Equivalence of aggregated Markov models of ion-channel gating

One cannot always distinguish different Markov models of ion-channel kinetics solely on the basis of steady-state kinetic data. If two generator (or transition) matrices are related by a similarity transformation that does not combine states with different conductances, then the models described by these generator matrices have the same observable steady-state statistics. This result suggests a procedure for expressing the model in a unique form, and sometimes reducing the number of parameters in a model. I apply the similarity transformation procedure to a number of simple models. When a model specifies the dependence of the rates of transition on an experimentally variable parameter such as the concentration of a ligand or the membrane potential, the class of equivalent models may be further restricted, but a model is not always uniquely determined even under these conditions. Voltage-step experiments produce non-stationary data that can also be used to distinguish models.     

On holographic models of memory

By means of the Volterra series an extension of the association properties of holography is proposed. Also, the significance of non-linearities embedded within the formal structure of a holographic-like associative memory is pointed out.     

Complex discrete dynamics from simple continuous population models

Nonoverlapping generations have been classically modelled as difference equations in order to account for the discrete nature of reproductive events. However, other events such as resource consumption or mortality are continuous and take place in the within-generation time. We have realistically assumed a hybrid ODE bidimensional model of resources and consumers with discrete events for reproduction. Numerical and analytical approaches showed that the resulting dynamics resembles a Ricker map, including the doubling route to chaos. Stochastic simulations with a handling-time parameter for indirect competition of juveniles may affect the qualitative behaviour of the model.     

Orientation and dynamics of transmembrane peptides: the power of simple models

In this review we discuss recent insights obtained from well-characterized model systems into the factors that determine the orientation and tilt angles of transmembrane peptides in lipid bilayers. We will compare tilt angles of synthetic peptides with those of natural peptides and proteins, and we will discuss how tilt can be modulated by hydrophobic mismatch between the thickness of the bilayer and the length of the membrane spanning part of the peptide or protein. In particular, we will focus on results obtained on tryptophan-flanked model peptides (WALP peptides) as a case study to illustrate possible consequences of hydrophobic mismatch in molecular detail and to highlight the importance of peptide dynamics for the experimental determination of tilt angles. We will conclude with discussing some future prospects and challenges concerning the use of simple peptide/lipid model systems as a tool to understand membrane structure and function.     

On the dynamics of a simple biochemical control circuit

The quantitative dynamics of a biochemical control circuit that regulates enzyme or protein synthesis by end-product feedback is analyzed. We first study a simplified repressible system, which is known to exhibit either a steady state or an oscillatory solution. By showing the analogy of thisn-dimensional system with a time-delay equation for a single variable the mechanism of the self-sustained oscillations becomes transparent. In a more sophisticated system we will find as well either steady state or oscillatory solutions. We determine the role of the parameters with respect to stability and frequency. The most general case will be treated by means of the concept of Lyapunov exponents.     

Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

On epigenetic regulation of the process of formation of long-term memory

The review summarizes current concepts on the role of several covalent post-translational chromatin modifications during memory formation in vertebrates and invertebrates. It describes a sequence of intracellular events from activation of receptors and signal pathways up to changes in the functional state of genome.     

Complex ecological models with simple dynamics: From individuals to populations

The aim of this work is to study complex ecological models exhibiting simple dynamics. We consider large scale systems which can be decomposed into weakly coupled subsystems. Perturbation Theory is used in order to get a reduced set of differential equations governing slow time varying global variables. As examples, we study the influence of the individual behaviour of animals in competition and predator-prey models. The animals are assumed to do many activities all day long such as searching for food of different types. The degree of competition as well as the predation pressure are dependent upon these activities. Preys are more vulnerable when doing some activities during which they are very exposed to predators attacks rather than for others during which they are hidden. We study the effect of a change in the average individual behaviour of the animals on interspecific relationships. Computer simulations of the whole sets of equations are compared to simulations of the reduced sets of equations.     

Estimating gap lifetime and memory from a simple model of forest canopy dynamics

Gap formation and closure represent important disturbance events in forests, but the processes involved are still poorly understood. We use models, which we and others previously developed, to make long-term predictions of tropical forest gap dynamics based on Barro Colorado Island data. We first fit the models to the data by comparing their discrete Fourier transforms, and we propose a definition for the lifetime of a gap and predict a large-gap lifetime typically to be less than 50 years. We find that the gap lifetime diverges logarithmically for large-gap sizes. We examine the ‘memory’ of spatial gap patterns via spatiotemporal correlations and find a correlation time of about 160 years, suggesting that present gap patterns could have long-lasting effects on forest spatial patterns.     

Statistical evaluation of ion-channel gating models based on distributions of log-likelihood ratios

The distributions of log-likelihood ratios (DeltaLL) obtained from fitting ion-channel dwell-time distributions with nested pairs of gating models (Xi, full model; Xi(R), submodel) were studied both theoretically and using simulated data. When Xi is true, DeltaLL is asymptotically normally distributed with predictable mean and variance that increase linearly with data length (n). When Xi(R) is true and corresponds to a distinct point in full parameter space, DeltaLL is Gamma-distributed (2DeltaLL is chi-square). However, when data generated by an l-component multiexponential distribution are fitted by l+1 components, Xi(R) corresponds to an infinite set of points in parameter space. The distribution of DeltaLL is a mixture of two components, one identically zero, the other approximated by a Gamma-distribution. This empirical distribution of DeltaLL, assuming Xi(R), allows construction of a valid log-likelihood ratio test. The log-likelihood ratio test, the Akaike information criterion, and the Schwarz criterion all produce asymmetrical Type I and II errors and inefficiently recognize Xi, when true, from short datasets. A new decision strategy, which considers both the parameter estimates and DeltaLL, yields more symmetrical errors and a larger discrimination power for small n. These observations are explained by the distributions of DeltaLL when Xi or Xi(R) is true.     

Complex dynamics of survival and extinction in simple population models with harvesting

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.     

Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials

Human immunodeficiency virus (HIV) dynamics represent a complicated variant of the text-book case of non-linear dynamics: predator-prey interaction. The interaction can be described as naturally reproducing T-cells (prey) hunted and killed by virus (predator). Virus reproduce and increase in number as a consequence of successful predation; this is countered by the production of T-cells and the reaction of the immune system. Multi-drug anti-HIV therapy attempts to alter the natural dynamics of the predator-prey interaction by decreasing the reproductive capability of the virus and hence predation. These dynamics are further complicated by varying compliance to treatment and insurgence of resistance to treatment. When following the temporal progression of viral load in plasma during therapy one observes a short-term (1-12 weeks) decrease in viral load. In the long-term (more than 12 weeks from the beginning of therapy) the reduction in viral load is either sustained, or it is followed by a rebound, oscillations and a new (generally lower than at the beginning of therapy) viral load level. Biomathematicians have investigated these dynamics by means of simulations. However the estimation of the parameters associated with the dynamics from real data has been mostly limited to the case of simplified, in particular linearized, models. Linearized model can only describe the short-term changes of viral load during therapy and can only predict (apparent) suppression. In this paper we put forward relatively simple models to characterize long-term virus dynamics which can incorporate different factors associated with resurgence: (Fl) the intrinsic non-linear HIV-1 dynamics, (F2) drug exposure and in particular compliance to treatment, and (F3) insurgence of resistant HIV-1 strains. The main goal is to obtain models which are mathematically identifiable given only measurements of viral load, while retaining the most crucial features of HIV dynamics. For the purpose of illustration we demonstrate an application of the models using real AIDS clinical trial data involving patients treated with a combination of anti-retroviral agents using a model which incorporates compliance data.     

What do simple models reveal about the population dynamics of a cooperatively breeding species?

Research on cooperatively breeding species has shown that their population dynamics differ from those of conventional breeders. Populations of cooperators are structured into groups, and group‐level Allee effects are likely common. We assess the ability of phenomenological models, lacking explicit group structure, to describe population dynamics in cooperative meerkats Suricata suricatta, and we assess potential Allee effects at the population level. Using maximum likelihood model fitting and information theoretic model selection, applied to time series data from a wild meerkat population, we find simple models that incorporate rainfall and conventional density dependence to be the most parsimonious of the models considered. Detecting no population‐level Allee effect, we conclude that explicit consideration of population structure will be key to understanding the mechanisms behind population dynamics in cooperatively breeding species.     

On the stability of delay equation models of simple human stretch reflexes

Human stretch reflexes (SRs) are often too weak and ineffectual to provide adequate postural regulation or rhythmic movement boosting (e.g. in ankle pushoff at the end of stance phase in fast running). Recent improvements in methods of artificially enhancing skeletomotor responses, especially in therapeutic regimens, should not be widely employed until the clonus-resisting stability properties of SRs are better understood. We formulate an idealized linear servo model of a segmentally-mediated SR system which includes the often ignored electromechanical coupling delay. For typical closed-loop (delay/gain) ratios, the model is shown to be unstable for all values of loop gain when operating as a position servo, but maximally stable when operating as a velocity servo. We claim that the velocity servo or more of its nonlinear relatives is a better model for some well studied SRs than, e.g., Houk's stiff muscle hypothesis. We also present evidence that even feeble and quickly saturating monosynaptic postural servos are always unstable if operated as pure position regulators.     

Cell assembly dynamics in detailed and abstract attractor models of cortical associative memory

Summary During the last few decades we have seen a convergence among ideas and hypotheses regarding functional principles underlying human memory. Hebb’s now more than fifty years old conjecture concerning synaptic plasticity and cell assemblies, formalized mathematically as attractor neural networks, has remained among the most viable and productive theoretical frameworks. It suggests plausible explanations for Gestalt aspects of active memory like perceptual completion, reconstruction and rivalry. We review the biological plausibility of these theories and discuss some critical issues concerning their associative memory functionality in the light of simulation studies of models with palimpsest memory properties. The focus is on memory properties and dynamics of networks modularized in terms of cortical minicolumns and hypercolumns. Biophysical compartmental models demonstrate attractor dynamics that support cell assembly operations with fast convergence and low firing rates. Using a scaling model we obtain reasonable relative connection densities and amplitudes. An abstract attractor network model reproduces systems level psychological phenomena seen in human memory experiments as the Sternberg and von Restorff effects. We conclude that there is today considerable substance in Hebb’s theory of cell assemblies and its attractor network formulations, and that they have contributed to increasing our understanding of cortical associative memory function. The criticism raised with regard to biological and psychological plausibility as well as low storage capacity, slow retrieval etc has largely been disproved. Rather, this paradigm has gained further support from new experimental data as well as computational modeling.     

Mechanisms of long-term memory

Long-term potentiation (LTP) of the hippocampus provides an excellent model on which to build hypotheses for the laying down of memories in the cerebral cortex. After repetitive activation, the primary happening seems to be the increase in transmitter sensitivity brought about by the increased Ca2+ in the recipient neurons. There may be secondary presynaptic changes. The extreme duration of the LTP may require structural changes of the synaptic spines. The hippocampus plays an essential role in the laying down of cognitive memories, the pathway to the frontal lobe being via the MD thalamus. The thalamo-cortical fibres activate stellate cells whose axons make climbing fibre-like ramifications up the apical dendrites of the pyramidal cells. On the Marr hypothesis repetitive conjunction of synaptic activation by these climbing fibres with synaptic activation by horizontal fibres on the apical dendrites produces prolonged potentiation of the horizontal fibre synapses, which is the neural basis of memory. Presumably this is a consequence of the raised Ca2+ of the apical dendrites, acting as it does on the hippocampal LTP. It will be considered how this elemental unit for cerebral memory can be developed into the varieties of cerebral memories that have been located by study of the regional cerebral blood flow during their retrieval. These sites for memory are in the frontal lobe, usually in the superior prefrontal area.     

On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton-Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.     

On the possibility of reduction of dissipative structure models to a simple form

A multicomponent reaction-diffusion system containing two different space scales (contrasting system) is considered. It is shown that if the system answers several conditions enumerated, it can be reduced to a bicomponent system which describes a dissipative structure either of a peak or of a step type. While in the first case the original and the final systems are equivalent only in the neighbourhood of zero, in the second case the equivalence is more general as the solution of these systems does not leave the region for which the procedure of reduction was developed.