Behavioral illusions: The Snark is a Boojum

Abstract

A behavioral illusion is a regularity of behavior that appears to reflect something about the functional characteristics of an organism when it does not. This illusion occurs when the methods appropriate to the study of an open-loop or zero feedback (Z)-system are used to study the behavior of what is, in fact, a closed-loop or negative feedback (N)-system. The situation is like the one described in Lewis Carroll’s The Hunting of the Snark, where the sought-after Snark—analogous to the actual organism function—looks just like the feared Boojum—analogous to the illusory one. This article describes examples of three different kinds of behavioral illusion and explains how researchers can avoid the mistake of taking a Boojum for a Snark by reorienting the study of behavior toward identifying the perceptual variables that organisms control and away from seeking regularities in their overt behavior.

Keywords

cognitive illusion perceptual control theory reinforcement illusion S–R illusion test for controlled variable

The behavior of organisms involves an interaction between organisms and their environment. Powers (1978) describes two possible ways in which these interactions can take place in terms of the nature of the organisms doing the interacting.¹ If organisms are open-loop, zero feedback systems—what Powers calls Z-systems (Figure 1a)—their interaction with the environment is reactive; their output, q_o, is ultimately a reaction to sensory input, q_i, that is caused by environmental variables, q_e. If, on the other hand, organisms are closed-loop, negative feedback systems—what Powers calls N-Systems (Figure 1b)—their interaction with the environment is more proactive; their output varies as necessary to keep sensory input in desired or reference states, protected from the disturbing effect of variables in the environment.

Figure 1.

Two ways an organism might interact with its environment: (a) The Z-system reacts, q_o, to sensory input, q_i, which is a function of environmental variables, q_e or (b) the N-system acts to maintain sensory input in reference state, q_i*, protected from the disturbing effects of environmental variables.

Examples of Z- and N-systems

Virtually all naturally occurring, nonliving, physical systems are Z-systems. A simple example is a ball rolling down an inclined plane. The ball and plane are the system and the environmental variable that is the ultimate cause of the behavior of this system is gravity. The laws of physics—the function k_e in Figure 1—convert gravity to the force, q_i, that acts on the ball. The output of the system is the movement of the ball down the plane, q_o. The system function, k_o, is Newton’s second law: F = ma; the input force accelerates the ball down the plane. The movement of the ball, q_o, is a reaction to the force, q_i, applied to the ball.

The most familiar example of an N-system is probably the home thermostat. Like the Z-System, environmental variables—such as outdoor temperature—are converted, by physical law, k_e, into the input to the thermostat, which is the temperature at the sensor. The thermostat converts input (sensed temperature) into output (the heat produced by a furnace) via the system function, k_o, which computes the difference between the sensed, q_i, and desired or reference, q_i* temperature. Unlike the Z-system, however, the output of the thermostat feeds back, via the feedback function, k_f, to influence the input variable. This feedback effect is negative in the sense that it reduces the difference between sensed and reference temperature. The result is that the heat output produced by the thermostat keeps sensed temperature (and, as a side-effect, the temperature in the room) in a reference state, protected from the disturbing effects of variations in the environmental variables, q_e.

Snark or Boojum?

The main goal of basic research in the behavioral sciences has been to understand the psychological processes that produce the behavior of organisms. These processes are represented by the function k_o in Figures 1a and 1b, which can be called the organism function. Behavioral research can therefore be viewed as an attempt to determine the nature of the organism function, which represents the mechanisms inside the organism that produce its visible behavior. The nature of the organism function is inferred from observation of regularities in the organism’s behavior using methods borrowed from the physical sciences—methods that assume that the system under study is a Z-system. These are the conventional methods of behavioral science (e.g., Levitin, 2002). Powers (1978) has shown that the behavioral regularities that are observed in studies that use these methods will be a true reflection of the organism function only if organisms are Z-systems. If organisms are N-systems, then taking these behavioral regularities as a reflection of the organism function is an example of what Powers (1978) called a behavioral illusion (p. 425).

A behavioral illusion occurs because the behavior of an N-system, when studied using conventional methods, looks just like that of a Z-system. The situation is like the one described by Lewis Carroll in his poem The Hunting of the Snark (1876) where the prized Snark—analogous to the actual organism function—looks just like the perilous Boojum—an irrelevant side effect of that function. If organisms are N-systems, then taking the behavioral regularities observed in studies using conventional methods as a true reflection of the organism function is equivalent to taking a Boojum for a Snark. In Carroll’s poem, the penalty for mistaking a Boojum for a Snark—a mistake made by the baker at the end of the poem—is to “softly and suddenly vanish away” (p. 54). The penalty for mistaking an illusory organism function for the actual one is somewhat less severe but, nevertheless, consequential for researchers who are trying to make progress in understanding how organisms “work.” If, indeed, organisms are N-systems, then researchers who have been using conventional methods to study the behavior of organisms have been basing their conclusions about the nature of the mechanisms that produce this behavior on data that can be misleading.

Systems analysis

Powers (1978) demonstrated the existence of behavioral illusions by doing a comparative analysis of the behavior of Z- and N-systems. This systems analysis involves deriving the equations that describe the behavior of each type of system in terms of functional relationships between the system and its environment. For simplicity we will use linear equations to describe these relationships.

Z-system analysis

The equation that describes the relationship between system and environment for the Z-system is straightforward:

q_{o} = k_{o} (k_{e} q_{e})

(1)

where k_eq_e is the sensory input, q_i, to the system; this input is a function, k_o, of environmental variables, q_e. So, equation (1) says that the observable output of the system, q_o, is a function of the sensory effects of environmental variables, k_eq_e. Since sensory inputs are assumed to be approximately proportional to environmental variables, equation (1) can be written as

q_{o} = k_{o} q_{e}

(1a)

This equation shows that variations in the output of a Z-system are ultimately caused by variations in environmental stimuli. Thus, a Z-system can be called a causal system and, according to equation 1a, the observed relationship between q_e and q_o reflects the organism function that characterizes how the system “works.”

Equation 1a describes the basic assumption about the relationship between system and environment made in conventional psychological experiments, which is that the organisms under study are Z-systems. In these experiments, q_e is the independent variable and q_o is the dependent variable and any observed relationship between these variables, when observed under properly controlled conditions, is presumed to reflect functional characteristics of the organism, represented by the function k_o.

N-system analysis

Two simultaneous equations are required to describe the behavior of an N-system:

q_{o} = k_{o} (q_{i}^{*} - q_{i})

(2)

q_{i} = k_{f} q_{o} + k_{e} q_{e}

(3)

Equation (2) is the organism function that describes how the system transforms input, q_i, into output, q_o; equation (3) is the environment function that describes the effect of environmental variables, q_e, including the effect of the system’s own output, on the system’s input. The variable q_i* in equation (2) defines the reference state of the input to the N-system; q_i* is the value of the input variable when output = 0.

Equations (2) and (3) must be solved simultaneously to see the relationship between system and environment for an N-system. The result is:

q_{o} = k_{f}^{- 1} (q_{i}^{*} - k_{e} q_{e})

(4)

This equation is equivalent to equation (1a), which describes the relationship between system and environment for a Z-system. The equivalence can be seen more easily by setting q_i*, the desired state of sensory input, to 0 and k_e, the effect of the environmental variable on the sensory variable, to 1, so that:

q_{o} = - k_{f}^{- 1} q_{e}

(4a)

This equation appears to describe a causal relationship between variations in environmental stimuli and system output that is analogous to the one shown in equation (1a) for a Z-system. But it differs from the Z-system equation in terms of the function that relates environmental variations to output variations. For the Z-system, the observed relationship between environmental variations and output variations is mediated by the organism function, k_o; for the N-system, the observed relationship between q_e and q_o is mediated by the inverse of the feedback function, k_f.

Equation (4a) describes the fact that, in a well-designed N-system, variations in output oppose (are negatively related to) the disturbing effects of environmental variables on the input variable, thus keeping it close to the reference state specified by q_i*. This process of producing outputs that keep sensed input in a reference state, protected from the effects of disturbance, is called control. Thus, an N-system is called a control system.

Is it a Z-system or an N-system?

Equations (1a) and (4a) show that the behavior observed in conventional behavioral research will look the same whether the organisms under study are Z- or N-systems; in either case, they will be seen to respond to variations in appropriately selected environmental stimuli (independent variables) with variations in their output (dependent variables). But equations (1a) and (4a) also show that the results of this research reflect different things about the organisms under study depending on whether they are Z- or N-systems. If organisms are Z-systems, then the results of conventional research tell you what you want to know: something about the nature of the internal processes that produced the observed behavior. If, however, organisms are N-systems, then the results of this research tell you mainly about characteristics of the environment in which the behavior was observed and virtually nothing about the processes that produced the behavior.

As noted above, conventional behavioral research assumes that organisms are Z-systems. But there is considerable evidence that organisms are actually N-systems (Marken, 1988, 2002; Powers, 1978, 1979). If organisms are, indeed, N-systems, then a large segment of the conventional behavioral research literature is misleading in the same way that visual illusions are misleading; the behavior seen in this research will appear to be that of a Z- rather than an N-system and, therefore, does not truly reflect the processes that produced the behavior. In the next section, we will present the results of conventional research studies that can be shown to be examples of this kind of illusion—a behavioral illusion.

Behavioral illusions

Failure to see that organisms are N-systems has resulted in three different types of behavioral illusion: (a) the S–R illusion, which is the illusion that behavior (R) is caused by environmental events (S); (b) the cognitive illusion, which is the illusion that behavior is an output caused by mental plans or programs; and (c) the reinforcement illusion, which is the illusion that behavior is selected by its consequences. In each case, the illusion can be shown to result from a failure to see that the behavior under study is that of an N- rather than a Z-system. Specifically, the illusion results from failure to see that the system under study is acting to control its own sensory input variables—controlled variables.

The S–R illusion

If the organism under study is an N-system, then any disturbances to a variable it is controlling will be associated with outputs that compensate for the net effects of those disturbances. If the researcher is unaware of the fact that the organism is an N-system—and is, therefore, unaware of the fact that the organism is controlling some perceptual variable—then disturbances can appear to be stimuli (S) that are causing the compensating outputs (R). The observed relationship between a disturbance and output will, therefore, appear to reflect something about the processes in the organism that transform stimulus input into response output when, in fact, they do not. This is the S–R illusion.

A tracking task

The S–R illusion can be demonstrated using a version of a simple compensatory tracking task.² The participant (P) in this task is asked to control the position of a spider, keeping it inside a box. The distance from the spider to the box is the controlled variable, q_i, and P is asked to keep this distance equal to zero (the reference state of q_i, q_i*). To do this, P must produce outputs (mouse movements, q_o) that compensate for a time-varying disturbance, q_e, that represents the spider’s efforts to get away from the box. This is a closed-loop task where P is acting as an N-system with respect to the distance between the spider and the box.

The typical results of this task are displayed in Figure 2, which shows the observed relationship between disturbance and output variations in two different 30-second phases of the experiment. In conventional behavioral research, these results would be seen as having been produced by a Z-system; the lines in Figure 2 would be taken to show how the disturbance (the apparent stimulus, S, or independent variable) causes output (R, or dependent variable) via the organism (in this case, the participant, P). The slopes of the lines would be taken to reflect the organism function, k_o, as per equation (1a). In this case, the organism function, k_o, would appear to reflect P’s sensitivity to variations in movements of the spider away from the box.

Figure 2.

Illusory S–R relationship seen in compensatory tracking task.

Since the apparent sensitivity of the organism, k_o, is proportional to the steepness of the slope of the lines in Figure 2, it appears that P’s sensitivity to the movements of the spider was much greater in Phase 1 than it was in Phase 2 of the experiment. But this is an illusion; P did not suddenly become more afraid of spiders in the middle of the experiment. The different slopes of the S–R or independent–dependent variable relationships seen in Figure 2 reflect a difference in P’s environment, not in P’s sensitivity. Specifically, what changes from Phase 1 to Phase 2 is the feedback function, k_f, that connects P’s output to the variable P is controlling—the controlled variable, q_i—which is the distance of the spider from the target box.

These results show that observed S–R relationships, even when observed under carefully controlled conditions, tell you little or nothing about the causal path from stimulus input to behavioral output if the organism under study is an N- rather than a Z-system (Kennaway, 2020).

Magnitude estimation

The S–R illusion described by equation (4a) also implies that the observed S–R relationship will have the same form as the inverse of the feedback function; if the feedback function is linear, then the observed relationship between q_o and q_e will be linear; if the feedback function is nonlinear, then the observed relationship between q_o and q_e will be nonlinear. An example of this nonlinear version of the S–R illusion can be seen in the magnitude estimation experiments developed by S. S. Stevens (1957).

In magnitude estimation experiments, a “standard” stimulus is presented, and the participant, P, is told a value to assign to it: the “modulus” value. Then the same stimulus with a different magnitude is presented and P is told to express its perceived magnitude as a second number. The ratio of the second number to the modulus is taken to be proportional to the ratio of the perceptual magnitudes of the stimuli. The obtained data led to Steven’s Power Law:

q_{o} = k q_{e}^{a}

(5)

where q_e is the physical stimulus magnitude and q_o is P’s numerical estimate of that magnitude.

If P were a Z-System, then equation (5)—the Power Law—would be equivalent to the organism function described in equation (1a); it would reflect a property of the person doing the magnitude estimation task. In this case, that property would be the way the perceptual system converts stimulus magnitude, q_e, into perceptual magnitude, Ψ(q_e), as shown in Figure 3a. However, a careful look at what P is doing in the magnitude estimation task suggests that P’s behavior is that of an N-system, in which case the Power Law is likely to be an example of a behavioral illusion.

Figure 3.

(a) Magnitude estimation as the output of a Z-system and (b) as the behavior of an N-system controlling the difference between the psychological value of both the stimulus, Ψ(q_e), and a numerical estimate of that stimulus, Ψ(q_o).

The participant in the magnitude estimation task is asked to control a relationship between the perceptual magnitude of two different stimuli—that of the stimulus being judged, Ψ(q_e), and that of the number given as the judgment of the magnitude of that stimulus,Ψ(q_o). The situation is shown in Figure 3b. The controlled variable in this task is Ψ(q_e) - Ψ(q_o)—the difference between the psychological magnitudes of stimulus and numerical judgment—and the reference for the state of this variable should be zero; the perceptual value of P’s numerical output should equal the perceptual magnitude of the stimulus. The stimulus is a disturbance to this controlled variable and P compensates for this disturbance by producing a numerical response that has a psychological magnitude that is equal to the psychological magnitude of the stimulus.

The function that transforms q_e into Ψ(q_e) is equivalent to the disturbance function, k_e, in equation (4) and the function that transforms q_o into Ψ(q_o) is equivalent to the feedback function, k_f, in equation (4a) that connects system output to the controlled variable. Both are psychophysical functions—functions that convert physical into psychological variables—so it is reasonable to assume that they are logarithmic, as shown in Figure 3b. Given this assumption, an N-system analysis of the magnitude estimation task leads to the following expected relationship between magnitude estimates, q_o, and stimulus inputs, q_e :

q_{o} = q_{e}^{k 1 / k 2}

(6)

where k1 and k2 are the coefficients of the log functions in Figure 3b. This equation is equivalent to Steven’s Power Law (equation 5) with a = k₁/k₂. As per equation (4a), equation (6) is the inverse of the logarithmic feedback function that relates q_o to q_i. This shows that the observed Power Law relationship between stimulus and response in a magnitude estimation task is likely to be a behavioral illusion, the observed Power Law being the inverse of the feedback connection between responses and the controlled variable rather than the logarithmic organism function that connects stimuli to responses.

The relationship shown here between Stevens’ Power Law and a logarithmic psychophysical function has been known for some time (MacKay, 1963). What is original in the present analysis is the use of this relationship to illustrate a behavioral illusion that can occur in research aimed at determining the internal organization of what turns out to be an N- rather than a Z-system.

The cognitive illusion

The cognitive illusion results from failure to see disturbances to a controlled variable and the actions that protect that variable from those disturbances so that observed variations in the controlled variable appear to be emitted outputs that are caused by cognitive processes inside the organism, such as mental plans or programs.

The cognitive illusion is based on another fact about N-system behavior that can be derived from equations (2) and (3). Solving these equations simultaneously for q_i rather than q_o results in the approximation:

q_{i} \approx q_{i}^{*}

(7).

This equation says that an N-system keeps the controlled variable, q_i, equal to the system’s reference specification, q_i*, for the state of that variable. This is true when the loop gain of the system is sufficiently high, and the dynamics of the system maintain stability. When this is the case, variations in the controlled variable will match variations in the mental (cognitive) specifications for that variable.

If an observer assumes that the organism under study is a Z-system, then observed variations in the variables it controls will appear to be outputs that are being produced by cognitive processes rather than inputs that are being maintained in variable reference states. The result will be the illusion that characteristics of what appear to be the observed outputs tell you something about the processes that produce them. This would be the case if, in fact, the behavior was that of a Z-system. But when the behavior is that of an N-system, observed characteristics of its behavior are only side-effects of the process of control.

Invariant velocity profiles

One example of the cognitive illusion is the invariant tangential velocity profile discovered by Atkeson and Hollerbach (1985). These profiles are found when people make simple movements, such as moving the tip of the index finger from one position to another. Figure 4a shows plots of the tangential velocity over time of movements made to targets at different distances from a starting position. Atkeson and Hollerbach found that, when normalized relative to peak velocity and movement time, the shape of these velocity profiles is invariant with respect to the different movement distances.

Figure 4.

(a) Observed invariant velocity profiles. (b) Velocity profiles produced by “Little Man” model.

The invariant velocity profiles shown in Figure 4a appear to be emitted outputs that are the result of cognitive processes in the behaving system that produce this invariance. Thus, the observed invariance is thought to reveal something about the nature of these processes. But it can be shown that this idea is based on an illusion.

Figure 4b shows tangential velocity profiles that were produced by an N-system model—the “Little Man” (Powers & Williams, 1992)—making movements over different distances equivalent to those made by the human participants in the Atkeson and Hollerbach (1985) study. The model that produced these profiles included no computational processes that were designed to produce such profiles. Figure 5a shows a screen shot of this model making a pointing movement toward a target object.

Figure 5.

(a) Snapshot of the “Little Man” hierarchical control model in action. (b) Functional diagram of control processes in the “little man.”

The movements shown in Figure 4b are produced by the hierarchy of control systems diagrammed in Figure 5b. The results of this modeling exercise show that the dynamics of human movement seen in the velocity profiles, which Atkeson and Hollerbach (1985) saw as the behavior of the participants’ apparent outputs, q_o, are likely a side-effect of the behavior of a perceptual input variable, q_i, that is being controlled relative to a varying reference specification, q_i*, as in equation (7).

Gait planarity

“Gait planarity” is a regularity that has been observed in the gait of animals that walk on two or four legs. The planarity referred to is not that of a plane in physical space, but a plane in an abstract configuration space. Seen from the side, an animal’s leg consists of three segments, the upper leg, the lower leg, and the foot. The angle that each of these segments makes with a reference direction, such as the vertical, varies over time. Plotting these three variables gives a path in three-dimensional space, which, for a repetitive motion like walking, forms a loop traced over and over. The degree to which this loop forms a plane is measured by a planarity index; the closer this index is to 0.0, the closer the loop is to lying on a plane.

Catavitello et al. (2018) found that the planarity index of the gaits in a wide variety of animals, from birds to elephants, is close to 0.0; this is the phenomenon of gait planarity. For various reasons, some of which may have to do with the concept of “synergy” (e.g., Klous et al., 2010), the finding of gait planarity was taken to imply that planar orbits must be produced by a mechanism for making planar orbits. That is, planar orbits are seen as outputs generated by cognitive processes.

We have studied planarity using a model, created some years ago by one of us (Richard Kennaway), for another purpose. It is a physical simulation of a six-legged walking robot (Archy), as shown in Figure 6. The robot contains a multilevel control system, by means of which it can walk towards food particles. The system is shown in Figure 7. The higher level controllers satisfy references for direction and distance to a food particle (Figure 7a). Their outputs are reference signals for lower level controllers that control the positions of feet that are lifted from the ground, relative to the body (Figure 7b). The controllers are all simple proportional–integral–derivative (PID) with fixed parameters chosen by its designer, by trial and error. So, we see that there is nothing in this mechanism that constitutes any sort of plan for producing planar trajectories in configuration space, yet planar trajectories are observed. This is shown in Figure 8, which shows that the planarity indexes for all joints in Archy’s gait are close to 0.0.

Figure 6.

Archy the robot.^a

Figure 7.

Main functional components of the Archy control systems (a) when feet are on the ground and (b) when feet are in the air.

Figure 8.

Planarity measures for Archy.

Our results contradict the idea that planarity is the result of a process of output generation. The specific example of the robot can easily be seen to have no intention, plan, or mechanism for producing them, yet it does produce them. They are epiphenomenal to the mechanism, a side-effect of the behavior of an N-system controlling its own sensory inputs.

Power law of movement

Another example of the cognitive illusion occurs in studies of a phenomenon known as the power law of movement, not to be confused with Stevens’ power law of psychophysics, which was discussed above. The power law of movement refers to the consistent finding of a power relationship between the velocity and curvature of movements made by humans and other organisms (Ivanenko et al., 2002; Lacquaniti et al., 1983; Viviani & Terzuolo, 1982; Zago et al., 2016). The power law can be stated as:

V = k * C^{β}

(8)

where V is the velocity and C is the degree of curvature at each instant during a movement. The “law” in “power law” comes from the fact that the exponent, β, of the power function is typically close to 1/3 or 2/3, depending on how velocity and curvature are measured. For simplicity, we will deal here only with the “2/3 Power Law,” which is found when velocity is measured as angular velocity and curvature is measured as the inverse of the radius of curvature.

The power law is assumed to reflect the processes involved in movement production and theories of movement production have been developed to account for the fact that movements tend to follow a power law (e.g., Gribble & Ostry, 1996; Karklinsky & Flash, 2015). The data in Figure 9 shows why this assumption cannot be correct. The graphs show behavior in a movement task where a participant, P, moved a cursor in an elliptical trajectory around a computer display using a mouse controller. The position of the cursor was being continuously disturbed in two dimensions by a low pass filtered noise waveform. To keep the cursor moving in an elliptical trajectory, P had to move the mouse to compensate for these disturbances while at the same time producing the elliptical cursor movement.

Figure 9.

Typical results of moving a cursor with a mouse while low pass disturbances are added to cursor movement. (a) Cursor and mouse movements made by a human participant. (b) Cursor and mouse movements made by a control model.

Figure 9a shows the behavior of a human participant making elliptical cursor movements (the solid line in Figure 9a) by moving the mouse (the dashed line in the figure). The cursor movements conform fairly closely to the 2/3 (.67) power law (observed β = .69); the mouse movements do not (observed β = .83). Since the mouse movements are the means used to produce the cursor movements, these results show that power law conforming movements reveal little or nothing about the processes that produced them.

The results in Figure 9a strongly suggest that the observed elliptical cursor movement is being produced by an N-system; the position of the cursor seems to be a controlled variable, q_i, that is being kept in a variable reference state, q_i*, per equation (7), and protected from disturbance, q_e, by the mouse movements, q_o (Kennaway & Marken, 2019). This hypothesis was tested by building a control model of the behavior seen in Figure 9a.

The model consists of two independent control systems: one controlling the position of the cursor in the X dimension, q_ix, and the other controlling the position of the cursor in the Y dimension, q_iy. The model assumes that the references for the X and Y position of the cursor, q_ix* and q_iy*, are being varied independently so as to specify elliptical movement of the cursor. These varying references were estimated from the data using methods described by Powers (1989).

The results of the modeling are shown in Figure 9b. As can be seen by comparing Figure 9b to Figure 9a, the model matched the participant’s behavior extremely well. Indeed, the correlation between model and human behavior (both mouse movement and elliptical cursor movements) was greater than .98. The model clearly captures the nature of the processes that produce the participant’s behavior even though there was nothing in the model like the open-loop, cognitive processes that are implied by the power law relationship between curvature and velocity.

The reinforcement illusion

The reinforcement illusion occurs in operant conditioning experiments where an organism’s output seems to be selected by its consequences. Because these consequences appear to strengthen outputs (in terms of increasing the probability of their occurrence) they have come to be called reinforcements. Powers (1971) demonstrated the reinforcement illusion using a reanalysis of the results of a shock avoidance experiment reported by Verhave (1959). In the experiment, rats had to press a lever a fixed number of times in a specified interval in order to avoid a shock. After sufficient training at a given interval setting, the rat approached an equilibrium rate of lever pressing that allowed it to avoid getting shocked. The results of the experiment are shown in Figure 10.

Figure 10.

Analysis of behavior in a shock avoidance experiment. Press rate as a function of shock interval for (a) 8 press requirement and (b) 1 press requirement (based on data from Powers, 1971).

The graphs show the observed (actual) press rate as a function of intershock interval length for rats working with two different requirements for the number of presses to be made per interval to avoid a shock: at least eight presses per interval for the rat on the left and one press per interval for the rat on the right in Figure 10. The data show that as the length of the intershock interval increases, the rate of pressing decreases. The reinforcement in this situation is shock avoidance and the data suggests that behavior (press rate) is being selected by its consequence (shock avoidance). However, this can be shown to be an illusion that results from failure to see that something about this consequence is being controlled by the rats.

Powers exposed the illusion by developing a model of the rats as N-systems controlling the probability of getting shocked. Shock probability was the perceptual input variable, q_i, controlled by the model and the reference for that variable, q_i*, was assumed to be zero. The output of the model, q_o, was lever press rate, which was assumed to be a normally distributed random variable. The main feature of the model is the nonlinear feedback function that converts press rate into the controlled variable, q_i.

The behavior of the best fitting model is shown by the open triangles in Figure 10. The root-mean-square (RMS) deviation of model press rate, q_o’, from observed press rate, q_o, was one press/minute for the rat with the eight press per interval requirement and 0.82 presses/minute for the rat with the one press per interval requirement. Though the fit of the model was quite good, Powers tried an alternate definition of the variable the rat was controlling to see if the fit of the model could be improved. This alternate model controlled shock rate rather than shock probability, where shock rate is simply shock probability divided by the interval within which the required number of presses must occur. This model was compared only to the data for the rat with the eight-press requirement. The RMS deviation of the best fitting version of the shock rate control model from the data was 3.24 presses/minute. Because this was a sufficiently worse fit than the one press/minute deviation for the shock probability control model, it was concluded that the rats were controlling the probability rather than the rate of shock and acting to keep that probability in a reference state of zero shocks/minute.

The relationship between intershock interval and response rate seen in Figure 10 makes it appear that the organism’s sensitivity to shock (in terms of response rate) declines nonlinearly as the intershock interval increases. However, the model suggests that this is an illusion; the decrease in response rate as a function of intershock interval does not reflect a change in the organism; it reflects the nonlinear feedback connection between the organism’s output and the perceptual variable it controls. The model shows that what is actually happening is that the rat is controlling shock probability at a reference of zero and acting to produce that result by varying its lever press rate appropriately to compensate for changes in the feedback function—the rate of lever pressing output required to keep the probability of shock nearly equal to zero.

Testing for controlled variables

The behavioral illusions described above occur when the system under study is thought to be a Z-system, whose behavior is caused by input variables, when it is actually an N-system, whose behavior is organized around the control of input variables—controlled variables. Therefore, the way to avoid succumbing to these illusions is by orienting the study of the behavior of living systems toward identifying the input variables around which their behavior is organized. This is done using an approach to behavioral research called the test for the controlled variable or TCV (Marken, 2005, 2014, 2020; Powers, 1973).

The TCV refers to a collection of methods that can be used to get an accurate picture of the variable(s) an organism is controlling (Marken, 2021; Runkel, 2004). The TCV is an iterative process that involves the formulation of hypotheses about the controlled variables around which the behavior under study might be organized. The source of these hypotheses is observation of the behavior from a control theory point of view (Marken, 2002), which involves looking at behavior with the understanding that it is likely to be that of an N-system. The next step is to determine what the effect of disturbances to the hypothesized controlled variable would be if it were not under control. For example, the velocity of a crosswind would be expected to affect the position of a car in its lane if that variable were not under control. If disturbances have far less than their expected effect on a hypothesized controlled variable, then that is evidence that the variable is under control. If, however, the disturbance has the expected effect, then the hypothesized variable is not under control and the test continues with a new hypothesis about the controlled variable. This process iterates until the experimenter finds a definition of the controlled variable that is protected from the effects of all disturbances that should affect it.

TCV in object interception

One line of research that has been carried out in a way that is largely consistent with the methodology suggested by the TCV is aimed at understanding the variables organisms control when they run to intercept moving objects. There are three main hypotheses regarding what these variables might be: linear optical trajectory (LOT), optical acceleration cancellation (OAC), and control of optical velocity (COV).

The LOT hypothesis is based on data that show temporal variations in the optical position of fly balls as seen by the fielder who caught them (McBeath et al., 1995, see Figure 4). These optical trajectories fall almost exactly on straight lines. Based on this data, the researchers hypothesized that the fielder was able to intercept the ball by moving to keep the optical trajectory of the ball moving in a straight line. The LOT hypothesis was, therefore, that the controlled variable when catching a fly ball is the ball’s optical trajectory and that this variable is maintained in the reference state “linear” by movements of the fielder; the fielder was thought to be controlling for a linear optical trajectory (LOT) of the image of the ball.

One obvious way to test this hypothesis is to apply disturbances that would affect the linearity of the optical trajectories if these trajectories were not under control. Shaffer et al. (2004) did this test by looking at the optical trajectories that were produced when dogs catch frisbees. The irregular movements of frisbees would produce highly nonlinear optical trajectories if the dogs were not controlling for keeping them straight; and apparently, they were not, as shown in Figure 11. The frisbee’s trajectory, which is a disturbance to a linear optical trajectory, was completely effective; the dog was clearly not controlling for a linear optical trajectory. The hypothesis that optical trajectory is the variable controlled when organisms intercept moving objects can, therefore, be rejected. The next step in the TCV is to develop a new hypothesis about the variable (or variables) that an organism controls when intercepting moving objects.

Figure 11.

Optical trajectory of dog catching frisbee, after Shaffer et al. (2004).

Two new hypotheses were tested simultaneously by Shaffer et al. (2013). The TCV was done using a model of object interception that is shown in Figure 12. It is a control system model of a person intercepting a toy helicopter flying in an irregular trajectory. The model consists of two control systems, each controlling a different perceptual variable. One system controls a variable that is a function of the vertical optical angle, α, of the toy helicopter and the other controls a variable that is a function of the lateral optical angle, β, of the helicopter.

Figure 12.

Control model of object interception. COV hypothesis shown as input functions.

The different hypotheses about the variables controlled when intercepting moving objects were implemented as different definitions of the perceptual variables controlled by the model. In one case, the model controlled the perceptions shown in Figure 12: vertical optical velocity, p_v = dα/dt, and lateral displacement, p_l = β. This was called the control of optical velocity or COV model. In the other case, the model controlled perceptions of vertical optical acceleration, p_v = d²α/dt, and lateral displacement, p_l = β; this was called the optical acceleration cancellation or OAC model. In both cases, the reference specifications for the states of all variables were zero.

The COV and OAV hypotheses were tested by comparing the behavior of the models to that of humans intercepting the toy helicopters. Regression analysis was used to determine how much of the variance in the observed ground tracks made when participants ran to intercept the helicopter (variations in output, q_o) was accounted for by the ground tracks produced by the best fitting COV and OAC models. The results showed that, averaging over 41 interception trials, the COV model accounted for an average of 93% of the variance in the observed ground tracks while the OAC model accounted for an average of only 75% of the variance in these tracks.

These results strongly suggest that object interception involves control of optical velocity (per the COV model) rather than optical acceleration (per the OAC model). The fit of the COV model to the ground tracks on four experimental trials is shown in Figure 13. Not only do these results suggest that optical velocity is the best definition of the variable controlled in object interception, but they also show that when you get an accurate definition of a controlled variable, the fit of an N-system (control) model to the data can be strikingly accurate, giving the researcher confidence that the model provides an accurate account of how the behavior actually “works.”

Figure 13.

Examples of object interception paths and correspondence of COV model behavior to the data (from Shaffer et al., 2013).

Is it N- or Z-system behavior?

Other approaches to testing for controlled variables are described by Marken (2021). But it is important to note that some version of the TCV should be the first step in the study of the behavior of living organisms if there is any possibility that the behavior under study is that of an N-system. If it is, then the use of the TCV will allow the researcher to avoid mistaking illusory for actual evidence of how the organism works (mistaking a Boojum for a Snark). If, on the other hand, the behavior is that of a Z-system, this will also be revealed by the TCV; when the TCV is used to study the behavior of what is actually a Z-system it will show that there is no controlled variable around which the behavior is organized. In that case, research can proceed using the conventional methods of psychology that are based on an open-loop causal model of behavior.

Discussion

We have described examples of three types of behavioral illusion—S–R, cognitive, and reinforcement—that result when the behavior of an N-system is taken for that of a Z-system. In all cases, the illusion is that the observed behavior tells you something about how the organism “works” when, in fact, it does not. In all cases, the way the organism appears to work is as an open-loop or Z-system when, in fact, it actually works as a closed-loop or N-system. The behavioral illusions described herein result from ignoring the existence of controlled variables—the variables around which the behavior of an N-system is organized; the appearance of the behavior as a response to stimuli, an output produced by cognitive processes, or a response selected by reinforcement is a side-effect of the controlling done by these N-systems.

We concluded by showing that these behavioral illusions—illusions equivalent to mistaking a Boojum (the apparent—illusory—organism function) for a Snark (the actual organism function)—can be avoided by being cognizant of the possibility that the organism under study may be an N- rather than a Z-system. We showed that it is possible to determine whether you are dealing with an N- rather than a Z-system by testing to see if the system’s behavior is organized around the control of perceptual aspects of its environment. This test for the controlled variable (TCV) involves looking for definitions of variable consequences of the organism’s actions that are being maintained in constant or varying reference states, protected from disturbance by the organism’s actions. Using an example of research aimed at testing for the variables controlled when intercepting moving objects, we showed that taking controlled variables into account not only makes it possible to avoid mistaking an N- for a Z-system but also provides an accurate explanation of how the system actually “works.”

Although the examples of behavioral illusions described here involve control of simple perceptual variables, we expect that the same illusions will be seen in behaviors involving control of more complex perceptions. For example, we would expect to see the S–R illusion in experiments, such as the classic “conformity” study by Asch (1956). In that study, an environmental variable (the independent variable), such as the size of a majority expressing an opinion contrary to that of the participant, appears to cause the participant’s behavior (the dependent variable), such as the probability of expressing agreement with the majority. Thus, the apparent causal relationship between independent and dependent variable is likely an example of the S–R illusion that results from ignoring the perceptual variable being controlled, which is probably something like the degree to which the participant is willing to be perceived as a deviant.

We would also expect to see the cognitive illusion in situations where the organism seems to be producing programmed or planned outputs. An example of such behavior is the fixed action pattern seen in the egg-rolling behavior of the greylag goose (Lorenz & Tinbergen, 1970). The apparently programmed sequence of actions that the goose uses to retrieve an egg that has fallen from the nest is likely a side effect of the goose’s efforts to control a perception of the pressure of the egg against the back of her bill (Marken, 2002).

The examples of behavioral illusions described in this article were selected because they very clearly demonstrate our main point, which is that the use of Z-appropriate methods to study behavior—the conventional methods of psychological research—will produce misleading results when used to study the behavior of what are actually N-systems. This conclusion follows directly from our analysis of the difference between Z- and N-systems. We consider this conclusion to apply to all Z- and N-systems, regardless of the complexity of the system or whether the overall system is a mix of both Z- and N-systems. In order to avoid mistaking the side effects of N-system behavior for that of a Z-system—that is, to avoid taking a Boojum (the illusory organism function) for a Snark (the true organism function)—we recommend that research aimed at understanding how behavior works start with some version of the TCV in order to see whether the behavior is that of a Z- or N-system and, if it is the latter, to determine the variable(s) around which its behavior is organized.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Richard Marken

Richard Kennaway

Tauseef Gulrez

Notes

Author biographies

Richard Marken is an experimental psychologist and human factors engineer specializing in the application of control systems models to understanding the purposeful behavior of living systems. He is currently retired but teaches part time at Antioch University and UCLA Extension. Recent publications include: (with M. S. Broder, J. M. Zambrano, & J. Lee), “Systematic Bias in Predictions of New Drugs’ Budget Impact: Analysis of a Sample of Recent US Drug Launches” in Current Medical Research and Opinion (2017), and (with D. Shaffer), “The Power Law as Behavioral Illusion: Reappraising the Reappraisals” in Experimental Brain Research (2018).

Richard Kennaway is an applied mathematician and computer scientist whose interests include natural and artificial control systems and biological modelling. He is a senior scientist at the John Innes Centre, Norwich, UK. Recent publications include: “Spatiotemporal Coordination of Cell Division and Growth During Organ Morphogenesis” in PLOS Biology (2019) and “Evolution of the Grass Leaf by Primordium Extension and Petiole-Lamina Remodeling” in Science (2021).

Tauseef Gulrez received his PhD. in robotics and artificial intelligence from Macquarie University, Sydney, Australia, in 2008. He was a senior research fellow in the cognitive engineering lab of the Australian Defence Force Academy (ADFA) before taking his present position in the department of computing and research at Syscon Australia. His main research interests include data fusion (AI methods), computer vision, robotics, and behavioural neuroscience. Recent publications include: (with S. Sandhu & W. Mansell), “Behavioral Anatomy of a Hunt: Using Dynamic Real-World Paradigm and Computer Vision to Compare Human User-Generated Strategies With Prey Movement Varying in Predictability” in Attention, Perception & Psychophysics (2020).

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