What Cascade Spreading Models Can Teach Us about the Brain

Abstract

The precise relationship between functional and structural connectivity in the brain is not well understood. Research in this area has, so far, mostly remained descriptive. In this issue of Neuron, Misić et al. (2015) forge a promising new direction by modeling the propagation of information as it relates to spatially constrained network properties. From these preliminary results a glimmer of hope in uncovering deep principles of brain organization begins to emerge.

Much progress in neuroscience today is driven in a large part by the use of tools, models, and ideas that are relatively new to the application of understanding the brain. In particular, our understanding of the structural and functional organization of the brain has benefitted from abstracting this organization as sets of nodes (i.e., cortical/subcortical regions) and edges (i.e., their mutual relationships) that form topologies, which can then be studied using analytical tools borrowed from the field of network science (Rubinov and Sporns, 2010). Using network models, neuroscientists have learned that the brain follows a ““small-world”” motif, in which regions with dense clustering of connections (supporting segregated/specialized processing) coexist with a reduced set of high-centrality regions (known as hubs) that facilitate rapid integration and distributed processing of information (Bassett and Bullmore, 2006). Similarly, comparative analysis of functional and anatomical brain topologies has revealed a tight connection between these two, yet highlighted the importance of indirect paths in shaping functional connectivity (FC) (Honey et al., 2009). Prior attempts at predicting FC from structural connectivity (SC) on the basis of Euclidean distance between nodes (Vértes et al., 2012), linear random walk models (Abdelnour et al., 2014), or structurally constrained nonlinear neuronal dynamic models (e.g., neural mass models [Honey et al., 2009], attractor models [Sporns et al., 2009]) have only been partially successful. Understanding the extent and mechanism by which SC—characterized by a sparse topology—helps to shape and constrain FC—characterized by a much denser topology—remains a key challenge for computational neuroscience (Goñi et al., 2014). In this issue of Neuron, Misić et al. (2015) propose a novel alternative approach—structurally constrained spreading models—to attempt to derive fundamental principles of how SC and FC can interact.

Cascade spreading models, commonly used in economics to model ““binary decisions with externalities,”” can help us understand how small perturbations (relative to the size of a system) can, in rare instances, spread to all or most other elements of the system (generate a cascade), despite these same systems’ proven immunity to similar size perturbations in the past (Watts, 2002). In particular, the model adopted by Misić and colleagues takes three inputs (Figure 1.A): a network topology, a threshold (θ) describing how easily influenced nodes are by the status of their neighbors, and a perturbation event. In its simplest form, a few principles dictate how the model behaves (Figure 1.B): (1) all nodes in the network must be in one of two states (active = 1 or inactive = 0); (2) at time t = 0 all nodes, except those representing the initial perturbation, are in the inactive state; (3) at each successive time interval the status of all nodes is updated according to a binary rule (node i becomes active if, and only if, the percentage of its neighbors in the active state is greater than θ; and (4) once a node becomes active, it remains active for the rest of the simulation. To apply this model to the human brain, the authors first created an anatomic network model of the brain using human high-resolution diffusion spectrum imaging. The vulnerability threshold of all nodes (θ) was then set so that any perturbation could cause global cascades (i.e., change the state of all other nodes). With these initial conditions in place, the authors then evaluated how different types of perturbations spread through the network and how resulting cascades relate to resting-state networks (RSNs); trying to uncover the mechanistic principles by which anatomy constrains the spreading of information across the brain.

Figure 1.

Schematic Depiction of Cascade Spreading Models. (A) Basic inputs to the model. (B) Example of how a global cascade forms in four steps following a single initial perturbation. At t = 0 only one node is active. At successive time points, nodes are updated synchronously according to the status of their neighbors, until the perturbation reaches all nodes in the network (i.e., global cascade). (C) Schematic depiction of a single-seed model at its initial and final stages. (D) Same as (C) for a double-seed collaborative model. (E) Same as (C) for a double-seed competitive model. In this case, nodes can take one of two possible active states (e.g., red or blue). (F) Same as (E), except that this time, the number of perturbing seeds can be greater than 2.

First, using single-seed perturbations (Figure 1.C; only one node is active at t = 0), the model predicted that events spread significantly faster to ipsilateral nodes. It also predicted that early spreading of cascades is mostly driven by a compact backbone of pathways that interconnect medial orbitofrontal cortex, posterior cingulate cortex, and bilateral insula, while later spreading is completed along more lateral pathways. This single-perturbation model also predicted significantly slower spread of cascades between sensory motor RSNs (e.g., visual, somatomotor) than between multimodal RSNs (e.g., default mode network, frontoparietal network). Next, the model was run with distributed cooperative perturbations—namely perturbations of the same type that simultaneously originate in more than one location—as inputs (Figure 1.D). This permitted the authors to investigate how the simultaneous occurrence of nonconflicting (i.e., noncancelling) events may affect the spreading velocity of cascades throughout the brain. Here, two main observations were made. First, greatest speed-ups are typically observed when perturbing seeds are located on different hemispheres or anatomical communities. Second, synergic relationships between the somatomotor and the visual and ventral attention networks, as well as between the visual and frontoparietal networks, were observed. When seeds are located in these pairs of networks, speed-ups in cascade spreading are the greatest. Altogether, these results high- light how, even in their simpler form, cascade spreading models were able to identify the key role of hub regions in disseminating information across the network; confirm the important role of shortest paths in shaping communication (Goñi et al., 2014) and enabling cooperative effects; and predict some basic, yet behaviorally significant, patterns of inter- play between different types of RSNs based solely on anatomical constrains.

Next, Misić and colleagues proposed an optimized version of the model that evaluates how noncooperative perturbations can compete for resources as they spread through the network (i.e., the brain). For this purpose, the model was augmented so that nodes could now be in three potential states (inactive, active state 1, and active state 2), and perturbations could originate in more than two locations (Figures 1.E and 1.F). In this configuration, the model predicted a tendency for regions from the default mode network, the dorsal attention network, and the frontoparietal network to become the areas where competing cascades converge, providing opportunities for integration of information. Moreover, the probability of pairs of regions becoming members of the same cascade proved a quite accurate predictor of FC during rest. These results not only highlight the power of cascade spreading models to predict FC in a manner comparable, and sometimes favorable, to previously proposed computational models, but they also strongly suggest that the anatomy of the brain directs events to converge in multimodal associative areas in frontal and parietal cortex regardless of where they originated. Working backward from this, it may be possible to build on this simulation scaffolding to start to derive rules or principles by which brains have evolved for the optimal balance of computational efficiency and adaptiveness to new tasks.

In summary, this new research from Misić and colleagues (2015) offers insight into how the human connectome shapes information spreading across the brain, and highlights the importance of specific anatomical design principles on shaping its functional organization. As with most initial explorations, the proposed models were applied in its simplest forms, and updating the complexity of the models in future studies may lead to further insights on the relationship between SC and FC. For ex- ample, in its current state, the integrative behavior of all nodes is determined solely by the topology of their neighborhood. Allowing the vulnerability threshold (θ) to vary across nodes according to a priori knowledge about their processing and integrative behaviors may increase the predictive value of the models. Additional flexibility to allow nodes to switch states more than once, for the model to take into consideration directionality information, and inclusion of subcortical regions (which have rich patterns of interconnectivity between themselves and the rest of the cortex) may also render the models more realistic and biologically grounded.

Nonetheless, the computational simplicity of these models, combined with their grounding in anatomy, their versatility to study multiple configurations of events, and their ease of interpretation, makes them amenable to many neuroscience questions. In the manuscript, the authors suggest their use to improve our understanding of sensory-motor integration by modeling simultaneous perturbation of these two systems. Perhaps other areas of relevance may include the prediction of how the brain may react to different types of anatomical insults (investigating how cascade spreads are affected by changes in the topology), the derivation of time constants for spontaneous changes in RSNs over time, or even the prediction of how epileptic events spread throughout the brain during and prior to seizure episodes. Better modeling of the dynamic spread of such events may help optimize surgical interventions for this and other conditions (Taylor et al., 2014).

Over the years, the study of the brain using noninvasive neuroimaging technologies has evolved from a region-centric approach toward a more distributed network-oriented conceptualization. In this updated view, the functional specificity of some brain regions is not denied, yet the emphasis is on understanding how cognitive processes such as emotions, motivation, or language emerge from the distributed synchronized activation of multiple regions with not so immutable roles (Mesulam, 1998). During both rest (Hutchison et al., 2013) and task (Gonzalez-Castillo et al., 2012, 2014; Orban et al., 2014), many brain regions go in and out of synchrony in an orchestrated, spatially structured manner. A simple visual stimulation task—which in the context of the work of Misić et al. (2015) could be thought of as a single-seed perturbation—has been shown to produce fMRI signal fluctuations time-locked with stimulation events in over 76% of imaged gray matter (i.e., a sizable cascade) (Gonzalez-Castillo et al., 2014). When the task also includes a decision- making and a motor component (i.e., a competitive multiseed scenario), activation extend reaches over 85% (Gonzalez-Castillo et al., 2014) or 98% (Gonzalez-Castillo et al., 2012) (which could be regarded as a global cascade) depending on spatial resolution and other scanning parameters. In their work, Misić and colleagues show that under specific threshold conditions—whose biological plausibility ought to be investigated in detail—the structural human connectome is wired to support the emergence of global cascades in response to well-localized initial perturbations. This is in agreement with the above-mentioned empirical observations.

In the future, more biologically grounded cascade spreading models could become a faithful squire to neuroscientists on the quest to uncover the underlying anatomical and functional mechanisms by which complex behaviors emerge in the human brain.