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Cooperative coding of continuous variables in networks with sparsity constraint

Fig 5

Loss evolution for different strengths of EI-balance.

(A) Loss evolution (dashed: analytical approximation (cf. Eq S30 and S41 in S1 Appendix, partly occluded; solid: network simulation) for balance strengths that are slightly weaker (orange), equal (blue) or slightly stronger (teal) than the critical balance, on a logarithmic scale. The slope of the decay is given by (see (B)), explicitly highlighted for the overdamped dynamics. The oscillation period of the underdamped dynamics is . In case of oscillations, the analytic approximation briefly reaches zero loss once in a period (sharp dips in dashed curve). In the network simulation there is also a pronounced oscillation, but there always remains a finite error. (B) Real part (decay rate , black/gray) and imaginary part (oscillation frequency times , red) of the complex frequency of the exponential loss evolution, scaled by . For weak EI-balance, measured by , there are two exponentially decaying modes (, black and gray curve). At the critical balance (blue dashed vertical line), there is only a single decay rate and no oscillation; the decay rate (in the overdamped case: of the relevant slower-decaying mode) is maximized. For stronger balance, network activity begins to oscillate (nonzero , red), and diverges once becomes negative. This happens approximately at , which is slightly larger than 1 because of the stabilizing effect of the contracting dynamics of the unbalanced network. Dashed vertical lines show the balance strengths scaled by for the curves in (A) (). Parameters: , , , and N = 200 for the network simulation.

doi: https://doi.org/10.1371/journal.pcbi.1012156.g005