A subthreshold MOS circuit for the Lotka-Volterra neural network producing the winners-share-all solution

Emergent properties of interacting populations of spiking neurons

Dynamic neuronal networks are a key paradigm of increasing importance in brain research, concerned with the functional analysis of biological neuronal networks and, at the same time, with the synthesis of artificial brain-like systems. In this context, neuronal network models serve as mathematical tools to understand the function of brains, but they might as well develop into future tools for enhancing certain functions of our nervous system. Here, we present and discuss our recent achievements in developing multiplicative point processes into a viable mathematical framework for spiking network modeling. The perspective is that the dynamic behavior of these neuronal networks is faithfully reflected by a set of non-linear rate equations, describing all interactions on the population level. These equations are similar in structure to Lotka-Volterra equations, well known by their use in modeling predator-prey relations in population biology, but abundant applications to economic theory have also been described. We present a number of biologically relevant examples for spiking network function, which can be studied with the help of the aforementioned correspondence between spike trains and specific systems of non-linear coupled ordinary differential equations. We claim that, enabled by the use of multiplicative point processes, we can make essential contributions to a more thorough understanding of the dynamical properties of interacting neuronal populations.

New accurate and flexible design procedure for a stable KWTA continuous time network

The classical continuous time recurrent (Hopfield) network is considered and adapted to K -winner-take-all operation. The neurons are of sigmoidal type with a controllable gain G, an amplitude m and interconnected by the conductance p. The network is intended to process one by one a sequence of lists, each of them with N distinct elements, each of them squeezed to [0,I] admission interval, each of them having an imposed minimum separation between elements z(min). The network carries out: 1) a matching dynamic process between the order of list elements and the order of outputs, and 2) a binary type steady-state separation between K and K+1 outputs, the former surpassing a +ξ threshold and the later falling under the -ξ threshold. As a result, the machine will signal the ranks of the K largest elements of the list. To achieve 1), the initial condition of processing phase has to be placed in a computable θ -vicinity of zero-state. This requires a resetting procedure after each list. To achieve 2) the bias current M has to be within a certain interval computable from circuit parameters. In addition, the steady-state should be asymptotically stable. To these goals, we work with high gain and exploit the sigmoid properties and network symmetry. The various inequality type constraints between parameters are shown to be compatible and a neat synthesis procedure, simple and flexible, is given for the tanh sigmoid. It starts with the given parameters N, K, I, z(min), m and computes simple bounds of p, G, ξ, θ, and M. Numerical tests and comments reveal qualities and shortcomings of the method.

Continuous attractors of Lotka-Volterra recurrent neural networks with infinite neurons

Continuous attractors of Lotka-Volterra recurrent neural networks (LV RNNs) with infinite neurons are studied in this brief. A continuous attractor is a collection of connected equilibria, and it has been recognized as a suitable model for describing the encoding of continuous stimuli in neural networks. The existence of the continuous attractors depends on many factors such as the connectivity and the external inputs of the network. A continuous attractor can be stable or unstable. It is shown in this brief that a LV RNN can possess multiple continuous attractors if the synaptic connections and the external inputs are Gussian-like in shape. Moreover, both stable and unstable continuous attractors can coexist in a network. Explicit expressions of the continuous attractors are calculated. Simulations are employed to illustrate the theory.

Analog integrated circuits for the Lotka-Volterra competitive neural networks

A subthreshold MOS integrated circuit (IC) is designed and fabricated for implementing a competitive neural network of the Lotka-Volterra (LV) type which is derived from conventional membrane dynamics of neurons and is used for the selection of external inputs. The steady-state solutions to the LV equation can be classified into three types, each of which represents qualitatively different selection behavior. Among the solutions, the winners-share-all (WSA) solution in which a certain number of neurons remain activated in steady states is particularly useful owing to robustness in the selection of inputs from a noisy environment. The measured results of the fabricated LV IC's agree well with the theoretical prediction as long as the influence of device mismatches is small. Furthermore, results of extensive circuit simulations prove that the large-scale LV circuit producing the WSA solution does exhibit a reliable selection compared with winner-take-all circuits, in the possible presence of device mismatches.

Global attraction and stability for Cohen–Grossberg neural networks with delays

We consider a class of Cohen-Grossberg neural networks with delays. We prove the existence and global asymptotic stability of an equilibrium point and estimate the region of existence. Furthermore, we show that the trajectories of the neural networks with positive initial data will stay in the positive region if the amplification function satisfies a divergent condition. We also establish the existence of a globally attracting compact set for more general networks. We estimate this compact set explicitly in terms of the network parameters from physiological and biological models. Our results can be applied to neural networks with a wide range of activation functions which are neither bounded nor globally Lipschitz continuous such as the Lotka-Volterra model. We also give some examples and simulations.

Dynamic stability conditions for Lotka-Volterra recurrent neural networks with delays

The Lotka-Volterra model of neural networks, derived from the membrane dynamics of competing neurons, have found successful applications in many "winner-take-all" types of problems. This paper studies the dynamic stability properties of general Lotka-Volterra recurrent neural networks with delays. Conditions for nondivergence of the neural networks are derived. These conditions are based on local inhibition of networks, thereby allowing these networks to possess a multistability property. Multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process. Under these nondivergence conditions, a compact set that globally attracts all the trajectories of a network can be computed explicitly. If the connection weight matrix of a network is symmetric in some sense, and the delays of the network are in L2 space, we can prove that the network will have the property of complete stability.

On the stability analysis of delayed neural networks systems

In this paper, the problems of stability of delayed neural networks are investigated, including the stability of discrete and distributed delayed neural networks. Under the generalization of dropping the Lipschitzian hypotheses for output functions, some stability criteria are obtained by using the Liapunov functional method. We do not assume the symmetry of the connection matrix and we establish that the system admits a unique equilibrium point in which the output functions do not satisfy the Lipschitz conditions and do not require them to be differential or strictly monotonously increasing. These criteria can be used to analyze the dynamics of biological neural systems or to design globally stable artificial neural networks.