Behavior changes over time. Our task is to understand it. The first step is to be able to describe it; the next step is to be able to predict it. In the past, the task of understanding behavior change was simplified to only describing and predicting the overall behavior, or mean of many momentary measures of behavior. For example, "On average the rate doubled when the condition was changed from A to B" or "The asymptotic rate to this schedule was 2.4 responses per second." (Where the dependent variable was the mean rate during five, 1 hour sessions following 25 or more sessions of exposure to the procedures.) These types of measures are labeled static. The future task of behavior analysis is to understand the moment to moment changes in behavior. These types of measures are labeled dynamical and must be displayed in two dimensions. A single number cannot characterize them. There are three broad categories of behavioral dynamics. Historical examples of their study have been fixed-interval performance (synchronous), the learning curve (transient), and "noise" (asynchronous).   There are a number of possible approaches to take when trying to study behavioral dynamics. The following analysis is based on two sets of simplifying assumptions and constraints. They were accepted for the most part in order to:

1. take advantage of a very well established analytical technology used in other disciplines to understand changing output,

2. simplify the task enough so that some progress could be made.

The first set of assumptions is depicted in the following 3 figures. It is simply the view that the output of an organism can be attributed to the input. Additionally, the distribution of an organism's rate changes over time can be attributed to the distribution of changes in reinforcement over time, and that both the input and output can be seen as changing frequencies. We are very familiar with this approach when we consider audio amplifiers. The inputs to and outputs from an animal can be conceptualized similarly.  If these relatively innocuous constraints are accepted, then a family of analysis techniques can be used as powerful tools to understand the dynamic aspects of behavior. The logic underlying the mathematical machinery of these analyses is very simple and is familiar to everyone.

First, predictions can be made based on arithmetic transitivity. If you have previously determined that the output is twice the input, then given the input you can predict that the output will be twice as large.

Second, many computations can be simplified by changing domains. For example, rather than multiplying two numbers to determine an answer, it is often easier to convert the original numbers to their logs, add the logs, and find the antilog of the answer. The following figure depicts virtually all of the important machinery of the present research. It shows how the relationship between a known input and a known output can be used to predict an unknown dynamical output given a known input.

The function used to adjust the input in order to predict an output (illustrated in green) is labeled a "transfer function." Larger version of figure above

The upper left section of the figure depicts the conversion of the pattern of the reinforcers in the reinforcer pulse input and the resulting response rates into the frequency domain and the division of each of the output frequencies by their respective input frequencies. The quotient is the transfer function. The upper right section depicts some arbitrary reinforcement pattern input, its conversion to the frequency domain and that representation multiplied by the transfer function, frequency by frequency, to produce the frequency domain representation of the predicted output. This is then converted to the time domain and compared to the actual output of the bird, bin by bin, to determine the quality of the prediction.

The following experimental procedure dramatically simplifies the determination of an animal's complete transfer function. Recall the conceptual similarity in the relationship between an input and an output of an animal and the input and output of an audio amplifier. If behavior is to be predicted for any possible frequency input, then the "distortion" for each possible input frequency must be empirically determined. This is obviously a daunting task. However, a simplifying realization can be made. Fourier's theorem shows that an impulse could be seen as containing all possible frequencies of sine waves. As a result, if a single impulse is presented to an audio amplifier, or an organism, then the output will also contain all possible frequencies plus the "distortion" added to each of those frequencies. In other words, an impulse input every hour will reveal how the system will react to any and all possible frequencies between a few seconds and an hour. The simplest procedure which could be used to determine an animal's transfer function is to present reinforcers in the form of a short reinforcement period or pulse. A transfer function covering the frequencies of greatest interest (i.e., behavior changes which take place more often than once per hour) can be approximated with a procedure providing extinction for a few minutes followed by a VI schedule for a few minutes followed by extinction for a few minutes, such that the entire cycle is about 1 hour in duration. Because a zero duration pulse cannot be implemented and a short pulse of VI is used. An imperfect sample of all possible frequencies is obtained. Some frequencies will be missed resulting in an imperfect transfer function, but it is a reasonable first approximation. Linear analysis, the most straightforward of the analysis techniques based on the transfer function, makes several additional simplifying assumptions in order to make the prediction task more tractable. These simplifying assumptions are very similar to the assumptions underlying linear regression. Linear regression assumes that the true regression is the same at all points, and that the regression does not change as the result of what immediately precedes a particular point on the x axis. The three major assumptions required for the use of linear analysis are similar in essence, they state that the behavior resultant of each pattern of inputs is independent of all other inputs. This independence refers to:

1. Independence from procedure

2. independence from prior history

3. independence from interaction of component procedures   Because of the novelty of this research program, a number of relatively simple but important procedural optimizations were revealed. First, if Fourier transforms are to be accomplished using FFT algorithms, then several constraints must be met. One of which is that one complete segment of a continuing pattern must be used. This requires that the response rate at the beginning of a trial must approximate the response rate at the end of the trial. Our original procedure of signaling each trial start with a blackout and providing the pulse at the beginning of the trial lead to a high response rate at the start of the trial. This implied that behavior would show a virtually zero duration ramp up to an unsignaled pulse. A second problem uncovered was that a single pulse procedure resulted in a mathematical artifact. The analysis produced no information at all at reciprocals of the pulse duration. This was equivalent to a situation where the size of a microphone caused a dead spot at all harmonics of a particular note. A foot long microphone would be dead for C sharp for example. This artifact would necessarily delete C sharp from every octave in a piece of music. The solution would be to use two microphones of different sizes and combine their output. The solution in the present case was to use a two pulse procedure where the two pulses were of different durations. What the first pulse could not detect the second pulse filled in. The third problem was one of less than ideal signal-to-noise ratios. An averaging procedure was used to increase the signal-to-noise ratio. The view was that larger or broader samples would diminish random error. The procedure first averaged across large samples (20 day averages), then averaged five of those samples in the frequency domain. Several tasks remain. An example of the practical and paradigmatic fecundity of the analysis is providing the tools to solve inverse questions. An example would be "what separation in pulses are necessary to cause an 80% decrement in rate? Alternatively, "how wide of a pulse would produce an 80% increment in rate"? To the degree that linear analysis can provide a general framework for these questions, it will contribute to a systematic account for them. A closed form analytical solution to the inverse problem is not yet available. The predicted behavior for each bird (based on that bird's transfer function) to each of five hypothetical schedules with increasingly larger extinction periods and to each of five hypothetical schedules with increasingly thin VI schedules were determined. A curve was fit to the increasing and decreasing series for each bird. The schedule which would produce an 80% sag in the behavior and the pulse reinforcement rate that would produce an 80% rise in the behavior was determined for each bird. Birds will be exposed to these procedures to determine the degree to which linear analysis can productively predict rates in highly constrained situations. Several implications of the assumptions underlying linear analysis were also examined. The procedures and results provided information bearing on the reasonableness of some of the assumptions underlying linear analysis.

In a multipulse procedure one or more increases in response rate occur following the reinforcement pulses. This "ringing" has several easy mentalistic explanations such as "the bird doesn't count because it's easier to peck to check to see if all the pulses have passed," but the cause is irrelevant to the prediction-making ability of linear analysis. The important point for linear analysis is whether a transfer function obtained on a single or multiple pulse procedure predicts the other. In this and the following dot plots the trial duration is represented across the X axis. The response rate is represented up the Y axis. The green bars indicate the position and duration of the VI pulses. Each dot represents the rate for an 8 second bin for one trial, for that position in the trial. The red line represents the mean rate for each portion of the trial. By noting the relative density of dots the modal behavior for each portion of the trial can easily be discerned. Additionally the dispersion of behavior is also clearly depicted. The dotted line designates "+1 Max IRI". It is the elapse of the longest IRI if it had been initiated at the end of the VI pulse period. If behavior were under the control of the time since the prior reinforcer it is the worst case confirmed-end-of-pulse. Animation of the daily performance for this bird under this procedure (4 Meg Quicktime movie) Animation of the daily performance for this bird under this procedure (1.3 Meg Quicktime movie)

As implemented, the multi trial procedure with a VI schedule pulse generates a slow increase across the latter half of each trial. This could be an anticipatory rise in rate controlled by the upcoming reinforcement pulse or it could be an intrinsic effect of the prior reinforcement pulse. Exposing the birds to a single trial per day would eliminate the anticipatory rise if it were a higher-order effect.  Animation of the daily performance for this bird under this procedure (1 Meg Quicktime movie) Animation of the daily performance for this bird under this procedure (0.8 Meg Quicktime movie)

As implemented, the procedure also generates a background rate of responding. This effect suggests that the responding shows less precise stimulus control than it would if responding were more costly or if some other concurrent activity were competing with the behavior to the pulse procedure.  Animation of the daily performance for this bird under this procedure (1.3 Meg Quicktime movie) Animation of the daily performance for this bird under this procedure (0.7 Meg Quicktime movie)

A linear analysis also predicts that two schedule pulses occurring on separate keys is exactly the same as two pulses on the same key. Linear analysis argues that the behavioral result of a procedure has no dependence on what the organisms experiences before, after, or at the same time.  Animation of the daily performance for this bird under this procedure (0.9 Meg Quicktime movie)  Send comments/criticisms/speculations to

Date Last Reviewed : January 10, 2002