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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 (physical-sciences 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

e-mail: 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/s11571-011-9157-x

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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 agent-based modeling,

where a system of agents, each following a set of (rela-

tively simple) rules, can give rise to emergent dynamics.

Agent-based 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

agent-based simulations.

Model

Our mathematical model is a system of two-dimensional,

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

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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

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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

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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

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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 phase-space 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

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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 never-ending oscillation of client and

therapist emotional state, which would not be a realistic

therapeutic outcome.

Utilizing the concept of a trapping domain from Poin-

care-Bendixson’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

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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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