Coherent feedback leads to robust background compensation in oscillatory and non-oscillatory homeostats

Abstract

When in a reaction kinetic integral controller a step perturbation is applied besides a constant background, the concentration of a controlled variable (described as A) will generally respond with decreased response amplitudes ΔA as backgrounds increase. The controller variable E will at the same time provide the necessary compensatory flux to move A back to its set-point. A typical example of decreased response amplitudes at increased backgrounds is found in retinal light adaptation. Due to remarks in the literature that retinal light adaptation would also involve a compensation of backgrounds we became interested in conditions how background compensation could occur. In this paper we describe novel findings how background influences can be robustly eliminated. When such a background compensation is active, oscillatory controllers will respond to a defined perturbation with always the same (damped or undamped) frequency profile, or in the non-oscillatory case, with the same response amplitude ΔA, irrespective of the background level. To achieve background compensation we found that two conditions need to apply: (i) an additional set of integral controllers (here described as I1 and I2) have to be employed to keep the manipulated variable E at a defined set-point, and (ii), I1 and I2 need to feed back to the A-E signaling axis directly through the controlled variable A. In analogy to a similar feedback applied in quantum control theory, we term these feedback conditions as ‘coherent feedback’. When analyzing retinal light adaptations in more detail, we find no evidence of the presence of background compensation mechanisms. Although robust background compensation, as described theoretically here, appears to be an interesting regulatory property, relevant biological or biochemical examples still need to be identified.

Background compensation revisited: Conserved phase response curves in frequency controlled homeostats with coherent feedback

“…Outline First I will describe the properties and breakdown conditions of two single-loop feedback oscillators based on motif 2 and motif 8 (for overview of the feedback motifs, see [31]). Then, two additional controllers (I 1 and I 2 ) will be added to each of the single-loop feedbacks such that I 1 and I 2 will feed directly back to the controlled variable (A) and thereby define a coherent feedback loop structure [18], which is found to rescue the oscillators from their breakdown conditions. Then I show that the coherent feedbacks lead to background compensation in their frequency resetting for both inflow or outflow step perturbations.…”

“…The setpoint of <A> is still given by Eq 9, while two setpoints for <E> are obtained from the rate equations of respectively I 1 and I 2 , i.e. [18, 29] …”

“…The interesting aspect of the coherent feedback scheme in Fig 4 is the fact that it can compensate for different but constant background perturbations. In our previous work [18] only outflow perturbations were considered, because the central feedback…”

“…In a previous study [18] we found that coherent feedback oscillators not only enable robust frequency homeostasis, but can also compensate for different but constant background perturbations. I extend here these earlier findings to include both inflow and outflow perturbations at the controlled variable, which show distinct phase dependencies in the oscillators’ frequency resetting.…”

“…The control of E by I 1 and I 2 leads in addition to frequency homeostasis [29]. In analogy to a similar feedback arrangement in quantum control theory [50,51] we have termed the feedback scheme in Fig 4 as 'coherent feedback' [18]. The rate equations are:…”