Neuropercolation: A Random Cellular Automata Approach to Spatio-temporal Neurodynamics

The 2-D Cluster Variation Method: Topography Illustrations and Their Enthalpy Parameter Correlations

One of the biggest challenges in characterizing 2-D image topographies is finding a low-dimensional parameter set that can succinctly describe, not so much image patterns themselves, but the nature of these patterns. The 2-D cluster variation method (CVM), introduced by Kikuchi in 1951, can characterize very local image pattern distributions using configuration variables, identifying nearest-neighbor, next-nearest-neighbor, and triplet configurations. Using the 2-D CVM, we can characterize 2-D topographies using just two parameters; the activation enthalpy (ε0) and the interaction enthalpy (ε1). Two different initial topographies (“scale-free-like” and “extreme rich club-like”) were each computationally brought to a CVM free energy minimum, for the case where the activation enthalpy was zero and different values were used for the interaction enthalpy. The results are: (1) the computational configuration variable results differ significantly from the analytically-predicted values well before ε1 approaches the known divergence as ε1→0.881, (2) the range of potentially useful parameter values, favoring clustering of like-with-like units, is limited to the region where ε0<3 and ε1<0.25, and (3) the topographies in the systems that are brought to a free energy minimum show interesting visual features, such as extended “spider legs” connecting previously unconnected “islands,” and as well as evolution of “peninsulas” in what were previously solid masses.

Scaling in Colloidal and Biological Networks

Scaling and dimensional analysis is applied to networks that describe various physical systems. Some of these networks possess fractal, scale-free, and small-world properties. The amount of information contained in a network is found by calculating its Shannon entropy. First, we consider networks arising from granular and colloidal systems (small colloidal and droplet clusters) due to pairwise interaction between the particles. Many networks found in colloidal science possess self-organizing properties due to the effect of percolation and/or self-organized criticality. Then, we discuss the allometric laws in branching vascular networks, artificial neural networks, cortical neural networks, as well as immune networks, which serve as a source of inspiration for both surface engineering and information technology. Scaling relationships in complex networks of neurons, which are organized in the neocortex in a hierarchical manner, suggest that the characteristic time constant is independent of brain size when interspecies comparison is conducted. The information content, scaling, dimensional, and topological properties of these networks are discussed.

Applying a Dynamical Systems Model and Network Theory to Major Depressive Disorder

Mental disorders like major depressive disorder can be modeled as complex dynamical systems. In this study we investigate the dynamic behavior of individuals to see whether or not we can expect a transition to another mood state. We introduce a mean field model to a binomial process, where we reduce a dynamic multidimensional system (stochastic cellular automaton) to a one-dimensional system to analyse the dynamics. Using maximum likelihood estimation, we can estimate the parameter of interest which, in combination with a bifurcation diagram, reflects the expectancy that someone has to transition to another mood state. After numerically illustrating the proposed method with simulated data, we apply this method to two empirical examples, where we show its use in a clinical sample consisting of patients diagnosed with major depressive disorder, and a general population sample. Results showed that the majority of the clinical sample was categorized as having an expectancy for a transition, while the majority of the general population sample did not have this expectancy. We conclude that the mean field model has great potential in assessing the expectancy for a transition between mood states. With some extensions it could, in the future, aid clinical therapists in the treatment of depressed patients.

Probabilistic Cellular Automata, Invariant Measures, and Perfect Sampling

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a one-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to also be a PCA. Last, we focus on the PCA majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure.

Pathologies in functional connectivity, feedback control and robustness: a global workspace perspective on autism spectrum disorders

We study the background to problems of functional connectivity in autism spectrum disorders within the neurocognitive framework of the global workspace model. This we proceed to do by observing network irregularities detracting from that of a well-formed small world network architecture. This is discussed in terms of pathologies in functional connectivity and lack of central coherence disrupting inter-network communication thus impairing effective cognitive action. A typical coherence-connectivity measure as a by-product of various neuroimaging results is considered. This is related to a model of feedback control in which a coherence function in the frequency domain is modified by an environmentally determined interaction parameter. With respect to the latter, we discuss the stability question that in theory may counterbalance inessential metabolic costs and incoherence of processing. We suggest that factors such as local overconnectivity and global underconnectivity, along with acute over-expenditure of metabolic costs give rise to instability within the connective core of the workspace.

Hierarchical random cellular neural networks for system-level brain-like signal processing

Sensory information processing and cognition in brains are modeled using dynamic systems theory. The brain's dynamic state is described by a trajectory evolving in a high-dimensional state space. We introduce a hierarchy of random cellular automata as the mathematical tools to describe the spatio-temporal dynamics of the cortex. The corresponding brain model is called neuropercolation which has distinct advantages compared to traditional models using differential equations, especially in describing spatio-temporal discontinuities in the form of phase transitions. Phase transitions demarcate singularities in brain operations at critical conditions, which are viewed as hallmarks of higher cognition and awareness experience. The introduced Monte-Carlo simulations obtained by parallel computing point to the importance of computer implementations using very large-scale integration (VLSI) and analog platforms.

MODELING GOAL-ORIENTED DECISION MAKING THROUGH COGNITIVE PHASE TRANSITIONS

Cognitive experiments indicate the presence of discontinuities in brain dynamics during high-level cognitive processing. Non-linear dynamic theory of brains pioneered by Freeman explains the experimental findings through the theory of metastability and edge-of-criticality in cognitive systems, which are key properties associated with robust operation and fast and reliable decision making. Recently, neuropercolation has been proposed to model such critical behavior. Neuropercolation is a family of probabilistic models based on the mathematical theory of bootstrap percolations on lattices and random graphs and motivated by structural and dynamical properties of neural populations in the cortex. Neuropercolation exhibits phase transitions and it provides a novel mathematical tool for studying spatio-temporal dynamics of multi-stable systems. The present work reviews the theory of cognitive phase transitions based on neuropercolation models and outlines the implications to decision making in brains and in artificial designs.

Evolutionary radiation and the spectrum of consciousness

Evolution is littered with polyphyletic parallelism: many roads lead to functional Romes. We propose consciousness embodies one such example, and represent it here with an equivalence class structure that factors the broad realm of necessary conditions information theoretic realizations of Baars' global workspace model. The construction suggests many different physiological systems can support rapidly shifting, highly tunable, and even simultaneous temporary assemblages of interacting unconscious cognitive modules. The discovery implies various animal taxa exhibiting behaviors we broadly recognize as conscious are, in fact, expressing different forms of the same underlying phenomenon. The variety of possibilities suggests minds today may be only a small surviving fraction of ancient evolutionary radiations--bush phylogenies of consciousness pruned by selection and chance extinction. Although few traces of the radiations may be found in the fossil record, exaptations and vestiges are scattered across the living mind.

Viewing brain processes as Critical State Transitions across levels of organization: Neural events in Cognition and Consciousness, and general principles

In this theoretical and speculative essay, I propose that insights into certain aspects of neural system functions can be gained from viewing brain function in terms of the branch of Statistical Mechanics currently referred to as "Modern Critical Theory" [Stanley, H.E., 1987. Introduction to Phase Transitions and Critical Phenomena. Oxford University Press; Marro, J., Dickman, R., 1999. Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press, Cambridge, UK]. The application of this framework is here explored in two stages: in the first place, its principles are applied to state transitions in global brain dynamics, with benchmarks of Cognitive Neuroscience providing the relevant empirical reference points. The second stage generalizes to suggest in more detail how the same principles could also apply to the relation between other levels of the structural-functional hierarchy of the nervous system and between neural assemblies. In this view, state transitions resulting from the processing at one level are the input to the next, in the image of a 'bucket brigade', with the content of each bucket being passed on along the chain, after having undergone a state transition. The unique features of a process of this kind will be discussed and illustrated.

Narrow-band oscillations in probabilistic cellular automata

Dynamical properties of neural populations are studied using probabilistic cellular automata. Previous work demonstrated the emergence of critical behavior as the function of system noise and density of long-range axonal connections. Finite-size scaling theory identified critical properties, which were consistent with properties of a weak Ising universality class. The present work extends the studies to neural populations with excitatory and inhibitory interactions. It is shown that the populations can exhibit narrow-band oscillations when confined to a range of inhibition levels, with clear boundaries marking the parameter region of prominent oscillations. Phase diagrams have been constructed to characterize unimodal, bimodal, and quadromodal oscillatory states. The significance of these findings is discussed in the context of large-scale narrow-band oscillations in neural tissues, as observed in electroencephalographic and magnetoencephalographic measurements.

Consciousness related neural events viewed as brain state space transitions

This theoretical and speculative essay addresses a categorical distinction between neural events of sensory-motor cognition and those presumably associated with consciousness. It proposes to view this distinction in the framework of the branch of Statistical Physics currently referred to as Modern Critical Theory (Stanley, Introduction to phase transitions and critical phenomena, 1987; Marro and Dickman, Nonequilibrium phase transitions in lattice, 1999). Based on established landmarks of brain dynamics, network configurations and their role for conveying oscillatory activity of certain frequencies bands, the question is examined: what kind of state space transitions can systems with these properties undergo, and could the relation between neural processes of sensory-motor cognition and those of events in consciousness be of the same category as is characterized by state transitions in non-equilibrium physical systems? Approaches for empirical validation of this view by suitably designed brain imaging studies, and for computational simulations of the proposed principle are discussed.

Collective consciousness and its pathologies: understanding the failure of AIDS control and treatment in the United States

We address themes of distributed cognition by extending recent formal developments in the theory of individual consciousness. While single minds appear biologically limited to one dynamic structure of linked cognitive submodules instantiating consciousness, organizations, by contrast, can support several, sometimes many, such constructs simultaneously, although these usually operate relatively slowly. System behavior remains, however, constrained not only by culture, but by a developmental path dependence generated by organizational history, in the context of market selection pressures. Such highly parallel multitasking – essentially an institutional collective consciousness – while capable of reducing inattentional blindness and the consequences of failures within individual workspaces, does not eliminate them, and introduces new characteristic malfunctions involving the distortion of information sent between workspaces and the possibility of pathological resilience – dysfunctional institutional lock-in. Consequently, organizations remain subject to canonical and idiosyncratic failures analogous to, but more complicated than, those afflicting individuals. Remediation is made difficult by the manner in which pathological externalities can write images of themselves onto both institutional function and corrective intervention. The perspective is applied to the failure of AIDS control and treatment in the United States.

Definitions of state variables and state space for brain-computer interface : Part 1. Multiple hierarchical levels of brain function

Neocortical state variables are defined and evaluated at three levels: microscopic using multiple spike activity (MSA), mesoscopic using local field potentials (LFP) and electrocorticograms (ECoG), and macroscopic using electroencephalograms (EEG) and brain imaging. Transactions between levels occur in all areas of cortex, upwardly by integration (abstraction, generalization) and downwardly by differentiation (speciation). The levels are joined by circular causality: microscopic activity upwardly creates mesoscopic order parameters, which downwardly constrain the microscopic activity that creates them. Integration dominates in sensory cortices. Microscopic activity evoked by receptor input in sensation induces emergence of mesoscopic activity in perception, followed by integration of perceptual activity into macroscopic activity in concept formation. The reverse process dominates in motor cortices, where the macroscopic activity embodying the concepts supports predictions of future states as goals. These macroscopic states are conceived to order mesoscopic activity in patterns that constitute plans for actions to achieve the goals. These planning patterns are conceived to provide frames in which the microscopic activity evolves in trajectories that adapted to the immediate environmental conditions detected by new stimuli. This circular sequence forms the action-perception cycle. Its upward limb is understood through correlation of sensory cortical activity with behavior. Now brain-machine interfaces (BMI) offer a means to understand the downward sequence through correlation of behavior with motor cortical activity, beginning with macroscopic goal states and concluding with recording of microscopic MSA trajectories that operate neuroprostheses. Part 1 develops a hypothesis that describes qualitatively the neurodynamics that supports the action-perception cycle and derivative reflex arc. Part 2 describes episodic, "cinematographic" spatial pattern formation and predicts some properties of the macroscopic and mesoscopic frames by which the embedded trajectories of the microscopic activity of cortical sensorimotor neurons might be organized and controlled.