Colloquium: Control of dynamics in brain networks

The ability to effectively control brain dynamics holds great promise for the enhancement of cognitive function in humans and the betterment of their quality of life. Yet, successfully controlling dynamics in neural systems is challenging, in part due to the immense complexity of the brain and the large set of interactions that can drive any single change. While some understanding has been gained of the control of single neurons, the control of large-scale neural systems—networks of multiply interacting components—remains poorly understood. Efforts to address this gap include the construction of tools for the control of brain networks, mostly adapted from control and dynamical systems theory. Informed by current opportunities for practical intervention, these theoretical contributions provide models that draw from a wide array of mathematical approaches. Recent developments are presented for effective strategies of control in dynamic brain networks, and potential mechanisms are also described that underlie such processes. Efforts are reviewed on the control of general neurophysiological processes with implications for brain development and cognitive function, as well as the control of altered neurophysiological processes in medical contexts such as anesthesia administration, seizure suppression, and deep-brain stimulation for Parkinson’s disease. This Colloquium is concluded with a forward-looking discussion regarding how emerging results from network control, especially approaches that deal with nonlinear dynamics or more realistic trajectories for control transitions, could be used to directly address pressing questions in neuroscience.

Figure 1

Model for adaptive cognitive control showing distinct mechanisms between different brain regions. Schematic of a neural network connecting the prefrontal cortex, which executes much of the “top-down” control actions to other brain regions. Another part of the brain, the anterior cingulate cortex, serves as a conflict monitoring mechanism that modulates the activity of control representations. Meanwhile, an adaptive gating mechanism regulates the updating of control representations in the prefrontal cortex through dopaminergic (DA) projections from the ventral tegmental area (VTA) that can also be facilitated through reinforcement learning (red asterisk). From [23].

Figure 2

Controlling a simple network. This small network can be controlled by an input vector ${\mathbf{u}}_{\mathcal{K}}=\mathbf{(}{u}_1(t),u_2(t){\mathbf{)}}^T$ (left), allowing us to move the network within the state space from its initial state to some desired final state (right). From [90].

Figure 3

Energetic costs of controllabilitry metrics. [111] proposed realistic control strategies that include the energetic costs of control (8). Average controllability describes transitions nearby on an energy landscape, while modal controllability describes transitions distant on this landscape.

Figure 4

Construction of a human brain structural network. (a) Diffusion imaging measures the direction of water diffusion in the human brain. (b) From these data, white matter streamlines can be reconstructed that connect brain regions. (c) An adjacency matrix representation of the structural connectivity: entries denote the estimated strength of white matter connectivity between brain regions. (d) The resulting brain network where nodes are brain regions, and where edges are the connection strengths between them.

Figure 5

Cognitive control hubs are differentially located across cognitive systems. (a) Hubs of average controllability are preferentially located in the default mode system. (b) Hubs of modal controllability are predominantly located in cognitive control systems, including both the frontoparietal and cingulo-opercular systems. (c) Hubs of boundary controllability are distributed throughout all systems, with the two predominant systems being ventral and dorsal attention systems. From [66].

Figure 6

Controllability metrics are positively correlated with age, with older youth displaying greater average and modal controllability than younger youth. Each data point represents the average strength of controllability metrics calculated on the brain network of a single individual, in a cohort of 882 healthy youth from ages 8 to 22 years. Brain networks were found to be optimized to support energetically easy transitions (average controllability) as well as energetically costly ones (modal controllability). There is a significant correlation between age and the ability to support this diverse range of dynamics: see inset or color (online) that denotes the age of the subjects. Note that modal controllability being a weighted sum of normalized eigenvectors is always capped at 1, hence its smaller range as compared to average controllability is not meaningful; rather, the relative differences between the values are meaningful here. From [134].

Figure 7

Burst suppression phenomenology. (a) A typical recording of burst suppression from a human subject anesthetized with propofol, a type of general anesthesia. The bursts manifest concurrently across the scalp (here, shown for left and right frontal electrodes). (b) Spectrogram for a frontal electrode during deep, but not burst suppression, general anesthesia. (c) At a deeper level of general anesthesia, burst suppression is achieved, and the spectrogram clearly displays epochs of quiescence. From [37].

Figure 8

Closed-loop stimulation for seizure suppression in a rat. Recordings from channels $a$, $b$, and $c$ in the cortex are filtered for spike detection, where signals exceeding the predetermined amplitude threshold are detected. These thresholded signals are used to trigger transcranial electric stimulation, which is applied through the scalp. From [16].

Figure 9

Schematics of patient electrophysiology and epileptic model. Left: Intracranial electrophysiology of patients with neocortical epilepsy. Each sensor (red dot) can be treated as a node within a functional network that uses magnitude squared coherence between sensors as network edges. Right: A model of the epileptic network, comprised of a seizure-generating system and a hypothesized regulatory system that controls the spread of pathologic seizure activity. From [84].

Figure 10

Synchronizability of structural brain networks and a negative correlation with age. (a) Schematic of a master stability function (MSF) for a generic network of oscillators, which gives the perturbative stability of a globally synchronous state ([113]). Such a state is stable when the MSF is negative for all positive eigenvalues of the graph Laplacian, hence the inverse spread of the Laplacian eigenvalues $1/{\sigma }^2(\{{\lambda }_i\})$ provides an estimate of synchronizability (or stability under synchrony); see [109]. (b) Synchronizability in structural brain networks estimated from diffusion imaging in a large cohort of 882 youth is found to be anticorrelated with mean average controllability as well as with age (see inset, or color online). From [134].

Figure 11

Motif structures that occur within networks. The motif structures studied by [142], through simulations of nonlinear biophysical neuronal models and their resulting controllability.

Figure 12

Example trajectory through state space. With external input (control signals), the system at state ${\mathbf{x}}_0$ is driven into the desired target state ${\mathbf{x}}_T$; without input the system’s passive dynamics leads to another state ${\mathbf{x}}_T$, where random brain regions are more active than others. From [18].

Figure 13

Setup for optogenetic control in a rat. Left: Fiber photometry setup showing a light path for fluorescence excitation and emission through a single 400 μm fiber optic implanted in the ventral tegmental area (VTA). Right: Recombinase-dependent viral targeting of GCaMP5 to VTA dopamine neurons. From [64].

Figure 14

Comparison between structural controllability and control using feedback vertex sets. (a) In structural controllability, the objective is to drive the network from an arbitrary initial state to any desired final state by acting on the network with an external signal $\mathbf{u}(t)$. The dynamics are considered to be well approximated by linear dynamics. (b) In feedback vertex set control, the objective is to drive the network from an arbitrary initial state to any desired dynamical attractor (e.g., a fixed point) by overriding the state of certain nodes. From [151].

Figure 15

Two-dimensional example of nonlocal trajectories. Example systems ${\dot{x}}_1=x_1+u_1(t)$, ${\dot{x}}_2=x_1$, where the curves indicate minimal-energy control trajectories for the given initial state (open symbol) and target states (solid symbols). Background arrows indicate the vector field in the absence of control. From [133].