Potential Research Directions
Geometric and Dynamic Properties of Brain Networks
Formally representing the brain as an ensemble of complex networks provides a theoretical framework and metrics for studying its architecture, developmental principles, evolution, dynamics, and the nature of its disorders. The graph of physical neuron connections defines the structural connectome, while graphs linking neurons of the same specialization define functional networks. The ensemble of functional networks has a complex multilayer structure. Current research in this field focuses on standard methods from complex network theory. Promising approaches also include spectral graph theory. Key research questions in this area:
Studying the influence of directionality and weightedness effects in the structural connectome on the spectral properties of the network.
Investigating the spectrum of the Laplacian for directed networks, its connection to the Dirac operator, and reconstructing directed networks from eigenvalue spectra.
Determining the architecture of the structural connectome using modern spectral analysis methods: non-backtracking matrices, belief propagation.
The relationship between the topological and metric structure of the connectome: to what extent do the topological features of subsystems constrain their spatial arrangement, and what strict limitations may exist?
Predicting the possible three-dimensional structure of the connectome based on functional interaction matrices.
Synchronization of excitations in functional networks. Dependence of synchronization on network architecture, noise levels, and other parameters. Emergence of chimeras and partial synchronization.
Spatial and functional identification of specific parts of the network spectrum and their possible connection to cyclic brain rhythms.
Interpreting eigenvalues in terms of the topological structure of the structural connectome.
Methods of *p*-adic analysis for studying the hierarchical structure of the brain.
Studying the eigenfunctions of Laplacians in the structural connectome and functional networks.
Localization/delocalization of excitations in the structural connectome and functional networks.
Spectral analysis methods for studying the eigenfunctions of the connectome; fractal dimensions.
Analyzing localization issues from a neurophysiological perspective.
Searching for possible delocalized non-ergodic phases.
The Brain as a Near-Critical System
Over the past decade, experimental evidence has accumulated suggesting that the brain operates in a "near-critical" state. Examples include power-law distributions of spatial and temporal scales of excitation avalanches and critical statistics in the spectrum of the structural connectome. However, a theoretical description of brain criticality is currently lacking. Key research questions in this area:
Analyzing standard critical phenomena in networks as applied to the structural connectome and functional networks: clustering, percolation, self-organized criticality, localization, synchronization.
Studying critical exponents of structural connectomes and functional networks using statistical physics methods.
Identifying natural order parameters for each critical phenomenon (from the perspective of standard phase transition classification).
Using singularity (bifurcation) theory to analyze criticality in brain networks.
Exploring potential applications of new types of critical phenomena formulated in recent decades:
Many-body localization.
Dynamical phase transitions (within fixed time intervals).
Measurement-induced phase transitions.
Page phase transitions.
Phase transitions in the ergodic regime.
Using different versions of entanglement entropy as an order parameter to distinguish phases.
Computing entanglement entropy, Wigner functions, various complexity indicators, and entanglement negativity for different brain states based on functional activity and clinical data (e.g., during wakefulness and sleep).
Formulating a Theory of Consciousness. Quantitative Definition of Consciousness.
A satisfactory theory of consciousness does not yet exist. Moreover, standard methods for analyzing disordered and non-equilibrium systems have not been fully applied to this problem. Key research questions in this area:
Describing the evolutionary emergence of consciousness as a result of a transition in semantic memory.
The possible role of measurements and measurement-induced phase transitions.
The potential emergence of self-organized measurement-induced phase transitions.
Attempting to define quantitative order parameters characterizing conscious and unconscious states:
Complexity indices (perturbation complexity index, purification).
Using statistical models, such as the Page transition, to describe stages of recovery from coma and anesthesia.
Modern coding theory methods:
Efficient information storage and retrieval.
The "information packing" problem.
The relationship between network topology and its optimal spatial organization.
Theoretical physics and mathematical aspects of episodic and semantic memory:
Using modern knot (link) theory, invariants, and homology to analyze brain entropy characteristics.
Applying category theory for efficient representation of structural connectome and functional network elements.
Dimensionality reduction and identification of "significant" elements without detailed analysis of the entire structure.
Neuron Specialization. Studying Mappings of Functional Network Ensembles to Stimulus Space
A large body of experimental data has been accumulated regarding the identification of neurons in the brain responsible for recognizing specific stimuli, objects, and events (place cells, border cells, head-direction cells, face-selective neurons, category-selective neurons, intention neurons, etc.). However, a theory of mappings from functional networks to stimulus space is currently lacking. Key research questions in this area:
Studying the geometry of stimulus space as a parameter space of a dynamical system.
Possible applications of category theory.
Investigating the space of mappings from functional networks to stimulus space.
Potential use of known embeddings of random networks into hyperbolic spaces.
Possible application of the holographic hypothesis for hyperbolic spaces in studying brain networks.
Studying Fisher information metrics and Berry connection in stimulus space.
Identifying singularities in the Fisher metric as indicators of phase transition points.
Describing the mapping of functional networks to stimulus space in terms of spin glass theory with frozen disorder and random matrix models.
Studying dynamics and critical phenomena in cognitive networks as a method for investigating states of consciousness.
Linguistic networks, association networks, etc.
Mapping functional networks to cognitive networks (semantic memory).
Connections to Other Research Directions in Complex Distributed Systems
Brain-computer interfaces.
Analysis of eye movements and visual perception and their relationship to impaired brain activity due to injury/disease/aging.
Methods for studying the brain and machine learning/deep learning tools.
Optimizing the recording of connectome data (working with big data).
Storing network information in the form of spectra, wavelets, and/or topological graph invariants.
Solving the inverse problem of data reconstruction (decompression).
Developing a theory of predictive coding and error correction for functional brain networks.
Possible connections to defining critical errors in multilayer networks (e.g., electrical grids).