Evolution of specificity in an immune network

A dynamic antigen response of the immune network is discussed, based on shape-space modelling. The present model extends the shape-space modelling by introducing the evolution of specificity of idiotypes. When the amount of external antigen increases, a measure of stability of the immune network is lost and thus the network can respond to the antigen. It is shown that specific and non-specific responses emerge as a function of antigen amounts. A specific response is observed with a fixed-point attractor, and a non-specific response is observed with a chaotic attractor for the lymphocyte population dynamics. The network topology also changes between fixed-point and chaotic attractors. For some antigen amounts, chaotic attractors will vanish or become long-lived super-transient states. A dynamic bell-shaped response function will thus emerge. The relevance of long-lived chaotic transient states embedded in fixed-point attractors is discussed with respect to immune functions.     

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.     

Multiple attractors in stage-structured population models with birth pulses

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.     

Les Attracteurs Inedits de l'hominisation

The recent discovery of a phenomenon of craniofacial growth, called craniofacial contraction, throws a new light on the process of hominization. The main interest of this discovery lies in a growth principle combining the different craniofacial units, that is to say, the neurocranium (neural skull), the chondrocranium (basal skull) and the splanchnocranium (visceral archs including the mandible). Until recent years, these different parts were considered as neighbouring element without any morphogenic or morphodynamic connection. But now, we know that the morphogenesis of the base of the skull governs that of the face. This basicranial morphogenesis is the occipital flexion. It generates morphogenic correlations with the face since embryogenesis. The ontogenic pathway of this phenomenon is the craniofacial contraction. It concerns embryonic dynamics connected with the spatial development of the embryonic neural system, the neural tube. These morphodynamics are common to each primate species, but they are differenciated by the amplitude of the embryonic contraction. We ask ourself the question: is hominization of the neurocephalic embryogenesis, that is the craniofacial contraction, plausible over a very long period, with gradual and chaotic evolutionary pathways, or, on the contrary, is the complexity of such an embryonic phenomenon, a limiting factor generating determined and predictible ontogenic thresholds? The study of extant and fossil primate skulls demonstrates that species are organized around 6 levels of embryonic contraction, which, starting from 60 millions years, evolve from the less to the most contracted skull. Among each ontogenic level, living and fossil species develop from the same embryonic system but between both levels, the embryos suddenly are reorganized. Therefore, I have defined an evolutive ontogenic unity, that is the fundamental ontogenesia. The cephalic pole has a fundamental ontogenesis, meaning that, beyond the diversities, we can see the same contraction in many living and extinct species. The ontogenic diversities are the result of the microevolution and are not predictible. In such a perspective, the ontogenic morphodynamics evolve with chaotic trajectories. But, between two embryonic levels, or two fundamental ontogeneses, evolutionary modalities are different. Eventually, from 60 millions years to XXth century, we observe the same phenomenon than during human ontogenesis; hominization of the cephalic pole is a craniofacial contraction. The evolutive pathway is stable, whatever the number of thresholds, the cranial shape changes but the ontogenic trajectory is preserved. This is a macroevolution because the embryonic system is reorganized. The logics of the phenomenon are an increasing dynamization, the human ontogenesis is the more unstable and the longer morphodynamics to stabilize the craniofacial contraction. To conclude, hominization is an iteration of an ontogenic process when embryos reach successive dynamic thresholds. The attractors are neither static, periodic, nor chaotic because the successive ontogenic trajectories are themselves in a stable evolutive trajectory, and the results with increasing contraction, complexified neocortical tissues and cephalocaudal reorganization are predictible. During hominization, irreversibility and innovations do not emerge with chaotic determinism, but with harmonic determinism in association with the correlations established between the embryonic tissues. When the system is destabilized, the embryonic systems do not forget the previous ontogenic pattern, on the contrary, they develop the pattern with new dynamical conditions. This sort of phenomenon is not described in the sciences of complexity. In the present case, we are in front of many millions years and the necessity to propose new concepts such as a new familly of attractors, namely the harmonic attractors.     

SIS epidemic attractors in periodic environments

The demographic dynamics are known to drive the disease dynamics in constant environments. In periodic environments, we prove that the demographic dynamics do not always drive the disease dynamics. We exhibit a chaotic attractor in an SIS epidemic model, where the demograhic dynamics are asymptotically cyclic. Periodically forced SIS epidemic models are known to exhibit multiple attractors. We prove that the basins of attraction of these coexisting attractors have infinitely many components.     

Novel tracking function of moving target using chaotic dynamics in a recurrent neural network model

Chaotic dynamics introduced in a recurrent neural network model is applied to controlling an object to track a moving target in two-dimensional space, which is set as an ill-posed problem. The motion increments of the object are determined by a group of motion functions calculated in real time with firing states of the neurons in the network. Several cyclic memory attractors that correspond to several simple motions of the object in two-dimensional space are embedded. Chaotic dynamics introduced in the network causes corresponding complex motions of the object in two-dimensional space. Adaptively real-time switching of control parameter results in constrained chaos (chaotic itinerancy) in the state space of the network and enables the object to track a moving target along a certain trajectory successfully. The performance of tracking is evaluated by calculating the success rate over 100 trials with respect to nine kinds of trajectories along which the target moves respectively. Computer experiments show that chaotic dynamics is useful to track a moving target. To understand the relations between these cases and chaotic dynamics, dynamical structure of chaotic dynamics is investigated from dynamical viewpoint.     

Chaotic neural network applied to two-dimensional motion control

Chaotic dynamics generated in a chaotic neural network model are applied to 2-dimensional (2-D) motion control. The change of position of a moving object in each control time step is determined by a motion function which is calculated from the firing activity of the chaotic neural network. Prototype attractors which correspond to simple motions of the object toward four directions in 2-D space are embedded in the neural network model by designing synaptic connection strengths. Chaotic dynamics introduced by changing system parameters sample intermediate points in the high-dimensional state space between the embedded attractors, resulting in motion in various directions. By means of adaptive switching of the system parameters between a chaotic regime and an attractor regime, the object is able to reach a target in a 2-D maze. In computer experiments, the success rate of this method over many trials not only shows better performance than that of stochastic random pattern generators but also shows that chaotic dynamics can be useful for realizing robust, adaptive and complex control function with simple rules.     

Change of dynamic regimes in the population of species with short life cycles: Results of an analytical and numerical study

There is a phenomenon of multiregimism found in the elementary mathematical model of population dynamics, meaning the possibility for different dynamic regimes to exist under the same conditions, with transition to these regimes dependent on the initial numerical values. The effect in question comes into existence in the model which has several different limiting regimes (attractors): equilibrium, regular fluctuations, and chaotic attractor. The revealed phenomenon of multiregimism lets us explain the initiation of fluctuations as well as disappearance of fluctuations. Adequacy of the model's dynamic regimes is depicted by their correlation with the actual dynamics of population size of bank vole (Myodes glareolus). It is shown that the impact of climatic factors on a reproductive process of a population noticeably extends the range of possible dynamic regimes and, in fact, leads to random migration over attraction basins of these regimes.     

Numerically evaluated functional equivalence between chaotic dynamics in neural networks and cellular automata under totalistic rules

Chaotic dynamics in a recurrent neural network model and in two-dimensional cellular automata, where both have finite but large degrees of freedom, are investigated from the viewpoint of harnessing chaos and are applied to motion control to indicate that both have potential capabilities for complex function control by simple rule(s). An important point is that chaotic dynamics generated in these two systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded patterns (attractors) in high-dimensional state space. An application of these chaotic dynamics to complex controlling is proposed based on an idea that with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the spatial structure of the maze is one of typical ill-posed problems, is solved with the use of chaos in both systems. Our computer simulations show that the success rate over 300 trials is much better, at least, than that of a random number generator. Our functional simulations indicate that both systems are almost equivalent from the viewpoint of functional aspects based on our idea, harnessing of chaos.     

The effect of seasonal harvesting on stage-structured population models

In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. We propose an exploited single-species model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Since birth pulse populations are often characterized with a discrete time dynamical system determined by its Poincaré map, we explore the consequences of harvest timing to equilibrium population sizes under seasonal dependence and obtain threshold conditions for their stability, and show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. Moreover, our results imply that the population can sustain much higher harvest rates if the mature fish is removed as early in the season (after the birth pulse) as possible. Further, the effects of harvesting effort and harvest timing on the dynamical complexity are also investigated. Bifurcation diagrams are constructed with the birth rate (or harvesting effort or harvest timing) as the bifurcation parameter, and these are observed to display rich structure, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, non-unique dynamics (meaning that several attractors coexist) and attractor crisis. This suggests that birth pulse, in effect, provides a natural period or cyclicity that makes the dynamical behavior more complex.This work is supported by National Natural Science Foundation of China (10171106)     

A Cayley tree immune network model with antibody dynamics

A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analysed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvistet al. (1989.Proc. natn. Acad. Sci. U.S.A. 86, 5074–5078), injection ofAb ₁ antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations ofAb ₁ andAb ₂ antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvistet al. but not accounted for previously.     

The effect of the Holling type II functional response on apparent competition

This article analyzes the classical 2-resource-1-consumer apparent competition community module with the Holling type II functional response. Two types of resource regulation (top-down vs. combined top-down and bottom-up) and two types of consumer behaviors (inflexible consumers with fixed preferences for resources vs. adaptive consumers) are considered. When resources grow exponentially and consumers are inflexible foragers, one resource is always outcompeted due to strong apparent competition. Density dependent resource growth relaxes apparent competition so that resources can coexist. As multiple attractors (either equilibria or limit cycles) coexist, population dynamics and community composition depend on initial population densities. Population dynamics change dramatically when consumers forage adaptively. In this case, the results both for top-down, and combined top-down and bottom-up regulation are similar and they show that species persistence occurs for a much larger set of parameter values when compared with inflexible consumers. Moreover, population dynamics will be chaotic when resource carrying capacities are high enough. This shows that adaptive consumer switching can destabilize population dynamics.     

Nonlinear transient computation as a potential "kernel trick" in cortical processing

Evidence has been found for the presence of chaotic dynamics at all levels of the mammalian brain. This has led to some searching questions about the potential role that nonlinear dynamics may have in neural information processing. We propose that chaos equips the brain with the equivalent of a kernel trick for solving hard nonlinear problems. The approach presented, which is described as nonlinear transient computation, uses the dynamics of a well known chaotic attractor. The paper provides experimental results to show that this approach can be used to solve some challenging pattern recognition tasks. The paper also offers evidence to suggest that the efficacy of nonlinear transient computation for nonlinear pattern classification is dependent only on the generic properties of chaotic attractors and is not sensitive to the particular dynamics of specific sub-regions of chaotic phase space. If, as this work suggests, nonlinear transient computation is independent of the particulars of any given chaotic attractor, then it could be offered as a possible explanation of how the chaotic dynamics that have been observed in brain structures contribute to neural information processing tasks.     

Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics.

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.     

On the robustness of complex heterogeneous gene expression networks

We analyze a continuous gene expression model on the underlying topology of a complex heterogeneous network. Numerical simulations aimed at studying the chaotic and periodic dynamics of the model are performed. The results clearly indicate that there is a region in which the dynamical and structural complexity of the system avoid chaotic attractors. However, contrary to what has been reported for Random Boolean Networks, the chaotic phase cannot be completely suppressed, which has important bearings on network robustness and gene expression modeling.     

Structural perturbations to population skeletons: transient dynamics, coexistence of attractors and the rarity of chaos

BACKGROUND: Simple models of insect populations with non-overlapping generations have been instrumental in understanding the mechanisms behind population cycles, including wild (chaotic) fluctuations. The presence of deterministic chaos in natural populations, however, has never been unequivocally accepted. Recently, it has been proposed that the application of chaos control theory can be useful in unravelling the complexity observed in real population data. This approach is based on structural perturbations to simple population models (population skeletons). The mechanism behind such perturbations to control chaotic dynamics thus far is model dependent and constant (in size and direction) through time. In addition, the outcome of such structurally perturbed models is [almost] always equilibrium type, which fails to commensurate with the patterns observed in population data. METHODOLOGY/PRINCIPAL FINDINGS: We present a proportional feedback mechanism that is independent of model formulation and capable of perturbing population skeletons in an evolutionary way, as opposed to requiring constant feedbacks. We observe the same repertoire of patterns, from equilibrium states to non-chaotic aperiodic oscillations to chaotic behaviour, across different population models, in agreement with observations in real population data. Model outputs also indicate the existence of multiple attractors in some parameter regimes and this coexistence is found to depend on initial population densities or the duration of transient dynamics. Our results suggest that such a feedback mechanism may enable a better understanding of the regulatory processes in natural populations.     

Complex dynamics in a three-level trophic system with intraspecies interaction

In this paper, we present a three-level trophic food chain, including intraspecies interaction. In contrast with other analyses, we consider the effect on the third trophic level by the first-level parameters. The model shows complex, as well as, chaotic oscillations. Bifurcation diagrams show period doubling route to chaos and crises. Also from the forward and backwards sections of the bifurcation diagrams, we find hysteresis. This result implies the coexistence of attractors for the same parameter values. In particular, we consider the coexistence of a chaotic and a P1 attractors. Our results show that the regulation in the food chain is not exclusive to either a food-prey or prey-predator interaction, but to a more subtle food-prey-predator interaction, where, for some parameter values, a food-prey or a prey-predator regulation may dominate the system's dynamics. Finally, we consider the impact of the intraspecies interaction in the overall dynamics of the food chain.     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep

Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calculation of dimensionality estimates the degrees of freedom of a signal. Nevertheless, it is difficult to decide from this kind of analysis whether a process is quasiperiodic or chaotic. Therefore, we performed a new analysis by calculating the first positive Lyapunov exponent L ₁ from sleep EEG data. Lyapunov exponents measure the mean exponential expansion or contraction of a flow in phase space. L ₁ is zero for periodic as well as quasiperiodic processes, but positive in the case of chaotic processes expressing the sensitive dependence on initial conditions. We calculated L ₁ for sleep EEG segments of 15 healthy men corresponding to the sleep stages I, II, III, IV, and REM (according to Rechtschaffen and Kales). Our investigations support the assumption that EEG signals are neither quasiperiodic waves nor a simple noise. Moreover, we found statistically significant differences between the values of L ₁ for different sleep stages. All together, this kind of analysis yields a useful extension of the characterization of EEG signals in terms of nonlinear dynamical system theory.