Mathematical model of the dynamics of psychotherapy.
ABSTRACT The success of psychotherapy depends on the nature of the therapeutic relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model. Previously Gottman et al. used a very similar approach (physicalsciences paradigm) for modeling and making predictions about husbandwife relationships. Given that this novel approach shed light on the dyadic interaction between couples, we have applied it to the study of the relationship between therapist and client. The results of our computations provide a new perspective on the therapeutic relationship and a number of useful insights. Our goal is to create a model that is capable of making solid predictions about the dynamics of psychotherapy with the ultimate intention of using it to better train therapists.

Article: Stability switches and double Hopf bifurcation in a twoneural network system with multiple delays
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ABSTRACT: Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a twoneural network system with the different types of delays involved in self and neighbor connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delaydependent stability regions are illustrated in the delayparameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasiperiodic solutions due to the Neimark Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.Cognitive Neurodynamics 12/2013; · 1.77 Impact Factor  SourceAvailable from: Dennis D Embry
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Page 1
RESEARCH ARTICLE
Mathematical model of the dynamics of psychotherapy
Larry S. Liebovitch•Paul R. Peluso•
Michael D. Norman•Jessica Su•John M. Gottman
Received: 23 March 2011/Revised: 3 May 2011/Accepted: 9 May 2011/Published online: 22 May 2011
? The Author(s) 2011. This article is published with open access at Springerlink.com
Abstract
nature of the therapeutic relationship between a therapist
and a client. We use dynamical systems theory to model
the dynamics of the emotional interaction between a ther
apist and client. We determine how the therapeutic end
point and the dynamics of getting there depend on the
parameters of the model. Previously Gottman et al. used a
very similar approach (physicalsciences paradigm) for
modeling and making predictions about husband–wife
relationships. Given that this novel approach shed light on
the dyadic interaction between couples, we have applied it
The success of psychotherapy depends on the
to the study of the relationship between therapist and client.
The results of our computations provide a new perspective
on the therapeutic relationship and a number of useful
insights. Our goal is to create a model that is capable of
making solid predictions about the dynamics of psycho
therapy with the ultimate intention of using it to better train
therapists.
Keywords
Dynamicalsystemstheory?Ordinarydifferentialequations?
Biological systems ? Psychotherapy
Nonlinear phenomena ? Dynamical systems ?
Introduction
One in four adults in the United States suffers with a
diagnosable mental disorder. These disorders are the
leading cause of disabilities and extract a physical and
emotional toll on these individuals, their families, and their
communities. Psychotherapy has been shown to be an
effective method for treating these disorders (Lambert and
Barley 2002; Kazdin 2008; Mozdzierz et al. 2009). Yet,
only one quarter of those with these disorders seek psy
chotherapy and one half drop out after the first session
(Muran et al. 2009). A therapist in possession of a better
understanding of psychotherapy would be able to improve
the success of therapy, reduce the client drop out rate, and
yield better ways to train novice therapists.
The success of psychotherapy depends on the nature of
the therapeutic relationship between a therapist and a client.
Studies have sought to identify the most essential elements
of this relationship. Although, those elements are not fully
understood, previous psychotherapy studies have reported
that the essential element is the personal relationship
between the therapist and client, rather than an abstract
L. S. Liebovitch
Charles E. Schmidt College of Science, Center for Complex
Systems and Brain Sciences, Center for Molecular Biology and
Biotechnology, Department of Psychology, Florida Atlantic
University, Boca Raton, FL 33431, USA
L. S. Liebovitch
Division of Mathematics and Natural Sciences,
Department of Physics, Queens College, City
University of New York, Flushing, NY 11367, USA
P. R. Peluso
College of Education, Department of Counselor Education,
Florida Atlantic University, Boca Raton, FL 33431, USA
M. D. Norman (&)
Charles E. Schmidt College of Science, Center for Complex
Systems and Brain Sciences, Florida Atlantic University,
Boca Raton, FL 33431, USA
email: mike.d.norman@gmail.com
J. Su
Charles E. Schmidt College of Science, Florida Atlantic
University, Boca Raton, FL 33431, USA
J. M. Gottman
The Gottman Institute, University of Washington,
Seattle, WA 98115, USA
123
Cogn Neurodyn (2011) 5:265–275
DOI 10.1007/s115710119157x
Page 2
theoretical framework used by the therapist (Lambert and
Barley 2002; Kazdin 2008; Mozdzierz et al. 2009; Muran
et al. 2009; Orlinsky and Howard 1977; Martin et al. 2000).
This suggests that some features of this relationship can be
represented by a model that describes how two people, a
dyad, react to themselves and to each other.
Previous studies of this relationship dyad have used the
social science paradigm of determining the functional
correlations between dependent and independent variables.
Here we use a physical science paradigm to investigate the
nature of this relationship. Just such a physical science
paradigm approach, based on rigorous mathematical
modeling, was pioneered by Gottman et al. (2002) to study
the interactions between husbands and wives and it proved
useful in understanding the stability (or instability) of their
marriages. They showed that this approach gives insights
into the dynamics of a marriage and has the power to make
specific successful predictions about whether the marriage
is stable or ends in divorce. We now modify and extend
their approach of husband–wife dyads to analyze and
understand the therapist–client dyad.
Our work is the first mathematically rigorous model used
inthestudypsychotherapy(withtheexceptionofGottman’s
work, mentioned above). Psychotherapy studies have been
done on a ‘case study’ basis or have used an intuitive
approach with no mathematical backbone. In our approach,
we formulate a rigorous mathematical model of the thera
pist–client relationship based on published empirical data
andourownexperience,determinethedynamicalproperties
of that model, and then compare those properties to the
known properties of the therapist–client relationship. We
will show that this approach yields new insights into the
therapeutic relationship. This model cannot, and is not
intended to, represent the full nature of the complex human
interaction in psychotherapy. However, the fact that it
does reveal important insights suggests that some simple
dynamical features may underlie the more complex behav
iors that emerge in the therapeutic relationship.
The long term goal is for such a theoretical mathemat
ical model to be used to describe and predict successful and
unsuccessful therapeutic relationships depending on the
parameters or conditions of the relationship. Thus, mod
eling the therapeutic relationship between therapists and
clients may allow researchers to be able to evaluate the
quality of the relationship and the effectiveness of specific
interventions that might create some significant therapeutic
gains with a predictability that has not yet been seen in the
clinical research literature. The information may allow
researchers to see how specific intervention strategies can
predict changes in clients as well as see how specific
intervention strategies actually produce changes in clients.
One of the fundamentally novel aspects of our type of
approach (beyond the presence of math) is the perspective
from which the model is developed. Rather than dissecting
the individual components of a system (the client and
therapist) in order to study them independently (i.e. com
piling an exhaustive survey of possible attributes), we
focus on reproducing the emergent dynamics of the rela
tionship that exists between the components. We look for
the set of (relevant) properties of the components that play
a dominant role in these relationships. It is the relationships
that inform the descriptions and properties of the compo
nents, not the other way around.
Mathematically, our model is based on coupled, ordin
ary, nonlinear differential equations. Differential equations
have previously been used to model interaction at many
scales, from human relationships [e.g. a love affair (Stro
gatz 1988, 1994)] to functional neurodynamics [e.g. neu
ronal populations (Ghosh et al. 2008)]. Our perspective
shares some commonalities with agentbased modeling,
where a system of agents, each following a set of (rela
tively simple) rules, can give rise to emergent dynamics.
Agentbased modeling has been successful in reproducing
emergent behaviors in large biological systems [e.g. bird
flocking (Reynolds 1987), the spread of epidemics/
dynamics of populations (Chowell et al. 2003)], social
systems [e.g. conflict (Cederman 2003), ethnic violence
(Lim et al. 2007)], and learning [e.g. the impact of emotion
on the strength of beliefs (Memon and Treur 2010)]. The
advantage of our dynamical systems approach using
ordinary differential equations is that we can analyze many
of the properties of our system analytically. Our model is
therefore much less computationally intensive than most
agentbased simulations.
Model
Our mathematical model is a system of twodimensional,
ordinary differential equations (ODEs) representing the
emotional valance of a therapist and client. Our ODEs were
based on Gottman et al.’s difference equations (Gottman
et al. 2002) as reformulated by Larry Liebovitch into dif
ferential equations (Liebovitch et al. 2008). We have used
a similar approach in models of conflict (Liebovitch et al.
2008) and gene regulatory networks (Liebovitch et al.
2009). The equations fordT
dtanddC
dtare
dT
dt¼ m1T þ b1þ c1FCðCÞ
dC
dt¼ m2C þ b2þ c2FTðTÞ:
ð1Þ
ð2Þ
These equations correspond to the dynamics of the thera
pist, T (Eq. 1), and client, C (Eq. 2), respectively, where
these variables are the emotional valence, or affect, of the
266Cogn Neurodyn (2011) 5:265–275
123
Page 3
therapist and client. For example, a positive value of T
would indicate the therapist is in a positive state, and a
negative value of T indicates the therapist is in a negative
state and similarly for the positive and negative values of C
for the client. The variables m1and m2represent the ther
apist’s and client’s (respectively) inertia to change and b1
and b2represent their emotional values when alone. The
parameters c1and c2are the coupling/reactivity strengths,
or scaling factors, of the influence functions. We assume
c1,2[0. The system is coupled via the influence functions
FC(C) and FT(T).
One of the major contributions of this model is the
formulation of these influence functions. These are the
blueprints for one actor’s emotional affect in response to
the other actor’s. Specifically, the influence that the client
has on the therapist FC(C) and the influence that the ther
apist has on the client FT(T). These functions are piecewise
linear segments in the differential equations reflecting the
dynamics of the therapist–client interaction. In other
words, they dictate how the two members of the dyad will
influence each other. Although these influence functions
are based on published empirical data and our own expe
rience, they are somewhat speculative in nature but none
theless can still provide a starting point for this exploratory
project. These functions are
8
:
FTðTÞ ¼
?3T þ 13:9
and the rationale for their choice is explored in ‘‘Influence
functions’’.
The dynamics of the system’s behavior can be analyzed
by identifying the critical points of the model, which can
represent the final steady state values that the dyad reach at
the conclusion of therapy. We then investigate for which
initial conditions of T and C the client and therapist will
reach the stable states. Finally, we will see how the
dynamics of their behavior depend on the parameters of the
model.
FCðCÞ ¼
0:5C þ 0:5
C þ 0:5
?0:5C þ 2
5T ? 0:1
0:5T ? 0:1
C?0
0\C?1
C[1
<
8
:
ð3Þ
T ?0
0\T ?4
T [4
<
ð4Þ
Influence functions
We now describe the influence functions and the empirical
basis for their functional form. How the therapist’s valence
depends on the client’s valence, FC(C), is shown in Fig. 1.
When the client’s affect is negative, the therapist will
exhibit more positive affect, though they may, under pro
longed exposure to clients negative affect, begin to exhibit
neutral and even negative affect in the face of extreme
negative behavior. This may even create a steady state in
the negative–negative space, which would effectively be
the death of therapy—a ‘‘black hole’’ from which the
therapeutic relationship dies (Bohart and Tallman 2010;
Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi
2002; Norcross 2010; Safran et al. 2002).
When the client is affectively neutral, therapists will
generally utilize strategies to elicit more positive emotions.
They will attempt to encourage clients, or try to get the
client to focus on their strengths and abilities, in the hopes
that this change of focus will change the clients affect
(Bohart and Tallman 2010). At the same time, therapists
may try to elicit any affect on the part of the client (which
may sometimes be negative). However, unless tied to a
broader strategy, this is generally born out of frustration
and may undermine the therapeutic alliance (Lambert and
Barley 2002; Gelso 2009; Horvath and Bedi 2002; Nor
cross 2010; Safran et al. 2002).
As the client’s affect moves from neutral to positive,
initially, the therapist will also exhibit more positive affect.
However, there is a point where, as the client’s affect
becomes more positive, the therapist may begin to exhibit
more neutral affect, as the therapist no longer needs to
actively encourage the client, but the positive affect sus
tains itself (Lambert and Barley 2002; Bohart and Tallman
2010; Gelso 2009; Gelso and Hayes 2002; Horvath and
Bedi 2002; Norcross 2010; Safran et al. 2002).
How the client’s valence depends on the therapist’s
valence, FT(T), is shown in Fig. 2. When a therapist
exhibits negative affect, the client is likely to react even
more negatively. The client may experience therapist
negative emotion as judgmental, or a signal of some dis
appointment in the client. This may be the result of
−5−4−3−2−1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Therapist Valence (T)
Client Valence (C)
Fig. 1 FC(C), Client’s influence function on the therapist
Cogn Neurodyn (2011) 5:265–275 267
123
Page 4
therapist frustration with either the pace of treatment, the
client’s unwillingness to change, or fears about the thera
pist’s own performance in conducting therapy (Bohart and
Tallman 2010; Anderson et al. 2010). It is reasonable to
suspect that this would be a part of a novice therapist’s
practice, but could also be reflective of therapists who may
be on the brink of burnout. This frustration may not even
be acknowledged by the therapist, but it may get picked up
by the client, and move the therapy towards the more
negative end of the graph. This is an indicator of a thera
peutic rupture, which in turn is a predictor of premature
termination from therapy (Muran et al. 2009; Norcross
2010). At the same time, there may be circumstances when
a display of negative emotion may be beneficial to the
therapeutic relationship. In particular, appropriate con
frontation or expressions of disappointment may be nec
essary feedback to the client. Again, the immediate result
may be a therapeutic rupture, but if it is done purposefully
or strategically, it may have a long term benefit for the
client. The success of this strategy depends a lot on the skill
of the therapist and the strength of the therapeutic rela
tionship (Lambert and Barley 2002; Gelso 2009; Gelso and
Hayes 2002; Horvath and Bedi 2002; Safran et al. 2002;
Anderson et al. 2010; Norcross 2002).
When the therapist is affectively neutral, most clients
are likely to be either slightly negative or neutral (partic
ularly early in the therapeutic process). Some clients may
not be influenced one way or another to a therapist’s
neutral affect, unless they find (i.e., project) it to be a signal
of therapist disinterest (e.g., the tabula rasa of psycho
analysis), at which point clients may react negatively
(Lambert and Barley 2002; Bohart and Tallman 2010;
Gelso 2009; Gelso and Hayes 2002; Horvath and Bedi
2002; Norcross 2010; Safran et al. 2002).
As the therapist’s affect moves from neutral to posi
tive, initially, the client may remain neutral, or slightly
negative (Lambert and Barley 2002; Bohart and Tallman
2010; Gelso 2009; Gelso and Hayes 2002; Horvath and
Bedi 2002; Safran et al. 2002; Norcross 2002). However,
as the therapist’s affect becomes more positive, the client
may respond positively by exhibiting more neutral affect
(Safran et al. 2002; Skovholt and Jennings 2004). This
could be a sign of the client either buying into the ther
apists message, or a sign that the client is beginning to
experience some positive results from the therapeutic
intervention. A positive steady state may emerge at this
point, where therapeutic gains may be maximized (Nor
cross 2010). However, as a consequence of extreme
expressions of positive affect on the part of the therapist,
the client might turn negative (i.e., get turned off, espe
cially if they perceive that it is disingenuous or too
pollyannish).
Analysis and solution
To truly appreciate the insights that the model provides,
one should look to the phase portraits that emerge from
numerically integrating the system from various initial
conditions. A phase portrait shows the directions and paths
of emotional change for the dyad. These changes are a
function of the dyad’s previous emotional state and the
parameters of the model. Phase portraits produced by the
model are explored in ‘‘Phase portraits’’.
For the purposes of simplifying the model as much as
possible while still preserving the dynamics, let us assume
for this numerical analysis a system with parameters
m1, m2= 1, b1, b2= 0, and c1, c2= 1. It’s important to
note that these particular parameter choices are evenly
matched. This would be the sign of novice therapist, as an
expert therapist might evoke more reactivity from a client
than a client evokes in the therapist (e.g. c2[c1).
It is also important at this point to understand the sig
nificance of a nullcline. The nullclines exist where the rate
of change of the emotional valence of the clientdC
therapistdT
dtequals zero. In order to explore the dynamics of
a system, one must start by finding the critical points, or
states, in that system. These critical points are found where
the nullclines intersect. In other words, the nullclines define
the points in the system where the rate of change of the
client and therapist both equal zero. With the values of the
parameters mentioned above, the nullclines
dC
dt¼ 0 become
T ¼ FCðCÞ
and
dtor
dT
dt¼ 0 and
ð5Þ
−5 −4−3 −2−1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Client Valence (C)
Therapist Valence (T)
Fig. 2 FT(T), Therapist’s influence function on the client
268Cogn Neurodyn (2011) 5:265–275
123
Page 5
C ¼ FTðTÞ
which are shown in Fig. 3. What this means is that for
these parameter values, the nullclines equate to the
influence functions. The critical points at the intersection
of these nullclines are
ð6Þ
ðC;TÞ1¼ ð?1:6;?0:3Þ and ðC;TÞ2¼ ð0:3;0:8Þ:
Linear stability analysis is used in order to analyze the
dynamics of this system. The stability of each critical point
of the model (critical points are defined by the intersection
of the nullclines; where the rate of change of both the
client’s and therapist’s emotional state equals zero), can be
analyzed using it’s corresponding Jacobian matrix. For this
system, with these parameters, stability analysis reveals
that the first critical point is a saddle (having one positive
and one negative eigenvalue) and that the second is an
attractor (having two negative eigenvalues). This attractor
is the stationary state in the system which defines the
values that the variables T and C reach once sufficient time
has passed (as long as they are not captured by the unstable
force of the saddle point which would result in C and T
going to ?1). The location of this stable state depends on
the parameters in the system.
The following equation can be used to generalize the
process of stability analysis in the region immediately
about a specific critical point.
?
The critical points emerge from the system as a result of
the parameter (and influence function) choices. They are
_T
_C
?
¼
m1
I2
I1
m2
??
T
C
??
ð7Þ
best visualized with a phase portrait, which shows the
dynamics of the system from various initial conditions. We
shall discuss phase portraits in ‘‘Phase portraits’’.
If we generalize our influence function segments to the
form FC(C) = M1C ? B1and FT(T) = M2T ? B2, then the
segments existing in the region of our critical point we are
analyzing can be used to define the slopes I1= c1M1and
I2= c2M2. In other words, I1 and I2 are the influence
function slopes (M1and M2) multiplied by the reactivity
scaling factors (c1and c2). Once again, this is useful for
understanding the dynamics of the relationship in the region
immediately surrounding the critical point being analyzed.
The influence functions are defined by fixed numbers
(with fixed ranges for the individual segments); therefore it
is beyond the scope of this model to have a purely ana
lytical form of these functions (which are treated compu
tationally). In order to approximate a general solution for
any given set of intersecting line segments, we must
incorporate the parameters that we set to zero in our
numerical analysis (b1and b2) and the ones we disregarded
when doing our stability analysis (B1and B2). Let us define
b1= c1B1? b1and b2= c2B2? b2. We can then express
our system (for any given pair of lines) as
_T ¼ m1T þ b1þ I1C
_C ¼ m2C þ b2þ I2T
The system has particular solutions
T ¼b2I1? b1m2
m1m2? I1I2
and
ð8Þ
C ¼b1I2? b2m1
m1m2? I1I2
and general solution
ð9Þ
T ¼ c1eutþ c2e? utþI1b2? m2b1
C ¼ c3eutþ c4e? utþb1I2? b2m1
m1m2? I1I2
m1m2? I1I2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where
and tr = m1? m2(the trace of the matrix from Eq. 7),
det = m1m2 I1I2 (the determinant of the matrix from
Eq. 7).
u ¼1
2½tr þ
tr2? 4det
p
?; ? u ¼1
2½tr ?
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
tr2? 4det
p
?
Phase portraits
The numerical integration function in Matlab, ODE 113,
was used to numerically integrate our system of equations
from different initial conditions.
The results of the integration for the parameters defined
in ‘‘Analysis and solution’’ are shown in Fig. 4. The
−5 −4−3−2−1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Client Valence (C)
Therapist Valence (T)
Fc(C)
FT(T)
Fig. 3 Influence functions, and therefore the nullclines (for the
parameter values m1,2= 1, b1,2= 0 and c1,2= 1)
Cogn Neurodyn (2011) 5:265–275 269
123
Page 6
trajectories go to the stable attractor at (C, T)2= (0.3, 0.8)
or are pulled towards and then pushed away from the
saddle point at (C, T)1= (1.6, 0.3). The evolution of
the system to the stable point would represent a successful
therapeutic endpoint. For example, if the client or therapist
are initially maximally positive (we’ve all encountered
people in this state from time to time), they both end up
mildly positive, which is the desired outcome.
In Fig. 4, where client and therapist have equal influence
with each other, the relationship will likely end up at the
positive attractor, as long as the therapist begins with
positive emotion. However, if the client starts therapy in a
very negative emotional state (C = 5 or 4) then the
therapist must be more positive in order to overcome
the movement towards the negative saddle point and into
the ‘‘black hole’’. Furthermore, with these parameters of
the model, if the client begins therapy with a mild negative
state (C = 1) or is neutral, the therapist can also match
the negative emotion and still attract the relationship
towards the positive stable steady state (approximately
T = 1 and C = 1). In addition, if the client starts therapy
with very positive affect (C = 2 to 5) the therapist can also
display some negative or neutral emotion and still draw the
relationship to the positive steady state. Going negative can
be a strategy for the therapist to either bring a client who is
mildly negative or neutral (C = 1 or 0) about therapy
into a positive space. It may also be a strategic method for
tamping down a client who is displaying highly (and pos
sibly unrealistically) positive emotions. Since, in this sce
nario, both the client and the therapist are equally
influential of the other, one can look at the other side of the
coin. Specifically, if the therapist initially is highly positive
(T = 5 to 2) and the client is either negative or positive, the
therapist will be drawn down toward the positive stable
steady state. The key, it seems, for this relationship to be
successful, is that (in most instances) the therapist must
avoid beginning with a negative affect. The only exception
is if the client is initially very negative, in which case, the
relationship will be pulled towards negative emotional
states, no matter what the therapist does.
Now, let’s explore what happens as we vary a parameter
of our model. We will say that the client is more reactive
than the therapist, meaning the client responds strongly to
the therapist (c1= 1, c2= 10), which could be the mark of
a skilled practitioner.
Figure 5 shows the phasespace plots of the trajectories
of the therapist and client if the client responds very
strongly to the therapist (c2= 10 rather than 1). This may
be an indication of a very influential or skilled therapist.
Just as in the first phase portrait (Fig. 4), there are two
critical points, one of which is a stable steady state attractor
where the client is very positive and the therapist is mod
erately positive, and the other is a saddle point.
There are some noteworthy results. The therapeutic
relationship is attracted toward the positive critical point
(steady state attractor) and the emotional state of the
relationship spirals, or oscillates up and down in time,
before reaching this final steady state. This seems to be in
line with clients’ frequent oscillations (i.e. ambivalence)
regarding change.
The client winds up more positive than the therapist, and
the relationship is attracted to the steady state rapidly (as
indicated by the tight spiraling). This seems to be a very
good outcome for therapy. There is also a cautionary finding
−5−4−3−2−1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Client Valence (C)
Therapist Valence (T)
Fig. 4 Phase portrait of the system integrated with parameter values
m1,2= 1, b1,2= 0 and c1,2= 1
−5−4 −3−2−1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Client Valence (C)
Therapist Valence (T)
Fig. 5 Phase portrait of the system integrated with parameter values
m1,2= 1, b1,2= 0, c1= 1 and c2= 10
270Cogn Neurodyn (2011) 5:265–275
123
Page 7
here that the client will likely respond very strongly to any
negative input from the therapist. As shown in Fig. 5, any
time that the therapist starts with a negative affect on the
therapist axis, the relationship will be directed towards the
saddle point and from there to increasingly negative values.
When the therapist starts with positive affect, the relation
ship will be directed towards the positive steady state. The
only exception to this is when the client affect starts very
negative (T = 5 to 3) and the initial therapist affect is
only neutral or slightly positive (T = 0 to 1). Thus, just as in
Fig. 4, as long as the therapist begins with positive emotion,
the client will reach this positive outcome, unless the client
starts very negative about treatment.
This model may be an ideal scenario for a brief therapy,
where change is swift (as indicated by the tight spiraling)
and the client is satisfied. Just as in Fig. 4, as long as the
therapist begins with positive emotion, the client will reach
this positive outcome, unless the client starts very negative
about treatment.
Discussion
Relevant parameters
The dynamics of our system at any given critical point can
be determined by Eq. 7.
If m1m2is positive (as is the case if m1= m2= 1),
and if I1and I2have opposite signs, the determinant is
positive. If I1and I2have the same sign, the sign of the
determinant depends on whether the product I1I2is large
enough to outweigh the positive influence of the inertial
terms. The point here is if I1and I2have the same sign, and
are large enough, then det\0 and we have a saddle.
As stated, the slope I of a nullcline at the intersection
point with another nullcline plays a large role in deter
mining the dynamics at that critical point. The inertia of the
actor m also has a significant role to play in the dynamics
as well. We will now explore the importance of the
determinant (det), trace (tr) and discriminant (tr2 4det)
of the coefficient matrix of the system to analyzing the
system’s dynamics at each of the critical points. If the
det\0, we get a saddle point. If the det[0 and tr2
4det[0, the critical point is a stable point. If tr2
4det\0, then the critical point is a stable spiral. Recall
that the coefficient matrix is determined by the parameters
of the system and the influence functions (Table 1).
Limit cycles
We now show that for this system, with these influence
functions, there is no limit cycle present. The reason a limit
cycle is not desirable in this model is simple: a limit cycle
would represent a neverending oscillation of client and
therapist emotional state, which would not be a realistic
therapeutic outcome.
Utilizing the concept of a trapping domain from Poin
careBendixson’s theorem, we show that the flow of tra
jectories along the edges of a closed, appropriately defined
region (what is important is the general shape of this
region, rather than the specific equations defining it’s
boundaries) such as the dashed line shown in Fig. 6 are
inwards, towards the stable attractor point. If the flow never
leaves the trapping domain, and we have shown that only a
stable fixed point exists within this domain, then no limit
cycle can exist. This does not preclude the existence of
stable spiral behavior within the trapping domain, however.
Once a trajectory of the system flows into the trapping
domain, it will not escape. The trapping domain is depicted
in Fig. 6.
For the attractor point (C, T)2, which exists at the
intersection of the nullclines (Eqs. 5 and 6) derived from
Eqs. 3 and 4 (where 0\C B 1 and 0\T B 4) at
ðC;TÞ2¼ ð0:3;0:8Þ;dT
dT
dt¼ m1T þ b1þ c1ðC þ 0:5Þ;
parameter values m1=  1, b1= 0 and c1= 1, this
becomes
dt¼ m1T þ b1þ c1FCðCÞ
and
becomes
standardforour
dT
dt¼ ?T þ C þ 0:5
ð10Þ
Likewise,dC
b2þ c2ð0:5T ? 0:1Þ and with the standard parameter
values m2= 1, b2= 0 and c2= 1, this becomes
dt¼ m2C þ b2þ c2FTðTÞ becomesdC
dt¼ m2C þ
dC
dt¼ ?C þ 0:5T ? 0:1
ð11Þ
What follows is one of a range of possible domain sizes,
but the concepts remain the same.
First, let us define, as generally as possible (without
being redundant), our trapping domain’s left boundary. For
C = 4 and T ¼ 1?y?4;dC
flow rightwards. To test this, we substitute these values into
Eq. 11. The rate of changedC
dtis positive and the flow is
rightwards or inwards toward the stable point.
Next, let us define our trapping domain’s upper
boundary. For C = 4 B x B 4 and T ¼ 4;dT
be negative and flow downwards. To test this, we substitute
dtwill always be positive and
dtwill always
Table 1 Relevant parameters
Signs of I1, I2
MagnitudeExpected behavior
SameLarge Saddle
SameSmallAttractor
Opposite SmallAttractor
OppositeLarge Spiral
Cogn Neurodyn (2011) 5:265–275271
123
Page 8
these values into Eq. 10. The rate of changedT
and the flow is downwards or inwards toward the stable
point.
Now, let us define our trapping domain’s right bound
ary. For C = 4 and T ¼ ?1?y?4;dC
negative and flow leftwards. To test this, we substitute
these values into Eq. 11. The rate of changedC
and the flow is leftwards or inwards toward the stable point.
The trapping domain’s lower boundary is created by the
saddle point in the negative–negative space (C, T)1=
1.6, 0.3. We shall define this lower boundary by the
lower dashed line seen in Fig. 6. It begins at C, T = 4, 1
and ends at C, T = 4, 1. The equation for this line seg
ment is T = 0.25C, where 4 B C B 4. Recall that a
saddle point is stable along one direction (attractive) and
unstable along the other (repulsive). The unstable region of
this saddle point is explicitly defined (as shown in
‘‘Analysis and solution’’) by our stability analysis and the
integration of the system as seen in the phase portrait
shown in Fig. 6. Any trajectory entering the trapping
domain from this lower boundary will have been pushed in
by the unstable force of the saddle point and will be pulled
in by the stable attractor. On the other side of this saddle
point (the lower left region of Fig. 6), trajectories are
pushed into the therapeutic ‘‘black hole’’ from which there
is no return.
No limit cycle can exist in this system because any
trajectory entering the trapping domain will never escape,
and within the trapping domain there only exists a single
critical point, which is a stable attractor. Any trajectory not
eventually caught by the attractive force of the stable point
that resides within the trapping domain will be subject to
the unstable force of the saddle point and descend into
further negativity.
dtis negative
dtwill always be
dtis negative
Bifurcation diagrams
We now present bifurcation diagrams showing how the
critical points of the system change when independently
changing the inertia, m2, of the client and the relative
strength of the influences between the therapist and client,
a = c2/c1.
Varying m2
Shown in Fig. 7 are the therapist values for the fixed points
as m2is varied from 3 to 3 in steps of 0.1. The other
parameters of the model are m1= 1, c1,2= 1, b1,2= 0.
For m2B 2.5, our saddle point ceases to exist. There is
also a ‘‘hump’’ created by the shifting stable point from
m2= 0.75 to m2= 0.25, with a peak at m2= 0.5.
The stable point and saddle exist at the same level of
emotional valence for the therapist at m2= 0 and for
0\m2\1 the stable point begins to descend into negative
emotional space for the therapist. At m2= 1, the stable
point ceases to exist and only the saddle remains for
increasing values of m2.
Shown in Fig. 8 are the client values for the fixed points
as m2is varied from 3 to 3 in steps of 0.1. The other
parameters of the model are m1= 1, c1,2= 1, b1,2= 0.
This bifurcation diagram shows characteristics similar to
our therapist plot. Specifically, the critical saddle point
ceasing to exist at about m2= 2.5. The stable point also
shows signs of the hump at m2= 0.5, but then continues
in an upward trend right until the stable point ceases to
exist at m2= 1. The saddle point continues to exist for
values of m2[2.5, just as in Fig. 7.
−5 −4−3−2 −1012345
−5
−4
−3
−2
−1
0
1
2
3
4
5
Client Valence (C)
Therapist Valence (T)
Fig. 6 Visualization of the trapping domain (dashed line)
−3−2−10123
−15
−10
−5
0
5
Client Inertia (m2)
Therapist Valence (T)
Fig. 7 Values of the attractor (.) and saddle (?) points of the
therapist as a function of m2
272Cogn Neurodyn (2011) 5:265–275
123
Page 9
It should be kept in mind that the inertia of an actor is
thought of as a dampening force on the dynamics of the
system (i.e. m2\0 typically), so m2values greater than
zero will be unusual. Another way of interpreting a non
negative m2value would be that the therapist is exhibiting
such a strong influence on the client, that the client has
negative inertia. It appears that the actor with the least
inertia (highest value of m2) will have the most positive
emotional outcome, assuming the initial conditions are
such that the trajectory goes to the stable point.
Varying a
Shown in Fig. 9 are the therapist values for the fixed points
as a = c2/c1is varied from 50 to 50 in steps of 1, by
increasing the client’s coupling strength, c2. The parameter
c1= 1 and is held constant throughout. The other param
eters of the model are m1,2= 1, b1,2= 0. For a\0,
which would indicate a negative coupling strength on the
client side (this is counterintuitive for a human dyad, much
as positive values of m are counterintuitive), the stable
point remains in slightly positive space for the therapist. A
peak in the therapist’s emotional state is seen at around
a = 2, which is exactly where the client’s emotional
dynamic changes from a steep increase to a more gradual
one.
Shown in Fig. 10 are the client values for the fixed
points as a is varied from 50 to 50 in steps of 1. The
parameter c1= 1 and is held constant throughout. The
other parameters of the model are m1,2= 1, b1,2= 0.
When a\0, we see that the stable attractor exists in the
negative space for the client, while the saddle is in positive
emotional space. When a[0 and the client’s reactivity
scaling factor is raised, we see that the client’s emotional
end state (the stable point) becomes more positive as well.
This illustrates the first of our conclusions (below) that the
person who is the most responsive ends up being the most
positive.
Conclusions
Through determining how the endpoints, stability, and
dynamics of the system depends on various parameters, we
have drawn a number of conclusions from this theoretical
framework.
The therapist or client who is the most responsive
to the other ends up being the most positive
If the client is more responsive to the affect of the therapist,
the client reaches a more positive affect than the therapist.
If the therapist is more responsive to the affect of the client,
the therapist reaches a more positive affect than the client.
This means that to achieve the most positive state for the
client, which defines successful therapy, the therapist’s
responses to the client should be moderated. This important
conclusion of the model may provide a dynamical basis for
understanding the intuitive empirical finding, over the last
century of psychotherapy, of the advantages to be gained
from the therapist presenting a low reactivity face to the
client. In terms of the mathematics of the model, increasing
the value of the influence function scaling factor c1,2will
result in an increase in the slope I1,2. This raises the
respective coordinate along its axis, thus improving the
emotional state of the person being influenced.
Recall that our fixed point is defined for T by Eq. 8 and
for C by Eq. 9. In order for this conclusion to hold true, the
−3 −2−10123
−30
−25
−20
−15
−10
−5
0
5
10
Client Inertia (m2)
Client Valence (C)
Fig. 8 Values of the attractor (.) and saddle (?) points of the client as
a function of m2
−50−40−30 −20 −100 1020 304050
−5
−4
−3
−2
−1
0
1
2
3
4
5
Therapist Valence (T)
Client Reactivity ( )
Fig. 9 Values of the attractor (.) and saddle (?) points of the
therapist as a function of a = c2/c1
Cogn Neurodyn (2011) 5:265–275273
123
Page 10
person’s uninfluenced emotional state (their emotional
state when alone) must not be negative (b1,2C 0).
The final stable state of the client may be approached
through emotional ups and downs
For many of the scenarios we’ve studied, the steady state is
reached through a spiral trajectory. This translates to the
client and therapist each going through up and down
emotional swings before reaching their final steady states.
These emotional swings are not necessarily backsliding on
the part of the client. Rather they are a direct result of the
dynamics driven by the therapist and client influence
functions, and therefore must be expected in these thera
peutic relationships.
A client who is less influenced by their own previous
state takes longer to reach their final stable state
A person who is less influenced by their own previous state
has a slowed approach to a final steady state attractor. The
smaller a person’s inertial term m, the more likely they are
to oscillate before reaching their steady state. Comple
mentarily, the lower the magnitude of the trace tr of the
system matrix, the slower the spiral decays.
A client who is less influenced by their own previous
state follows a similar trajectory to one that is more
responsive to their therapist
Responding more weakly to one’s own previous emotional
state yields a similar pattern of dynamics as responding
more strongly to the other person. We define types of phase
portraits by the number and type of critical points that
exist.
Suppose a point is close to being a saddle point. The
determining factor for that is whether det = m1m2
I1I2\0. If we’re on the tipping point between being a
saddle and an attractor, we can assume that I1and I2have
the same sign, so the only question is whether they are big
enough. In this case, increasing I1I2will have the same
effect as decreasing m1m2, that is to say, increasing the
influences has the same effect as weakening your response
to your previous emotional state.
Now suppose a point is on the brink between being a
stable point and a spiral. The determining factor is whether
(m1 m2)2? 4I1I2\0. If we’re on the tipping point
between being an attractor and a spiral, we can assume that
I1and I2have opposite signs. In this case, increasing the
influence functions has the same effect as bringing m1and
m2closer together.
Summary
As noted above, this model cannot, and is not intended to,
represent the full nature of the complex human interaction
in psychotherapy. However, the fact that it does reveal
important insights about therapist neutrality, client emo
tional swings, and the reciprocal roles of inertia and
influence between therapist and client, are consistent with
therapists’ empirical experiences. This may suggest that
some simple dynamical features may underlie the more
complexbehaviorsthat emerge
relationship.
inthetherapeutic
Future work
The present exploratory work is already quite a significant
conceptual leap in trying to develop a new approach that
may shed light on the dynamics of psychotherapy. We hope
that it will serve as a firm starting point to further develop
new theoretical and empirical studies. Theoretically,
Gottman et al. (2002) found that changes in the influence
functions with time were essential features of how rela
tionships in marriages were improved (or worsened). We
want to explore how changing these influence functions,
during the course of therapy, can improve the therapeutic
outcome. Experimentally, again as Gottman et al. (2002)
did, we want to video record psychotherapy sessions, code
the time dependent valence of the therapist and client, use
that data to determine the best fit parameters of the model
system, and then determine which of those parameters best
correlates with independent measures of the success of
therapy. We are especially interested in learning how the
−50 −40−30−20−100 102030 40 50
−5
0
5
10
Client Valence (C)
Client Reactivity ( )
Fig. 10 Values of the attractor (.) and saddle (?) points of the client
as a function of a = c2/c1
274 Cogn Neurodyn (2011) 5:265–275
123
Page 11
influence functions and dynamics differ between inexpe
rienced and experienced therapists.
The long term goal is to understand enough about the
dynamics of psychotherapy to suggest which approaches
are likely to be the most beneficial for the client and how
those approaches can be empirically tested. This may also
lead to new ways to train therapists to utilize those best
approaches.
Acknowledgments
(Director of Research, CNRS, Marseille; Associate Professor, Center
for Complex Systems and Brain Sciences, Florida Atlantic Univer
sity) for his support on this project. This material is based upon work
supported by the National Science Foundation under Grant No.
0638662.
We would like to thank Dr. Viktor Jirsa
Open Access
Creative Commons Attribution Noncommercial License which per
mits any noncommercial use, distribution, and reproduction in any
medium, provided the original author(s) and source are credited.
This article is distributed under the terms of the
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