Calculation of the Volterra kernels of non-linear dynamic systems using an artificial neural network

The Volterra series is a well-known method of describing non-linear dynamic systems. A major limitation of this technique is the difficulty involved in the calculation of the kernels. More recently, artificial neural networks have been used to produce black box models of non-linear dynamic systems. In this paper we show how a certain class of artificial neural networks are equivalent to Volterra series and give the equation for the nth order Volterra kernel in terms of the internal parameters of the network. The technique is then illustrated using a specific non-linear system. The kernels obtained by the method described in the paper are compared with those obtained by a Toeplitz matrix inversion technique. Received: 4 June 1993/Accepted in revised form: 2 March 1994     

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.     

Estimation of Thalamocortical and Intracortical Network Models from Joint Thalamic Single-Electrode and Cortical Laminar-Electrode Recordings in the Rat Barrel System

A new method is presented for extraction of population firing-rate models for both thalamocortical and intracortical signal transfer based on stimulus-evoked data from simultaneous thalamic single-electrode and cortical recordings using linear (laminar) multielectrodes in the rat barrel system. Time-dependent population firing rates for granular (layer 4), supragranular (layer 2/3), and infragranular (layer 5) populations in a barrel column and the thalamic population in the homologous barreloid are extracted from the high-frequency portion (multi-unit activity; MUA) of the recorded extracellular signals. These extracted firing rates are in turn used to identify population firing-rate models formulated as integral equations with exponentially decaying coupling kernels, allowing for straightforward transformation to the more common firing-rate formulation in terms of differential equations. Optimal model structures and model parameters are identified by minimizing the deviation between model firing rates and the experimentally extracted population firing rates. For the thalamocortical transfer, the experimental data favor a model with fast feedforward excitation from thalamus to the layer-4 laminar population combined with a slower inhibitory process due to feedforward and/or recurrent connections and mixed linear-parabolic activation functions. The extracted firing rates of the various cortical laminar populations are found to exhibit strong temporal correlations for the present experimental paradigm, and simple feedforward population firing-rate models combined with linear or mixed linear-parabolic activation function are found to provide excellent fits to the data. The identified thalamocortical and intracortical network models are thus found to be qualitatively very different. While the thalamocortical circuit is optimally stimulated by rapid changes in the thalamic firing rate, the intracortical circuits are low-pass and respond most strongly to slowly varying inputs from the cortical layer-4 population.     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part I: Computational study

Parametric and non-parametric modeling methods are combined to study the short-term plasticity (STP) of synapses in the central nervous system (CNS). The nonlinear dynamics of STP are modeled by means: (1) previously proposed parametric models based on mechanistic hypotheses and/or specific dynamical processes, and (2) non-parametric models (in the form of Volterra kernels) that transforms the presynaptic signals into postsynaptic signals. In order to synergistically use the two approaches, we estimate the Volterra kernels of the parametric models of STP for four types of synapses using synthetic broadband input–output data. Results show that the non-parametric models accurately and efficiently replicate the input–output transformations of the parametric models. Volterra kernels provide a general and quantitative representation of the STP.     

Recolonisation by diffusion can generate increasing rates of spread

Diffusion is one of the most frequently used assumptions to explain dispersal. Diffusion models and in particular reaction-diffusion equations usually lead to solutions moving at constant speeds, too slow compared to observations. As early as 1899, Reid had found that the rate of spread of tree species migrating to northern environments at the beginning of the Holocene was too fast to be explained by diffusive dispersal. Rapid spreading is generally explained using long distance dispersal events, modelled through integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, i.e. decaying slower than any exponential. We show here that classical reaction-diffusion models of the Fisher-Kolmogorov-Petrovsky-Piskunov type can produce patterns of colonisation very similar to those of IDEs, if the initial population is EU at the beginning of the considered colonisation event. Many similarities between reaction-diffusion models with EU initial data and IDEs with EU kernels are found; in particular comparable accelerating rates of spread and flattening of the solutions. There was previously no systematic mathematical theory for such reaction-diffusion models with EU initial data. Yet, EU initial data can easily be understood as consequences of colonisation-retraction events and lead to fast spreading and accelerating rates of spread without the long distance hypothesis.     

Cycles in cannibalistic egg-larval interactions

A model of a cannibalistic larval-egg interaction such as occurs in Tribolium is developed which leads to a system of nonlinear Volterra integral equations. I determine the local stability properties of the unique equilibrium point of the model. A Hopf bifurcation analysis shows that the model always undergoes a subcritical bifurcation when stability is lost. Numerical solutions confirm the presence of multiple attractors over a range of parameter values. The form of the cycles observed in the numerical solutions is analogous to that observed in laboratory populations of Tribolium.     

Stability in a class of cyclic epidemic models with delay

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.     

Seed dispersal patterns in a temperate forest during a mast event: performance of alternative dispersal kernels

Seed dispersal patterns were studied in a north-western Spain temperate forest community to assess the performances of alternative dispersal kernels during two years with ecologically contrasting scenarios; a non-mast year, and a mast year of the dominant canopy species, beech Fagus sylvatica. Dispersal kernels were fitted under a Bayesian modeling framework. Both simple and mixture kernels were considered for the five more abundant tree species (Corylus avellana, Crataegus monogyna, F. sylvatica, Ilex aquifolium and Taxus baccata). Mixture kernels provided a better fit for almost all species, and the log-normal performed best for T. baccata. No relationship between dispersal syndromes and the best dispersal kernel function emerged. However, we found temporal changes in the shape of the dispersal kernels that seemed to be related to variation in relative fruit production among species and the resulting changes in the responses of dispersal vectors. This reveals a potential role for disperser-mediated indirect effects in terms of introducing temporal variation in species spread. In this sense, our results highlight the need to consider single species seed dispersal as a community process. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A biosensor for detecting changes in cognitive processing based on nonlinear systems analysis.

A new type of biosensor, based on hippocampal slices cultured on multielectrode arrays, and using nonlinear systems analysis for the detection and classification of agents interfering with cognitive function is described. A new method for calculating first and second order kernel was applied for impulse input-spike output datasets and results are presented to show the reliability of the estimations of this parameter. We further decomposed second order kernels as a sum of nine exponentially decaying Laguerre base functions. The data indicate that the method also reliably estimates these nine parameters. Thus, the state of the system can now be described with a set of ten parameters (first order kernel plus nine coefficients of Laguerre base functions) that can be used for detection and classification purposes.     

Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion

Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv ₁ of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv ₁. Presented at the 1974 SIAM Fall Meeting.     

Integro-differential equations and the stability of neural networks with dendritic structure

 We analyse the effects of dendritic structure on the stability of a recurrent neural network in terms of a set of coupled, non-linear Volterra integro-differential equations. These, which describe the dynamics of the somatic membrane potentials, are obtained by eliminating the dendritic potentials from the underlying compartmental model or cable equations. We then derive conditions for Turing-like instability as a precursor for pattern formation in a spatially organized network. These conditions depend on the spatial distribution of axo-dendritic connections across the network. Received: 2 November 1994/Accepted in revised form: 7 March 1995     

Homology modeling and dynamics of the extracellular domain of rat and human neuronal nicotinic acetylcholine receptor subtypes α4β2 and α7

In recent years, it has become clear that the neuronal nicotinic acetylcholine receptor (nAChR) is a valid target in the treatment of a variety of diseases, including Alzheimer’s disease, anxiety, and nicotine addiction. As with most membrane proteins, information on the three-dimensional (3D) structure of nAChR is limited to data from electron microscopy, at a resolution that makes the application of structure-based design approaches to develop specific ligands difficult. Based on a high-resolution crystal structure of AChBP, homology models of the extracellular domain of the neuronal rat and human nAChR subtypes α4β2 and α7 (the subtypes most abundant in brain) were built, and their stability assessed with molecular dynamics (MD). All models built showed conformational stability over time, confirming the quality of the starting 3D model. Lipophilicity and electrostatic potential studies performed on the rat and human α4β2 and α7 nicotinic models were compared to AChBP, revealing the importance of the hydrophobic aromatic pocket and the critical role of the α-subunit Trp—the homolog of AChBP-Trp 143—for ligand binding. The models presented provide a valuable framework for the structure-based design of specific α4β2 nAChR subtype ligands aimed at improving therapeutic and diagnostic applications. Figure Electrostatic surface potential of the binding site cavity of the neuronal nicotinic acetylcholine receptor (nAChR). Nicotinic models performed with the MOLCAD program: a rat α7, b rat α4β2, c human α7, d human α4β2. All residues labeled are part of the α7 (a,c) or α4 (b,d) subunit with the exception of Phe 117, which belongs to subunit β2 (d). Violet Very negative, blue negative, yellow neutral, red very positive     

Properties of normal and mutant polypeptide fragments from the dimer self-association sites of human red cell spectrin

We have examined the properties and interactions of expressed polypeptide fragments from the N-terminus of the α-chain and the C-terminus of the β-chain of human erythroid spectrin. Each polypeptide comprises one complete structural repeating unit, together with the incomplete repeat that interacts with its partner when spectrin tetramers are formed. The shared repeat thus generated is made up of two helices from the C-terminal part of the β-chain and one helix from the N-terminus of the α-chain. Three mutant β-chain fragments with amino acid substitutions in the incomplete terminal repeat were also studied. The α- and β-chain fragments were both substantially monomeric, as shown by sedimentation equilibrium. Circular dichroism analysis and thermal denaturation profiles revealed that the complete repeat present in each fragment had entered the stable tertiary fold. Unexpectedly, the conformational stability of the folded β-chain repeat was found to be grossly perturbed by the mutations, all of them well beyond its C-terminal boundary; possible explanations for this phemomenon are considered. Sedimentation equilibrium showed that in equimolar mixtures the wild-type α- and β-chain peptides formed a 1:1 complex. Mixing curves, observed by circular dichroism, revealed that association was accompanied by an increase in α-helicity. From continuous-variation profiles an association constant in the range 1–2×10⁶ M^–1 was inferred. The association was unaffected by the apparently unstructured anionic tail of 54 residues, found at the C-terminus of the spectrin β-chain. Of the three mutations in the β-chain fragment, one (an Ala→Val replacement in the A helix segment of the incomplete repeat) had a relatively small effect on the association with the α-chain fragment, whereas Trp→Arg mutations in the A and in the remote B helix segments were much more deleterious. These observations are consistent with the relative severities of the haemolytic conditions associated with the mutations. Received: 10 August 1998 / Revised version: 13 October 1998 / Accepted: 13 October 1998     

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush–Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.     

Stability and pattern formation for competing populations with asymmetric nonlocal coupling

We consider a model of two competing species with asymmetric nonlocal coupling in a competition for resources. The nonlocal coupling is via convolution integrals and the asymmetry is via convolution kernel functions which are not even functions of their arguments. The nonlocality is due to species mobility, so that at any fixed point in space the competition for resources depends not just on the populations at that point but on a suitably weighted average of the populations. We introduce two parameters, δ  , describing the extent of the coupling, with δ=0$\delta =0$ corresponding to local coupling, and α  , describing the extent of the asymmetry, with α=0$\alpha =0$ corresponding to symmetric nonlocal interactions. We consider the case where the model admits a stable coexistence equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We consider two specific kernel functions, (i) an asymmetric Gaussian and (ii) an asymmetric stepfunction. We compute the stability boundary as a function of α, and for δ beyond the stability boundary we determine unstable wavenumber bands. We compute nonlinear patterns for δ significantly beyond the stability boundary. Patterns consist of arrays of islands, regions of nonzero population, separated by either near-deadzones where the populations are small, but nonzero, or by deadzones where populations are exponentially small and essentially extinct. We find solutions consisting of propagating traveling waves of islands, solutions exhibiting colony formation, where a colony is formed just ahead of an island and eventually grows as the parent island decays, and modulated traveling waves, where competition between the two species allows propagation and inhibits colony formation. We explain colony formation and the modulated traveling waves as due to a positive feedback mechanism associated with small variations in the amplitude of the parent island.     

Optimal modeling of an arterial system

The aortic blood flow is described by a set of nonlinear hyperbolic partial differential equations that account for mass and momentum conservation, and nonlinear models for the mechanical properties of the artery. Identification is used for determining the wave speed, arterial taper, and cross section: these parameters reflect the elastic characteristics of the aorta wall and control the pulsatile response. The differential equations were numerically integrated by the Lax-Wendroff scheme of Abarbanel and Goldberg [J. Comput. Phys. 10:1–21 (1972)] that avoids nonlinear oscillations. The Gauss-Newton technique was used for the parameter identification. By reference to reported elocity and pressure input-output pairs, a parameter vector is found such that the distance in the L₂ norm between the predicted outputs and the measured functions is minimal. Calculations of the velocity and pressure waves show excellent compatibility of the model with reported experimental data: starting from arbitrary parameter estimates, which yield grossly distorted waveforms, the error is typically reduced to 7–8%. Introduction of viscoelastic behaviour for the arterial wall in the form of a Volterra integral for the cross section does not lead to significant improvement. Numerical examples are presented which prove the convergence, accuracy, and stability of the algorithm. Emphasis is placed on the computational feasibility of the proposed system identification.     

Analytic steady-state space use patterns and rapid computations in mechanistic home range analysis

Mechanistic home range models are important tools in modeling animal dynamics in spatially complex environments. We introduce a class of stochastic models for animal movement in a habitat of varying preference. Such models interpolate between spatially implicit resource selection analysis (RSA) and advection-diffusion models, possessing these two models as limiting cases. We find a closed-form solution for the steady-state (equilibrium) probability distribution u* using a factorization of the redistribution operator into symmetric and diagonal parts. How space use is controlled by the habitat preference function w depends on the characteristic width of the animals’ redistribution kernel: when the redistribution kernel is wide relative to variation in w, u* ∝ w, whereas when it is narrow relative to variation in w, u* ∝ w ². In addition, we analyze the behavior at discontinuities in w which occur at habitat type boundaries, and simulate the dynamics of space use given two-dimensional prey-availability data, exploring the effect of the redistribution kernel width. Our factorization allows such numerical simulations to be done extremely fast; we expect this to aid the computationally intensive task of model parameter fitting and inverse modeling.      

Global stability of single-species diffusion volterra models with continuous time delays

In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature. This work was performed under the auspices of G.N.F.M., C.N.R. (Italy) and within the activity of the Evolution Equations and Applications group, M.P.I. (Italy). I thank the Department of Applied Mathematics, Shizuoka University, Japan, which enabled me to visit Urbino.