Restrictions and Stability of Time-Delayed Dynamical Networks

This paper deals with the global stability of time-delayed dynamical networks. We show that for a time-delayed dynamical network with non-distributed delays the network and the corresponding non-delayed network are both either globally stable or unstable. We demonstrate that this may not be the case if the network's delays are distributed. The main tool in our analysis is a new procedure of dynamical network restrictions. This procedure is useful in that it allows for improved estimates of a dynamical network's global stability. Moreover, it is a computationally simpler and much more effective means of analyzing the stability of dynamical networks than the procedure of isospectral network expansions introduced in [Isospectral graph transformations, spectral equivalence, and global stability of dynamical networks. Nonlinearity, 25 (2012) 211-254]. The effectiveness of our approach is illustrated by applications to various classes of Cohen-Grossberg neural networks.Comment: 32 pages, 9 figure

The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality

We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity. In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria. More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.Air Force Office of Scientific Research (90-0128

Synchronized Oscillations During Cooperative Feature Linking in a Cortical Model of Visual Perception

A neural network model of synchronized oscillator activity in visual cortex is presented in order to account for recent neurophysiological findings that such synchronization may reflect global properties of the stimulus. In these recent experiments, it was reported that synchronization of oscillatory firing responses to moving bar stimuli occurred not only for nearby neurons, but also occurred between neurons separated by several cortical columns (several mm of cortex) when these neurons shared some receptive field preferences specific to the stimuli. These results were obtained not only for single bar stimuli but also across two disconnected, but colinear, bars moving in the same direction. Our model and computer simulations obtain these synchrony results across both single and double bar stimuli. For the double bar case, synchronous oscillations are induced in the region between the bars, but no oscillations are induced in the regions beyond the stimuli. These results were achieved with cellular units that exhibit limit cycle oscillations for a robust range of input values, but which approach an equilibrium state when undriven. Single and double bar synchronization of these oscillators was achieved by different, but formally related, models of preattentive visual boundary segmentation and attentive visual object recognition, as well as nearest-neighbor and randomly coupled models. In preattentive visual segmentation, synchronous oscillations may reflect the binding of local feature detectors into a globally coherent grouping. In object recognition, synchronous oscillations may occur during an attentive resonant state that triggers new learning. These modelling results support earlier theoretical predictions of synchronous visual cortical oscillations and demonstrate the robustness of the mechanisms capable of generating synchrony.Air Force Office of Scientific Research (90-0175); Army Research Office (DAAL-03-88-K0088); Defense Advanced Research Projects Agency (90-0083); National Aeronautics and Space Administration (NGT-50497

Letter to the Editor: Physiological Interpretation of the Self-Organizing Map Algorithm

Air Force Office of Scientific Research (F49620-92-J-0499); Office of Naval Research (N00014-92-J-4015, N00014-91-J-4100

Brain Learning, Attention, and Consciousness

The processes whereby our brains continue to learn about a changing world in a stable fashion throughout life are proposed to lead to conscious experiences. These processes include the learning of top-down expectations, the matching of these expectations against bottom-up data, the focusing of attention upon the expected clusters of information, and the development of resonant states between bottom-up and top-down processes as they reach an attentive consensus between what is expected and what is there in the outside world. It is suggested that all conscious states in the brain are resonant states, and that these resonant states trigger learning of sensory and cognitive representations. The model which summarize these concepts are therefore called Adaptive Resonance Theory, or ART, models. Psychophysical and neurobiological data in support of ART are presented from early vision, visual object recognition, auditory streaming, variable-rate speech perception, somatosensory perception, and cognitive-emotional interactions, among others. It is noted that ART mechanisms seem to be operative at all levels of the visual system, and it is proposed how these mechanisms are realized by known laminar circuits of visual cortex. It is predicted that the same circuit realization of ART mechanisms will be found in the laminar circuits of all sensory and cognitive neocortex. Concepts and data are summarized concerning how some visual percepts may be visibly, or modally, perceived, whereas amoral percepts may be consciously recognized even though they are perceptually invisible. It is also suggested that sensory and cognitive processing in the What processing stream of the brain obey top-down matching and learning laws that arc often complementary to those used for spatial and motor processing in the brain's Where processing stream. This enables our sensory and cognitive representations to maintain their stability a.s we learn more about the world, while allowing spatial and motor representations to forget learned maps and gains that are no longer appropriate as our bodies develop and grow from infanthood to adulthood. Procedural memories are proposed to be unconscious because the inhibitory matching process that supports these spatial and motor processes cannot lead to resonance.Defense Advance Research Projects Agency; Office of Naval Research (N00014-95-1-0409, N00014-95-1-0657); National Science Foundation (IRI-97-20333

Birth of a Learning Law

Defense Advanced Research Projects Agency; Office of Naval Research (N00014-95-1-0409, N00014-95-1-0657, N00014-92-J-1309

State estimation for delayed neural networks

Copyright [2005] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this letter, the state estimation problem is studied for neural networks with time-varying delays. The interconnection matrix and the activation functions are assumed to be norm-bounded. The problem addressed is to estimate the neuron states, through available output measurements, such that for all admissible time-delays, the dynamics of the estimation error is globally exponentially stable. An effective linear matrix inequality approach is developed to solve the neuron state estimation problem. In particular, we derive the conditions for the existence of the desired estimators for the delayed neural networks. We also parameterize the explicit expression of the set of desired estimators in terms of linear matrix inequalities (LMIs). Finally, it is shown that the main results can be easily extended to cope with the traditional stability analysis problem for delayed neural networks. Numerical examples are included to illustrate the applicability of the proposed design method

Neural Control of Interlimb Oscillations I: Human Bimanual Coordination

How do humans and other animals accomplish coordinated movements? How are novel combinations of limb joints rapidly assembled into new behavioral units that rnove together in in-phase or anti-phase movement patterns during complex movement tasks? A neural central pattern generator (CPG) model simulates data from human bimanual coordination tasks. As in the data, anti-phase oscillations at low frequencies switch to in-phase oscillations at high frequencies, in-phase oscillation occur both at low and high frequencies, phase fluctuations occur at the anti-phase in-phase transition, a "seagull effect" of larger errors occurs at intermediate phases, and oscillations slip toward in-phase and anti-phase when driven at intermediate phases. These oscillations and bifurcations are emergent properties of the CPG model in response to volitional inputs. The CPC model is a version of the Ellias-Grossberg oscillator. Its neurons obey Hodgkin-Huxley type equations whose excitatory signals operate on a faster time scale than their inhibitory signals in a recurrent on-center off-surround anatomy. When an equal cornmand or GO signal activates both model channels the model CPC: can generate both in-phase and anti-phase oscillations at different GO amplitudes. Phase transitions frorn either in-phase to anti-phase oscillations, or from anti-phase to in- phase oscillations, can occur in different pararncter ranges, as the GO signal increases.Air Force Office of Scientific Research (F49620-92-J-0499, 90-0083, F49620-92-J-0225, 90-0128); Office of Naval Research (N00014-92-J-1309); Army Research Office (DAAL03-0088); National Science Foundation (IRI-90-24877

Neural Control of Interlimb Oscillations I: Human Bimanual Coordination

How do humans and other animals accomplish coordinated movements? How are novel combinations of limb joints rapidly assembled into new behavioral units that rnove together in in-phase or anti-phase movement patterns during complex movement tasks? A neural central pattern generator (CPG) model simulates data from human bimanual coordination tasks. As in the data, anti-phase oscillations at low frequencies switch to in-phase oscillations at high frequencies, in-phase oscillation occur both at low and high frequencies, phase fluctuations occur at the anti-phase in-phase transition, a "seagull effect" of larger errors occurs at intermediate phases, and oscillations slip toward in-phase and anti-phase when driven at intermediate phases. These oscillations and bifurcations are emergent properties of the CPG model in response to volitional inputs. The CPC model is a version of the Ellias-Grossberg oscillator. Its neurons obey Hodgkin-Huxley type equations whose excitatory signals operate on a faster time scale than their inhibitory signals in a recurrent on-center off-surround anatomy. When an equal cornmand or GO signal activates both model channels the model CPC: can generate both in-phase and anti-phase oscillations at different GO amplitudes. Phase transitions frorn either in-phase to anti-phase oscillations, or from anti-phase to in- phase oscillations, can occur in different pararncter ranges, as the GO signal increases.Air Force Office of Scientific Research (F49620-92-J-0499, 90-0083, F49620-92-J-0225, 90-0128); Office of Naval Research (N00014-92-J-1309); Army Research Office (DAAL03-0088); National Science Foundation (IRI-90-24877