Finding actions that satisfy the constraints imposed by both external inputs and internal representations is central to decision making. We demonstrate that some important classes of constraint satisfaction problems (CSPs) can be solved by networks composed of homogeneous cooperative-competitive modules that have connectivity similar to motifs observed in the superficial layers of neocortex. The winner-take-all modules are sparsely coupled by programming neurons that embed the constraints onto the otherwise homogeneous modular computational substrate. We show rules that embed any instance of the CSP's planar four-color graph coloring, maximum independent set, and sudoku on this substrate and provide mathematical proofs that guarantee these graph coloring problems will convergence to a solution. The network is composed of nonsaturating linear threshold neurons. Their lack of right saturation allows the overall network to explore the problem space driven through the unstable dynamics generated by recurrent excitation. The direction of exploration is steered by the constraint neurons. While many problems can be solved using only linear inhibitory constraints, network performance on hard problems benefits significantly when these negative constraints are implemented by nonlinear multiplicative inhibition. Overall, our results demonstrate the importance of instability rather than stability in network computation and offer insight into the computational role of dual inhibitory mechanisms in neural circuits.

Models of cortical neuronal circuits commonly depend on inhibitory feedback to control gain, provide signal normalization, and selectively amplify signals using winner-take-all (WTA) dynamics. Such models generally assume that excitatory and inhibitory neurons are able to interact easily because their axons and dendrites are colocalized in the same small volume. However, quantitative neuroanatomical studies of the dimensions of axonal and dendritic trees of neurons in the neocortex show that this colocalization assumption is not valid. In this letter, we describe a simple modification to the WTA circuit design that permits the effects of distributed inhibitory neurons to be coupled through synchronization, and so allows a single WTA to be distributed widely in cortical space, well beyond the arborization of any single inhibitory neuron and even across different cortical areas. We prove by nonlinear contraction analysis and demonstrate by simulation that distributed WTA subsystems combined by such inhibitory synchrony are inherently stable. We show analytically that synchronization is substantially faster than winner selection. This circuit mechanism allows networks of independent WTAs to fully or partially compete with other.

Learning and decision making in the brain are key processes critical to survival, and yet are processes implemented by nonideal biological building blocks that can impose significant error. We explore quantitatively how the brain might cope with this inherent source of error by taking advantage of two ubiquitous mechanisms, redundancy and synchronization. In particular we consider a neural process whose goal is to learn a decision function by implementing a nonlinear gradient dynamics. The dynamics, however, are assumed to be corrupted by perturbations modeling the error, which might be incurred due to limitations of the biology, intrinsic neuronal noise, and imperfect measurements. We show that error, and the associated uncertainty surrounding a learned solution, can be controlled in large part by trading off synchronization strength among multiple redundant neural systems against the noise amplitude. The impact of the coupling between such redundant systems is quantified by the spectrum of the network Laplacian, and we discuss the role of network topology in synchronization and in reducing the effect of noise. We discuss range of situations in which the mechanisms we model arise in brain science and draw attention to experimental evidence suggesting that cortical circuits capable of implementing the computations of interest here can be found on several scales. Finally, simulations comparing theoretical bounds to the relevant empirical quantities show that the theoretical estimates we derive can be tight.

The neocortex has a remarkably uniform neuronal organization, suggesting that common principles of processing are employed throughout its extent. In particular, the patterns of connectivity observed in the superficial layers of the visual cortex are consistent with the recurrent excitation and inhibitory feedback required for cooperative-competitive circuits such as the soft winner-take-all (WTA). WTA circuits offer interesting computational properties such as selective amplification, signal restoration, and decision making. But these properties depend on the signal gain derived from positive feedback, and so there is a critical trade-off between providing feedback strong enough to support the sophisticated computations while maintaining overall circuit stability. The issue of stability is all the more intriguing when one considers that the WTAs are expected to be densely distributed through the superficial layers and that they are at least partially interconnected. We consider how to reason about stability in very large distributed networks of such circuits. We approach this problem by approximating the regular cortical architecture as many interconnected cooperative-competitive modules. We demonstrate that by properly understanding the behavior of this small computational module, one can reason over the stability and convergence of very large networks composed of these modules. We obtain parameter ranges in which the WTA circuit operates in a high-gain regime, is stable, and can be aggregated arbitrarily to form large, stable networks. We use nonlinear contraction theory to establish conditions for stability in the fully nonlinear case and verify these solutions using numerical simulations. The derived bounds allow modes of operation in which the WTA network is multistable and exhibits state-dependent persistent activities. Our approach is sufficiently general to reason systematically about the stability of any network, biological or technological, composed of networks of small modules that express competition through shared inhibition.

The richness and complexity of recurrent cortical circuits is an inexhaustible source of inspiration for thinking about high-level biological computation. In past theoretical studies, constraints on the synaptic connection patterns of threshold-linear networks were found that guaranteed bounded network dynamics, convergence to attractive fixed points, and multistability, all fundamental aspects of cortical information processing. However, these conditions were only sufficient, and it remained unclear which were the minimal (necessary) conditions for convergence and multistability. We show that symmetric threshold-linear networks converge to a set of attractive fixed points if and only if the network matrix is copositive. Furthermore, the set of attractive fixed points is nonconnected (the network is multiattractive) if and only if the network matrix is not positive semidefinite. There are permitted sets of neurons that can be coactive at a stable steady state and forbidden sets that cannot. Permitted sets are clustered in the sense that subsets of permitted sets are permitted and supersets of forbidden sets are forbidden. By viewing permitted sets as memories stored in the synaptic connections, we provide a formulation of long-term memory that is more general than the traditional perspective of fixed-point attractor networks. There is a close correspondence between threshold-linear networks and networks defined by the generalized Lotka-Volterra equations.