Modeling functional resting-state brain networks through neural message passing on the human connectome

Abstract

In this work, we propose a natural model for information flow in the brain through a neural message-passing dynamics on a structural network of macroscopic regions, such as the human connectome (HC). In our model, each brain region is assumed to have a binary behavior (active or not), the strengths of interactions among them are encoded in the anatomical connectivity matrix defined by the HC, and the dynamics of the system is defined by the Belief Propagation (BP) algorithm, working near the critical point of the network. We show that in the absence of direct external stimuli the BP algorithm converges to a spatial map of activations that is similar to the Default Mode Network (DMN) of the brain, which has been defined from the analysis of functional MRI data. Moreover, we use Susceptibility Propagation (SP) to compute the matrix of long-range correlations between the different regions and show that the modules defined by a clustering of this matrix resemble several Resting State Networks (RSN) determined experimentally. Both results suggest that the functional DMN and RSNs can be seen as simple consequences of the anatomical structure of the brain and a neural message-passing dynamics between macroscopic regions. With the new model, we explore predictions on how functional maps change when the anatomical brain network suffers structural alterations, like in Alzheimer’s disease and in lesions of the Corpus Callosum. The implications and novel interpretations suggested by the model, as well as the role of criticality, are discussed.

Introduction

Despite the efforts of the scientific community, the relationship between the structure and function of the brain is still an open question in Neuroscience. On one hand, the combination of diffusion spectrum imaging and tractography algorithms allows the determination of a macroscopic representation of the structural network of the brain (Hagmann et al., 2008). On the other hand, from functional magnetic resonance imaging (fMRI), it is possible to determine the resting-state functional connections between the brain regions at the macroscopic level (Greicius, Krasnow, Reiss, & Menon, 2003). Many studies in cognitive neurosciences have shown that dynamical models based on predictive processing and associated message passing (e.g., Kalman filtering, predictive coding, belief propagation and variational message passing) have a wide explanatory power for brain function, especially in relation to cortical hierarchies and neurophysiology (Fox & Friston, 2012). However, at a macroscopic level, it has been more difficult to find a plausible model able to quantitatively describe how functional networks emerge from the structural organization of the brain.

Recent studies have shown a recurrent finding of a set of brain regions that appear active in almost all brain states, which has been called the brain Default Mode Network (Raichle, 2015). Based on traditional experimental studies that related brain regions with different functions, several authors have explained the Default Mode Network (DMN) as a brain system that is preferentially active when individuals are not focused on external stimuli. The DMN has been claimed to be involved in processes with emotional support given activations of the Ventral Medial Prefrontal cortex (Simpson, Drevets, Snyder, Gusnard, & Raichle, 2001). Activations in the medial temporal lobe suggested the involvement in retrieval of memories from past experiences (Buckner et al., 2008, Vincent et al., 2006) while those in the Medial Prefrontal Cortex Dorsal suggested self-referring mental activity (Gusnard & Raichle, 2001). These two subsystems converged in the Posterior Cingulate Cortex which is a well-known hub for information integration in the brain (Buckner et al., 2008). In general, the evidence indicates that the activity in the DMN never shuts down, but it is just modulated during the resting-state (Raichle, 2015). Additionally, its functional elements can be differentially affected during the execution of a task by the nature of the action (for example, the presence or absence of emotional components) (Andrews-Hanna, Reidler, Sepulcre et al., 2010, Gusnard and Raichle, 2001). However, it is still unclear the functional role (if any) of the DMN as a whole, or even why it is present in almost all brain states. Clearly, there is no reason why the DMN should have a function, as it is just a description of functional connectivity in terms of covariance patterns, extracted with a spatial independent component analysis (Beckmann, DeLuca, Devlin, & Smith, 2005). As such, it may be better to ask how these patterns emerge from first principles (Raichle, 2015).

The appearance of anatomical and functional brain networks brought to neuroscience the introduction of concepts and techniques from Network Theory and Complex Systems. For example, in the last decade many studies have shown that structural networks of the brain are characterized by high cluster index and modularity combined with a high efficiency and a short path length (Bullmore and Sporns, 2009, Rubinov and Sporns, 2010, Sporns, 2011). Similar results have also been found when studying functional networks (see e.g., Fallani et al., 2010, Papo et al., 2014, Sanabria-Diaz et al., 2013, Sporns, 2018, and references therein). Specifically, network theory has brought into neuroscience the discussion about whether brain functioning arises from functional specialization or functional integration. Studies at the level of cortical hierarchies have provided a strong basis for balancing the importance of both mechanisms, establishing the context for a long-standing treatment of distributed processing in the brain (Fox and Friston, 2012, Friston, 2004, Mastrandrea et al., 2017, Telesford et al., 2011). Although this positions network science as an adequate approach to study the structure–function relation, even at the macroscopic level, most of the current studies are descriptive in nature and theoretical explanations of the relationship between the properties of the networks and normal or pathological brain states are not straightforward (Sporns, 2013).

As a first attempt to understand brain function from these perspectives, in 1996 Per Bak proposed that the complexity of brain function should be associated with the existence of a critical dynamics in a sense similar to that found in a second order phase transition (Bak, 1996). The first clear evidence of critical dynamics in the brain was given by the experiments of Beggs and Plenz in 2003 (Beggs & Plenz, 2003). They identified a mode of spontaneous activity in cortical networks from organotypic cultures and acute slices, called “neuronal avalanches”. They found that this neural activity has numerous points of contact with Self-Organized Criticality theory. For example, the distribution of avalanche sizes, from all cortical networks, showed no characteristic scale and could be described by power laws, resembling those found near the critical point of a second order phase transition in systems such as sandpile avalanches, and magnetism. Years later, Dante Chialvo and colleagues showed that also fMRI data gathered in humans in the absence of external stimuli, exhibits these scale-free properties (Chialvo, 2010, Expert et al., 2011), supporting a picture consistent with a brain at rest being near a critical point (Tagliazucchi, Balenzuela, Fraiman, & Chialvo, 2012).

Second order phase transitions have been largely studied in Statistical Physics, where multiple computational models have emerged in the line of these previous empirical findings (Chialvo, 2010, Deco and Jirsa, 2012, Fraiman et al., 2009, Kitzbichler et al., 2009). In this respect, the Ising model, a well-known paradigm in Physics, has been used in the modeling of neural networks (Amit, 1989, Amit et al., 1985, Das et al., 2014), and more specifically to describe the macroscopic brain activity in brain networks at the appropriate critical point (Deco et al., 2012, Fraiman et al., 2009, Marinazzo et al., 2014). Recently, the work of Haimovici, Tagliazucchi, Balenzuela, and Chialvo (2013) demonstrated, using the Greenberg–Hastings model on the structural network defined by the Human Connectome, that the brain functional dynamics observed experimentally can be replicated, just by tuning the model to a region near its critical point.

In this work, we also use the Ising model to describe the macroscopic activity in the anatomical regions of the structural network defined by the Human Connectome (HC). However, different to previous studies, we propose the use of a message-passing algorithm called Belief Propagation (BP) to simulate the actual transmission of information in the brain. After the seminal introduction of BP in information theory (Pearl, 1988), this algorithm has been widely used in fields such as inference for Bayesian and Neural networks, error correcting codes, mobile communication, and probabilistic image processing, since it computes exact marginals distributions in acyclic graphs and approximate solutions for more general graphs, for which many variants have also been developed (Nishiyama & Watanabe, 2009).

Recently, a more general definition that unifies many types of message-passing algorithms on complex graph neural networks, has been named neural message passing, finding first applications in quantum chemistry and machine learning (Gilmer et al., 2017, Lanchantin et al., 2019). In neuroscience, there is also a recent interest for using BP as a model for neuronal processing within cortical microcircuits, related to the self-criticality of brain’s Bayesian computations (Friston, Parr, & de Vries, 2017) and for making inference in biological networks (Parr, Markovic, Kiebel, & Friston, 2019). In physics, the BP dynamics was introduced as a computationally efficient alternative to Monte-Carlo methods on Random Networks by Yedidia (Yedidia, Freeman, & Weiss, 2003). The fixed point solutions of these algorithms can be identified with the saddle point solution of the Bethe approximation of the corresponding free energy of the system (Yedidia et al., 2003). In this case, the dynamics followed by the algorithm is seen as an empirical strategy to reach the stationary states of the system, independently of the real (usually stochastic) dynamics defining the behavior of the system. Therefore, there have been only rare discussions about the correspondence between the dynamics of the algorithm and the dynamics of the physical system (Lage-Castellanos, Mulet, & Ricci-Tersenghi, 2014). One exception here is the necessary connection between neuronal dynamics, when cast as a gradient flow on variational (c.f., Bethe) free energy. This is necessarily associated with critical slowing and instability. This is due to the fact that the curvature of free energy minima is – by definition – small, engendering critical dynamics. This has some interesting interpretations in terms of fundamental principles and of the search for flat minima — from a machine learning perspective (Friston, Breakspear, & Deco, 2012).

In this work, we modify this perspective by assuming that the actual mechanism of interaction between brain macroscopic regions is well described by the basic rules of the BP algorithm. The intuition behind this idea was advanced by Friston and colleagues (Friston et al., 2017), who showed that a BP scheme can be constructed to find the attracting (stationary) point of a set of differential equations, which correspond to a gradient descent on marginal variational free energy. Moreover, in the context of a neuronal network within the cortical layer, they provided an interpretation for log expectations of hidden BP states in terms of postsynaptic depolarization of neuronal populations. Then, BP entails the convolution form of neural mass models of population activity that have been proposed to model electrophysiological responses (for example, Jansen & Rit, 1995). We based our proposal on the theoretical connection established a few years ago by Ott and Stoop between the continuous dynamics of a network of Hopfield neurons and the BP (Ott & Stoop, 2007). This means that we can interpret the BP application to a neuronal network as an estimate of the macroscopic electrical activity (voltage-firing rate) of every node in the HC by modeling their interaction with exchanging messages. In particular, in the absence of external stimulation, where initial activations can be random, the BP algorithm would predict the spontaneous functional brain state emerging from the structural properties of the network, i.e., due to brain architecture.

Another message-passing algorithm, the Susceptibility Propagation (SP), allows the determination of long-range correlations between the state (activity) of nodes that arise around the critical point of the network (Mora, 2007). We will interpret these correlations as predictors of the functional connectivity mediated by the electrophysiological activity of the brain. Despite the lack of a direct validation for these connectivity matrices, we can explore their network properties and compare them with experimental data. In particular, we explore their modularity structure and show that the modules obtained from our correlation matrices are very similar to the resting-state Networks (RSN) that have been previously found by spatio-temporal Independent Component Analysis (ICA) of resting-state fMRI data (Power et al., 2011, Sporns and Betzel, 2016).

In summary, we hope to provide an explanation for the emergence of the RSNs based upon belief propagation and neuronal message passing, under the assumption that the brain is an organ of prediction. As a second goal, we perform a preliminary application of the model to studying the changes in the resting-state functional activity/connectivity – obtained with the message-passing algorithms – when the structural network presents different anomalies or disruptions. In particular, we exploreAlzheimer’s disease and lesions of the Corpus Callosum, by introducing in the HC changes consistent with studies of anatomical connectivity in subjects with these conditions.