Generic Psychotherapy

Relationships and Non-Linear Dynamics

William Dubin, Ph.D.

 


Introduction

The self psychologists have elaborated on the classical discussions of the kinds of relationships between the patient and the analyst Shane, Shane, & Gales 1997. The Shanes and Gales describe three major kinds of patient-analyst "relational configurations".

  1. The Old Self with the Old Other
  2. The Old Self with the New Other
  3. The New Self with The New Other

The Old  Self with  the Old Other

They described his configuration as follows:

"By this pattern  we refer  to a current relationship with the analyst wherein both the analyst and the self are perceived do by the patient, with or without self reflection, has figures predominantly organized time the basis of traumatic relational experiences with significant tramatogenic others from the past.  We conceptual trauma broadly as including experiences of  loss, deprivation, intrusion, or neglect  as well as threatened, perceived, or actual danger do with physical well- being of the self. Such experiences may occur over a the time, may constitute short- term or even single episodes; the defining features are the sense of helplessness and of being overwhelmed, accompanied by other dysphoric affects and bodily experiences, and leading to sequestering strategy, which we conceptualize has self- protective, are invoked to attempt   mastery of the trauma and to defend against future trauma.  The patient's ensuing relationship patterns tend to remain fixed and inflexible as ways of being with an other that had been experienced as relatively adaptive and sometime in the past, as the best of only available way to the individual under traumatic circumstances, which no longer pertain in the present." (Shane et al 1997 p. 103)

The Old Self the New Other

" By this pattern we refer to a current analytic relationship wherein the patient continues to  field caught in old, traumatized ways, incapable of feeling differently about himself or herself, but nevertheless is beginning to experience the analyst has a person different from the past, traumatizing others, a novel other, without prototype in the patient's life, not categorize predominantly on unassimilated conscious, the conscious, or relatively sequestered ( unconscious) patterns from the past.  The analyst may appear, for example, as soothing, comforting, understanding, and regulating, and insight providing, it as one who is able to be and to remain with the patient in painful and frightening experiences.  However, the patient continues, too few himself in a dysphoric manner. This relational configuration represents a transitional shift from and self old other, on the way to a new self  with new other patient-- analyst relational configuration, but this transitional position may persist in analysis for long time.  Moreover, the configuration may alter in either direction so that the patient in the analysis may revert to an old self with old other patient -- analyst relational configuration rather than move forward in the developmental trajectory. than a" ( Shane et al 1997 p. 104)

The New Self with New Other

By this pattern we refer to an analytic relationship describing the patient's emergent achievement of a balance between accommodation and assimilation, a balanced equilibrium, the in developing capacity for organizing current experiences of   self and self with other in new and different ways instead of persistently organizing self and self with other based on a recategorization predominantly influenced by tramatogenic experience from the past.   The new self is an integrated self,   with affectivity, agency, historicity, and coherence; moreover, the new self is a consolidated self, with capacities for intersubjective and interpersonal relatedness, for mature self- reflective awareness, with a solidly endowed sense of reality, capacities for pleasure and intimacy, and the ability for invoking creative self-elaborative of and adaptive self- protective strategies that do not disrupt the self 's consolidation.   There is integrated and consolidated self has been attained in the analysis deposit to new experiences with the analyst has a novel other than a consolidated, secure tie to the analyst has been achieved in the analytic ambiance of safety, comfort, and intimacy sharing. (Shane et al 1997 p. 104-105)

Explore the interactive applet below to summarize some of the possibilities.

Shane, et al make the important point that the fundamental underlying psychological dynamics are not linear. It may be helpful to take an excursion into some mathematical concepts that are at the base of linearity and non-linearity.   Unfortunately most people who are interested in psychological concepts are put of by math, in part because they do not think it is relevant, and to a large extent because they have had bad experiences learning math. My concern about giving these mathematical illustrations is that some of my readers will say to themselves, "I never understand math", and simply click away to another more easily understood web site.  The interactive exercises given below are designed to give an intuitive feel of these concepts.  It is important to feel free to mess up and to not understand.  If you do not understand, push the Reset button. Start again following the instructions and reading the results out loud to yourself. Keep trying or come back at another time.  If weird things happen push Reset and try again.


Exploring Non-Linear Dynamics

Before we can speak about non-linear dynamics it is important to have an idea of what linear means.  Very simply, linear refers to a straight line.  In this case if you plot the function, i.e. y = x + 2, on a graph, it produces a straight line.  Please refer to the Applet below. Click on the pull down menu and then click on Linear. Fill in a starting value. Press Step once  you will see a value below where you see "Linear sequence".  This was the starting value that you selected.  Now press Step again. Note the next value on the list, and the line on the graph. Then press Run and follow a series of points. Note that the graph forms a straight line. This is what is meant by linear. A change in the horizontal direction produces a  change of constant magnitude in the vertical direction.  To see a  graph of a non-linear function, i.e. y = x squared, click on the pull down menu and select Square. Enter a starting value of 1.1. Click on Step a few times and watch the shape the plot forms. Note that it is an upward curving line.   This is a non-linear function. A little change in the horizontal direction is associated with a increasingly large change in the vertical direction.  If anything you do not follow happens or if you want to try something else push Reset and start all over again. Repeat your experiments until you get a feel how the linear and square functions work. Note that these plots are not the usual simple plots of a linear or a square function. The following rule is being applied for all of the functions given below:

  1. Enter a Starting Value.
  2. Compute the function and put result on the output sequence list.
  3. Make the result the next Starting Value.
  4. Repeat the sequence as many times as you want, or until you reach some limit.

There are several useful psychological truths that can be illustrated in these exercises.   The first has to do with the idea of the domain of a given function. For example see what happens when you try to take the square root of a negative number. Take the following steps: 

  1. Click on the drop down menu and click on Square Root.
  2. Enter -4 as the Starting Value.
  3. Click on Step.

To try this experiment out  Go back to the simulator.

You get an error because there are no real square roots of negative numbers.   Remember that a square root of a number is the number when multiplied by itself that gives the original number. The square root of 4 is 2 because 2 times 2 is 4. Now see what happens if you try to take the square root of -4.  If you would try to take -2 as the square root, see what happens when you multiply -2 times -2 equals. Of course the answer is plus 4, not minus 4. If you would try plus 2 times plus 2 you would also get plus 4. So no real number can be found which is the square root of -4. Thus, in the domain of real numbers, you can not find the square root of a negative number. The function, square root, only operates in the realm of real numbers ranging from 0 to positive infinity. This is just one example of the more general idea that just about all truths are valid within a limited domain.  Trying to apply a truth outside its suitable realm produces confusion, nonsense, frustration, and misunderstanding.  Examples of how this might be applied psychologically are the concepts of always and never.  Let's take a mundane example, a couple is fighting.  One says to the other, "You are always so mean to me".  The terms always and never leave no room for exception.  In most cases the statements are simply not true. There are times when the spouse is not mean and there are situations where math is not utterly impenetrable. As was stated above my concern about giving these mathematical illustrations is that some of my readers will say to themselves, "I never understand math", and simply click away to another more easily understood web site. The task it to continue even though you do not as yet understand. The experience of moving from confusion to having a flash of insight can be abrupt. It is an excellent example of non-linear dynamics.

As an aside, the situation is even more complicated.  When I tried to take the square root of -4  on my scientific calculator I got the result (0,2).  I thought I would get an error message.  The result, (0,2), is a complex number in the form of (real number, imaginary number).  The calculator operated in a broader domain than the underlying Java program that runs the applets.  The Java recognizes real numbers.  It chokes on imaginary numbers. Recall integers are whole numbers, 1, 2, 3... Real numbers fill in all the fractional spaces between the integers, i.e. 1.1, 1.2, 1.3...  The realm of complex numbers transcends the domain of real numbers.   This is a good example of getting caught in a mind set, and having it expanded by mathematical explorations.

Square

Let's get back to some other experiments that you can perform to get a feel of non-linear dynamics.  Try the following experiments with the Square function. However, imagine the outcome before you try it.  Also remember the simulator gives what the mathematicians call the orbit of a function. That is a list of the results of repeatedly applying the function to the result that was previously obtained. It is useful to think of the orbit as a time sequence.  In the case of the Square function, if you started with 2 (That would be time 0.) and squared it you would get 4. (That would be time 1.) When you press The Step button the 4 is squared to make the next Starting Value 16, (That would be time 2). and so forth. The orbit shows you where the sequence goes from one time to the next.

  1. Where does the orbit go when you start with a number greater that 1?
  2. What happens when the starting number is exactly 1?
  3. Try a number greater than -1 and less than 1.
  4. Try a negative number less than -1.

Keep playing with the simulator until you are clear on what is happening under each of the four starting conditions.  Return to the simulator.

One of the grand ideas that emerges from the above experiments is the notion of sensitivity to initial condition.  The outcome is very different depending on where you start.  Under linear conations things go smoothly wherever you start.  The same is not true with non-linear behavior.  In this sense most human behavior is non-linear.  Where you start out truly makes a difference. (1) When the starting value is greater  than 1 the sequence moves toward + infinity. (2) When you start at 1 the result is always 1. (3) When you start with a value greater than -1 and less than +1 the orbit goes to 0, and (4) negative numbers less than -1 also go to + infinity.  In addition there is the idea of critical thresholds where thing can go dramatically one way or another depending where one starts.  Situations remain the same, escalate, or dissipate depending where they start. The space between escalation and dissipation is as fine as you can slice it. For example, the difference between .99999 and 1.00001 is very small, but the divergence in outcome is potentially infinite.  How does all of this relate to psychology?

Think of a couple that has a disagreement. The rule that they play by is tit for tat.  I will do to you what you do to me and I will up the ante each time.  I am angry with you. You return my anger with greater intensity than I showed to you in the first place. Following the tit for tat rule I respond with even greater anger. The situation escalates.  On the other hand, I respond to the disagreement with understanding. You respond to my understanding with compromise.  The disagreement diminishes and ultimately goes away. The initial reaction has an enormous impact on the outcome.  In this case there is an arbitrarily fine line between a mess and an agreement.  It should be noted that people usually do not think they are behaving on the basis of rules.  Rules are abstractions that usually are seen from outside observation.  Internally one feels that he or she is just responding to a situation that the other is imposing upon him. One does not think that he or she is following a rule.

Square Root

The square root function is different.  Consider what will happen in the following two cases before you start actually using the simulator.

  1. Enter a Starting Value greater that 1.
  2. Enter a starting value between 0 and something less than 1.

Go to the simulator.

Were you surprised.  Wherever you start, the orbit eventually goes to 1.  It does not matter what the initial condition is, repeatedly applying the square root to itself ultimately arrives at 1.  The mathematicians call this phenomenon an attractor.   The orbit is attracted to a particular value. One might think of it as being caught up in the repetition compulsion.  No matter what I do, I wind up in the same old place.

Log

The log function is less interesting.  The log reduces the starting value until it is less than 1.  The psychological metaphor would be some one who tends to minimize everything.  Conversely, the square function exaggerates most everything, and blows it up.

Chaos

Clearly the most interesting and complex of these functions is the "chaos" function.  Actually  the equation, F(x) = cx(1-x), is called the logistic function.  Here c is a constant, and x is a variable that can be given a value between 0 and 1. The trajectory of the orbit is determined by the starting value, x, and the constant, c.  The following experiments will reveal the surprising intricacy of the logistic function. Use .5 as the Starting Value in the following experiments. Later on you can experiment with other Starting Values if you like. See if you can figure out why it might be called the "chaos" function. Set c to each of the following values, and run the simulator until you see some pattern.  Try to say what the pattern is.

    c = .5,  c = 1,  c = 1.5,   c = 2,  c = 3,  c = 3.2,  c = 3.5,  c = 3.55,  c = 3.83,

For  c = 4, use a starting value of 5.01, then try it again with a Starting Value of .5. Finally, use  c = 5.

Return to the simulator.

If your have an interest in exploring these computer experiments more fully, see Devaney 1990 . He gives the following results:

  1. For c = .5 all orbits tend to 0.
  2. For c = 1, all orbits tend to 0, but very slowly.  Push Run several times.
  3. For c = 1.5, all orbits tend to 1/3.
  4. For c = 2, all orbits tend to 1/2.
  5. For c = 3, all orbits tend to 2/3, but very slowly.  Moreover, these orbits oscillate from one side of 2/3 to the other as they approach 2/3.
  6. For c = 3.2, all orbits approach the period 2 cycle .5130... and .799456...
  7. For c = 3.5, all orbits approach a period 4 cycle given by .38282..., .82694..., .50088..., and .87500...
  8. For c = 3.55 all orbits approach a period 8 cycle.
  9. For c =  3.83 all orbits approach a period 3 cycle.
  10. For c = 4, there is no pattern whatsoever most Starting Values, whoever a Starting value of .5 becomes 0 very quickly.
  11. For c = 5, all orbits "apparently" tend to - infinity, but there are many that do not.

As you can see, this very simple equation produces a great variety of results.   Sometimes it quickly moves to a stable point.  Other times it oscillates between 2, 3, 4, or more points, and there are conditions then its behavior becomes utterly unpredictable. That, by the way, is why it is called the chaos function. Sometimes the cutoff point between one pattern and another is abrupt.  A slight change in the value of c can make a dramatic difference.  Psychological systems are obviously much more complex.  So it unlikely that the logistic equation can actually be used to model physiological behavior. I am using it as a metaphor and as a way to stimulate thinking. Nevertheless, ample intricacy can be found following simple straightforward rules.  An important concept here is that a time series is being followed. The series starts with a set of initial conditions. Some operation is performed on the starting values at time t(0). The result of these operations gives a new starting value at time t(1). This process continues to time t(2), time t(3)... In real life this sort of thing is happening continuously. We live in the immediate present. We react to what is happening consciously and unconsciously in the present. These reactions update our internal and external states. We respond to the new sate. And on and on... This is a dynamic view of what is continuously happening.

Psychologically the sense of self seems to function somewhat like c in the logistic equation.  The self-sense seems to modulate our emotions and behavior.  My inner feeling about myself can result in my behavior being quite stable, or shift back and forth between fixed alternatives. Under other conditions, it may gyrate utterly unpredictably from one state to another.  The understanding the controlling importance of the self-sense and attempting to influence it is one of the truly generic factors in psychotherapy. 

For example, think of c as how good you think about yourself. If you are feeling good about yourself, you are much more likely to be able to handle situations smoothly. However, if you are feeling unworthy your reactions and view of the world is more likely to be unstable. In the Shane_introduction above they spoke of different self-organizations, the old self and the new self. What happens to cause the flip from one state to the other is an absorbing question. Some times patients seem endlessly caught in one state or the other. On other occasions an imperceptibly slight ruffle will provoke a major shift. This is a fine example of non-linear dynamics. A slight change in conditions can produce a dramatic alteration in outcome. If a patient comes to a session in the state, resentful old self with unresponsive old other, the quality of the analyst's response can produce a dramatic shift in the patient's condition. If the analyst responds empathetically, the patient can shift to seeing the analyst as a new other, and or seeing himself as a new self. Conversely, if the analyst slightly misses the empathic mark, the patient may become more entrenched in his or her old miserable self who is even more convinced that the analyst will never understand. Each of us has an inner meaning comparator, a sense of goodness of fit. As Mark Twain said, "The difference between the right word and a word that is not quite right, is the difference between lightning and a lightening bug". The "just right" or "just wrong" response by the analyst can have a powerful effect on the patient's response.

The functions given above are abstract mathematical operations. Logical operations move us a step closer to inner experience. The Boolean logic functions: and, inclusive or, and exclusive or are worth considering. People often think in either or -- black or white terms. A patient expressed it wonderfully. "I think in 'completelys'. John is completely good or he is completely bad." The one view precludes the other. This is a common example of exclusive or thinking. The logical functions have two initial conditions, or premises, and a conclusion. Learning about logical operators can be done abstractly and or concretely. It is probably easier to start concretely. The following exercise asks you to say whether or not pairs characteristics are true about yourself. Check true or false about each characteristic. Then click on the Result button to see the logical function. There is no need for concern your answers stay in your own computer. They are not broadcast over the Internet. As an aside, note the effect of the interaction between the two items. Also sense your feelings and attitude about having or not having the attributes. To get a feel for the logical functions try to predict the result before you push the Results button.

 

To see this interactive Applet concretely illustrating Boolean functions upgrade your browser.

Possibly having to answer either true or false for some of the items was troublesome. If you had the urge to want to qualify some of your answers by saying, "Sometimes I am angry or depressed. Having to say true or false does not due justice to the actual situation", you would be right. Either or, black or white logic simply does not obtain in most real life situations. Indeed, learning to think in both and rather than either or terms is a step in the direction of maturity.

After you have done the concrete exercise you should have a pretty good feel for how the logical operators work. The following abstract exercise should be straightforward.

To see this interactive Applet abstractly illustrating Boolean functions upgrade your browser.