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The neocortex has a remarkably uniform neuronal organization, suggesting thatcommon principles of processing are employed throughout its extent. Inparticular, the patterns of connectivity observed in the superficial layers ofthe visual cortex are consistent with the recurrent excitation and inhibitoryfeedback required for cooperative-competitive circuits such as the softwinner-take-all (WTA). WTA circuits offer interesting computational propertiessuch as selective amplification, signal restoration, and decision making. But,these properties depend on the signal gain derived from positive feedback, andso there is a critical trade-off between providing feedback strong enough tosupport the sophisticated computations, while maintaining overall circuitstability. We consider the question of how to reason about stability in verylarge distributed networks of such circuits. We approach this problem byapproximating the regular cortical architecture as many interconnectedcooperative-competitive modules. We demonstrate that by properly understandingthe behavior of this small computational module, one can reason over thestability and convergence of very large networks composed of these modules. Weobtain parameter ranges in which the WTA circuit operates in a high-gainregime, is stable, and can be aggregated arbitrarily to form large stablenetworks. We use nonlinear Contraction Theory to establish conditions forstability in the fully nonlinear case, and verify these solutions usingnumerical simulations. The derived bounds allow modes of operation in which theWTA network is multi-stable and exhibits state-dependent persistent activities.Our approach is sufficiently general to reason systematically about thestability of any network, biological or technological, composed of networks ofsmall modules that express competition through shared inhibition.

Collective stability of networks of winner-take-all circuits