I wonder if we are confusing the broader community by calling this integral control. The essence is that the steady equations are structured such that they imply regulated variable = reference. This may be because there is a specific variable whose derivative is equal to reference minus regulated variable (an explicit integrator) or you can do a change of variables to reach that specific form, but the general implication I wrote is much more general and easy enough to explain to a biologist with a couple of examples. Why do we need to see an integrator explicitly, other than making a connection to control theory?
I'm thinking out loud on the blog, and I'll readily admit I'm not sure myself. I guess what I'm thinking is the following. For linear systems that converge to a steady state, a hidden integral controller is necessary and sufficient for a regulated variable to converge to a reference. And for differentiable nonlinear systems, the same is true about any linearization. In this regard, integral control is fundamental and points to a pattern to look for in a regulatory network.
Do you disagree? It feels important to show that simple proportional controllers can't achieve perfect adaptation.
Also, I don't think I understand the change of variables you are proposing here. Could you spell it out in more detail?
I don't disagree at all; I'm also thinking out loud. It is fundamentally important and indeed fascinating to map biological phenomena to man-made control theory. Especially if you are trying to synthesize a circuit for homeostasis, like Mustafa's antithetic feedback, trying to imitate an integrator provides very important design guidance. I guess I had in mind a biologist or endocrinologist who is still traumatized by a calculus requirement and who wants to run for the door at the first utterance of the word integral. To those who want a simple way to judge homeostasis in a biological system, a crash course in PI may not be necessary; it is more direct to show them how to look for an implication in the system structure that, at steady state, the regulated variable must be equal to set point. I would state that criterion and make it concrete with a couple of examples like the antithetic feedback (very easy to represent with simple equations that have the desired property) and perhaps also a thermostat with PI control to say this is how engineers do it and in fact antithetic feedback is a biological analog of that. That order would intrigue people more about control theory and its biological connections than first attempting to teach PI and manipulating the equations of antithetic feedback to display an integral, when that's not necessary to see inherent implication that steady state implies reference equals set point. Biologists are also not too keen on changes of variables, as the new ones may not mean anything tangible to them.
No integrals in sight. And this system can only converge to a fixed point if the CA_blood converges to setpoint.
On the other hand, this equation _explicitly_ has what any control theorist would identify as an integrator, no change of variables needed. And any machine learning person would understand why the Vitamin_D equation looks like gradient descent. I feel like this example has something for everyone.
I think even a calculus-averse person would intuitively grasp the idea of regulation on the basis of accumulated past history of errors rather than on just the current error signal. Although I would also suggest that biologists are more comfortable with calculus than what we give them credit for.
Under mild assumptions, internal model of disturbance is necessarily embedded in a system that rejects that disturbance; see e.g. Section III here: 10.1109/CDC.2018.8619624 My point was, is it critical to look for a change of variables that explicitly displays the internal model? Often, no change of variables is necessary (e.g. Calcium example Ben wrote about) or it is very simple (z1-z2 in the antithetical feedback https://www.cell.com/fulltext/S2405-4712(16)00005-3#fig1). But, it can also be cumbersome and not illuminating (e.g., change of variables leading to (34) in 10.1109/CDC.2018.8619624).
I understand the point about the internal model principle (eg., disturbance zeroes compensated by controller poles, which requires an integrator). But bringing this out explicitly may also require change of variables, like the example in Eduardo’s paper on IMP.
Elegant representation of elegant biology. As a cardiologist, a similar analysis of blood pressure or heart rate would be fun. Sneak in some calculus if you can
A lucid overview! Wish you had been on the faculty here in 1975 when I entered medical school at Minnesota. I seem to recall that basics of calcium metabolism were taught using a sometimes murky motif.
I wonder if we are confusing the broader community by calling this integral control. The essence is that the steady equations are structured such that they imply regulated variable = reference. This may be because there is a specific variable whose derivative is equal to reference minus regulated variable (an explicit integrator) or you can do a change of variables to reach that specific form, but the general implication I wrote is much more general and easy enough to explain to a biologist with a couple of examples. Why do we need to see an integrator explicitly, other than making a connection to control theory?
I'm thinking out loud on the blog, and I'll readily admit I'm not sure myself. I guess what I'm thinking is the following. For linear systems that converge to a steady state, a hidden integral controller is necessary and sufficient for a regulated variable to converge to a reference. And for differentiable nonlinear systems, the same is true about any linearization. In this regard, integral control is fundamental and points to a pattern to look for in a regulatory network.
Do you disagree? It feels important to show that simple proportional controllers can't achieve perfect adaptation.
Also, I don't think I understand the change of variables you are proposing here. Could you spell it out in more detail?
I don't disagree at all; I'm also thinking out loud. It is fundamentally important and indeed fascinating to map biological phenomena to man-made control theory. Especially if you are trying to synthesize a circuit for homeostasis, like Mustafa's antithetic feedback, trying to imitate an integrator provides very important design guidance. I guess I had in mind a biologist or endocrinologist who is still traumatized by a calculus requirement and who wants to run for the door at the first utterance of the word integral. To those who want a simple way to judge homeostasis in a biological system, a crash course in PI may not be necessary; it is more direct to show them how to look for an implication in the system structure that, at steady state, the regulated variable must be equal to set point. I would state that criterion and make it concrete with a couple of examples like the antithetic feedback (very easy to represent with simple equations that have the desired property) and perhaps also a thermostat with PI control to say this is how engineers do it and in fact antithetic feedback is a biological analog of that. That order would intrigue people more about control theory and its biological connections than first attempting to teach PI and manipulating the equations of antithetic feedback to display an integral, when that's not necessary to see inherent implication that steady state implies reference equals set point. Biologists are also not too keen on changes of variables, as the new ones may not mean anything tangible to them.
I think I can write out calcium homeostasis without ever saying integral or derivative:
Ca_blood[t+1] = Ca_blood[t] + Ca_bone[t] + Ca_intestine[t] - Ca_demand[t]
Ca_bone[t] = K1*PTH[t]
Ca_intestine[t] = K2*Vitamin_D[t]
Vitamin_D[t+1] = Vitamin_D[t]+ K3*PTH[t]
PTH[t] = K4*(setpoint-Ca_blood[t])
No integrals in sight. And this system can only converge to a fixed point if the CA_blood converges to setpoint.
On the other hand, this equation _explicitly_ has what any control theorist would identify as an integrator, no change of variables needed. And any machine learning person would understand why the Vitamin_D equation looks like gradient descent. I feel like this example has something for everyone.
I think even a calculus-averse person would intuitively grasp the idea of regulation on the basis of accumulated past history of errors rather than on just the current error signal. Although I would also suggest that biologists are more comfortable with calculus than what we give them credit for.
Setting things up so that regulated variable = reference is the definition of homeostasis. Integral action is not necessarily required for this.
Under mild assumptions, internal model of disturbance is necessarily embedded in a system that rejects that disturbance; see e.g. Section III here: 10.1109/CDC.2018.8619624 My point was, is it critical to look for a change of variables that explicitly displays the internal model? Often, no change of variables is necessary (e.g. Calcium example Ben wrote about) or it is very simple (z1-z2 in the antithetical feedback https://www.cell.com/fulltext/S2405-4712(16)00005-3#fig1). But, it can also be cumbersome and not illuminating (e.g., change of variables leading to (34) in 10.1109/CDC.2018.8619624).
I understand the point about the internal model principle (eg., disturbance zeroes compensated by controller poles, which requires an integrator). But bringing this out explicitly may also require change of variables, like the example in Eduardo’s paper on IMP.
Elegant representation of elegant biology. As a cardiologist, a similar analysis of blood pressure or heart rate would be fun. Sneak in some calculus if you can
A lucid overview! Wish you had been on the faculty here in 1975 when I entered medical school at Minnesota. I seem to recall that basics of calcium metabolism were taught using a sometimes murky motif.
I'm still slogging through learning physiology. I empathize with how a course could get irrevocably complicated.