Time-dependent regimes of a tourism-based social–ecological system: Period-doubling route to chaos

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.     

Neural chaos and schizophrenia

Recent data indicate that random-like processes are related to the defects in the organization of semantic memory in schizophrenia which is more disorganized and less definable than those of controls with more semantic links and more bizarre and atypical associations. These aspects of schizophrenic cognition are similar to characteristics of chaotic nonlinear dynamical systems. In this context, the hypothesis tested in this study is that dynamic changes of electrodermal activity (EDA) as a measure of brain and autonomic activity may serve as a characteristic which can be used as an indicator of possible neural chaotic process in schizophrenia. In the present study, bilateral EDA in rest conditions were measured in 40 schizophrenic patients and 40 healthy subjects. Results of nonlinear and statistical analysis indicate left-side significant differences of positive largest Lyapunov exponents in schizophrenia patients compared to the control group. This might be interpreted that the neural activity during rest in schizophrenic patients is significantly more chaotic than in the control group. The relationship was confirmed by surrogate data testing. These data suggest that increased neural chaos in patients with schizophrenia may influence brain processes that can cause random-like disorganization of mental processes.     

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.     

Control of peptide product sizes by the energy-dependent protease ClpAP

Processive proteases can unfold proteins and cleave them into fragments of a characteristic size. The detailed mechanism by which product sizes are controlled is still in question. One possible mechanism for the control of product sizes would be translocation of unfolded polypeptides to the protease active sites in units of defined length. We have investigated the mechanism by which ClpAP, an energy-dependent protease from Escherichia coli, controls the sizes of its peptide products. We show that ClpAP generates peptide products with a distribution of sizes that has a pronounced peak at a peptide length of 6-8 amino acid residues. This product size distribution, which is similar to that observed previously for the proteasome, is robust to perturbations that interfere with translocation or proteolysis. To explain these results, we propose a mechanism in which translocation alternates with proteolysis, allowing peptides of more or less uniform length to be cleaved processively from a translocating substrate. To estimate the rate and energy efficiency of ClpAP-catalyzed measurements of product sizes, we apply information theory to quantify how precisely the product sizes are controlled. This analysis may also prove to be useful in characterizing the mechanisms of other proteases and nucleases, such as the proteasome and Dicer, which control the sizes of their products.     

The incorporation of epigenetics in artificial gene regulatory networks

Artificial gene regulatory networks are computational models that draw inspiration from biological networks of gene regulation. Since their inception they have been used to infer knowledge about gene regulation and as methods of computation. These computational models have been shown to possess properties typically found in the biological world, such as robustness and self organisation. Recently, it has become apparent that epigenetic mechanisms play an important role in gene regulation. This paper describes a new model, the Artificial Epigenetic Regulatory Network (AERN) which builds upon existing models by adding an epigenetic control layer. Our results demonstrate that AERNs are more adept at controlling multiple opposing trajectories when applied to a chaos control task within a conservative dynamical system, suggesting that AERNs are an interesting area for further investigation.     

Density-dependent birth rate, birth pulses and their population dynamic consequences

 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 a single-species model with stage structure for the dynamics in a wild animal population for which births occur in a single pulse once per time period. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker functions or Beverton-Holt functions, and obtain the threshold conditions for their stability. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Received: 13 June 2001 / Revised version: 7 September 2001 / Published online: 8 February 2002     

Age dependent dispersal is not a simple process: Density dependence, stability, and chaos

I present both discrete and continuous models for a single species with overlapping generations, density dependence, and movement rates that vary with age. In the discrete time and space model, I show that if the strongly density dependent age class(es) are stationary, the spatially structured model may exhibit instabilities or even chaos which are not present in the corresponding model without spatial structure. I argue that the conditions which lead to these diffusive instabilities are likely to be met in natural populations. Thus it is important to consider the interaction between age structure and spatial structure in both experimental and theoretical work. In particular, the conditions leading to chaos are more common than would be predicted in models which ignore structure.     

Terminal chaos for information processing in neurodynamics

New nonlinear phenomenon — terminal chaos caused by failure of the Lipschitz condition at equilibrium points of dynamical systems is introduced. It is shown that terminal chaos has a well organized probabilistic structure which can be predicted and controlled. This gives an opportunity to exploit this phenomenon for information processing. It appears that chaotic states of neurons activity are associated with higher level of cognitive processes such as generalization and abstraction.     

Pathological tremors as diffusional processes

 Two types of pathological tremors, essential and Parkinsonian, are studied using dynamical systems theory. It is shown that pathological tremors can be characterized as diffusional processes. The time-scale range for the diffusional scaling law to be valid starts from about one to several tens of the mean oscillation period. This time-scale range contrasts sharply with the predictable time scale for deterministic chaos, which is usually only a small fraction of the mean oscillation period. The diffusions in pathological tremors are usually anomalous. A number of quantities are designed to characterize the diffusions in the tremor. Their relevance to potential clinical applications is discussed. It is argued that in order to discriminate between Parkinsonian and essential tremors, quantities not of purely dynamical origin may be more useful, since purely dynamical quantities emphasize more the dynamical similarities between the two types of tremors. Received: 30 May 2001 / Accepted in revised form: 6 November 2001     

Limitations of perturbative techniques in the analysis of rhythms and oscillations

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.     

Construction and characterization of a highly regulable expression vector, pLAC11, and its multipurpose derivatives, pLAC22 and pLAC33

A number of different expression vectors have been developed to facilitate the regulated overproduction of proteins in Escherichia coli and related bacteria. Some of the more popular ones include pKK223-3, pKK233-2, pTrc99A, and the pET family of expression vectors. These vectors were designed to be regulable and can be grown under conditions that repress protein production or under conditions that induce protein production. Unfortunately, however, numerous researchers have found that these vectors produce significant amounts of protein even when grown under repressed conditions. We describe here a new expression vector, pLAC11, which was designed to be more regulable and thus more tightly repressible when grown under repressed conditions. The tight regulation of pLAC11 was achieved by utilizing the O3 auxiliary operator, CAP binding site, promoter, and O1 operator that occur in the wild-type lac control region. The pLAC11 vector can be used to conduct physiologically relevant studies in which the cloned gene is expressed at levels comparable to that obtainable from the chromosomal copy of the gene in question. In experiments in which a bacterial cell contained both a null allele in the chromosome and a second copy of the wild-type allele on pLAC11, we observed that cells grown under repressed conditions exhibited the null phenotype while cells grown under induced conditions exhibited the wild-type phenotype. Two multipurpose derivatives of pLAC11, pLAC22, and pLAC33 have also been constructed to fulfill different experimental needs.     

In Defense of Some `Cartesian' Assumptions Concerning the Brain and Its Operation

I argue against a growing radical trend in current theoretical cognitive science that moves from the premises of embedded cognition, embodied cognition, dynamical systems theory and/or situated robotics to conclusions either to the effect that the mind is not in the brain or that cognition does not require representation, or both. I unearth the considerations at the foundation of this view: Haugeland's bandwidth-component argument to the effect that the brain is not a component in cognitive activity, and arguments inspired by dynamical systems theory and situated robotics to the effect that cognitive activity does not involve representations. Both of these strands depend not only on a shift of emphasis from higher cognitive functions to things like sensorimotor processes, but also depend on a certain understanding of how sensorimotor processes are implemented - as closed-loop control systems. I describe a much more sophisticated model of sensorimotor processing that is not only more powerful and robust than simple closed-loop control, but for which there is great evidence that it is implemented in the nervous system. The is the emulation theory of representation, according to which the brain constructs inner dynamical models, or emulators, of the body and environment which are used in parallel with the body and environment to enhance motor control and perception and to provide faster feedback during motor processes, and can be run off-line to produce imagery and evaluate sensorimotor counterfactuals. I then show that the emulation framework is immune to the radical arguments, and makes apparent why the brain is a component in the cognitive activity, and exactly what the representations are in sensorimotor control.     

Driving reservoir models with oscillations: a solution to the extreme structural sensitivity of chaotic networks

A large body of experimental and theoretical work on neural coding suggests that the information stored in brain circuits is represented by time-varying patterns of neural activity. Reservoir computing, where the activity of a recurrently connected pool of neurons is read by one or more units that provide an output response, successfully exploits this type of neural activity. However, the question of system robustness to small structural perturbations, such as failing neurons and synapses, has been largely overlooked. This contrasts with well-studied dynamical perturbations that lead to divergent network activity in the presence of chaos, as is the case for many reservoir networks. Here, we distinguish between two types of structural network perturbations, namely local (e.g., individual synaptic or neuronal failure) and global (e.g., network-wide fluctuations). Surprisingly, we show that while global perturbations have a limited impact on the ability of reservoir models to perform various tasks, local perturbations can produce drastic effects. To address this limitation, we introduce a new architecture where the reservoir is driven by a layer of oscillators that generate stable and repeatable trajectories. This model outperforms previous implementations while being resistant to relatively large local and global perturbations. This finding has implications for the design of reservoir models that capture the capacity of brain circuits to perform cognitively and behaviorally relevant tasks while remaining robust to various forms of perturbations. Further, our work proposes a novel role for neuronal oscillations found in cortical circuits, where they may serve as a collection of inputs from which a network can robustly generate complex dynamics and implement rich computations.     

Prospects for advancing the understanding of complex biochemical systems

The application of mathematical theories to understanding the behaviour of complex biochemical systems is reviewed. Key aspects of behaviour are identified as the flux through particular pathways in a steady state, the nature and stability of dynamical states, and the thermodynamic properties of systems. The first of these is dealt primarily in theories of metabolic control, and metabolic control analysis (MCA) is an important example. The valid application of this theory is limited to steady-state systems, and the cases where the essential features of control can be derived from calibration experiments which perturb the state of the system by a sufficiently small amount from its operating point. In practice, time-dependent systems exist, it is not always possible to know a priori whether applied perturbations are sufficiently small, and important features of control may lie farther from the operating point than the application of the theory permits. The nature and stability of dynamical and thermodynamical states is beyond the scope of MCA. To understand the significance of these limitations fully, and to address the dynamical and thermodynamical properties, more complete theories are required. Non-linear systems theory offers the possibility of studying important questions regarding control of steady and dynamical states. It can also link to thermodynamic properties of the system including the energetic efficiency of particular pathways. However, its application requires a more detailed characterisation of the system under study. This extra detail may be an essential feature of the study of non-equilibrium states in general, and non-ideal pathways in particular. Progress requires considerably more widespread integration of theoretical and experimental approaches than currently exists.     

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.     

Melanopsin-Derived Visual Responses under Light Adapted Conditions in the Mouse dLGN

A direct projection from melanopsin-expressing intrinsically photosensitive retinal ganglion cells (ipRGCs) reaches the primary visual thalamus (dorsal lateral geniculate nucleus; dLGN). The significance of this melanopsin input to the visual system is only recently being investigated. One unresolved question is the degree to which neurons in the dLGN could use melanopsin to track dynamic changes in light intensity under light adapted conditions. Here we set out to address this question. We were able to present full field steps visible only to melanopsin by switching between rod-isoluminant ‘yellow’ and ‘blue’ lights in a mouse lacking cone function (Cnga3^-/-). In the retina these stimuli elicited melanopsin-like responses from a subset of ganglion cells. When presented to anaesthetised mice, we found that ~25-30% of visually responsive neurones in the contralateral dLGN responded to these melanopsin-isolating steps with small increases in firing rate. Such responses could be elicited even with fairly modest increases in effective irradiance (32% Michelson contrast for melanopsin). These melanopsin-driven responses were apparent at bright backgrounds (corresponding to twilight-daylight conditions), but their threshold irradiance was strongly dependent upon prior light exposure when stimuli were superimposed on a spectrally neutral ramping background light. While both onset and offset latencies were long for melanopsin-derived responses compared to those evoked by rods, there was great variability in these parameters with some cells responding to melanopsin steps in <1 s. These data indicate that a subset of dLGN units can employ melanopsin signals to detect modest changes in irradiance under photopic conditions.     

Trophic structure and dynamical complexity in simple ecological models

We study the dynamical complexity of five non-linear deterministic predator–prey model systems. These simple systems were selected to represent a diversity of trophic structures and ecological interactions in the real world while still preserving reasonable tractability. We find that these systems can dramatically change attractor types, and the switching among different attractors is dependent on system parameters. While dynamical complexity depends on the nature (e.g., inter-specific competition versus predation) and degree (e.g., number of interacting components) of trophic structure present in the system, these systems all evolve principally on intrinsically noisy limit cycles. Our results support the common observation of cycling and rare observation of chaos in natural populations. Our study also allows us to speculate on the functional role of specialist versus generalist predators in food web modeling.