Positive feedback

Positive feedback

Iteration of simple positive feedback equation. In this example, h = 3 and M = 4 (top), 6, (middle) or 8 (bottom row). Left column: continuous functions, iteration superposed. Right column: x-values as a function of iteration round. Red line: y = Mx^h/ (1+Mx^h); blue line: y = {(Mx^h/ (1+Mx^h) - x} / x; black line: iteration, starting at x = 1.

If Mx_n^h/(1+Mx_n^h) – x = 0 has one or more real solutions, as in the bottom row, the system is bistable. The values to which the system will eventually converge are at x = 0 and at the highest zero-crossing of blue line. The lowest zero-crossing is the threshold.

For each value of h > 1, there is a value of M where the system goes from monostable to bistable. At this point the threshold is equal to the highest stable. Plotted are M (blue, primary y-axis, not a straight line) and M^1/h (green, secondary y-axis) and the value of the threshold (red, secondary y-axis).

(Is the maximum in the green curve in any way significant? M relates to the free energy of binding: the higher M, the tighter the binding of the complex. Its h^th root relates to the contribution of the 'individual subunits')

last edited by MJS on March 28, 2002