STATISTICAL MECHANICS AND DYNAMICAL COMPLEX SYSTEMS – Research
Typical recent studies of the Statistical Mechanics and Dynamical Systems Laboratory are:
1) Analysis and modelling of the complex fractal architecture of the neuron axons spanning the human brain
Based on the classical Magnetic Resonance Imaging (MRI) technique it is now possible to obtain high resolution brain images reaching scales as low as ~0.2 mm. Diffusion Tensor MRI tomography (DTIMRI) allows for capturing the 3D motion of the water diffusion in the brain area by using the anisoptropic movement of water molecules to capture the positions of the axons connecting the neurons, see Fig. 1 for reconstruction of the neuron axons network from MRI recordings. Based on thorough analysis of MRI images from may healthy brains we have demonstrated selfsimilarity in the form of a hierarchical architecture of the connectivity matrix of the neuron axons network, which is proper to the healthy state of the brain, while its fractal dimensions were calculated to df~2.5. Using fractal and multifractal analyses we show that these complex connectivity patterns cannot be a result of simple or even chaotic processes because rare, unexpected configurations are present, indicative of the influence of long range exchange mechanisms. A further challenge is to extend the already existing models to cover the cases of pathological brain architectures (Alzheimer, Parkinson and Schizophrenia), a problem which has not been undertaken thus far.
Fig. 1. Typical neuron axons network representation from DTIMRI images
In parallel with the analysis of the detailed structure of the neuron axons network, we use numerical simulations to illustrate brain dynamics, using precise network connectivity matrices as recorded by the MRI analysis. The hierarchical connectivity in the brain, we consider as a key factor for the development of the intricate spatiotemporal patterns covering multiple scales, necessary for the emergence of cognition. Coupling schemes between a large number of neurons follow three main connectivity patterns: alltoall (global) coupling, nearest neighbours (or local ) coupling and nonlocal coupling where each neuron is linked with a finite number of neighbours. The last coupling pattern under special conditions produces the socalled ”chimera states”, where some of the oscillators (neurons) oscillate in phase while others are incoherent, see Fig. 2.
Fig. 2. Exemplary chimera state in 1D network (ring) with nonlocal connectivity. We note that synchronous and asynchronous regions coexist simultaneously in the network.
The coexistence of coherent and incoherent states have been associated with partial brain functionality, as in simple brain tasks which engage only a few modules of the brain or in unihemispheric sleep. We believe that the presence of chimera states is far more important than proposed for the unihemispheric sleep and is directly related to brain functionality during, at least, simple tasks.
Fig. 3 Typical chimera states in 2D: coherent stripe (left), incoherent spiral fronts (middle) and singlespot chimera (right).
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Related Movies: Chimera States
 Chimera state on a ring network (1D) with fractal connectivity
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (SPOT)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (STRIPES)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (SQUARES)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (GRID_OF_SPOTS)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (LINES_OF_SPOTS)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (LINES_AND_GAMMAS)
 Chimera state on a 2D torus network with Leaky IntegrateandFire dynamics (ARROWS)
 Chimera states in Brain Dynamics – Healthy Subject
 Chimera states in Brain Dynamics – Patient with Tumor(SubjectBBB)
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2) Reaction Diffusion Processes on Complex Networks
Abstract networks can be used for the description of the dynamics in reactive processes and they can give valuable information about the evolution and the products of each process. The abstract reactive networks are constructed as follows: The phase space of the variables (concentrations in reactive systems) is partitioned into a finite number of segments, which constitute the nodes of the abstract network. Transitions between the nodes are dictated by the dynamics of the reactive process and provide the links between the nodes. These are weighted networks, since each link weight reflects the transition rate between the corresponding statesnodes. With this construction the network properties mirror the dynamics of the underlying process and one can investigate the system properties by studying the corresponding abstract network.
As an example, the abstract network of the Lattice Limit Cycle (LLC) model was analysed and its transition matrix elements were computed via Kinetic (Dynamic) Monte Carlo simulations. For this model it was shown that the degree distribution follows a power law with exponent 1, while the average clustering coefficient c(N) scales with the network size N as with a power law exponent 1.46. The computed exponents classify the LLC abstract reactive network into the scalefree networks. This conclusion corroborates earlier investigations demonstrating the formation of fractal spatial patterns in LLC reactive dynamics due to stochasticity and to the clustering of homologous species. The present construction of abstract networks (based on the partition of the phase space) is generic and can be implemented with appropriate adjustments in many dynamical systems and in time series analysis.

Fig. 4: Representative snapshots of the LLC system from 2D Kinetic Monte Carlo simulations. An initially homogeneous system (t=0 MCS) develops clusters and inhomogeneous distribution of species (t=10000 MCS).

Fig. 4 : Reactiondiffusion process on a 3D fractal support using Kinetic Monte Carlo Simulations. Left: Deterministic Sierpinski sponge, Right: Random Sierpinski sponge
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Related Movies: ReactionDiffusion on Complex Networks
 ReactionDiffusion process on 2D lattice
 ReactionDiffusion process on a 2D (spiral)
 ReactionDiffusion process on a fractal lattice
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3) Bioinformatics
The complexity of the primary structure of human DNA is explored using methods from nonequilibrium statistical mechanics, dynamical systems theory and information theory. A collection of statistical analyses are performed on the DNA data and the results are compared across species and with sequences derived from different stochastic processes. Although detailed balance seems to hold at the level of a binary alphabet in genomic data, it fails when all four basepairs are considered, suggesting spatial asymmetry and irreversibility. Furthermore, the block entropy does not increase linearly with the block size, reflecting the long range nature of the correlations in the human genomic sequences. To probe locally the spatial structure of the chain we study local quantities, such as the exit distances from a specific symbol, the distribution of recurrence distances and the Hurst exponent, all of which show power law tails and long range characteristics. These results suggest that human DNA can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment.
Fig. 5 : Exemplary Information Transfer Analysis between a sequence and its shift by n symbols in the 4letter representation. Line with circles depicts the DNA sequence, line with crosses a random sequence and line with diamonds a modelgenerated sequence.
Complexity measures are also used to compare the genomic characteristics of different organisms belonging to distinct classes spanning the evolutionary tree: higher eukaryotes, amoebae, unicellular eukaryotes and bacteria. We have demonstrated that the conditional probability matrix for the fourletter and ATCG alphabet is markedly asymmetric in eukaryotes while it is nearly symmetric in bacterial genomes. Overall, the conditional probability, the fluxes, the block entropy content and the exit distance distributions can be used as markers, discriminating between eukaryotic and prokaryotic DNA, allowing in many cases to discern details related to finer classes. In all cases the reduction from four letters to two mask some important statistical and spatial properties, with the ATCG alphabet having higher ability of discrimination than the AGCT one.
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