Large amplification in stage-structured models: Arnol’d tongues revisited

The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnold tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnold tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.     

Coexistence of Tonic Spiking Oscillations in a Leech Neuron Model

The leech neuron model studied here has a remarkable dynamical plasticity. It exhibits a wide range of activities including various types of tonic spiking and bursting. In this study we apply methods of the qualitative theory of dynamical systems and the bifurcation theory to analyze the dynamics of the leech neuron model with emphasis on tonic spiking regimes. We show that the model can demonstrate bi-stability, such that two modes of tonic spiking coexist. Under a certain parameter regime, both tonic spiking modes are represented by the periodic attractors. As a bifurcation parameter is varied, one of the attractors becomes chaotic through a cascade of period-doubling bifurcations, while the other remains periodic. Thus, the system can demonstrate co-existence of a periodic tonic spiking with either periodic or chaotic tonic spiking. Pontryagins averaging technique is used to locate the periodic orbits in the phase space.     

On the concept of attractor for community-dynamical processes II: the case of structured populations

In Part I of this paper Jacobs and Metz (2003) extended the concept of the Conley-Ruelle, or chain, attractor in a way relevant to unstructured community ecological models. Their modified theory incorporated the facts that certain parts of the boundary of the state space correspond to the situation of at least one species being extinct and that an extinct species can not be rescued by noise. In this part we extend the theory to communities of physiologically structured populations. One difference between the structured and unstructured cases is that a structured population may be doomed to extinction and not rescuable by any biologically relevant noise before actual extinction has taken place. Another difference is that in the structured case we have to use different topologies to define continuity of orbits and to measure noise. Biologically meaningful noise is furthermore related to the linear structure of the community state space. The construction of extinction preserving chain attractors developed in this paper takes all these points into account.     

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.     

Wave separation versus bandpass filtering: a comparative non-linear analysis of brain α-EEG signals with and without psychotropic drug treatment

Attempts to demonstrate low-dimensional attractor behaviour in the analysis of electroencephalographic (EEG) signals meet with difficulties that in part stem from a departure from single-system dynamics. In order to address this problem, the -waves can be extracted by digital filtering or by wave separation; these two techniques are compared in order to specify the conditions in which finite impulse response (FIR) bandpass filters can be used. The comparison was made using 18 EEG records of 3 min duration under resting conditions (6 subjects, 3 records per subject: prior to apomorphine administration, then 90 min and 150 min post-treatment). No presence of low-dimensional dynamic episodes in -signals was observed without digital processing. Sixty 5 s sections showing attractor behaviour were found after filtering and twenty five 5 s sections after wave separation. The mean correlation dimension was calculated for each experimental condition and for 4 subjects, in order to observe the temporal profile of the drug. When attractors were found after wave separation, bandpass filtering then also showed attractor behaviour, with the same temporal profile. However, the reverse is not true: attractors were found after bandpass filtering that were not present after wave separation; in this case the results deserve confirmation, although the temporal profiles for all cases in which attractors were found after filtering remained comparable.     

Synaptic Potentiation Facilitates Memory-like Attractor Dynamics in Cultured In Vitro Hippocampal Networks

Collective rhythmic dynamics from neurons is vital for cognitive functions such as memory formation but how neurons self-organize to produce such activity is not well understood. Attractor-based computational models have been successfully implemented as a theoretical framework for memory storage in networks of neurons. Additionally, activity-dependent modification of synaptic transmission is thought to be the physiological basis of learning and memory. The goal of this study is to demonstrate that using a pharmacological treatment that has been shown to increase synaptic strength within in vitro networks of hippocampal neurons follows the dynamical postulates theorized by attractor models. We use a grid of extracellular electrodes to study changes in network activity after this perturbation and show that there is a persistent increase in overall spiking and bursting activity after treatment. This increase in activity appears to recruit more “errant” spikes into bursts. Phase plots indicate a conserved activity pattern suggesting that a synaptic potentiation perturbation to the attractor leaves it unchanged. Lastly, we construct a computational model to demonstrate that these synaptic perturbations can account for the dynamical changes seen within the network.     

On the concept of attractor for community-dynamical processes I: the case of unstructured populations

We introduce a notion of attractor adapted to dynamical processes as they are studied in community-ecological models and their computer simulations. This attractor concept is modeled after that of Ruelle as presented in [11] and [12]. It incorporates the fact that in an immigration-free community populations can go extinct at low values of their densities.Mathematics Subject Classification (2000):37b20, 37c50, 37c70     

Wave-separation in complex systems. Application to brain-signals

A departure from single-system dynamics, that may arise in characterizing self-organized dynamics of complex systems, is dealt with by using the Karhunen-Loève expansion of the trajectory matrix to decompose an experimental signal in a sum of spectral features. For an electroencephalographic -signal, a separation of waves and extraction of additive sub-signals are achieved, each sub-signal covering a well-bounded and physiologically meaningful frequency range. From the subsignals, an attractor that vanishes on phase-randomizing the data is characterized, under conditions where none was found for the recorded signal.     

Nuclear androdioecy and gynodioecy

We formulate two single-locus Mendelian models, one for androdioecy and the other one for gynodioecy, each with 3 parameters: t the male (female) fertility rate of males (females) to hermaphrodites, s the fraction of the progeny derived from selfing; and g the fitness of inbreeders. Each model is expressed as a transformation of a 3 dimensional zygotic algebra, which we interpret as a rational map of the projective plane. We then study the dynamics for the evolution of each reproductive system; and compare our results with similar published models. In this process, we introduce a general concept of fitness and list some of its properties, obtaining a relative measure of population growth, computable as an eigenvalue of a mixed mating transformation for a population in equilibrium. Our results concur with previous models of the evolution of androdioecy and gynodioecy regarding the threshold values above which the sexual polymophism is stable, although the previous models assume constant the fraction of ovules from hermaphrodites that are self pollinated, while we assume constant the fraction of the progeny derived from selfing. A stable androdioecy requires more stringent conditions than a stable gynodioecy if the amount of pollen used for selfing is negligible in comparison with the total amount of pollen produced by hermaphrodites. Otherwise, both models are identical. We show explicitly that the genotype fitnesses depend linearly on their frequencies. Simulations show that any population not at equilibrium always converges to the equilibrium point of higher fitness. However, at intermediate steps, the fitness function occasionally decreases.     

Accurate Path Integration in Continuous Attractor Network Models of Grid Cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal''s position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat''s velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.     

Place Cell Rate Remapping by CA3 Recurrent Collaterals

Episodic-like memory is thought to be supported by attractor dynamics in the hippocampus. A possible neural substrate for this memory mechanism is rate remapping, in which the spatial map of place cells encodes contextual information through firing rate variability. To test whether memories are stored as multimodal attractors in populations of place cells, recent experiments morphed one familiar context into another while observing the responses of CA3 cell ensembles. Average population activity in CA3 was reported to transition gradually rather than abruptly from one familiar context to the next, suggesting a lack of attractive forces associated with the two stored representations. On the other hand, individual CA3 cells showed a mix of gradual and abrupt transitions at different points along the morph sequence, and some displayed hysteresis which is a signature of attractor dynamics. To understand whether these seemingly conflicting results are commensurate with attractor network theory, we developed a neural network model of the CA3 with attractors for both position and discrete contexts. We found that for memories stored in overlapping neural ensembles within a single spatial map, position-dependent context attractors made transitions at different points along the morph sequence. Smooth transition curves arose from averaging across the population, while a heterogeneous set of responses was observed on the single unit level. In contrast, orthogonal memories led to abrupt and coherent transitions on both population and single unit levels as experimentally observed when remapping between two independent spatial maps. Strong recurrent feedback entailed a hysteretic effect on the network which diminished with the amount of overlap in the stored memories. These results suggest that context-dependent memory can be supported by overlapping local attractors within a spatial map of CA3 place cells. Similar mechanisms for context-dependent memory may also be found in other regions of the cerebral cortex.     

The importance of transients' dynamics in spatially extended populations

Recent theoretical works on the dynamics of metapopulations have highlighted the existence of very long transients (supertransients) with abrupt changes in behaviour which occur following perturbation of the system away from its attractor. If this phenomenon is common in natural systems, populations that do not oscillate can begin to fluctuate wildly without any change in the environmental conditions. However, the frequency of occurrence of supertransients is currently poorly understood even in model systems. Here we explore their occurrence in metapopulation models which relax the important assumption of global synchrony of events implicit in all the coupled map lattice models for which supertransients have so far been demonstrated. We find supertransients in all the models but always only for a very restricted range of parameter combinations. However, we also report for the first time another type of longer-lived transient (mesotransients) that occurs on shorter time-scales than supertransients and is found for a much wider set of conditions. We argue that these medium-term changes in the dynamics of populations can be of more ecological relevance than the long-term changes of supertransients.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Complex Formation of E. coli RNA Polymerase with Bacteriophage T2 DNA: Long-Range Effects in DNA

Conformation behavior of phase T2 DNA in the process of its interaction with it E. coli RNA polymerase was studied using spin labeling technique. T2 DNA was modified by the spin-labeled imidazole at OH-groups of glucosylated cytidine residues. It was shown that the binding of RNA polymerase under the conditions favoring the formation of open promoter complexes induces specific conformational changes in the spin-labeled DNA. The observed conformational changes encompass not only the promoter regions of DNA which are involved in direct contacts with RNA polymerase molecules but extend over remote DNA sites (long-range effect). In relation to this effect, current theoretical models of DNA dynamics are discussed.     

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.