Attractor States Research Articles

Domain control of nonlinear networked systems and applications to complex disease networks

The control of complex nonlinear dynamical networks is an ongoing challenge in diverse contexts ranging from biology to social sciences. To explore a practical framework for controlling nonlinear dynamical networks based on meaningful physical and experimental considerations, we propose a new concept of the domain control for nonlinear dynamical networks, i.e., the control of a nonlinear network in transition from the domain of attraction of an undesired state (attractor) to the domain of attraction of a desired state. We theoretically prove the existence of a domain control. In particular, we offer an approach for identifying the driver nodes that need to be controlled and design a general form of control functions for realizing domain controllability. In addition, we demonstrate the effectiveness of our theory and approaches in three realistic disease-related networks: the epithelial-mesenchymal transition (EMT) core network, the T helper (Th) differentiation cellular network and the cancer network. Moreover, we reveal certain genes that are critical to phenotype transitions of these systems. Therefore, the approach described here not only offers a practical control scheme for nonlinear dynamical networks but also helps the development of new strategies for the prevention and treatment of complex diseases.

Compensatory interactions to stabilize multiple steady states or mitigate the effects of multiple deregulations in biological networks.

Complex diseases can be modeled as damage to intracellular networks that results in abnormal cell behaviors. Network-based dynamic models such as Boolean models have been employed to model a variety of biological systems including those corresponding to disease. Previous work designed compensatory interactions to stabilize an attractor of a Boolean network after single node damage. We generalize this method to a multinode damage scenario and to the simultaneous stabilization of multiple steady state attractors. We classify the emergent situations, with a special focus on combinatorial effects, and characterize each class through simulation. We explore how the structural and functional properties of the network affect its resilience and its possible repair scenarios. We demonstrate the method's applicability to two intracellular network models relevant to cancer. This work has implications in designing prevention strategies for complex disease.

Cell Fate Decision as High-Dimensional Critical State Transition.

Cell fate choice and commitment of multipotent progenitor cells to a differentiated lineage requires broad changes of their gene expression profile. But how progenitor cells overcome the stability of their gene expression configuration (attractor) to exit the attractor in one direction remains elusive. Here we show that commitment of blood progenitor cells to the erythroid or myeloid lineage is preceded by the destabilization of their high-dimensional attractor state, such that differentiating cells undergo a critical state transition. Single-cell resolution analysis of gene expression in populations of differentiating cells affords a new quantitative index for predicting critical transitions in a high-dimensional state space based on decrease of correlation between cells and concomitant increase of correlation between genes as cells approach a tipping point. The detection of “rebellious cells” that enter the fate opposite to the one intended corroborates the model of preceding destabilization of a progenitor attractor. Thus, early warning signals associated with critical transitions can be detected in statistical ensembles of high-dimensional systems, offering a formal theory-based approach for analyzing single-cell molecular profiles that goes beyond current computational pattern recognition, does not require knowledge of specific pathways, and could be used to predict impending major shifts in development and disease.

The global dynamical complexity of the human brain network.

How much information do large brain networks integrate as a whole over the sum of their parts? Can the dynamical complexity of such networks be globally quantified in an information-theoretic way and be meaningfully coupled to brain function? Recently, measures of dynamical complexity such as integrated information have been proposed. However, problems related to the normalization and Bell number of partitions associated to these measures make these approaches computationally infeasible for large-scale brain networks. Our goal in this work is to address this problem. Our formulation of network integrated information is based on the Kullback-Leibler divergence between the multivariate distribution on the set of network states versus the corresponding factorized distribution over its parts. We find that implementing the maximum information partition optimizes computations. These methods are well-suited for large networks with linear stochastic dynamics. We compute the integrated information for both, the system’s attractor states, as well as non-stationary dynamical states of the network. We then apply this formalism to brain networks to compute the integrated information for the human brain’s connectome. Compared to a randomly re-wired network, we find that the specific topology of the brain generates greater information complexity.

Maximum Entropy Production Is Not a Steady State Attractor for 2D Fluid Convection

Multiple authors have claimed that the natural convection of a fluid is a process that exhibits maximum entropy production (MEP). However, almost all such investigations were limited to fixed temperature boundary conditions (BCs). It was found that under those conditions, the system tends to maximize its heat flux, and hence it was concluded that the MEP state is a dynamical attractor. However, since entropy production varies with heat flux and difference of inverse temperature, it is essential that any complete investigation of entropy production allows for variations in heat flux and temperature difference. Only then can we legitimately assess whether the MEP state is the most attractive. Our previous work made use of negative feedback BCs to explore this possibility. We found that the steady state of the system was far from the MEP state. For any system, entropy production can only be maximized subject to a finite set of physical and material constraints. In the case of our previous work, it was possible that the adopted set of fluid parameters were constraining the system in such a way that it was entirely prevented from reaching the MEP state. Hence, in the present work, we used a different set of boundary parameters, such that the steady states of the system were in the local vicinity of the MEP state. If MEP was indeed an attractor, relaxing those constraints of our previous work should have caused a discrete perturbation to the surface of steady state heat flux values near the value corresponding to MEP. We found no such perturbation, and hence no discernible attraction to the MEP state. Furthermore, systems with fixed flux BCs actually minimize their entropy production (relative to the alternative stable state, that of pure diffusive heat transport). This leads us to conclude that the principle of MEP is not an accurate indicator of which stable steady state a convective system will adopt. However, for all BCs considered, the quotient of heat flux and temperature difference F / Δ T —which is proportional to the dimensionless Nusselt number—does appear to be maximized.

Changing Patterns of Scaffolding and Autonomy During Individual Music Lessons: A Mixed Methods Approach

Learning implies change. Inside and outside schools, we aim for students to change into people with more skills and knowledge as well as with a growing sense of agency and responsibility. Furthermore, education itself is subject to constant change. In this article, we examine change in 3 case studies in the context of individual music lessons from a complex dynamic systems approach. Three beginning string instrument students and their teacher were followed for 18 months (28 lessons per dyad) by means of video observations. We combine cluster analyses on longitudinal data with qualitative illustrations of moment-to-moment teacher–student transactions to gain insight into how and why change in teaching and learning occurs. The 3 cases show distinct profiles of change: the development of strong suboptimal attractor states, the emergence of optimal attractor states, and a profile that is characterized by falling back into suboptimal attractor states. We discuss how optimal and suboptimal learning and teaching trajectories can unfold over time and how a mixed methods approach can provide valuable new insights for both practice and future research.

Developing Dynamic Field Theory Architectures for Embodied Cognitive Systems with cedar.

Embodied artificial cognitive systems, such as autonomous robots or intelligent observers, connect cognitive processes to sensory and effector systems in real time. Prime candidates for such embodied intelligence are neurally inspired architectures. While components such as forward neural networks are well established, designing pervasively autonomous neural architectures remains a challenge. This includes the problem of tuning the parameters of such architectures so that they deliver specified functionality under variable environmental conditions and retain these functions as the architectures are expanded. The scaling and autonomy problems are solved, in part, by dynamic field theory (DFT), a theoretical framework for the neural grounding of sensorimotor and cognitive processes. In this paper, we address how to efficiently build DFT architectures that control embodied agents and how to tune their parameters so that the desired cognitive functions emerge while such agents are situated in real environments. In DFT architectures, dynamic neural fields or nodes are assigned dynamic regimes, that is, attractor states and their instabilities, from which cognitive function emerges. Tuning thus amounts to determining values of the dynamic parameters for which the components of a DFT architecture are in the specified dynamic regime under the appropriate environmental conditions. The process of tuning is facilitated by the software framework cedar, which provides a graphical interface to build and execute DFT architectures. It enables to change dynamic parameters online and visualize the activation states of any component while the agent is receiving sensory inputs in real time. Using a simple example, we take the reader through the workflow of conceiving of DFT architectures, implementing them on embodied agents, tuning their parameters, and assessing performance while the system is coupled to real sensory inputs.

A Cache System Design for CMPs with Built-In Coherence Verification

This work reports an effective design of cache system for Chip Multiprocessors (CMPs). It introduces built-in logic for verification of cache coherence in CMPs realizing directory based protocol. It is developed around the cellular automata (CA) machine, invented by John von Neumann in the 1950s. A special class of CA referred to as single length cycle 2-attractor cellular automata (TACA) has been planted to detect the inconsistencies in cache line states of processors’ private caches. The TACA module captures coherence status of the CMPs’ cache system and memorizes any inconsistent recording of the cache line states during the processors’ reference to a memory block. Theory has been developed to empower a TACA to analyse the cache state updates and then to settle to an attractor state indicating quick decision on a faulty recording of cache line status. The introduction of segmentation of the CMPs’ processor pool ensures a better efficiency, in determining the inconsistencies, by reducing the number of computation steps in the verification logic. The hardware requirement for the verification logic points to the fact that the overhead of proposed coherence verification module is much lesser than that of the conventional verification units and is insignificant with respect to the cost involved in CMPs’ cache system.

An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network.

BackgroundBoolean network modeling has been widely used to model large-scale biomolecular regulatory networks as it can describe the essential dynamical characteristics of complicated networks in a relatively simple way. When we analyze such Boolean network models, we often need to find out attractor states to investigate the converging state features that represent particular cell phenotypes. This is, however, very difficult (often impossible) for a large network due to computational complexity.ResultsThere have been some attempts to resolve this problem by partitioning the original network into smaller subnetworks and reconstructing the attractor states by integrating the local attractors obtained from each subnetwork. But, in many cases, the partitioned subnetworks are still too large and such an approach is no longer useful. So, we have investigated the fundamental reason underlying this problem and proposed a novel efficient way of hierarchically partitioning a given large network into smaller subnetworks by focusing on some attractors corresponding to a particular phenotype of interest instead of considering all attractors at the same time. Using the definition of attractors, we can have a simplified update rule with fixed state values for some nodes. The resulting subnetworks were small enough to find out the corresponding local attractors which can be integrated for reconstruction of the global attractor states of the original large network.ConclusionsThe proposed approach can substantially extend the current limit of Boolean network modeling for converging state analysis of biological networks.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-016-0338-4) contains supplementary material, which is available to authorized users.

Dynamic Organization of Hierarchical Memories

In the brain, external objects are categorized in a hierarchical way. Although it is widely accepted that objects are represented as static attractors in neural state space, this view does not take account interaction between intrinsic neural dynamics and external input, which is essential to understand how neural system responds to inputs. Indeed, structured spontaneous neural activity without external inputs is known to exist, and its relationship with evoked activities is discussed. Then, how categorical representation is embedded into the spontaneous and evoked activities has to be uncovered. To address this question, we studied bifurcation process with increasing input after hierarchically clustered associative memories are learned. We found a “dynamic categorization”; neural activity without input wanders globally over the state space including all memories. Then with the increase of input strength, diffuse representation of higher category exhibits transitions to focused ones specific to each object. The hierarchy of memories is embedded in the transition probability from one memory to another during the spontaneous dynamics. With increased input strength, neural activity wanders over a narrower state space including a smaller set of memories, showing more specific category or memory corresponding to the applied input. Moreover, such coarse-to-fine transitions are also observed temporally during transient process under constant input, which agrees with experimental findings in the temporal cortex. These results suggest the hierarchy emerging through interaction with an external input underlies hierarchy during transient process, as well as in the spontaneous activity.

The Impact of Cortical Lesions on Thalamo-Cortical Network Dynamics after Acute Ischaemic Stroke: A Combined Experimental and Theoretical Study.

The neocortex and thalamus provide a core substrate for perception, cognition, and action, and are interconnected through different direct and indirect pathways that maintain specific dynamics associated with functional states including wakefulness and sleep. It has been shown that a lack of excitation, or enhanced subcortical inhibition, can disrupt this system and drive thalamic nuclei into an attractor state of low-frequency bursting and further entrainment of thalamo-cortical circuits, also called thalamo-cortical dysrhythmia (TCD). The question remains however whether similar TCD-like phenomena can arise with a cortical origin. For instance, in stroke, a cortical lesion could disrupt thalamo-cortical interactions through an attenuation of the excitatory drive onto the thalamus, creating an imbalance between excitation and inhibition that can lead to a state of TCD. Here we tested this hypothesis by comparing the resting-state EEG recordings of acute ischaemic stroke patients (N = 21) with those of healthy, age-matched control-subjects (N = 17). We observed that these patients displayed the hallmarks of TCD: a characteristic downward shift of dominant α-peaks in the EEG power spectra, together with increased power over the lower frequencies (δ and θ-range). Contrary to general observations in TCD, the patients also displayed a broad reduction in β-band activity. In order to explain the genesis of this stroke-induced TCD, we developed a biologically constrained model of a general thalamo-cortical module, allowing us to identify the specific cellular and network mechanisms involved. Our model showed that a lesion in the cortical component leads to sustained cell membrane hyperpolarization in the corresponding thalamic relay neurons, that in turn leads to the de-inactivation of voltage-gated T-type Ca2+-channels, switching neurons from tonic spiking to a pathological bursting regime. This thalamic bursting synchronises activity on a population level through divergent intrathalamic circuits, and entrains thalamo-cortical pathways by means of propagating low-frequency oscillations beyond the restricted region of the lesion. Hence, pathological stroke-induced thalamo-cortical dynamics can be the source of diaschisis, and account for the dissociation between lesion location and non-specific symptoms of stroke such as neuropathic pain and hemispatial neglect.

A dynamical framework to relate perceptual variability with multisensory information processing.

Multisensory processing involves participation of individual sensory streams, e.g., vision, audition to facilitate perception of environmental stimuli. An experimental realization of the underlying complexity is captured by the “McGurk-effect”- incongruent auditory and visual vocalization stimuli eliciting perception of illusory speech sounds. Further studies have established that time-delay between onset of auditory and visual signals (AV lag) and perturbations in the unisensory streams are key variables that modulate perception. However, as of now only few quantitative theoretical frameworks have been proposed to understand the interplay among these psychophysical variables or the neural systems level interactions that govern perceptual variability. Here, we propose a dynamic systems model consisting of the basic ingredients of any multisensory processing, two unisensory and one multisensory sub-system (nodes) as reported by several researchers. The nodes are connected such that biophysically inspired coupling parameters and time delays become key parameters of this network. We observed that zero AV lag results in maximum synchronization of constituent nodes and the degree of synchronization decreases when we have non-zero lags. The attractor states of this network can thus be interpreted as the facilitator for stabilizing specific perceptual experience. Thereby, the dynamic model presents a quantitative framework for understanding multisensory information processing.

Dissecting Embryonic Stem Cell Self-Renewal and Differentiation Commitment from Quantitative Models.

To model quantitatively embryonic stem cell (ESC) self-renewal and differentiation by computational approaches, we developed a unified mathematical model for gene expression involved in cell fate choices. Our quantitative model comprised ESC master regulators and lineage-specific pivotal genes. It took the factors of multiple pathways as input and computed expression as a function of intrinsic transcription factors, extrinsic cues, epigenetic modifications, and antagonism between ESC master regulators and lineage-specific pivotal genes. In the model, the differential equations of expression of genes involved in cell fate choices from regulation relationship were established according to the transcription and degradation rates. We applied this model to the Murine ESC self-renewal and differentiation commitment and found that it modeled the expression patterns with good accuracy. Our model analysis revealed that Murine ESC was an attractor state in culture and differentiation was predominantly caused by antagonism between ESC master regulators and lineage-specific pivotal genes. Moreover, antagonism among lineages played a critical role in lineage reprogramming. Our results also uncovered that the ordered expression alteration of ESC master regulators over time had a central role in ESC differentiation fates. Our computational framework was generally applicable to most cell-type maintenance and lineage reprogramming.

Self‐adjustment in rivers: Evidence for least action as the primary control of alluvial‐channel form and process

AbstractLeast action principle (LAP) in rivers is demonstrated by maximum flow efficiency (MFE) and is the foundation of variational mechanics based on energy and work rather than Newtonian force and momentum. Empirical evidence shows it to be the primary control for the adjustment of alluvial channels. Because most rivers flow with imposed water and sediment loads down valley gradients they have largely inherited, they self‐regulate energy expenditure to match the work they are required to do to remain stable. Overpowered systems develop a variety of channel patterns to expend excess energy and remain stable. Australia offers an opportunity to study low‐energy rivers closely adjusted to very low continental gradients. The anabranching Marshall and single‐thread Plenty Rivers flow down nearly straight channels with average H numbers [ratio between excess bed shear and width/depth (W/D) ratio] close to the optimum of 0.3 for stationary equilibrium. Ridge‐form divisions of the original channel width create anabranches that radically alter W/D ratios relative to bed shear, the same being true for short‐wide islands on the large low‐gradient Yangtze River in China. In contrast, Mount Chambers Creek in Australia's tectonically more active Flinders Ranges is accreting an alluvial fan with unstable distributary channels exhibiting H numbers well below the optimum. LAP also explains profound biases in Earth's stratigraphic record. Because meandering is an energy‐shedding mechanism, sinuous rivers sequester relatively little sediment resulting in all sequences being just a few tens of metres thick. In contrast, low‐energy braided disequilibrium systems can sequester sediment piles over a kilometre in thickness and tens of kilometres wide. LAP provides a new paradigm for river research by identifying the attractor state controlling river channel evolution. It links advances in theoretical physics to fluvial geomorphology, stratigraphy and hydraulic engineering and opens opportunities for diverse investigations in Earth system science. Copyright © 2016 John Wiley & Sons, Ltd.

Hippocampal Attractor Dynamics Predict Memory-Based Decision Making

Memories are thought to be retrieved by attractor dynamics if a given input is sufficiently similar to a stored attractor state [1-5]. The hippocampus, a region crucial for spatial navigation [6-12] and episodic memory [13-18], has been associated with attractor-based computations [5, 9], receiving support from the way rodent place cells "remap" nonlinearly between spatial representations [19-22]. In humans, nonlinear response patterns have been reported in perceptual categorization tasks [23-25]; however, it remains elusive whether human memory retrieval isdriven by attractor dynamics and what neural mechanisms might underpin them. To test this, we used a virtual reality [7, 11, 26-28] task where participants learned object-location associations within two distinct virtual reality environments. Participants were subsequently exposed to four novel intermediate environments, generated by linearly morphing the background landscapes of the familiar environments, while tracking fMRI activity. We show that linear changes in environmental context cause linear changes in activity patterns in sensory cortex but cause dynamic, nonlinear changes in both hippocampal activity pattern and remembered locations. Furthermore, the sigmoidal response in the hippocampus scaled with the strength of the sigmoidal pattern in spatial memory. These results indicate that mnemonic decisions in an ambiguous novel context relate to putative attractor dynamics in the hippocampus, which support the dynamic remapping of memories.

Physical mechanism of mind changes and tradeoffs among speed, accuracy, and energy cost in brain decision making: Landscape, flux, and path perspectives**Project supported by the National Natural Science Foundation of China (Grant Nos. 21190040, 91430217, and 11305176).

Cognitive behaviors are determined by underlying neural networks. Many brain functions, such as learning and memory, have been successfully described by attractor dynamics. For decision making in the brain, a quantitative description of global attractor landscapes has not yet been completely given. Here, we developed a theoretical framework to quantify the landscape associated with the steady state probability distributions and associated steady state curl flux, measuring the degree of non-equilibrium through the degree of detailed balance breaking for decision making. We quantified the decision-making processes with optimal paths from the undecided attractor states to the decided attractor states, which are identified as basins of attractions, on the landscape. Both landscape and flux determine the kinetic paths and speed. The kinetics and global stability of decision making are explored by quantifying the landscape topography through the barrier heights and the mean first passage time. Our theoretical predictions are in agreement with experimental observations: more errors occur under time pressure. We quantitatively explored two mechanisms of the speed-accuracy tradeoff with speed emphasis and further uncovered the tradeoffs among speed, accuracy, and energy cost. Our results imply that there is an optimal balance among speed, accuracy, and the energy cost in decision making. We uncovered the possible mechanisms of changes of mind and how mind changes improve performance in decision processes. Our landscape approach can help facilitate an understanding of the underlying physical mechanisms of cognitive processes and identify the key factors in the corresponding neural networks.

A Robust Feedforward Model of the Olfactory System.

Most natural odors have sparse molecular composition. This makes the principles of compressed sensing potentially relevant to the structure of the olfactory code. Yet, the largely feedforward organization of the olfactory system precludes reconstruction using standard compressed sensing algorithms. To resolve this problem, recent theoretical work has shown that signal reconstruction could take place as a result of a low dimensional dynamical system converging to one of its attractor states. However, the dynamical aspects of optimization slowed down odor recognition and were also found to be susceptible to noise. Here we describe a feedforward model of the olfactory system that achieves both strong compression and fast reconstruction that is also robust to noise. A key feature of the proposed model is a specific relationship between how odors are represented at the glomeruli stage, which corresponds to a compression, and the connections from glomeruli to third-order neurons (neurons in the olfactory cortex of vertebrates or Kenyon cells in the mushroom body of insects), which in the model corresponds to reconstruction. We show that should this specific relationship hold true, the reconstruction will be both fast and robust to noise, and in particular to the false activation of glomeruli. The predicted connectivity rate from glomeruli to third-order neurons can be tested experimentally.

Is cortical connectivity optimized for storing information?

Cortical networks are thought to be shaped by experience-dependent synaptic plasticity. Theoretical studies have shown that synaptic plasticity allows a network to store a memory of patterns of activity such that they become attractors of the dynamics of the network. Here we study the properties of the excitatory synaptic connectivity in a network that maximizes the number of stored patterns of activity in a robust fashion. We show that the resulting synaptic connectivity matrix has the following properties: it is sparse, with a large fraction of zero synaptic weights ('potential' synapses); bidirectionally coupled pairs of neurons are over-represented in comparison to a random network; and bidirectionally connected pairs have stronger synapses on average than unidirectionally connected pairs. All these features reproduce quantitatively available data on connectivity in cortex. This suggests synaptic connectivity in cortex is optimized to store a large number of attractor states in a robust fashion.

Corticospinal Modulations during Bimanual Movement with Different Relative Phases.

The purpose of this study was to investigate corticospinal modulation of bimanual (BM) movement with different relative phases (RPs). The participants rhythmically abducted and adducted the right index finger (unimanual (UM) movement) or both index fingers (BM movement) with a cyclic duration of 1 s. The RP of BM movement, defined as the time difference between one hand movement and the other hand movement, was 0°, 90°, or 180°. Motor evoked potentials (MEPs) in the right flexor dorsal interosseous muscle elicited by transcranial magnetic stimulation (TMS) were obtained during UM or BM movement. Corticospinal excitability in the first dorsal interosseous muscle during BM movement with 90° RP was higher than that during UM movement or BM movement with 0° or 180° RP. The correlation between muscle activity level and corticospinal excitability during BM movement with 90° RP was smaller than that during UM movement or BM movement with 0° or 180° RP. The higher corticospinal excitability during BM movement with 90° RP may be caused by the greater effort expended to execute a difficult task, the involvement of interhemispheric interaction, a motor binding process, or task acquisition. The lower dependency of corticospinal excitability on the muscle activity level during BM movement with 90° RP may reflect the minor corticospinal contribution to BM movement with an RP that is not in the attractor state.

Relative stability of network states in Boolean network models of gene regulation in development

Progress in cell type reprogramming has revived the interest in Waddington’s concept of the epigenetic landscape. Recently researchers developed the quasi-potential theory to represent the Waddington’s landscape. The Quasi-potential U(x), derived from interactions in the gene regulatory network (GRN) of a cell, quantifies the relative stability of network states, which determine the effort required for state transitions in a multi-stable dynamical system. However, quasi-potential landscapes, originally developed for continuous systems, are not suitable for discrete-valued networks which are important tools to study complex systems. In this paper, we provide a framework to quantify the landscape for discrete Boolean networks (BNs). We apply our framework to study pancreas cell differentiation where an ensemble of BN models is considered based on the structure of a minimal GRN for pancreas development. We impose biologically motivated structural constraints (corresponding to specific type of Boolean functions) and dynamical constraints (corresponding to stable attractor states) to limit the space of BN models for pancreas development. In addition, we enforce a novel functional constraint corresponding to the relative ordering of attractor states in BN models to restrict the space of BN models to the biological relevant class. We find that BNs with canalyzing/sign-compatible Boolean functions best capture the dynamics of pancreas cell differentiation. This framework can also determine the genes' influence on cell state transitions, and thus can facilitate the rational design of cell reprogramming protocols.