Jack Davis and David Wirtshafter on Settling Points and Parsimony
WIRTSHAFTER, DAVID, AND DAVIS, JOHN D.
Set points, settling points, and the control of body weight. Physiology and Behavior 19: 75-78, 1977.
Comment by Nori Geary, May 2018
John D (Jack) Davis (1928-2018) was a formidable scientist. He contributed important data, innovative methodologies and quantitative models that advanced understanding of food intake and weight regulation in rats (GP Smith, 2001, 2018; J Smith, 2001). Jack was a founding member of SSIB and received SSIB’s Distinguished Career Award in 2007. This SSIB classic concerns one of his minor – but frequently cited – papers, written with his then student David Wirtshafter. The paper is a thought experiment examining the potential of what they called a “settling-point” system, which involved negative-feedback control without a set point, to explain the phenomena of body-weight regulation. They first asserted that weight regulation, as it was then understood, could be achieved by a negative feedback system with or without a set point, then argued that the simpler settling-point system is preferable on the grounds of parsimony, and finally noted that because a set-point for body weight is a neural process, only its discovery should justify adoption of a set-point hypothesis. Forty years on, it remains to be discovered.
Their paper is not quantitative, but assumes some understanding of the mathematical operation of feedback systems. This may be less familiar territory than in the 1970s, when control theory was au courant in physiology and behavior. Therefore, to help crystallize Wirtshafter and Davis’s (1977) points, I first introduce the models they considered in a slightly more detail than they did.
2. The settling-point and set-point models of weight regulation
Control theory is a mathematical characterization of dynamical systems (Åström and Murray, 2016). Development of a control-theory model usually begins by arranging the components of the system to be modeled in a block diagram. Figure 1 shows the block diagram of Wirtshafter and Davis’s (1977) “settling point” system. The component names are shown black upper-case letters inside the boxes and explained in the caption; each component represents a process or function.
Figure. 1. Wirtshafter and Davis’s (1977) settling-point control block diagram. The components are shown in black. S is a sensory process that stimulate eating; its strength is assumed to be a constant, s. This signal sums ( Σ1 ) with the feedback process H to produce ε1. ε1 provides input to process G, which integrates ε1 and all other controls energy intake and energy expenditure so as to produce a particular body weight (BW). G also provides feedback, i.e., an input to H. In this reduced model, H and G simply modify their inputs by constants of proportionality, h and g. The successive dynamic transformations in the feedback loop are represented as functions of time (t) in blue equations adjacent to the connecting lines. These equations were not shown by Davis and Wirtshafter (1977).
The net performance of the system is characterized by transfer functions, which express outputs divided by the inputs. In Wirtshafter and Davis’s (1977) model, the overall transfer function is:
Wirtshafter and Davis (1977) stipulated that each process’s output is equal to its input times a constant, a common assumption in control engineering (Åström and Murray, 2016). I added equations describing the actions of each process in blue adjacent to the connecting arrows in Figure 1, with the values of the constants in lower-case letters and the variables expressed as functions of time, e.g., ε1(t). It is simplest to consider the rate of change body weight, d/dt BW, in kg/d or some such unit. First, in an initial, equilibrium situation in which body weight is stable, the sensory input s and the feedback H(t) are equal and d/dt BW = 0. Then, at time t = 0, s increases; e.g., perhaps more palatable, calorically dense food is offered. The dynamic changes in the system equations can be solved with the aid of Laplace transformations. The solution indicates that weight changes according to a negative exponential function:
Where e is Euler’s number, the base of natural logarithms. Integrating the equation yields the equation for total body weight change from time 0 to t, BW(t):
The important point is that as time (t) goes by, the exponential term (e-ght) approaches 0, and body weight approaches a final stable value, or settling point. Note that the level of the settling point is given by s/h, the ratio of the magnitudes of sensory stimulus s and strength parameter, or gain, of the feedback process, h. If these values change, the settling point will also change. For example, an increase in the sensory input from 0 to 2 will increase body weight twice as much as an increase from 0 to 1. Similarly, if the strength parameter h is doubled due to some neuroendocrine process, the change in body weight produced by a change in s will be halved.
The alternative to a the simple linear negative-feedback or “settling-point” model that Wirtshafter and Davis (1977) considered was a linear proportional (P) set-point model. Figure 2 shows its block diagram. The essential feature is that the feedback signal is compared to an external reference signal, r in Figure 2, known as the set point, to produce an error signal, ε2(t). Note that the set point is a system input, whereas the settling point is a system output. In linear P control, the G process produces a signal equal to the difference between r and hW(t) times a constant. It is also possible to use integrals of ε2(t) (I control), derivatives of it (D control), combinations (PI control), or non-linear functions; linear PI control is most common in engineering (Åström and Murray, 2016). The transfer equations describing steady-state values for set-point models can be more complex than for a simple proportional negative-feedback system due to the additional external input (the set point signal, r) and an additional summing point. The important point in the present context is that with a constant challenge, such as a tonic change in s, a P-control set-point system reaches a steady-state level that is different from the desired, or set-point, level. This is clear from the facts that the system in not activated until there is an error and that its response is proportional to the error. Increasing the strength of the feedback signal (i.e., increasing h) decreases, but does not eliminate, the steady-state error.
Figure. 2. A set-point control block diagram. The components in the lower left have been added to the settling-point diagram: an externally determined reference signal r (the set point) that sums with the feedback process H to produce ε2, the error signal. For linear proportional control, ε2 is the sum of H and r times a constant, u. ε2 then sums with S to produce ε1. Again, successive dynamic transformations are represented as functions of time (t) in blue.
3.The models compared
The heart of Wirtshafter and Davis’s (1977) paper is the question, did available data provide any reason to prefer a set-point account of body weight regulation to a settling-point account? As described above, a settling-point model achieves weight stability under constant conditions, and a P-control set-point model does not achieve regulation at the set point in the face of a tonic challenge. These considerations favor the more parsimonious settling-point account – if the observed weight is not the set-point weight, why call it that?
What about the apparent regulation of lower body weights by rats with lateral hypothalamic (LH) lesions? One possibility is that the lesions reduce r, the set point. Wirtshafter and Davis (1977) pointed out that the identical weight changes could be effected in the settling-point model by decreasing the strength of the G process or increasing the H process. Furthermore, like the LH-lesioned rat, forced deviations from the now-preferred lower weight would elicit compensatory changes re-establishing that lower weight. The apparent regulation of higher body weights by rats with ventromedial hypothalamic (VMH) lesions could be dealt with in a parallel way. Furthermore, the “finickiness” of VMH-lesioned rats was easily explicable by the settling-point model. That is, if the palatability of the diet offered is reduced enough, VMH-lesioned rats maintain the same weight as control rats. This can be achieved in the settling-point model simply by reducing the strength of the S process, whereas the set-point model offers no ready explanation.
The phenomena of obesity related to hedonic eating and dietary obesity are easily explicable by both models, via changes in the S process, as mentioned above.
The final issue considered by Wirtshafter and Davis (1977) concerned weight changes associated with hibernation, migration, and pregnancy. These changes could easily be produced in the settling-point model by changes in the H process related to seasonal or endocrine signals or by addition of biasing signals that would add at the summing point shown in Figure 1. Davis and Wirtshafter (1978) considered further data and reached similar judgements.
In short, in all the cases discussed, the use of a set point to explain the data seemed to Wirtshafter and Davis (1977) to be “unnecessary and unparsimonious” (p 77). Because the principle of parsimony is well accepted in science and philosophy in cases where data are insufficient (Baker, 2016; Einstein, 1930; Stewart, 1993; Thanukos, 2008), this is a worthy argument. Others also pointed out that set points were unnecessary to explain the phenomena of obesity (e.g., Booth et al., 1976; Peck, 1976), but none did so in the didactic manner of Wirtshafter and Davis (1977). The data remain insufficient for a decision, so the debate engendered by Wirtshafter and Davis (1977) continues (albeit with frequent errors in understanding the nature of the settling-point control; Hall and Guo, 2017; Rosenbaum and Liebel, 2016; Speakman et al., 2011). Three interesting threads deserve comment.
First, Tam et al. (2009) presented a partially parameterized a mixed model based on some body-weight and plasma leptin data in mice (leptin was assumed to be the body-weight feedback signal). Settling-point control was used at most weights, in part to fit data on dietary obesity. At very low weights, however, they felt that the strong defense against life-threatening weight loss demanded PI set-point control. The addition of I control, in which the accumulated error also contributes to ε2(t) in Figure 2, maintains the regulated variable much nearer the set point. But a simple alternative would be to hypothesize simply a settling-point model in which the H process becomes much stronger at low weight. (If H is stronger at low weights than at healthy or high weights, the process would be non-linear. Biology is rarely linear, so this seems not to be an important problem.)
Next, Hall (Hall, 2006, and many subsequent publications) constructed and parameterized a detailed model of the dynamic changes in fat mass and fat-free mass (which sum to body weight) that result from changes in the intakes of protein, carbohydrate and fat in humans. This model is the current state of the art. At present, however, it does not include feedbacks from body composition onto the control of eating. But Polidori et al. (2016) have begun to develop these as well. These authors measured body composition every 52 d for 1 y in patients with type 2 diabetes mellitus who were chronically treated with a sodium glucose co-transporter inhibitor that increases urinary glucose excretion, thus resulting in insensible weight loss. They then used the Hall model to estimate the changes in energy intake that would produce the observed body-composition changes and fit these estimates to a negative-feedback process. The modelled control signal was proportional to the sum of the rate of change of body weight plus the absolute change in body weight during the successive intervals. Note that the model does not include a set point, but would require a modest change to add a rate term to Wirtshafter and Davis’s (1977) proportional-control process.
Finally, the curious difficulty of obese patients to maintain weight loss (Heymsfield and Wadden, 2017; MacLean et al., 2015) and the apparent parallel in some rat dietary-obesity experiments (Levin et al., 1987, 2002; Rolls et al., 1980) has attracted a great deal of attention. While these phenomena have not been formally modelled, they appear to be more parsimoniously fit by assuming a change in the set point than by alterations in the settling-point model. That, however, is another story.
I thank Gerard P. Smith, MD, David Wirtshafter, PhD, and Barry Levin, MD, for helpful comments on earlier drafts.
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Wirtshafter D, Davis JD. Set points, settling points, and the control of body weight. Physiol Behav 19:75-78, 1977.