The BCM rule feedforward connection shows a hysteresis in pre- and postsynaptic stimulation.

<p><i>(A)</i> The predicted probability distributions <i>p</i>[<i>S</i>] (in colour-code) for the system from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004031#pcbi.1004031.g004" target="_blank">Fig. 4A</a> is strongly influenced by varying the postsynaptic stimulation. Black lines indicate the values of <i>S</i> (treated as continuous variable) for which synapse creation and deletion are equally probable. These points correspond to stable (continuous line) or unstable (dashed line) fixed points of the dynamical system following the net probability flow of our system and indicate the existence of local extrema in the long-term equilibrium probability distribution. Two bifurcations lead to an appearance and disappearance of a bistability, which indicates a possible hysteresis. <i>(B)</i> The same applies for varying the presynaptic activity, although the second bifurcation on the right hand side does not reveal for continuous <i>S</i>, but takes place in the discrete case as both sign changes happen between two consecutive states. <i>(C)</i> Simulation reveals the predicted hysteresis: postsynaptic stimulation was increased stepwise such that <i>v</i>_<i>i</i>(<i>S</i> = 0) increased by steps of 0.01 until it reached 1.0 and then decreased again. For each stimulation, the average number of synapses was calculated separately for the in- and decreasing direction and later averaged over all stimulation cycles (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004031#sec004" target="_blank">Methods</a>). The blue curve depicts the average number of synapses in the increasing and the green curve the decreasing direction. <i>(D)</i> Altering presynaptic activity in the same way also yields a hysteresis loop. (Parameters: BCM rule with synaptic scaling with <i>μ</i> = 0.2, <i>θ</i> = 0.08; <i>υ</i>_<i>tss</i> = 0.1; <i>κ</i> = 9.0, structural plasticity <i>P</i> = 12; ln <i>p</i>_<i>build</i> = −16, <i>a</i> = 2.0, <i>ρ</i> = 0.125, in <i>A, C</i>: <i>v</i>_<i>j</i> = 0.656, in <i>B, D</i>: <i>v</i>_<i>i</i>(<i>S</i> = 0) = 0.2975)