Attractor States Research Articles

Dynamical features of stimulus integration by interacting cortical columns

In multi-stable perception and notably binocular rivalry, transitions between alternative perceptual states occur over a wide range of time-scales [1], but the distribution of dominance periods exhibits peculiar and consistent properties. For example, the tight correlation between mean and variance of this distribution represents a scalar property and a form of Weber's law [2]. Although several neural mechanisms for Weber's law have been proposed [3,4], here we focus on an alternative framework that involves interactions within a finite number of neural assemblies (such as cortical columns) encoding a distributed neural representation of a given perceptual state [5,6]. In this framework, each neural assembly shows nonlinear attractor dynamics (i.e., a double-well energy landscape) and switches spontaneously between inactive and active states due to endogenous noise. Energy landscape and, consequently, switching rates of each assembly are modulated by its external visual input and also by recurrent input from other assemblies [1,7]. A transition between one perceptual state and another is modelled as a two-stage process: first, the successive activation of assemblies due to external visual input (stochastic integration) and, second, the snowballing activation of additional assemblies due to recurrent interactions (global attractor dynamics) [1,6]. Whereas every local activation reflects a gain of evidence in favour of the associated perceptual state, the high-dimensional dynamics towards a global attractor state reflects the crossing of the perceptual threshold. We present our progress towards an analytic treatment of this modelling framework. After reducing the non-linear input dependence of local assemblies to escape rates (Kramer's escape problem), the stochastic integration of a population of assemblies becomes a well-known birth-death process (Ehrenfest process). From the master equation of a finite number of independent assemblies, we recursively obtain exact analytical expressions for all moments of the first-passage time (FPT) distribution (if the perceptual threshold is modelled as an absorbing barrier). A scalar property holds in two regimes: a low-threshold, drift-dominated regime in which the threshold is independent of input and the FPT distribution is less skewed, and a high-threshold, fluctuation-dominated regime in which the threshold depends on input and the FPT distribution is more skewed. Finally, we consider synaptic couplings between cortical columns resorting to a mean-field approximation. In this framework, network dynamics of bistable assemblies are captured replacing constant transition rates by state-dependent rates. For strong enough couplings, more than one global attractor may coexist, and stochastic integration can be viewed as a slow descent across a rough surface until perceptual threshold is crossed [1]. In a low-threshold limit, we derive a uniform shift (time-warp) of the FPT distribution, allowing us to generalize the analytic expressions found earlier for independent assemblies. We conclude that this modelling framework contributes to the development of a general theory for multi-modular networks of neurons.

Attractor states, control parameters, and co-adaptation in instructed L2 inferential comprehension: a design-based research study of a critical reading intervention

Previous L1 and L2 research on inferential comprehension has tended to follow a quantitative orientation. By contrast, L2 research on critical reading is qualitative and tends to ignore inferences. This paper presents a qualitative, design-based study of a critical reading intervention focused on promoting generative rhetorical inferences and investigating co-adaptation and emergence of new meaning-making capacities. Complexity theory (CT) constructs were used to research processes of co-adaptation between the participants' comprehension and the teacher-researcher's understanding of learning and instructional needs. Identification of attractor states and control parameters in classroom discourse were used to explore unpredicted factors influencing the participants' inferential comprehension and further refine the intervention. The results indicate that rhetorical genre knowledge acted as a control parameter driving the students' comprehension to attractor states characterized by implausible inferences, and that this knowledge explains the emergence of pragmatic meaning (rhetorical inferences) from semantic meaning. The paper illustrates the usefulness of CT constructs in doing design-based research qualitatively in a manner that informs both theory and practice.

Attractor states, control parameters, and co-adaptation in instructed L2 inferential comprehension: a design-based research study of a critical reading intervention

Previous L1 and L2 research on inferential comprehension has tended to follow a quantitative orientation. By contrast, L2 research on critical reading is qualitative and tends to ignore inferences. This paper presents a qualitative, design-based study of a critical reading intervention focused on promoting generative rhetorical inferences and investigating co-adaptation and emergence of new meaning-making capacities. Complexity theory (CT) constructs were used to research processes of co-adaptation between the participants' comprehension and the teacher-researcher's understanding of learning and instructional needs. Identification of attractor states and control parameters in classroom discourse were used to explore unpredicted factors influencing the participants' inferential comprehension and further refine the intervention. The results indicate that rhetorical genre knowledge acted as a control parameter driving the students' comprehension to attractor states characterized by implausible inferences, and that this knowledge explains the emergence of pragmatic meaning (rhetorical inferences) from semantic meaning. The paper illustrates the usefulness of CT constructs in doing design-based research qualitatively in a manner that informs both theory and practice.

Recurrent motifs as resonant attractor states in the narrative field: a testable model of archetype

At the most basic level, archetypes represented Jung's attempt to explain the phenomenon of recurrent myths and folktale motifs (Jung 1956, 1959, para. 99). But the archetype remains controversial as an explanation of recurrent motifs, as the existence of recurrent motifs does not prove that archetypes exist. Thus, the challenge for contemporary archetype theory is not merely to demonstrate that recurrent motifs exist, since that is not disputed, but to demonstrate that archetypes exist and cause recurrent motifs. The present paper proposes a new model which is unlike others in that it postulates how the archetype creates resonant motifs. This model necessarily clarifies and adapts some of Jung's seminal ideas on archetype in order to provide a working framework grounded in contemporary practice and methodologies. For the first time, a model of archetype is proposed that can be validated on empirical, rather than theoretical grounds. This is achieved by linking the archetype to the hard data of recurrent motifs rather than academic trends in other fields.

Формирование самоорганизующихся отображений сенсорных сигналов на непрерывные нейросетевые аттракторы

In this article, we describe a simple binary neuron system, which implements a self-organized map. The system consists of R input neurons (R receptors), and N output neurons of a recurrent neural network. The neural network has a quasi-continuous set of attractor states (one-dimensional “bump attractor”). Due to the dynamics of the network, each external signal (i.e. activity state of receptors) imposes transition of the recurrent network into one of its stable states (points of its attractor). That makes our system different from the “winner takes all” construction of T.Kohonen. In case, when there is a one-dimensional cyclical manifold of external signals in R-dimensional input space, and the recurrent neural network presents a complete ring of neurons with local excitatory connections, there exists a process of learning of connections between the receptors and the neurons of the recurrent network, which enables a topologically correct mapping of input signals into the stable states of the neural network. The convergence rate of learning and the role of noises and other factors affecting the described phenomenon has been evaluated in computational simulations.

Accuracy and response-time distributions for decision-making: linear perfect integrators versus nonlinear attractor-based neural circuits

Animals choose actions based on imperfect, ambiguous data. “Noise” inherent in neural processing adds further variability to this already-noisy input signal. Mathematical analysis has suggested that the optimal apparatus (in terms of the speed/accuracy trade-off) for reaching decisions about such noisy inputs is perfect accumulation of the inputs by a temporal integrator. Thus, most highly cited models of neural circuitry underlying decision-making have been instantiations of a perfect integrator. Here, in accordance with a growing mathematical and empirical literature, we describe circumstances in which perfect integration is rendered suboptimal. In particular we highlight the impact of three biological constraints: (1) significant noise arising within the decision-making circuitry itself; (2) bounding of integration by maximal neural firing rates; and (3) time limitations on making a decision. Under conditions (1) and (2), an attractor system with stable attractor states can easily best an integrator when accuracy is more important than speed. Moreover, under conditions in which such stable attractor networks do not best the perfect integrator, a system with unstable initial states can do so if readout of the system’s final state is imperfect. Ubiquitously, an attractor system with a nonselective time-dependent input current is both more accurate and more robust to imprecise tuning of parameters than an integrator with such input. Given that neural responses that switch stochastically between discrete states can “masquerade” as integration in single-neuron and trial-averaged data, our results suggest that such networks should be considered as plausible alternatives to the integrator model.

Neuronal Avalanches Differ from Wakefulness to Deep Sleep – Evidence from Intracranial Depth Recordings in Humans

Neuronal activity differs between wakefulness and sleep states. In contrast, an attractor state, called self-organized critical (SOC), was proposed to govern brain dynamics because it allows for optimal information coding. But is the human brain SOC for each vigilance state despite the variations in neuronal dynamics? We characterized neuronal avalanches – spatiotemporal waves of enhanced activity - from dense intracranial depth recordings in humans. We showed that avalanche distributions closely follow a power law – the hallmark feature of SOC - for each vigilance state. However, avalanches clearly differ with vigilance states: slow wave sleep (SWS) shows large avalanches, wakefulness intermediate, and rapid eye movement (REM) sleep small ones. Our SOC model, together with the data, suggested first that the differences are mediated by global but tiny changes in synaptic strength, and second, that the changes with vigilance states reflect small deviations from criticality to the subcritical regime, implying that the human brain does not operate at criticality proper but close to SOC. Independent of criticality, the analysis confirms that SWS shows increased correlations between cortical areas, and reveals that REM sleep shows more fragmented cortical dynamics.

Signaling networks, network pathology and computational chemotherapy

Cellular signal transduction pathways are highly interconnected and form a complex network where many signaling molecules interact with each other in a complicated manner [1-2]. The complexity of the network structure suggests that the signal transduction network may conduct sophisticated information processing and decision making for cell fate determination rather than simple passing of an outer signal to the interior of the cell [3-4]. Cell fate, the stable cellular phenotype, is determined by the characteristic stable state of the complex network that is defined by a set of state values containing the activity states of all molecules in the network. Thus, systems biological investigation is crucial to understand the cell fate decision mechanism encoded in the nonlinear dynamics of the signal transduction network [5-6]. The recent study by Choi et al [7] shows that state space modeling and attractor landscape analysis of complex molecular networks can reveal a hidden cell fate decision mechanism and identify key network components that together could synergistically force cancer cells to undergo apoptosis. Choi et al investigated the regulatory mechanisms underlying p53 dynamics and its function in modulating cell fate outcome at the system level. In this study, the state transition dynamics of the p53 regulatory network were analyzed using Boolean network modeling and attractor landscape analysis where each network state is represented as a point in the cellular state space and the network state eventually converges to a fixed point (or a set of points) called the attractor state which represents the most stable state with the lowest potential energy in the state space. The analysis results revealed that the most critical network components, which determine both p53 dynamics and the cell fate outcome in response to DNA damage, are feedbacks and interactions between p53, Mdm2, Wip1, Cyclin G and ATM. Disruption of the above critical feedback controls was found to not only result in change of p53 dynamics, i.e. pulsing vs. sustained elevation, but also alteration of cell fate outcome. The attractor landscape analysis was then employed to investigate the DNA damage response of a representative breast cancer cell line, MCF7, and the effect of nutlin-3, a well-known inhibitor of Mdm2, in comparison with normal cells. Limited efficacy of nutlin-3 was indicated by its induction of small basin of attraction to apoptotic attractor in the state-space landscape with p53 dynamics mostly being oscillatory. In contrast, the analysis revealed that treatment of nutlin-3 in combination with inhibition of Wip1 would engender strong synergistic effect in activating p53-mediated apoptosis, as such combined perturbations resulted in a larger basin of attraction to apoptotic attractors with sustained elevation of p53 (Figure ​(Figure1).1). The predicted synergistic effect was validated by single-cell imaging experiment, using a fluorescent p53 reporter line of MCF7. The experimental data suggest that a combinatorial treatment of nutlin-3 and Wip1 inhibitor is a more effective strategy for inducing apoptosis in response to DNA damaging chemotherapeutics. This study demonstrates that system-level analysis of p53 network dynamics and its regulation using attractor landscape can be employed to understand the complex cell fate decision mechanism and identify novel therapeutic strategies for treating cancer. This study also suggests a possible paradigm shift to ‘network pathology’ where we use network information instead of molecular information for personalized diagnosis and treatment of complex disease. The idea of network pathology is that the different mutation profile of each patient can be reflected in the modified network structure for individualized state space analysis. It further suggests the new concept of ‘computational chemotherapy’ that can provide an optimal therapeutic strategy for each individual patient on the basis of network pathology. Figure 1 The state space analysis of the p53 network and the resulting potential energy landscapes for MCF7 cells after different inhibitory treatments that modify the network and induce cell death in the presence of DNA damage

Uncertainty and certainty in cellular dynamics

Uncertainty and certainty in cellular dynamics

Feedback Inhibition Enables Theta-Nested Gamma Oscillations and Grid Firing Fields

Cortical circuits are thought to multiplex firing rate codes with temporal codes that rely on oscillatory network activity, but the circuit mechanisms that combine these coding schemes are unclear. We establish with optogenetic activation of layer II of the medial entorhinal cortex that theta frequency drive to this circuit is sufficient to generate nested gamma frequency oscillations in synaptic activity. These nested gamma oscillations closely resemble activity during spatial exploration, are generated by local feedback inhibition without recurrent excitation, and have clock-like features suitable as reference signals for multiplexing temporal codes within rate-coded grid firing fields. In network models deduced from our data, feedback inhibition supports coexistence of theta-nested gamma oscillations with attractor states that generate grid firing fields. These results indicate that grid cells communicate primarily via inhibitory interneurons. This circuit mechanism enables multiplexing of oscillation-based temporal codes with rate-coded attractor states.

Stimulus number, duration and intensity encoding in randomly connected attractor networks with synaptic depression

Randomly connected recurrent networks of excitatory groups of neurons can possess a multitude of attractor states. When the internal excitatory synapses of these networks are depressing, the attractor states can be destabilized with increasing input. This leads to an itinerancy, where with either repeated transient stimuli, or increasing duration of a single stimulus, the network activity advances through sequences of attractor states. We find that the resulting network state, which persists beyond stimulus offset, can encode the number of stimuli presented via a distributed representation of neural activity with non-monotonic tuning curves for most neurons. Increased duration of a single stimulus is encoded via different distributed representations, so unlike an integrator, the network distinguishes separate successive presentations of a short stimulus from a single presentation of a longer stimulus with equal total duration. Moreover, different amplitudes of stimulus cause new, distinct activity patterns, such that changes in stimulus number, duration and amplitude can be distinguished from each other. These properties of the network depend on dynamic depressing synapses, as they disappear if synapses are static. Thus, short-term synaptic depression allows a network to store separately the different dynamic properties of a spatially constant stimulus.

The CRISP theory of hippocampal function in episodic memory

Over the past four decades, a “standard framework” has emerged to explain the neural mechanisms of episodic memory storage. This framework has been instrumental in driving hippocampal research forward and now dominates the design and interpretation of experimental and theoretical studies. It postulates that cortical inputs drive plasticity in the recurrent cornu ammonis 3 (CA3) synapses to rapidly imprint memories as attractor states in CA3. Here we review a range of experimental studies and argue that the evidence against the standard framework is mounting, notwithstanding the considerable evidence in its support. We propose CRISP as an alternative theory to the standard framework. CRISP is based on Context Reset by dentate gyrus (DG), Intrinsic Sequences in CA3, and Pattern completion in cornu ammonis 1 (CA1). Compared to previous models, CRISP uses a radically different mechanism for storing episodic memories in the hippocampus. Neural sequences are intrinsic to CA3, and inputs are mapped onto these intrinsic sequences through synaptic plasticity in the feedforward projections of the hippocampus. Hence, CRISP does not require plasticity in the recurrent CA3 synapses during the storage process. Like in other theories DG and CA1 play supporting roles, however, their function in CRISP have distinct implications. For instance, CA1 performs pattern completion in the absence of CA3 and DG contributes to episodic memory retrieval, increasing the speed, precision, and robustness of retrieval. We propose the conceptual theory, discuss its implications for experimental results and suggest testable predictions. It appears that CRISP not only accounts for those experimental results that are consistent with the standard framework, but also for results that are at odds with the standard framework. We therefore suggest that CRISP is a viable, and perhaps superior, theory for the hippocampal function in episodic memory.

Non-genetic adaptive dynamics for cellular robustness

Heterogeneity in molecular expressions is non-longer seen as a nuisance. Recent works have shown that it is required for cellular diversity, in terms of variable responses and for cell fate decisions (Selvarajoo, 2012, 2013). This commentary discusses on the importance of non-genetic heterogeneity for evolutionary robustness. Darwinian Theory of natural selection, involving heritable traits and mutations, has been long appreciated. However, there are other concepts that have received little attention or a lack of trust. One such theory is Lamarckism, which suggests that acquired characteristics of living systems can be passed over to its offspring without genetic modification. In other words, information regarding the part of a living organism that is often used and unused will help shape the offspring. This hypothesis, originally posed by Jean-Baptiste Lamarck in the early 19th century, has been long debated. With technological advances in biological experimentations, Lamarckism is gaining interest in the 21st century. This, in part, is due to the appreciation of epigenetics or non-genetic variability observed in living cells. We know epigenetics allow cells to flexibly adapt to changing environments within a lifetime. Lamarckism, on the other hand, suggests the passing on of such flexibility in a “programmed” manner to the offspring. This phenomenon has been tested on different cell types in recent years. Originally, Sui Huang and colleagues (Chang et al., 2008) studied cell-to-cell variability in mouse hematopoietic progenitor cells, and found the presence of metastable gene expression (attractor) states. Firstly, they showed the expression of Sca-1 protein between cells was heterogeneous and followed a Gaussian-like distribution. Subsequently, they sorted cells using flow cytometry into low, medium and high Sca-1 concentration and re-cultured the cells. Remarkably, the offspring regained the original parental distribution of Sca-1 after 48 h (Figure ​(Figure1A,1A, Chang et al., 2008). This result suggested the existence of “pre-programmed” states in cells that are regained despite perturbing the cells from their physiological state, independent of selection. Figure 1 Non-genetic heterogeneity in living cells. Re-culturing cells sorted into (A) low, medium and high expressions of Sca-1 in mouse hematopoietic cells, (B) SL, Basal, and Luminal expressed breast carcinoma cells, returns to original parental metastable ... When the sorted cells were perturbed with distinct stimuli, cells with low Sca-1 were more prone for erythroid lineage whereas cells with high Sca-1 were myeloid prone. Thus, non-genetic molecular heterogeneity has a role for diversity in cell differentiation process. In another work investigating heterogeneity in cancer cells, similar metastable attractor phenomenon was observed (Gupta et al., 2011). Parental population of heterogeneous breast cancer lines were sorted into three subpopulations of mammary epithelial cell states characterized by distinct cell-surface markers: stem-like, basal, and luminal (Figure ​(Figure1B,1B, Gupta et al., 2011). Culturing the sorted “homogenous” cells over six days gradually returned the offspring to their parental heterogeneous populations. Again, this result demonstrated the presence of strong non-genetic equilibrium state. More recently, Sui Huang and colleagues investigated the non-genetic ability or plasticity of cancer cells to therapeutic intervention (Pisco et al., 2013). They investigated the cause for drug resistance in HL60 leukemic cells. Cancer cells can become drug resistant due to the role of certain proteins such as the multidrug resistance protein 1 (MDR1), which is responsible to guide a variety of drugs out of the cells. Untreated leukemic cells showed single distribution of MDR1 expressions with a large majority of cells possessing low expressions and only about 1–2% showing high MDR1 expressions required for drug resistant levels [Figure 1A in Pisco et al. (2013)]. Interestingly, treatment with the drug, vincristine (VINC), showed emergence of bimodal MDR1 distributions, where now a certain number of cells possessed increased MDR1 expressions crucial for drug resistance both with respect to time and with increasing doses of VINC (Figure ​(Figure1C,1C, Pisco et al., 2013). This emergent behavior is also independent of selection. Re-culturing cells derived from specifically low and high MDR1 expressions both produced offspring that possessed the original parental single modal MDR1 distribution, however, with different relaxation times. Overall, the experiments from these studies clearly provide evidence that non-genetic heterogeneity is crucial for the adaptability and survivability of living cells to perturbation within their lifetime, and this trait is passed on to their offspring. It is, therefore, conceivable that living systems are not only governed by Darwinian evolutionary principles (longer term), but also through soft inheritance described by Lamarckism (shorter term) for managing survival and adaptation over different time scales.

A hexapod walker using a heterarchical architecture for action selection

Moving in a cluttered environment with a six-legged walking machine that has additional body actuators, therefore controlling 22 DoFs, is not a trivial task. Already simple forward walking on a flat plane requires the system to select between different internal states. The orchestration of these states depends on walking velocity and on external disturbances. Such disturbances occur continuously, for example due to irregular up-and-down movements of the body or slipping of the legs, even on flat surfaces, in particular when negotiating tight curves. The number of possible states is further increased when the system is allowed to walk backward or when front legs are used as grippers and cannot contribute to walking. Further states are necessary for expansion that allow for navigation. Here we demonstrate a solution for the selection and sequencing of different (attractor) states required to control different behaviors as are forward walking at different speeds, backward walking, as well as negotiation of tight curves. This selection is made by a recurrent neural network (RNN) of motivation units, controlling a bank of decentralized memory elements in combination with the feedback through the environment. The underlying heterarchical architecture of the network allows to select various combinations of these elements. This modular approach representing an example of neural reuse of a limited number of procedures allows for adaptation to different internal and external conditions. A way is sketched as to how this approach may be expanded to form a cognitive system being able to plan ahead. This architecture is characterized by different types of modules being arranged in layers and columns, but the complete network can also be considered as a holistic system showing emergent properties which cannot be attributed to a specific module.

Qualia Could Arise from Information Processing in Local Cortical Networks

Re-entrant feedback, either within sensory cortex or arising from prefrontal areas, has been strongly linked to the emergence of consciousness, both in theoretical and experimental work. This idea, together with evidence for local micro-consciousness, suggests the generation of qualia could in some way result from local network activity under re-entrant activation. This paper explores the possibility by examining the processing of information by local cortical networks. It highlights the difference between the information structure (how the information is physically embodied), and the information message (what the information is about). It focuses on the network’s ability to recognize information structures amongst its inputs under conditions of extensive local feedback, and to then assign information messages to those structures. It is shown that if the re-entrant feedback enables the network to achieve an attractor state, then the message assigned in any given pass of information through the network is a representation of the message assigned in the previous pass-through of information. Based on this ability the paper argues that as information is repeatedly cycled through the network, the information message that is assigned evolves from a recognition of what the input structure is, to what it is like, to how it appears, to how it seems. It could enable individual networks to be the site of qualia generation. The paper goes on to show networks in cortical layers 2/3 and 5a have the connectivity required for the behavior proposed, and reviews some evidence for a link between such local cortical cyclic activity and conscious percepts. It concludes with some predictions based on the theory discussed.

SIMPLE CYCLIC MOVEMENTS AS A DISTINCT AUTISM FEATURE - COMPUTATIONAL APPROACH

Diversity of symptoms in autism dictates a broad definition of Autism Spectrum of Disorders(ASD). Each year percentage of children diagnosed with ASD is growing. One common diag-nostic feature in individuals with ASD is the tendency to atypical simple cyclic movements.The motor brain activity seems to generate periodic attractor state that is hard to escape.Despite numerous studies scientists and clinicians do not know exactly if ASD is a result ofa simple but general mechanism, or a complex set of mechanisms, both on neural, molecularand system levels. Simulations using biologically relevant neural network model presentedhere may help to reveal simplest mechanisms that may be responsible for specific behavior.Abnormal neural fatigue mechanisms may be responsible for motor as well as many if notall other symptoms observed in ASD.

Boundary Surface of 5-Valued Memory

The subject of research in this paper is multiple-valued (MV) memory cell—particularly the morphology of boundary surface of five-valued memory. By accepting the values of parasitic accumulation elements on the chip, very complicated morphology of the boundary surfaces occurs, which separates various attractors from each other. This is due to the occurrence of undesirable oscillations—a stable limit cycles, which makes it impossible to control memory. These dynamic attractors are so dominant that their regions of attraction even surround regions of attraction of static attractors—required logic levels of memory. Therefore, in the realization of the MV memory on the chip is necessary to know the values of the parasitic elements, because their presence may cause a malfunction of the memory. In this case, only calculation and displaying the boundary surface provides exact answers related to operation of the MV memory.

Transtibial Amputee Joint Motion has Increased Attractor Divergence During Walking Compared to Non-Amputee Gait

The amputation and subsequent prosthetic rehabilitation of a lower leg affects gait. Dynamical systems theory would predict the use of a prosthetic device should alter the functional attractor dynamics to which the system self-organizes. Therefore, the purpose of this study was to compare the largest Lyapunov exponent (a nonlinear tool for assessing attractor dynamics) for amputee gait compared to healthy non-amputee individuals. Fourteen unilateral, transtibial amputees and fourteen healthy, non-amputee individuals ambulated on a treadmill at preferred, self-selected walking speed. Our results showed that the sound hip (p = 0.013), sound knee (p = 0.05), and prosthetic ankle (p = 0.023) have significantly greater largest Lyapunov exponents than healthy non-amputees. Furthermore, the prosthetic ankle has a significantly greater (p = 0.0.17) largest Lyapunov exponent than the sound leg ankle. These findings indicate attractor states for amputee gait with increased divergence. The increased attractor divergence seems to coincide with decreased ability for motor control between the natural rhythms of the individual and those of the prosthetic device. Future work should consider the impact of different prostheses and rehabilitation on the attractor dynamics.

Stability analysis and observational measurement in 5-D Brans–Dicke cosmology

This Letter is a study of the effects of higher dimensional gravity and Brans–Dicke (BD) scalar field on cosmic acceleration in 5-D BD cosmological model. We assume a flat cosmological model in which the matter content of the universe is either cold dark matter or radiation. In a framework to study attractor solutions in the phase space we simultaneously constrain the model parameters with the observational data for distance modulus. The phase space analysis illustrates that the universe begins from an unstable state in the past and eventually reaches an asymptotically stable state (attractor). We examine the model by performing Hubble parameter test in addition to statefinder diagnosis. We also reconstruct the equation of state parameter, the scale factor in 3-D space and along extra dimension. The results show that due to the presence of extra dimension and Brans–Dicke scalar field in the model, the universe undergoes a period of acceleration.

Coordinated Optimization of Visual Cortical Maps (II) Numerical Studies

In the juvenile brain, the synaptic architecture of the visual cortex remains in a state of flux for months after the natural onset of vision and the initial emergence of feature selectivity in visual cortical neurons. It is an attractive hypothesis that visual cortical architecture is shaped during this extended period of juvenile plasticity by the coordinated optimization of multiple visual cortical maps such as orientation preference (OP), ocular dominance (OD), spatial frequency, or direction preference. In part (I) of this study we introduced a class of analytically tractable coordinated optimization models and solved representative examples, in which a spatially complex organization of the OP map is induced by interactions between the maps. We found that these solutions near symmetry breaking threshold predict a highly ordered map layout. Here we examine the time course of the convergence towards attractor states and optima of these models. In particular, we determine the timescales on which map optimization takes place and how these timescales can be compared to those of visual cortical development and plasticity. We also assess whether our models exhibit biologically more realistic, spatially irregular solutions at a finite distance from threshold, when the spatial periodicities of the two maps are detuned and when considering more than 2 feature dimensions. We show that, although maps typically undergo substantial rearrangement, no other solutions than pinwheel crystals and stripes dominate in the emerging layouts. Pinwheel crystallization takes place on a rather short timescale and can also occur for detuned wavelengths of different maps. Our numerical results thus support the view that neither minimal energy states nor intermediate transient states of our coordinated optimization models successfully explain the architecture of the visual cortex. We discuss several alternative scenarios that may improve the agreement between model solutions and biological observations.