Structure-Function Relationship of the Brain: A comparison between the 2D Classical Ising model and the Generalized Ising model

There is evidence that the functional patterns of the brain observed at rest using fMRI are sustained by a structural architecture of axonal fiber bundles. As neuroimaging techniques advance with time, the relationship between structure and function has become the object of many studies in neuroscience. As recently suggested, the well defined connectivity structure found in the brain can be used to understand the self organization of the brain at rest, as well as to infer the functional connectivity patterns of the brain using different models, such as the Kuramoto model which studies synchronization, and the 2-dimensional classical Ising model, which studies the global dynamics of the brain at the critical temperature. These models were successful in capturing the underlying properties of the brain. To extend this understanding, our objective is to develop the generalized Ising model following the lesson from the 2-dimensional Ising model, as it could be simulated using the anatomical structure of the brain. This model can then be used to study functional information integration and segregation in the brain at rest. Thus the primary research question would be: can the generalized Ising model explain the functional behaviour of the resting brain at the critical temperature ? Preliminary analyses were carried out to determine the critical temperature of the models and to compare the correlation distributions. Further analyses were carried out using graph theory considering the brain as a network. By observing the results obtained from our simulations, it can be inferred that there is a temperature that is different from the critical temperature of the model at which the generalized Ising model shows a match with the empirical functional connectivity. At that temperature, the generalized Ising model could be used to study the global dynamics, as well as the local dynamics of the brain

Calculating the Dimensionality of the Brain, and Other Applications of an Optimized Generalized Ising Model in Predicting Brain\u27s Spontaneous Functions

Understanding a system as complex as the human brain is a very demanding task. Directly working with structural and functional neuroimaging data has led to most of the understanding we have gained about the human brain. However, performing only the direct statistical comparisons on the empirical function and the structure does not fully explain the observed long-range functional correlations. Therefore, implementations of mathematical models to gain further understanding of the relationship between the structure and function of the brain is critical. Additionally, spontaneous functions of the brain can only be predicted using computer simulated models; which will be pivotal for studying the patients with accidental brain injuries. Therefore, this research aims to present an optimized computer simulated model not only to further understand the structure-function relationship of the brain but also to predict the functional changes when anatomy is altered. Based on prior work, 2-dimensional classical Ising model stands out among the other models in modeling the functions of the brain due to its simplicity. Hence, a 2-dimensional Ising model was simulated on a structural connectome (generalized Ising model) that acts as a proxy for the anatomical connectivity in the brain. Simulations allowed the prediction of functional connectivity using the structure, at criticality. It also enabled the introduction of a novel methodology to calculate the ”dimensionality” of the brain. Our results showed the dimensionality of a healthy brain is two when it is defined using the information flow in the brain. Further research illustrated the dependency of dimensionality on the diffusion tractography method used to obtain the structural connectome. It was also concluded that an optimized generalized Ising model has to be simulated using a structural connectome generated by deterministic tractography to acquire the best predictions of empirical function. Additional investigations into a more generalized version of the Ising model—Potts model with different number of spin states—illustrated that increasing the number of spin states does not increase the predictability. It also supported the hypothesis that the model could be simulating the digital nature of direct neural activity rather than the indirect activity measured by brain imaging

A method for independent component graph analysis of resting-state fMRI

Introduction: Independent component analysis (ICA) has been extensively used for reducing task-free BOLD fMRI recordings into spatial maps and their associated time-courses. The spatially identified independent components can be considered as intrinsic connectivity networks (ICNs) of non-contiguous regions. To date, the spatial patterns of the networks have been analyzed with techniques developed for volumetric data. Objective: Here, we detail a graph building technique that allows these ICNs to be analyzed with graph theory. Methods: First, ICA was performed at the single-subject level in 15 healthy volunteers using a 3T MRI scanner. The identification of nine networks was performed by a multiple-template matching procedure and a subsequent component classification based on the network neuronal properties. Second, for each of the identified networks, the nodes were defined as 1,015 anatomically parcellated regions. Third, between-node functional connectivity was established by building edge weights for each networks. Group-level graph analysis was finally performed for each network and compared to the classical network. Results: Network graph comparison between the classically constructed network and the nine networks showed significant differences in the auditory and visual medial networks with regard to the average degree and the number of edges, while the visual lateral network showed a significant difference in the small-worldness. Conclusions: This novel approach permits us to take advantage of the well-recognized power of ICA in BOLD signal decomposition and, at the same time, to make use of well-established graph measures to evaluate connectivity differences. Moreover, by providing a graph for each separate network, it can offer the possibility to extract graph measures in a specific way for each network. This increased specificity could be relevant for studying pathological brain activity or altered states of consciousness as induced by anesthesia or sleep, where specific networks are known to be altered in different strength

Role of Dimensionality in Predicting the Spontaneous Behavior of the Brain Using the Classical Ising Model and the Ising Model Implemented on a Structural Connectome

© Pubuditha M. Abeyasinghe et al. 2018. There is accumulating evidence that spontaneous fluctuations of the brain are sustained by a structural architecture of axonal fiber bundles. Various models have been used to investigate this structure-function relationship. In this work, we implemented the Ising model using the number of fibers between each pair of brain regions as input. The output of the Ising model simulations on a structural connectome was then compared with empirical functional connectivity data. A simpler two-dimensional classical Ising model was used as the baseline model for comparison purpose. Thermodynamic properties, such as the magnetic susceptibility and the specific heat, illustrated a phase transition from an ordered phase to a disordered phase at the critical temperature. Despite the differences between the two models, the lattice Ising model and the Ising model implemented on a structural connectome (the generalized Ising model) exhibited similar patterns of global properties. To study the behavior of the generalized Ising model around criticality, calculation of the dimensionality and critical exponents was performed for the first time, by introducing a new concept of distance based on structural connectivity. Same value inside the fitting error was found for the dimensionality in both models suggesting similar behavior of the models around criticality

The emergence of integrated information, complexity, and \u27consciousness\u27 at criticality

© 2020 by the authors. Integrated Information Theory (IIT) posits that integrated information (F) represents the quantity of a conscious experience. Here, the generalized Ising model was used to calculate F as a function of temperature in toy models of fully connected neural networks. A Monte-Carlo simulation was run on 159 normalized, random, positively weighted networks analogous to small five-node excitatory neural network motifs. Integrated information generated by this sample of small Ising models was measured across model parameter spaces. It was observed that integrated information, as an order parameter, underwent a phase transition at the critical point in the model. This critical point was demarcated by the peak of the generalized susceptibility (or variance in configuration due to temperature) of integrated information. At this critical point, integrated information was maximally receptive and responsive to perturbations of its own states. The results of this study provide evidence that F can capture integrated information in an empirical dataset, and display critical behavior acting as an order parameter from the generalized Ising model

A comparison of diffusion tractography techniques in simulating the generalized Ising model to predict the intrinsic activity of the brain

Diffusion tractography is a non-invasive technique that is being used to estimate the location and direction of white matter tracts in the brain. Identifying the characteristics of white matter plays an important role in research as well as in clinical practice that relies on finding the relationship between the structure and function of the brain. An Ising model implemented on a structural connectivity (SC) has proven to explain the spontaneous fluctuations in the brain at criticality using brain’s structure depicted by white matter tracts. Since the SC is the only input of the model, identifying the tractography technique which provides a SC that delivers the highest prediction of the brain’s intrinsic activity via the generalized Ising model (GIM) is essential. Hence an Ising model is simulated on SCs generated using two different acquisition schemes (single and multi-shell) and two different tractography approaches (deterministic and probabilistic) and analyzed at criticality across 69 healthy subjects. Results showed that by introducing the GIM, predictability of the empirical correlation matrix increases on average from 0.2 to 0.6 compared to the predictability using the empirical connectivity matrix directly. It is also observed that the SC generated using deterministic tractography without fractional anisotropy resulted in the highest correlation coefficient value of 0.65 between the simulated and empirical correlation matrices. Additionally, calculated dimensionalities per simulation illustrated that the dimensionality depends upon the method of tractography that has been used to extract the SC

Correction to: A comparison of diffusion tractography techniques in simulating the generalized Ising model to predict the intrinsic activity of the brain (Brain Structure and Function, (2021), 226, 3, (817-832), 10.1007/s00429-020-02211-6)

In the original publication of the article, the affiliation of the third author is incorrect. It should read as IRCCS SDN, Istituto Di Ricerca Diagnostica E Nucleare, Via E. Gianturco 113, 80143 Naples, Italy The original article has been corrected

The Emergence of Integrated Information, Complexity, and ‘Consciousness’ at Criticality

Integrated Information Theory (IIT) posits that integrated information ( Φ ) represents the quantity of a conscious experience. Here, the generalized Ising model was used to calculate Φ as a function of temperature in toy models of fully connected neural networks. A Monte–Carlo simulation was run on 159 normalized, random, positively weighted networks analogous to small five-node excitatory neural network motifs. Integrated information generated by this sample of small Ising models was measured across model parameter spaces. It was observed that integrated information, as an order parameter, underwent a phase transition at the critical point in the model. This critical point was demarcated by the peak of the generalized susceptibility (or variance in configuration due to temperature) of integrated information. At this critical point, integrated information was maximally receptive and responsive to perturbations of its own states. The results of this study provide evidence that Φ can capture integrated information in an empirical dataset, and display critical behavior acting as an order parameter from the generalized Ising model

Consciousness and the dimensionality of DOC patients via the generalized ising model

The data from patients with severe brain injuries show complex brain functions. Due to the difficulties associated with these complex data, computational modeling is an especially useful tool to examine the structure–function relationship in these populations. By using computational modeling for patients with a disorder of consciousness (DoC), not only we can understand the changes of information transfer, but we also can test changes to different states of consciousness by hypothetically changing the anatomical structure. The generalized Ising model (GIM), which specializes in using structural connectivity to simulate functional connectivity, has been proven to effectively capture the relationship between anatomical structures and the spontaneous fluctuations of healthy controls (HCs). In the present study we implemented the GIM in 25 HCs as well as in 13 DoC patients diagnosed at three different states of consciousness. Simulated data were analyzed and the criticality and dimensionality were calculated for both groups; together, those values capture the level of information transfer in the brain. Ratifying previous studies, criticality was observed in simulations of HCs. We were also able to observe criticality for DoC patients, concluding that the GIM is generalizable for DoC patients. Furthermore, dimensionality increased for the DoC group as compared to healthy controls, and could distinguish different diagnostic groups of DoC patients

A method for independent component graph analysis of resting-state fMRI

© 2017 The Authors. Brain and Behavior published by Wiley Periodicals, Inc. Introduction: Independent component analysis (ICA) has been extensively used for reducing task-free BOLD fMRI recordings into spatial maps and their associated time-courses. The spatially identified independent components can be considered as intrinsic connectivity networks (ICNs) of non-contiguous regions. To date, the spatial patterns of the networks have been analyzed with techniques developed for volumetric data. Objective: Here, we detail a graph building technique that allows these ICNs to be analyzed with graph theory. Methods: First, ICA was performed at the single-subject level in 15 healthy volunteers using a 3T MRI scanner. The identification of nine networks was performed by a multiple-template matching procedure and a subsequent component classification based on the network “neuronal” properties. Second, for each of the identified networks, the nodes were defined as 1,015 anatomically parcellated regions. Third, between-node functional connectivity was established by building edge weights for each networks. Group-level graph analysis was finally performed for each network and compared to the classical network. Results: Network graph comparison between the classically constructed network and the nine networks showed significant differences in the auditory and visual medial networks with regard to the average degree and the number of edges, while the visual lateral network showed a significant difference in the small-worldness. Conclusions: This novel approach permits us to take advantage of the well-recognized power of ICA in BOLD signal decomposition and, at the same time, to make use of well-established graph measures to evaluate connectivity differences. Moreover, by providing a graph for each separate network, it can offer the possibility to extract graph measures in a specific way for each network. This increased specificity could be relevant for studying pathological brain activity or altered states of consciousness as induced by anesthesia or sleep, where specific networks are known to be altered in different strength