Marko Puljic

Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions

We model the dynamical behavior of the neuropil, the densely interconnected neural tissue in the cortex, using neuropercolation approach. Neuropercolation generalizes phase transitions modeled by percolation theory of random graphs, motivated by properties of neurons and neural populations. The generalization includes (1) a noisy component in the percolation rule, (2) a novel depression function in addition to the usual arousal function, (3) non-local interactions among nodes arranged on a multi-dimensional lattice. This paper investigates the role of non-local (axonal) connections in generating and modulating phase transitions of collective activity in the neuropil. We derived a relationship between critical values of the noise level and non-locality parameter to control the onset of phase transitions. Finally, we propose a potential interpretation of ontogenetic development of the neuropil maintaining a dynamical state at the edge of criticality.

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

We outline the basic principles of neuropercolation, a generalized percolation model motivated by the dynamical properties of the neuropil, the densely interconnected neural tissue structure in the cortex. We apply the mathematical theory of percolation in lattices to analyze chaotic dynamical memories and their related phase transitions. This approach has several advantages, including the natural introduction of noise that is necessary for system stability, a greater degree of biological plausibility, a more uniform and simpler model description, and a more solid theoretical foundation for neural modeling. Critical phenomena and scaling properties of a class of random cellular automata (RCA) are studied on the lattice \(\mathbb Z^{2}\). In addition to RCA, we study phase transitions in mean-field models, as well as in models with axonal, non-local interactions. Relationship to the Ising universality class and to Toom cellular automata is thoroughly analyzed.

Activation clustering in neural and social networks

Questions related to the evolution of the structure of networks have received recently a lot of attention in the literature. But what is the state of the network given its structure? For example, there is the question of how the structures of neural networks make them behave? Or, in the case of a network of humans, the question could be related to the states of humans in general, given the structure of the social network. The models based on stochastic processes developed in this article, do not attempt to capture the fine details of social or neural dynamics. Rather they aim to describe the general relationship between the variables describing the network and the aggregate behavior of the network. A number of nontrivial results are obtained using computer simulations.

Random graph theory and neuropercolation for modeling brain oscillations at criticality

Mathematical approaches are reviewed to interpret intermittent singular space-time dynamics observed in brain imaging experiments. The following aspects of brain dynamics are considered: nonlinear dynamics (chaos), phase transitions, and criticality. Probabilistic cellular automata and random graph models are described, which develop equations for the probability distributions of macroscopic state variables as an alternative to differential equations. The introduced modular neuropercolation model is motivated by the multilayer structure and dynamical properties of the cortex, and it describes critical brain oscillations, including background activity, narrow-band oscillations in excitatory-inhibitory populations, and broadband oscillations in the cortex. Input-induced and spontaneous transitions between states with large-scale synchrony and without synchrony exhibit brief episodes with long-range spatial correlations as observed in experiments.

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.

Phase transitions in a probabilistic cellular neural network model having local and remote connections

Inspired by a neuronal architecture, we show how to produce dynamical behaviors in a special kind of probabilistic cellular neural network system. We demonstrate that the spatial and temporal behavior of neural activity undergoes sudden changes if the connection structure and noise component are varied. We characterize quantitatively phase transitions using the activation and cluster size. We indicate the potential role our present results may play in developing the theory of computation using non-convergent neurodynamic principles, called neurpercolation.

Metastability of Mean Field Neuropercolation – The Role of Inhibitory Populations

Critical properties of dynamic models of neural populations are studied. Based on the classical work of Erdős-Renyi on the evolution of random graphs and motivated by the properties of the cortical tissue, a new class of random cellular automata called neuropercolation has been introduced recently. This work analyzes the role of inhibitory populations in generating multistable dynamics near criticality. The results are interpreted in the context of experimentally observed meta-stable behavior in the cortex.

Pattern-based computing via sequential phase transitions in hierarchical mean field neuropercolation

In this work, we describe operational principles of a pattern-based computing paradigm based on the neuropercolation model, which can be used as associative memory supporting sensory processing and pattern recognition. Neuropercolation extends the concept of phase transitions to interactive populations exhibiting frequent transients in their spatio-temporal dynamics, which can be viewed as manifestations of an asynchronous computer working with a sequence of meta-stable spatial patterns, in a bid to unravel the limitations of Turing computing principles. The model is motivated by the structural and dynamical properties of large-scale neural populations in the cerebral cortex and it implements basic building blocks of neurodynamics following the hierarchy of Freeman K-sets.The introduced mean-field approximation allows rigorous mathematical analysis of the emergent dynamics, which is the major novel contribution of this work. Specifically, we derive exact conditions for the onset of non-zero background activity, for the transition from steady state to narrow-band (limit cycle) oscillations, and for the transition from narrow-band to broad-band (chaotic) dynamics. We describe an array of connected oscillators, which exhibits transient synchronization episodes manifesting meta-stable collective states. The corresponding meta-stable spatial amplitude patterns are destabilized by inputs or spontaneously and jump to another pattern, yielding a sequence of transient patterns. These patterns are shaped by the connections between the nodes modifiable through learning. The sequence of patterns manifest the steps of the computation, which embody the meaning of the input data in the context of the system past experiences.

Chaotic behavior in probabilistic cellular neural networks

Experiments conducted in brains by electroencephalographic and magnetoencephalographic techniques reveal widespread coherent oscillations. The oscillations over multiple frequency bands overlap and result in signals with broad spectra. Previous studies showed that various frequencies can be modeled by probabilistic cellular automata with coupled inhibitory and excitatory interactions. In this work we show that coupled oscillator layers can create broad-spectrum chaotic oscillations with power spectral densities over long times segments converging to Brown noise features. Models of cortical neurodynamics provide an interpretation of the observed phenomena.

Modeling learning and strategy formation as phase transitions in cortical networks

Learning in the mammalian brain is commonly modeled through changing synaptic connections in cortical networks. Dynamical brain models indicate that learning leads to the formation of limit cycle oscillations across cortical areas and that the oscillatory regimes re-emerge when the learnt input is presented to the system. In this work, learning is modeled using a graph-theoretical model, which captures salient characteristics of the learning process. We introduce a random graph that combines a torus with lattice edges and additional random edges, which have power law length distribution. On this graph, we consider bootstrap percolation with excitatory and inhibitory vertices. Theoretical and numerical studies indicate the presence of various dynamical regimes on these graphs. Here, the transitions between fixed-point and limit cycle attractors are analyzed. We link this transition to changes in cortical networks during category learning, which have been observed in animal experiments using electro-cortiograph (ECoG) arrays over sensory cortices. We discuss how learning leads to categorization and strategy formation, and how the theoretical modeling results can be used for designing learning and adaptation in computationally aware intelligent machines.