the relaxation times. These have however not been a topic of psychological research or psychotherapy research. Nevertheless, some signatures of attracting behavior can be observed in available time series. Time series have been analyzed for autoregressive properties using the autocorrelation function and, in the context of dyadic interaction, using the cross-correlation function (CCF), so that we can make use of the connection between autoregression and point attractors. The presence of autoregression dynamics with significant autocorrelation coefficients at lags of a few seconds points to the presence of fixed-point attractors with relaxation rates at a similar range of several seconds. For example, in an actigraphic study of motor activity of n ¼ 100 schizophrenia patients (Walther, Ramseyer, Horn, Strik, &
Tschacher,2014), significant autocorrelations had a mean duration of approximately 7.8 s, and this duration varied with the symptoms of patients. In patients with affective disorders (unpublished data), this duration extended over 10.8 s. In cross- correlations of dyadic behavior, we found an average of 6.0 s in healthy interactants (Tschacher et al.,2018) and the same duration in a psychotherapy course monitored over an extended period of time (Ramseyer & Tschacher,2016) (Fig.8.3).
In sum, synchrony research yields empirical reasons to assume that dyadic behavior is characterized both by coupling and by asymptotic stability. Stability points to the presence of attractors that can be described by relaxation times and the position of theirfixed points.
It is interesting to explore the psychological meaning of synchrony—is it asso- ciated with properties of interest such as common factors of therapeutic interventions and with therapy outcome? Research so far has supported that nonverbal synchrony may have such properties, as it was found linked with the quality of the therapeutic alliance (Ramseyer & Tschacher, 2011,2016), with the client’s self-efficacy and with positive therapy outcome. In healthy conversing dyads, higher synchrony
8.3 Empirical Studies on Synchrony and Social Coupling 109
entailed increased positive affect and decreased negative affect (Tschacher, Rees, &
Ramseyer,2014). A majority of studies has thus confirmed the generally prosocial impact of nonverbal synchrony, afinding that supports our assumptions expressed in Sect.8.2.
The mathematical models in Sects. 8.1 and 8.2 have introduced the coupling constantsa,b. In Sect.8.3we reported that therapist-client coupling, or synchrony, can be computed using two empirical time series, where the coupling is operationalized by the cross-correlation of the time series. Cross-correlations are computed using a time lagLbetween the two time series. The time lagLof cross- correlated time series is obviously linked to the phase shiftϑof two oscillatory state variables, as shown in Fig.8.2. But how?—can we connect the oscillatory approach to the cross-correlational approach, and can we estimate the coupling constants using the cross-correlations wefind in empirical data?
To explore this open question, we will proceed as follows:
We will connect the central measure of the minimal model, the phase shift as elaborated in Sect. 8.2, with the coupling constants of the minimal model. This yields an equation that containsδ and the coupling constants. The mathematical treatment of this step is in Info-Box8.4. According to Info-Box 8.4, the relation between the phase shift δ and the coupling constants a, b is—in a very good approximation (cf.8.46)–this:
Fig. 8.3 Cross-correlation function (CCF) of client and therapist movement behavior in psycho- therapy (adapted from Ramseyer & Tschacher,2016). Red graph, the average CCF over 27 therapy sessions of a client-therapist dyad. Blue (light blue) graphs, the single (average) CCFs of surrogate data. For a duration of about 6 s, the real CCFs exceed the surrogate CCFs
tan ϑ¼ 23=2bj ja1=2 gth
gclþbg2th=kcl ð8:46Þ Besidesaandb, three more constants enter this relation: the personality constants of therapist and clientgth,gcland the client’s inverse relaxation timekcl. The relation (8.46) has been derived under the assumption of negative and smalla, i.e.jajb.
Equation (8.46) however provides us with only one equation for the two unknowns a,b.A second equation is needed, which is provided by (8.57) that relatesjajb toω. Thus to calculatea,b, we need the empirically measured quantitiesδandω. This information can be obtained from the properly normalized cross-correlation function that we denote byC L^ð Þ. This is treated in Info-Box8.5.
When these steps have been accomplished, one can turn to empirical work with measured time series in Chap.9, where we explore the attractors of one-dimensional time series. Based on the empirical cross-correlations of two-dimensional time series, we also explore the attractors arising from the coupling between therapist and client.
Info-Box 8.4: Relationship Between Phase Shift and Coupling Constants We will start with the phase shift ϑ that was established in Eq. (8.26) as
tan ϑ¼ωγ.ωis the frequency by which the two state variables oscillate, andγ is a damping constant, as can be concluded from an inspection of Info-Box8.2, Eqs. (8.16) and (8.22). Conferring to Eq. (8.22), we put
ω¼ 1
4a2X2clþ2abX2th
1
2 ð8:47Þ
γ¼1
2aXcl ð8:48Þ
In these equations,Xcl,Xthrepresent the stationary solutions of the client’s and therapist’s state variablesxcl,xthin analogy to Eqs. (8.3) and (8.4c):
Xth¼ gthkth
kthaXcl
ð8:49Þ
Xcl ¼gclþbX2th kcl
ð8:50Þ Stability of the solution requires that the damping is negative,γ<0, which means because ofXcl>0, that
(continued)
8.3 Empirical Studies on Synchrony and Social Coupling 111
Info-Box 8.4 (continued)
a<0 ð8:51Þ
The frequencyω(see8.47) must be real, which means 1
4a2X2clþ2abX2th
>0 ð8:52Þ
This requires thatab<0, and therefore because of (8.51)
b>0 ð8:53Þ
as well as
1
4a2X2cl <j jbXa 2th ð8:54Þ Upon closer examination, we see that this is true for |a|b. Thus from (8.47) follows that
ωð2j jba Þ12Xth ð8:55Þ As in our model the effect of the client on the therapist is small (ais small), we therefore may substitute (8.49) simply by
Xth¼gth ð8:56Þ
with gth as the therapist’s long-term mean of the state variable X, in other words the therapist’s trait or“personality.”With this, (8.55) becomes
ωð2j jba Þ12gth ð8:57Þ and (8.50) becomes
Xcl ¼gclþbg2th
kcl : ð8:58Þ
We can now insertγ(8.48) with (8.58) andω(8.57) into Eq. (8.26):
tanϑ¼ð2j jba Þ1=2gth
1=2a gðclþbg2th=kclÞ, or after transformation, Eq. (8.46).
(continued)
Info-Box 8.4 (continued)
Let us discuss this result (8.46). If wefind that the therapist has a notable effect on the client, the termbg2th=kcl is exactly what the therapeutic intervention has added to the dysfunctional mean client stategcl. This means the therapeutic impact is high when the therapist’s coupling constant, the therapist’s trait, and the client’s relaxation time (sinceτcl¼1/kcl) are high.
Info-Box 8.5: Relationship Between Phase Shift and Cross-Correlations Let us turn to the question how we may derive the phase shift ϑ and the frequency ω from the cross-correlations we find empirically. We stated in Sect.8.2with regard to the“minimal model”thatϑis proportional to the time lag L¼ϑ=ω by which the client is lagging behind the therapist, i.e., that the therapist is leading in the interaction. In the following,h. . .idenotes thefirst moment of a sample of time series values, i.e., its temporal mean. We write the cross-correlation function (time lagL) of the client’s and the therapist’s time series as
C Lð Þ ¼hXthð ÞXt clðtþLÞi ð8:59Þ With the help of
Xthð Þ ¼t hXthð Þt i þξð Þ,t h i ¼ξ 0 ð8:60Þ Xclð Þ ¼t hXclð Þt i þζð Þ,t h i ¼ζ 0, ð8:61Þ we can rewrite the cross-correlation as
C Lð Þ ¼ hXthihXcli
|fflfflfflfflfflffl{zfflfflfflfflfflffl}
const:
þhξð Þζt ðtþLÞi
|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
~ C Lð Þ
ð8:62Þ
where C L~ð Þ
L¼0! ð8:63Þ
Therefore we can determineC L~ð Þusing Fig.8.4. To do this, we subtract the mean of all cross-correlationshC(L)ifrom the value of the ordinate,C L~ð Þ.
Note thatξ(t) andζ(t) are empirically measured (or measurable) quantities giving rise to the empirically determined cross-correlationC L~ð Þ. To see how we can determine the phase shiftϑand the frequencyω, we resort to our model that we have treated in detail in Info-Box8.3:
(continued)
8.3 Empirical Studies on Synchrony and Social Coupling 113
Info-Box 8.5 (continued)
ξð Þ ¼t 2cζ0Dcos ð Þeωt Kt ð8:64Þ ζð Þ ¼t 2ζ0cos ðωtϑÞeKt ð8:65Þ We insert (8.64) and (8.65) withtreplaced byt+LinC L~ð Þ(cf. (8.62)) and obtain for small enough damping
C~ðLÞ ¼const:hcosðωtÞ;cos;ðωðtþLÞ ϑÞi ð8:66Þ Because of cos (ω(t+L)ϑ),C L~ð Þis a periodic function ofLwith period 2π/ω. To evaluate (8.66) we have to evaluate the averageh. . .iin (8.66) by the integral (withT¼2π/ω)
1=T ðT
0
ðcosðωtÞcosðωðtþLÞ ϑÞÞdt,
whose evaluation yields
1=2cos ðωLϑÞ ð8:67Þ
Thus the theoretically determined cross-correlationC~theoryð ÞL is given by
C~theoryð Þ ¼L const:cos ðωLϑÞ ð8:68Þ
Now we are in the position to determine the parametersϑandω, by a best fit between the theoretical cross-correlation (8.68) and the empirical cross- correlationC L~ð Þ(8.62). We choose (8.68) in a way that its maximum coincides with the empirically foundLmaxof Fig.8.4.
Fig. 8.4 Cross-correlations
~
C Lð Þin relation to lags,L
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).