Negative feedback can be destabilising

It is quite obvious that the subjection of every variable of a system to negative feedback is mandatory for the stable maintenance of a stationary state. Were a system under no feedback-control, it would "drift away" according to varying inputs or random fluctuations of its state variables. Were a system under exclusive positive feedback-control, it would of course be unstable. However, negative feedback can lead to expanding oscillations, a source of instability. Indeed, a linear system of 3 or more variables under control of a single negative feedback circuit, as pictured in Figure 1 (as detailed in section 2, to this graph corresponds a unique matrix, which defines the dynamics of the linear system, with zero as an initial condition), cannot be stable around the origin, whatever the weight of the feedback circuit (the proof is direct, based on the study of the characteristic polynomial). The critical length of a feedback circuit is 2: negative feedback involving 1 or 2 state variables provides stability per se to an isolated linear system, while it doesn't when longer circuits are involved. Intuitively, the corrections to the variations of a variable come "too late", and give rise to an ever-expanding series of "over-corrections", a phenomenon commonly known as hunting. Of course, real-world systems oscillations do no keep expanding, because these systems are not linear and there is a saturation in synthesis rates (such non-linearities are actually necessary for the occurrence of asymptotically-stable limit cycles). Such long negative-feedback circuits seem to be the basis for cicardian clocks [3] and mitotic oscillations [4], and it has been shown that they could generate oscillations in MAP-kinase cascades [5]; a model system for a biological clock [6], which has been implemented in the prokaroyte E. coli, oscillates only when the weight of the negative feedback circuit is strong enough (Figure 1b in [6]). The necessity of the presence of a negative feedback circuit (of length strictly greater than 1) in the generation of oscillations resulting in a stable limit-cycle has in fact already been shown in [7] and [8] (boolean systems whose feedback circuits are all negative have no steady-states, but stable cyclic trajectories [9]).

Figure 1: Interaction graph associated to a 3-dimensional matrix possessing a single, negative feedback circuit; if such a matrix is the Jacobian matrix of a differential system at a stationary state, the stationary state is unstable.