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PubMed2001

Epidemic spread and bifurcation effects in two-dimensional network models with viral dynamics.

We extend a previous network model of viral dynamics to include host populations distributed in two space dimensions. The basic dynamical equations for the individual viral and immune effector densities within a host are bilinear with a natural threshold condition. In the general model, transmission between individuals is governed by three factors: a saturating function g( small middle dot) describing emission as a function of originating host virion level; a four-dimensional array B that determines transmission from each individual to every other individual; and a nonlinear function F, which describes the absorption of virions by a host for a given net arrival rate. A summary of the properties of the viral-effector dynamical system in a single host is given. In the numerical network studies, individuals are placed at the mesh points of a uniform rectangular grid and are connected with an m(2)xn(2) four-dimensional array with terms that decay exponentially with distance between hosts; g is linear and F has a simple step threshold. In a population of N=mn individuals, N0 are chosen randomly to be initially infected with the virus. We examine the dependence of maximal population viral load on the population dynamical parameters and find threshold effects that can be related to a transcritical bifurcation in the system of equations for individual virus and host effector populations. The effects of varying demographic parameters are also examined. Changes in alpha, which is related to mobility, and contact rate beta also show threshold effects. We also vary the density of (randomly chosen) initially infected individuals. The distribution of final size of the epidemic depends strongly on N0 but is invariably bimodal with mass concentrated mainly near either or both ends of the interval [1,N]. Thus large outbreaks may occur, with small probability, even with only very few initially infected hosts. The effects of immunization of various fractions of the population on the final size of the epidemic are also explored. The distribution of the final percentage infected is estimated by simulation. The mean of this quantity is obtained as a function of immunization rate and shows an almost linear decline for immunization rates up to about 0.2. When the immunization rate is increased past 0.2, the extra benefit accrues more slowly. We include a discussion of some approximations that illuminate threshold effects in demographic parameters and indicate how a mean-field approximation and more detailed studies of various geometries and rates of immunization could be a useful direction for future analysis.

Algorithms↗
PubMed2001

A dynamical model for the analysis and acceleration of learning in feedforward networks.

A dynamical system model is derived for feedforward neural networks with one layer of hidden nodes. The model is valid in the vicinity of flat minima of the cost function that rise due to the formation of clusters of redundant hidden nodes with nearly identical outputs. The derivation is carried out for networks with an arbitrary number of hidden and output nodes and is, therefore, a generalization of previous work valid for networks with only two hidden nodes and one output node. The Jacobian matrix of the system is obtained, whose eigenvalues characterize the evolution of learning. Flat minima correspond to critical points of the phase plane trajectories and the bifurcation of the eigenvalues signifies their abandonment. Following the derivation of the dynamical model, we show that identification of the hidden nodes clusters using unsupervised learning techniques enables the application of a constrained application (Dynamically Constrained Back Propagation-DCBP) whose purpose is to facilitate prompt bifurcation of the eigenvalues of the Jacobian matrix and, thus, accelerate learning. DCBP is applied to standard benchmark tasks either autonomously or as an aid to other standard learning algorithms in the vicinity of flat minima. Its application leads to significant reduction in the number of required epochs for convergence.

Acceleration↗
PubMed2001

From topology to dynamics in biochemical networks.

Abstract formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements N--> infinity. We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scale-free networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scale-free biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells. (c) 2001 American Institute of Physics.

Journal Article↗
PubMed2006

A neural network approach to dynamic task assignment of multirobots.

In this paper, a neural network approach to task assignment, based on a self-organizing map (SOM), is proposed for a multirobot system in dynamic environments subject to uncertainties. It is capable of dynamically controlling a group of mobile robots to achieve multiple tasks at different locations, so that the desired number of robots will arrive at every target location from arbitrary initial locations. In the proposed approach, the robot motion planning is integrated with the task assignment, thus the robots start to move once the overall task is given. The robot navigation can be dynamically adjusted to guarantee that each target location has the desired number of robots, even under uncertainties such as when some robots break down. The proposed approach is capable of dealing with changing environments. The effectiveness and efficiency of the proposed approach are demonstrated by simulation studies.

Algorithms↗
PubMed2004

Constraining the topology of neural networks to ensure dynamics with symmetry properties.

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron (MLP) network and the radial basis function (RBF) network. In global modeling problems it becomes crucial to impose conditions for symmetry in order to stand a chance of reproducing symmetry-related phenomena. Sufficient conditions are given for MLP and RBF networks to have a set of fixed points that are symmetrical with respect to the origin of the phase space. In the case of MLP networks such conditions reduce to the absence of bias parameters and the requirement of odd activation functions. This turns out to be important from a dynamical point of view since some phenomena are only observed in the context of symmetry, which is not a structurally stable property. The results are illustrated using bench systems that display symmetry, such as the Duffing-Ueda oscillator and the Lorenz system.

Algorithms↗
PubMed2006

Emergence of coherence in complex networks of heterogeneous dynamical systems.

We present a general theory for the onset of coherence in collections of heterogeneous maps interacting via a complex connection network. Our method allows the dynamics of the individual uncoupled systems to be either chaotic or periodic, and applies generally to networks for which the number of connections per node is large. We find that the critical coupling strength at which a transition to synchrony takes place depends separately on the dynamics of the individual uncoupled systems and on the largest eigenvalue of the adjacency matrix of the coupling network. Our theory directly generalizes the Kuramoto model of equal strength all-to-all coupled phase oscillators to the case of oscillators with more realistic dynamics coupled via a large heterogeneous network.

Computer Communication Networks↗
PubMed2002

Coexistence of regular and irregular dynamics in complex networks of pulse-coupled oscillators.

For general networks of pulse-coupled oscillators, including regular, random, and more complex networks, we develop an exact stability analysis of synchronous states. As opposed to conventional stability analysis, here stability is determined by a multitude of linear operators. We treat this multioperator problem exactly and show that for inhibitory interactions the synchronous state is stable, independent of the parameters and the network connectivity. In randomly connected networks with strong interactions this synchronous state, displaying regular dynamics, coexists with a balanced state exhibiting irregular dynamics. External signals may switch the network between qualitatively distinct states.

Biological Clocks↗
PubMed2004

The next step in systems biology: simulating the temporospatial dynamics of molecular network.

As a result of the time- and context-dependency of gene expression, gene regulatory and signaling pathways undergo dynamic changes during development. Creating a model of the dynamics of molecular interaction networks offers enormous potential for understanding how a genome orchestrates the developmental processes of an organism. The dynamic nature of pathway topology calls for new modeling strategies that can capture transient molecular links at the runtime. The aim of this paper is to present a brief and informative, but not all-inclusive, viewpoint on the computational aspects of modeling and simulation of a non-static molecular network.

Algorithms↗
PubMed2000

Multi-step-ahead prediction using dynamic recurrent neural networks.

A method for the development of empirical predictive models for complex processes is presented. The models are capable of performing accurate multi-step-ahead (MS) predictions, while maintaining acceptable single-step-ahead (SS) prediction accuracy. Such predictors find applications in model predictive controllers and in fault diagnosis systems. The proposed method makes use of dynamic recurrent neural networks in the form of a nonlinear infinite impulse response (IIR) filter. A learning algorithm is presented, which is based on a dynamic gradient descent approach. The effectiveness of the method for accurate MS prediction is tested on an artificial problem and on a complex, open-loop unstable process. Comparative results are presented with polynomial Nonlinear AutoRegressive with eXogeneous (NARX) predictors, and with recurrent networks trained using teacher forcing. Validation studies indicate that excellent generalization is obtained for the range of operational dynamics studied. The research demonstrates that the proposed network architecture and the associated learning algorithm are quite effective in modeling the dynamics of complex processes and performing accurate MS predictions.

Algorithms↗
PubMed2002

Spatiotemporal forward solution of the EEG and MEG using network modeling.

Dynamic systems have proven to be well suited to describe a broad spectrum of human coordination behavior such synchronization with auditory stimuli. Simultaneous measurements of the spatiotemporal dynamics of electroencephalographic (EEG) and magnetoencephalographic (MEG) data reveals that the dynamics of the brain signals is highly ordered and also accessible by dynamic systems theory. However, models of EEG and MEG dynamics have typically been formulated only in terms of phenomenological modeling such as fixed-current dipoles or spatial EEG and MEG patterns. In this paper, it is our goal to connect three levels of organization, that is the level of coordination behavior, the level of patterns observed in the EEG and MEG and the level of neuronal network dynamics. To do so, we develop a methodological framework, which defines the spatiotemporal dynamics of neural ensembles, the neural field, on a sphere in three dimensions. Using magnetic resonance imaging we map the neural field dynamics from the sphere onto the folded cortical surface of a hemisphere. The neural field represents the current flow perpendicular to the cortex and, thus, allows for the calculation of the electric potentials on the surface of the skull and the magnetic fields outside the skull to be measured by EEG and MEG, respectively. For demonstration of the dynamics, we present the propagation of activation at a single cortical site resulting from a transient input. Finally, a mapping between finger movement profile and EEG/MEG patterns is obtained using Volterra integrals.

Algorithms↗
PubMed1998

Learning persistent dynamics with neural networks.

Persistence is one of the most common characteristics of real-world time series. In this work we investigate the process of learning persistent dynamics by neural networks. We show that for chaotic times series the network can get stuck for long training periods in a trivial minimum of the error function related to the long-term autocorrelation in the series. Remarkably, in these cases the transition to the trained phase is quite abrupt. For noisy dynamics the training process is smooth. We also consider the effectiveness of two of the most frequently used decorrelation methods in avoiding the problems related to persistence. Copyright 1997 Elsevier Science Ltd.

Journal Article↗
PubMed2003

A dynamic recurrent neural network for multiple muscles electromyographic mapping to elevation angles of the lower limb in human locomotion.

This paper describes the use of a dynamic recurrent neural network (DRNN) for simulating lower limb coordination in human locomotion. The method is based on mapping between the electromyographic signals (EMG) from six muscles and the elevation angles of the three main lower limb segments (thigh, shank and foot). The DRNN is a fully connected network of 35 hidden units taking into account the temporal relationships history between EMG and lower limb kinematics. Each EMG signal is sent to all 35 units, which converge to three outputs. Each output neurone provides the kinematics of one lower limb segment. The training is supervised, involving learning rule adaptations of synaptic weights and time constant of each unit. Kinematics of the locomotor movements were recorded and analysed using the opto-electronic ELITE system. Comparative analysis of the learning performance with different types of output (position, velocity and acceleration) showed that for common gait mapping velocity data should be used as output, as it is the best compromise between asymptotic error curve, rapid convergence and avoidance of bifurcation. Reproducibility of the identification process and biological plausibility were high, indicating that the DRNN may be used for understanding functional relationships between multiple EMG and locomotion. The DRNN might also be of benefit for prosthetic control.

Adult↗
PubMed1994

Waves and stimulus-modulated dynamics in an oscillating olfactory network.

The temporal dynamics of electrical activity in an olfactory organ, the procerebral lobe of the terrestrial mollusc Limax maximus, is studied. The lobe exhibits intrinsic oscillations in its field potential. Intracellular recordings show that the lobe contains two classes of neurons, both with activity phase-locked to the oscillation. Neurons in one class produce periodic bursts of spikes while those in the other class fire infrequently but receive strong, periodic inhibition whose onset coincides with the burst. The large-scale activity of these neurons is imaged in preparations stained with voltage-sensitive dyes. We observe waves of electrical activity that span the width of the lobe and travel its full length along a longitudinal axis. Simultaneous optical and intracellular recordings show that the form of the wave reflects the electrical activity of both classes of neurons. The application of natural odor stimuli causes the electrical activity along the lobe to transiently switch from the state with propagating waves to one with spatially uniform oscillations. The behavioral and computational relevance of this change in global timing is discussed.

Animals↗
PubMed1979

Structure and dynamics of neural network oscillators.

Techniques are given to represent oscillating neural networks by asynchronous logical switching networks, and to analyze the oscillating networks using a directed graph called a state transition diagram. Consideration is restricted to network oscillators containing no rhythm determining pacemaker neurons, and no neurons exhibiting self-limiting properties such as post-inhibitory rebound or accumulating refractoriness. In the state transition diagrams, stable oscillations are associated with a particular geometric configuration called a cyclic attractor (the heavy cycle in Fig. 2). We show that given the network connectivity it is possible to predict autonomous dynamic behaviour, as well as behaviour following hyperpolarizing or depolarizing inputs to neurons of the network. Conversely, given information about patterns of firing activity during cycles and transients of neural networks, the network connectivity can be predicted. The theoretical techniques can be used to generate a census of network structures capable of generating stable oscillations. Several representative network oscillators are discussed in the context of previous theoretical and experimental studies of the structure of neural network oscillators. Although the number of theoretically possile network oscillators capable of generating sustained oscillations is very large, the techniques which are given should be useful in the design of experiments capable of distinguishing between equally plausible hypotheses.

Action Potentials↗
PubMed2004

Neural network modelling of dynamic cerebral autoregulation: assessment and comparison with established methods.

A time lagged recurrent neural network (TLRN) was implemented to model the dynamic relationship between arterial blood pressure (ABP) and cerebral blood flow velocity (CBFV) and its performance was compared to classical linear model such as transfer function analysis, Aaslid's dynamic autoregulation model, and the Wiener-Laguerre moving average filter. A simple linear regression was also tested as a naive estimator. In 16 normal subjects, CBFV was continuously recorded with Doppler ultrasound and ABP with the Finapres device during six repeated thigh cuff manoeuvres. Using mean beat-to-beat values of ABP as input and CBFV as output, the performance of each method was assessed by the model's predicted velocity correlation coefficient and normalized mean square error (MSE). Cross-validation was performed using three thigh cuff manoeuvres for the training data set and the other three for the validation set. The four methods studied performed significantly better than the zero-order naive estimator. The TLRN performed better than transfer function analysis, but was not significantly different from the time-domain techniques, despite showing the minimum predictive MSE. CBFV step responses could be extracted from the TLRN showing the presence of non-linear behaviour both in terms of amplitude and directionality.

Adaptation, Physiological↗
PubMed2004

Scaling of directed dynamical small-world networks with random responses.

A dynamical model of small-world networks, with directed links which describe various correlations in social and natural phenomena, is presented. Random responses of sites to the input message are introduced to simulate real systems. The interplay of these ingredients results in the collective dynamical evolution of a spinlike variable S(t) of the whole network. The global average spreading length (s) and average spreading time (s) are found to scale as p(-alpha)ln(N with different exponents. Meanwhile, S(t) behaves in a duple scaling form for N>>N(*): S approximately f(p(-beta)q(gamma)t), where p and q are rewiring and external parameters, alpha, beta, and gamma are scaling exponents, and f(t) is a universal function. Possible applications of the model are discussed.

Disease Transmission, Infectious↗
PubMed2007

Thermodynamically feasible kinetic models of reaction networks.

The dynamics of biological reaction networks are strongly constrained by thermodynamics. An holistic understanding of their behavior and regulation requires mathematical models that observe these constraints. However, kinetic models may easily violate the constraints imposed by the principle of detailed balance, if no special care is taken. Detailed balance demands that in thermodynamic equilibrium all fluxes vanish. We introduce a thermodynamic-kinetic modeling (TKM) formalism that adapts the concepts of potentials and forces from irreversible thermodynamics to kinetic modeling. In the proposed formalism, the thermokinetic potential of a compound is proportional to its concentration. The proportionality factor is a compound-specific parameter called capacity. The thermokinetic force of a reaction is a function of the potentials. Every reaction has a resistance that is the ratio of thermokinetic force and reaction rate. For mass-action type kinetics, the resistances are constant. Since it relies on the thermodynamic concept of potentials and forces, the TKM formalism structurally observes detailed balance for all values of capacities and resistances. Thus, it provides an easy way to formulate physically feasible, kinetic models of biological reaction networks. The TKM formalism is useful for modeling large biological networks that are subject to many detailed balance relations.

Cell Physiological Phenomena↗
PubMed2000

Intrinsic dynamics in neuronal networks. I. Theory.

Many networks in the mammalian nervous system remain active in the absence of stimuli. This activity falls into two main patterns: steady firing at low rates and rhythmic bursting. How are these firing patterns generated? Specifically, how do dynamic interactions between excitatory and inhibitory neurons produce these firing patterns, and how do networks switch from one firing pattern to the other? We investigated these questions theoretically by examining the intrinsic dynamics of large networks of neurons. Using both a semianalytic model based on mean firing rate dynamics and simulations with large neuronal networks, we found that the dynamics, and thus the firing patterns, are controlled largely by one parameter, the fraction of endogenously active cells. When no endogenously active cells are present, networks are either silent or fire at a high rate; as the number of endogenously active cells increases, there is a transition to bursting; and, with a further increase, there is a second transition to steady firing at a low rate. A secondary role is played by network connectivity, which determines whether activity occurs at a constant mean firing rate or oscillates around that mean. These conclusions require only conventional assumptions: excitatory input to a neuron increases its firing rate, inhibitory input decreases it, and neurons exhibit spike-frequency adaptation. These conclusions also lead to two experimentally testable predictions: 1) isolated networks that fire at low rates must contain endogenously active cells and 2) a reduction in the fraction of endogenously active cells in such networks must lead to bursting.

Action Potentials↗