Linear inhibitory plasticity fails to stabilize weights for high postsynaptic firing rates.

A. Schematic of a feedforward inhibitory motif. A single postsynaptic excitatory neuron with rate νE receives input from NE excitatory presynaptic neurons, with firing rate ρE and weight wEE and NI inhibitory presynaptic neurons, with firing rate νI and weight wEI. The inhibitory neurons receive external excitatory input with rate ρI and input from the presynaptic excitatory neurons via wIE. B. Plasticity curve of E-to-E weights (, blue) as a function of the postsynaptic rate νE. The postsynaptic LTD/LTP threshold is set to 1. C. E-to-E weight change () as a function of the presynaptic excitatory rate ρE for different I-to-E weights wEI ranging from 0 to 1.5. The presynaptic LTD/LTP threshold is shown for wEI = 0.75 (vertical dashed line). D. Plasticity curves of E-to-E (, blue) and I-to-E (, red) weights as a function of the postsynaptic rate νE. The excitatory and inhibitory LTD/LTP thresholds are identical (). The black cross marks the postsynaptic rate where the plasticity curves cross beyond which the weight dynamics become unstable. E. Phase portrait of the dynamics of E-to-E (wEE) and I-to-E (wEI) weights. Gray arrows indicate the direction of weight evolution over time, points represent three different weight initializations, , and green lines represent the weight evolution for each case. The two colored points represent initial weights in F. Black line indicates the line attractor and the dashed line separates stable from unstable initial conditions (Methods, Eq 20). F. E-to-E (wEE, blue) and I-to-E (wEI, red) weights as a function of time for stable (solid lines, ) and unstable (dashed lines, ) initial conditions.