Emotion As a Thermostat: Representing Emotion Regulation Using a Damped Oscillator Model
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Emotion Copyright 2005 by the American Psychological Association 2005, Vol. 5, No. 2, 208–225 1528-3542/05/$12.00 DOI: 10.1037/1528-3542.5.2.208 Emotion as a Thermostat: Representing Emotion Regulation Using a Damped Oscillator Model Sy-Miin Chow and Nilam Ram Steven M. Boker University of Virginia University of Notre Dame Frank Fujita Gerald Clore Indiana University South Bend University of Virginia The authors present in this study a damped oscillator model that provides a direct mathematical basis for testing the notion of emotion as a self-regulatory thermostat. Parameters from this model reflect individual differences in emotional lability and the ability to regulate emotion. The authors discuss concepts such as intensity, rate of change, and acceleration in the context of emotion, and they illustrate the strengths of this approach in comparison with spectral analysis and growth curve models. The utility of this modeling approach is illustrated using daily emotion ratings from 179 college students over 52 consecutive days. Overall, the damped oscillator model provides a meaningful way of representing emotion regulation as a dynamic process and helps identify the dominant periodicities in individuals’ emotions. Keywords: differential equation, oscillator, dynamic, emotion regulation, spectral analysis Building on popular belief in the “blue Monday” phenomenon, cyclic change are still lacking. Even though researchers have Larsen and Kasimatis (1990; see also Huttenlocher, 1992; Reid, begun to incorporate longitudinal designs into the study of affect 2000) presented early evidence advocating the existence of a (e.g., Diener, Fujita, & Smith, 1995; Eid & Diener, 1999), the weekly cycle in individuals’ daily mood fluctuations. Using spec- focus on a linear notion of change is still prominent in these tral analysis, these researchers found a strong weekly rhythm in a studies. With the exceptions of spectral analysis (e.g., Larsen & group of college students’ average hedonic level. In broader con- Kasimatis, 1990) and other related frequency-domain and time- texts, rhythmicity has also been examined in relation to diurnal series analyses, most of the dominant longitudinal methodologies changes in mood and circadian activity (e.g., Larsen, 1985a; Mur- (e.g., growth curves and hierarchical linear models; Bryk & Rau- ray, Allen, Trinder, & Burgess, 2002; Rusting & Larsen, 1998) and denbush, 1987; McArdle & Epstein, 1987; Meredith & Tisak, seasonal affective disorder (e.g., Johansson et al., 2003). As more 1990) have been used to represent linear change, even though several emerging evidence has shown clear associations between these nonlinear extensions of these methodologies have been proposed cyclicities and other key aspects of life (e.g., Brown, 2000; Pet- in the psychometric literature (e.g., Browne & du Toit, 1991). tengill, 1993), a closer examination of the nature and determinants In a recent theoretical model of mood regulation, Larsen (2000; of these dynamic processes is imperative. see also Carver & Scheier, 1982, 1990) proposed that mood Despite ample research suggesting the existence of different regulation is, by nature, a dynamic process. A weekly mood cycle, physiological and affective cycles in everyday life (e.g., Brown, for instance, imparts that emotions are dynamic—they exhibit 2000; Stone, 1985), methodologies amenable to the modeling of specific patterns of change over the course of the week. Following Larsen’s (2000) lead, we present in this article a damped oscillator model (Boker & Graham, 1998; Nesselroade & Boker, 1994) that Sy-Miin Chow, Nilam Ram, and Gerald Clore, Department of Psychol- provides a direct mathematical basis for testing this mood regula- ogy, University of Virginia; Steven M. Boker, Department of Psychology, University of Notre Dame; Frank Fujita, Department of Psychology, Indi- tion model. In our presentation, we discuss the basic elements of ana University South Bend. the model, including intensity, rate of change, and change in the Part of the writing of this article took place while while Sy-Miin Chow rate of change as they pertain to the study of emotion. We also was at the University of Notre Dame. Sy-Miin Chow and Nilam Ram highlight some of the similarities and differences between this acknowledge the support provided by National Institute on Aging Grants differential equation modeling approach and other more widely R01 AG18330 and T32 AG20500-01, both awarded to John R. Nessel- known approaches, such as growth curve model and spectral roade. We thank John Nesselroade, Jack McArdle, other members of the analysis. Finally, we illustrate through an empirical example the Institute for Developmental and Health Research Methodology at the potential utility of the approach as a tool for understanding emo- University of Virginia, and Anthony Ong for helpful comments on earlier versions of this article. tion processes (e.g., interindividual differences in patterns of day- Correspondence concerning this article should be addressed to Sy-Miin to-day emotional variability; see Fleeson, Malanos, & Achille, Chow, who is now at the Department of Psychology, University of Notre 2002; Nesselroade & Baltes, 1979, for the rationales for such an Dame, 108 Haggar Hall, Notre Dame, IN 46556. E-mail: [email protected] examination). 208 EMOTION AS A THERMOSTAT 209 Intensity, Rate of Change, and Acceleration in Emotion the individual’s emotion. The nature of a process as conveyed by the amount of acceleration is somewhat analogous to Davidson’s The temporal aspect of emotion and its relatedness to individual (1998) concept of the “rise time” of an emotion, and he theorized differences in emotion regulation have been discussed by several that abnormalities in the rise time (or other temporal characteris- researchers (e.g., Davidson, 1998; Larsen, 2000). The rate of tics) of an individual’s emotion may be indicative of emotion change and acceleration represent different temporal characteris- dysfunctions. tics of affective processes. Before presenting the mathematical The concept of acceleration in emotion can be illustrated more foundations of the damped oscillator model, we begin by clarifying concretely using the following example. The speed shown on a car some basic terms and concepts pertinent to the differential equa- speedometer (representing the rate of change) indicates the pro- tion model, including intensity (or level), rate of change, and jected distance traveled by the car in one unit of time (e.g., 60 change in the rate of change (or acceleration). mph). However, the car may be speeding up or slowing down. This change in the rate of change (i.e., acceleration or braking) is Level or Intensity indicated by how quickly the needle on the speedometer is moving. Intensity (denoted as Y in the model equation to come), in the In the context of emotion, suppose one must give an important context of the differential equation model presented here, repre- presentation the next day. It is possible, perhaps probable, that sents the magnitude of displacement or deviation in an individual’s one’s anxiety would increase. Assuming that this change in anxiety emotion compared with some baseline affect level.1 As certain level is assessed on an hourly basis, the amount of increase in environmental cues or objects elicit a particular kind of emotion anxiety per hour would constitute the rate of change in anxiety. (e.g., sadness) from an individual, the extent to which the individ- Initially, there may be a steady increase in anxiety level that ual’s level of emotion deviates from his or her baseline level persists for hours. However, as the presentation hour approaches, represents the intensity of that emotion. High intensities represent one’s anxiety might really kick in, increasing more rapidly (accel- large distances from baseline (regardless of direction), and low erating) as the moment of truth arrives. Such changes in the rate of intensities represent small distances from baseline. Here, intensity change are explicitly included in the forthcoming model. is conceptualized and used explicitly to characterize states, rather To further illustrate the relationships between acceleration, rate than as a personality trait. Stable individual differences in emotion of change, and intensity, as they might apply to emotion processes, intensity observed across different situations and contexts (e.g., we ask the reader to consider the scenario depicted in Figure 1. Moskowitz, 1982) may, in contrast, reflect a trait-like disposition This figure, representing the progression of sadness, was generated of affect intensity (Larsen, 1985b; Larsen, Diener, & Emmons, 1986). using the damped oscillator model (the corresponding mathemat- ical equation will be presented shortly). Initially (i.e., at Time 0), the individual’s level of sadness (represented with a solid line) is Rate of Change perturbed and deviates momentarily from his or her usual set point The rate of change (Y) represents the magnitude of change in (represented using a dashed line).2 Over time, this heightened level intensity over one unit of time (e.g., day to day, pretest vs. posttest, of sadness dissipates, and the individual slowly returns to his or her year to year, etc.). In other words, rate of change describes the usual baseline (Y 0), overshooting a few times before settling change in emotion intensity in relation to time (i.e., first derivative into an equilibrium state. Instead of visualizing merely the rela- of emotion intensity with respect to