Solving constraint-satisfaction problems with distributed neocortical-like neuronal networks

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 CSPs 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 non-saturating 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 non-linear multiplicative inhibition. Overall, our results demonstrate the importance of instability rather than stability in network computation, and also offer insight into the computational role of dual inhibitory mechanisms in neural circuits.Comment: Accepted manuscript, in press, Neural Computation (2018

Collective stability of networks of winner-take-all circuits

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. We consider the question of 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 multi-stable 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.Comment: 7 Figure

Competition through selective inhibitory synchrony

Models of cortical neuronal circuits commonly depend on inhibitory feedback to control gain, provide signal normalization, and to 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 co-localized 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 co-localization assumption is not valid. In this paper 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 non-linear contraction analysis, and demonstrate by simulation that distributed WTA sub-systems 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 each other.Comment: in press at Neural computation; 4 figure

Computation in Dynamically Bounded Asymmetric Systems

Previous explanations of computations performed by recurrent networks have focused on symmetrically connected saturating neurons and their convergence toward attractors. Here we analyze the behavior of asymmetrical connected networks of linear threshold neurons, whose positive response is unbounded. We show that, for a wide range of parameters, this asymmetry brings interesting and computationally useful dynamical properties. When driven by input, the network explores potential solutions through highly unstable ‘expansion’ dynamics. This expansion is steered and constrained by negative divergence of the dynamics, which ensures that the dimensionality of the solution space continues to reduce until an acceptable solution manifold is reached. Then the system contracts stably on this manifold towards its final solution trajectory. The unstable positive feedback and cross inhibition that underlie expansion and divergence are common motifs in molecular and neuronal networks. Therefore we propose that very simple organizational constraints that combine these motifs can lead to spontaneous computation and so to the spontaneous modification of entropy that is characteristic of living systems

Adaptive Neural Models of Queuing and Timing in Fluent Action

Temporal structure in skilled, fluent action exists at several nested levels. At the largest scale considered here, short sequences of actions that are planned collectively in prefrontal cortex appear to be queued for performance by a cyclic competitive process that operates in concert with a parallel analog representation that implicitly specifies the relative priority of elements of the sequence. At an intermediate scale, single acts, like reaching to grasp, depend on coordinated scaling of the rates at which many muscles shorten or lengthen in parallel. To ensure success of acts such as catching an approaching ball, such parallel rate scaling, which appears to be one function of the basal ganglia, must be coupled to perceptual variables, such as time-to-contact. At a fine scale, within each act, desired rate scaling can be realized only if precisely timed muscle activations first accelerate and then decelerate the limbs, to ensure that muscle length changes do not under- or over-shoot the amounts needed for the precise acts. Each context of action may require a much different timed muscle activation pattern than similar contexts. Because context differences that require different treatment cannot be known in advance, a formidable adaptive engine-the cerebellum-is needed to amplify differences within, and continuosly search, a vast parallel signal flow, in order to discover contextual "leading indicators" of when to generate distinctive parallel patterns of analog signals. From some parts of the cerebellum, such signals controls muscles. But a recent model shows how the lateral cerebellum, such signals control muscles. But a recent model shows how the lateral cerebellum may serve the competitive queuing system (in frontal cortex) as a repository of quickly accessed long-term sequence memories. Thus different parts of the cerebellum may use the same adaptive engine system design to serve the lowest and the highest of the three levels of temporal structure treated. If so, no one-to-one mapping exists between levels of temporal structure and major parts of the brain. Finally, recent data cast doubt on network-delay models of cerebellar adaptive timing.National Institute of Mental Health (R01 DC02852

Isoperimetric Partitioning: A New Algorithm for Graph Partitioning

Temporal structure is skilled, fluent action exists at several nested levels. At the largest scale considered here, short sequences of actions that are planned collectively in prefronatal cortex appear to be queued for performance by a cyclic competitive process that operates in concert with a parallel analog representation that implicitly specifies the relative priority of elements of the sequence. At an intermediate scale, single acts, like reaching to grasp, depend on coordinated scaling of the rates at which many muscles shorten or lengthen in parallel. To ensure success of acts such as catching an approaching ball, such parallel rate scaling, which appears to be one function of the basal ganglia, must be coupled to perceptual variables such as time-to-contact. At a finer scale, within each act, desired rate scaling can be realized only if precisely timed muscle activations first accelerate and then decelerate the limbs, to ensure that muscle length changes do not under- or over- shoot the amounts needed for precise acts. Each context of action may require a different timed muscle activation pattern than similar contexts. Because context differences that require different treatment cannot be known in advance, a formidable adaptive engine-the cerebellum-is needed to amplify differences within, and continuosly search, a vast parallel signal flow, in order to discover contextual "leading indicators" of when to generate distinctive patterns of analog signals. From some parts of the cerebellum, such signals control muscles. But a recent model shows how the lateral cerebellum may serve the competitive queuing system (frontal cortex) as a repository of quickly accessed long-term sequence memories. Thus different parts of the cerebellum may use the same adaptive engine design to serve the lowest and highest of the three levels of temporal structure treated. If so, no one-to-one mapping exists between leveels of temporal structure and major parts of the brain. Finally, recent data cast doubt on network-delay models of cerebellar adaptive timing.National Institute of Mental Health (R01 DC02582

Data-driven Feature Tracking for Event Cameras

Because of their high temporal resolution, increased resilience to motion blur, and very sparse output, event cameras have been shown to be ideal for low-latency and low-bandwidth feature tracking, even in challenging scenarios. Existing feature tracking methods for event cameras are either handcrafted or derived from first principles but require extensive parameter tuning, are sensitive to noise, and do not generalize to different scenarios due to unmodeled effects. To tackle these deficiencies, we introduce the first data-driven feature tracker for event cameras, which leverages low-latency events to track features detected in a grayscale frame. We achieve robust performance via a novel frame attention module, which shares information across feature tracks. By directly transferring zero-shot from synthetic to real data, our data-driven tracker outperforms existing approaches in relative feature age by up to 120% while also achieving the lowest latency. This performance gap is further increased to 130% by adapting our tracker to real data with a novel self-supervision strategy