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We examine the interplay of nonlinearity of a dynamical system and thermal fluctuation of its environment in the "physical limit" of small damping and slow diffusion in a semiclassical context and show that the trajectories of c-number variables exhibit dynamical chaos due to the thermal fluctuations of the bath.
A compact glance at the history and impact of mathematical models in development provides background for predicting the fate of dynamical systems modeling in developmental psychology. Dynamic models are considered and the articles by Thelen and Ulrich (1991) and van Geert (1991) are summarized. Deterministic and probabilistic models are compared and some cautions are presented along with a consideration of forms that can be attained by linear models in comparison to those well known in logistic difference models. Reactions to the research by Rabinowitz, Grant, Howe, and Walsh; Kreindler and Lumsden; and Cooney and Troyer are given, with some concluding remarks on survival of dynamical systems modeling in developmental psychology.
A bird's eye view of the role of nontrivial dynamical phenomena in physiological systems is presented. Many levels in the physiological hierarchy are affected. It appears that physiology has a particularly high affinity to chaos. The old central place of physiology in the circle of sciences can perhaps be regained if the two disciplines join resources. Even physics may turn out to be dependent on the chaotic micro dynamics of the brain.
To date, mathematical models of the dynamics of infectious disease have consistently focused on understanding the long-term behavior of the interacting components, where the steady state solutions are paramount. However for most acute infections, the longterm behavior of the pathogen population is of little importance to the host and population health. We introduce the notion of transient pathology, where the short-term dynamics of interaction between the immune system and pathogens is the principal focus. We identify the amplifying effect of the absence of a fully operative immune system on the pathogenesis of the initial inoculum, and its implication for the acute severity of the infection. We then formalize the underlying dynamics, and derive two measures of transient pathogenicity: the peak of infection (maximum pathogenic load) and the time to peak of infection, both crucial to understanding the early dynamics of infection and its consequences for early intervention.
Increasing evidence suggests that the brain utilizes distributed codes that can only be analyzed by simultaneously recording the activity of multiple neurons. This paper introduces a new methodology for studying neural ensemble recordings. The method uses a novel representation to provide complementary information about the stimuli which are contained in the temporal pattern of the spike sequence. By using this procedure, a high correlation of synchronized events with stimuli times is apparent. To quantify the results and to compare the performance of this method against the most traditional raster plot, we have used Fano factor and cross-correlation analysis. Our results suggest that several consecutive spikes from different neurons within an extended time window may encode behaviorally relevant information. We propose that this new representation, in addition to the other approaches currently used (standard raster plots, multivariate statistical methods, neuronal networks, information theory, etc.), can be a useful procedure to describe population spike dynamics.
This study shows how sensory-action sequences of imitating finite state machines (FSMs) can be learned by utilizing the deterministic dynamics of recurrent neural networks (RNNs). Our experiments indicated that each possible combinatorial sequence can be recalled by specifying its respective initial state value and also that fractal structures appear in this initial state mapping after the learning converges. We also observed that the sequences of mimicking FSMs are encoded utilizing the transient regions rather than the invariant sets of the evolved dynamical systems of the RNNs.
The results of the study reported in Brain Research in 1995 by Isaacson et al. [Isaacson, R.L., Varner, J.A., Baars, J.-M., de Wied, D., 1995. The effects of pregnenolone sulfate and ethylestrenol on retention of a passive avoidance task. Brain Res. 689, 79-84] have been re-examined with special emphasis placed on the distributions of latencies found in the passive avoidance task using rats. This study used two retention tests, one 24 h after training the other at 48 h after training. In the first experiment in that study a range of doses of two anabolic steroids, pregnenolone sulfate and ethylestrenol, were given s.c. just after the footshock training trial. In experiment 2 a similar range of doses of both steroids was given to the rats 1 h before the first retention test. Placing emphasis on the distributions rather than measures of central tendencies revealed that, in contrast to the vehicle treated animals, the anabolic steroid treated animals exhibited bimodal distributions of response latencies. These differences between control and hormone treated animals were observed in both experiments. The new information was interpreted in terms of non-linear dynamics including some aspects of Chaos theory.
The placozoan Trichoplax adhaerence is one of the most primitive multi-cellular organisms, and moves about accompanying perpetual changes in its shape. Changes in position, locomotion velocity and the outer shape of the organism were monitored quantitatively with use of a computer image analysis, and their dynamic patterns in free locomotion and upon feeding were analyzed in terms of non-linear dynamics. The organism changed its behavioral patterns discontinuously in response to various concentrations of yeast extracts (food). (1) At low concentrations, the organism moved fast with perpetual random changes in shape. Both locomotion velocity and shape changes exhibited 1/f fluctuations. (2) At high concentrations, the shape of the organism as well as the locomotion exhibited oscillations with periods of about 8 min. These limit cycle oscillations bifurcated into the period 2 at the highest concentration tested. The organism flattened more strongly and the locomotion was more reduced on the whole at higher concentrations. (3) At the intermediate concentrations, two patterns as monitored above appeared: one pattern continued for a while and switched to the other abruptly. (4) The average square displacement of the organism increased linearly with time in all cases, indicating that the locomotion is a Brownian movement. In this way, the feeding behaviors by the placozoan are organized as successive co-operative transitions among non-linear dynamic states.
The perceived direction of a grating moving behind an elongated aperture is biased towards the aperture's long axis. This "barber pole" illusion is a consequence of integrating one-dimensional (1D) or grating and two-dimensional (2D) or terminator motion signals. In humans, we recorded the ocular following responses to this stimulus. Tracking was always initiated at ultra-short latencies (approximately 85 ms) in the direction of grating motion. With elongated apertures, a later component was initiated 15-20 ms later in the direction of the terminator motion signals along the aperture's long axis. Amplitude of the later component was dependent upon the aperture's aspect ratio. Mean tracking direction at the end of the trial (135-175 ms after stimulus onset) was between the directions of the vector sum computed by integrating either terminator motion signals only or both grating and terminator motion signals. Introducing an elongated mask at the center of the "barber pole" did not affect the latency difference between early and later components, indicating that this latency shift was not due to foveal versus peripheral locations of 1D and 2D motion signals. Increasing the size of the foveal mask up to 90% of the stimulus area selectively reduced the strength of the grating motion signals and, consequently, the amplitude of the early component. Conversely, reducing the contrast of, or indenting the aperture's edges, selectively reduced the strength of terminator motion signals and, consequently, the amplitude of the later component. Latencies were never affected by these manipulations. These results tease apart an early component of tracking responses, driven by the grating motion signals and a later component, driven by the line-endings moving at the intersection between grating and aperture's borders. These results support the hypothesis of a parallel processing of 1D and 2D motion signals with different temporal dynamics.
Social insect colonies have many of the properties of adaptive networks. The simple rules governing how local interactions among individuals translate into group behaviors are found across social groups, giving social insects the potential to have a profound impact on our understanding of the interplay between network dynamics and social evolution.
A fundamental challenge for any general theory of neural circuits is how to characterize the structure of the space of all possible circuits over a given model neuron. As a first step in this direction, this letter begins a systematic study of the global parameter space structure of continuous-time recurrent neural networks (CTRNNs), a class of neural models that is simple but dynamically universal. First, we explicitly compute the local bifurcation manifolds of CTRNNs. We then visualize the structure of these manifolds in net input space for small circuits. These visualizations reveal a set of extremal saddle node bifurcation manifolds that divide CTRNN parameter space into regions of dynamics with different effective dimensionality. Next, we completely characterize the combinatorics and geometry of an asymptotically exact approximation to these regions for circuits of arbitrary size. Finally, we show how these regions can be used to calculate estimates of the probability of encountering different kinds of dynamics in CTRNN parameter space.
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A model of cell cycle kinetics is proposed, which includes unequal division of cells, and a nonlinear dependence of the fraction of cells re-entering proliferation on the total number of cells in the cycle. The model is described by a nonlinear functional-integral equation. It is analyzed using the operator semigroup theory combined with classical differential equations approach. A complete description of the asymptotic behavior of the model is provided for a relatively broad class of nonlinearities. The nonnegative solutions either tend to a stable steady state, or to zero. The simplicity of the model makes it an interesting step in the analysis of dynamics of nonlinear structure populations.
Biological time series often contain passing episodes of nearly periodic dynamics. In chaotic systems, such transient periodicity can reflect the existence of semiperiodic saddles--nonstable invariant sets--contained in the attractor. Motion in the vicinity of such objects has a prominent periodic component. In addition, trajectories can become temporarily trapped in these neighbourhoods before exiting. The immediate dynamical antecedents (low-order preimages) of transient periodicity are well defined and, along with the saddles to which they map, correspond to regions of enhanced predictability under nonlinear forecasting. This suggests that it may be possible to forecast the onset of transient periodicity in systems for which overall predictability is low. The present paper reviews these concepts and applies them to biological phenomena at different levels of organization.
We report on the first realization of a single bosonic Josephson junction, implemented by two weakly linked Bose-Einstein condensates in a double-well potential. In order to fully investigate the nonlinear tunneling dynamics we measure the density distribution in situ and deduce the evolution of the relative phase between the two condensates from interference fringes. Our results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions. Additionally, we confirm a novel nonlinear effect known as macroscopic quantum self-trapping, which leads to the inhibition of large amplitude tunneling oscillations.
Randomly connected Boolean networks have been used as mathematical models of neural, genetic, and immune systems. A key quantity of such networks is the number of basins of attraction in the state space. The number of basins of attraction changes as a function of the size of the network, its connectivity and its transition rules. In discrete networks, a simple count of the number of attractors does not reveal the combinatorial structure of the attractors. These points are illustrated in a reexamination of dynamics in a class of random Boolean networks considered previously by Kauffman. We also consider comparisons between dynamics in discrete networks and continuous analogues. A continuous analogue of a discrete network may have a different number of attractors for many different reasons. Some attractors in discrete networks may be associated with unstable dynamics, and several different attractors in a discrete network may be associated with a single attractor in the continuous case. Special problems in determining attractors in continuous systems arise when there is aperiodic dynamics associated with quasiperiodicity of deterministic chaos.
The behaviour of three biologically inspired networks is investigated: Immune Idiotypic Network (IIN), Hopfield Network (HN) and Coupled Map Lattice (CML), mainly the effect on the dynamics induced by connecting the network in a frustrated way. Frustration occurs when the global structure is such that local connectivity patterns responsible for stable behaviour are intertwined, leading to mutually competing attractors and chaotic itinerancy among brief appearance of these attractors. Frustration destabilizes the network and provokes an unpredictable "wavering" among the stable dynamic regimes which characterize the same network when it is interconnected in a non-frustrated way. As the main contribution of this paper, an immune idiotypic network in which the prevailing behaviour is oscillatory is studied in detail. It is shown how connecting an elementary three-clone network in a frustrating way transforms the oscillatory regime into a chaotic one. This chaotic regime is further analysed and several interesting aspects are discussed such as the variable homogeneity, the intrinsic chaotic itinerancy among brief oscillatory regimes and the strong unpredictability. In addition, dynamical regimes obtained by frustrating the connectivity of HN and CML are presented and the similarities as well as the differences with the IIN dynamics are emphasized. Common to all these networks is the description of the frustrated chaos as a succession of attempts to relax the network into one of the oscillatory regimes given by a weaker and non-frustrated connectivity, an impossible achievement making the dynamics rambling over brief but repelling orbits.