Regimes and mechanisms of transient amplification in abstract and biological neural networks
Fig 5
A, Representative examples of non-normal amplification defined by the timescale of the transient response of the nonlinear network—period, Δt, for which ‖r(t)‖ ≥ 1: “weak” (Δt ≤ 500 ms); “short transient” (500 < Δt < 2000 ms); and “long transient” (Δt ≥ 2000 ms). Grey dotted line indicates ‖r‖ = 1. B, Timescale of the response in the nonlinear network (as in panel A), parametrised by the norms of the spectrum and feedforward structure. Yellow indicates timescale longer than 10 seconds. Boxes correspond to the values used for the plots in panel A (colour coded): (feedforward norm, spectrum norm) = (100, 700), (500, 500), and (700, 100) for weak, short transient, and long transient, respectively. C, Maximum norm of the dynamical response per initial condition for different percentages of the norm assigned to the spectrum, ranging from a matrix whose entire norm is assigned to the spectrum (yellow; 100% case, normal matrix) to a matrix whose entire norm is assigned to the feedforward part (dark red; 0% case, nilpotent matrix). The network size is N = 200 in all panels. Both eigenspectrum and feedforward structures are random uniform.