A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

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

Delay-Induced Transient Oscillations in a Two-Neuron Network

Finite transmission times between neurons, referred to as delays, may appear in hardware implementation of neural networks. We analyze the dynamics of a two-neuron network in which the delay modifies the transient and not the long-term behavior of the network. We show that the delay causes some trajectories to oscillate transiently before reaching stationary behavior and the duration of these transients increases exponentially with the delay. Such a phenomeno deteriorates network performance

Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks

Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the typical length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry \e_c=0.33, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a Random Map with reversal symmetry. The time-scale after which cycles are reached grows exponentially with system size $N$, and the exponent vanishes in the symmetric limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$, where $h_i$ is the local field experienced by the neuron $i$. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger is the asymmetry, the faster is the relaxation of $E$, and the higher is the asymptotic value reached. $E$ reachs very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge

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

Improving Associative Memory in a Network of Spiking Neurons

In this thesis we use computational neural network models to examine the dynamics and functionality of the CA3 region of the mammalian hippocampus. The emphasis of the project is to investigate how the dynamic control structures provided by inhibitory circuitry and cellular modification may effect the CA3 region during the recall of previously stored information. The CA3 region is commonly thought to work as a recurrent auto-associative neural network due to the neurophysiological characteristics found, such as, recurrent collaterals, strong and sparse synapses from external inputs and plasticity between coactive cells. Associative memory models have been developed using various configurations of mathematical artificial neural networks which were first developed over 40 years ago. Within these models we can store information via changes in the strength of connections between simplified model neurons (two-state). These memories can be recalled when a cue (noisy or partial) is instantiated upon the net. The type of information they can store is quite limited due to restrictions caused by the simplicity of the hard-limiting nodes which are commonly associated with a binary activation threshold. We build a much more biologically plausible model with complex spiking cell models and with realistic synaptic properties between cells. This model is based upon some of the many details we now know of the neuronal circuitry of the CA3 region. We implemented the model in computer software using Neuron and Matlab and tested it by running simulations of storage and recall in the network. By building this model we gain new insights into how different types of neurons, and the complex circuits they form, actually work. The mammalian brain consists of complex resistive-capacative electrical circuitry which is formed by the interconnection of large numbers of neurons. A principal cell type is the pyramidal cell within the cortex, which is the main information processor in our neural networks. Pyramidal cells are surrounded by diverse populations of interneurons which have proportionally smaller numbers compared to the pyramidal cells and these form connections with pyramidal cells and other inhibitory cells. By building detailed computational models of recurrent neural circuitry we explore how these microcircuits of interneurons control the flow of information through pyramidal cells and regulate the efficacy of the network. We also explore the effect of cellular modification due to neuronal activity and the effect of incorporating spatially dependent connectivity on the network during recall of previously stored information. In particular we implement a spiking neural network proposed by Sommer and Wennekers (2001). We consider methods for improving associative memory recall using methods inspired by the work by Graham and Willshaw (1995) where they apply mathematical transforms to an artificial neural network to improve the recall quality within the network. The networks tested contain either 100 or 1000 pyramidal cells with 10% connectivity applied and a partial cue instantiated, and with a global pseudo-inhibition.We investigate three methods. Firstly, applying localised disynaptic inhibition which will proportionalise the excitatory post synaptic potentials and provide a fast acting reversal potential which should help to reduce the variability in signal propagation between cells and provide further inhibition to help synchronise the network activity. Secondly, implementing a persistent sodium channel to the cell body which will act to non-linearise the activation threshold where after a given membrane potential the amplitude of the excitatory postsynaptic potential (EPSP) is boosted to push cells which receive slightly more excitation (most likely high units) over the firing threshold. Finally, implementing spatial characteristics of the dendritic tree will allow a greater probability of a modified synapse existing after 10% random connectivity has been applied throughout the network. We apply spatial characteristics by scaling the conductance weights of excitatory synapses which simulate the loss in potential in synapses found in the outer dendritic regions due to increased resistance. To further increase the biological plausibility of the network we remove the pseudo-inhibition and apply realistic basket cell models with differing configurations for a global inhibitory circuit. The networks are configured with; 1 single basket cell providing feedback inhibition, 10% basket cells providing feedback inhibition where 10 pyramidal cells connect to each basket cell and finally, 100% basket cells providing feedback inhibition. These networks are compared and contrasted for efficacy on recall quality and the effect on the network behaviour. We have found promising results from applying biologically plausible recall strategies and network configurations which suggests the role of inhibition and cellular dynamics are pivotal in learning and memory