Fig 1.
Building nonlinear brain networks.
(a) Subject-specific structural brain networks are built based on a parcellation of the brain into 83 anatomically defined brain regions (network nodes) with connections between regions given by the density of streamlines linking them. (b) The dynamics of each region are represented by a single Wilson-Cowan oscillator, and these oscillators are coupled according to the structural connectivity of a single subject. (c) Brain states are quantified by calculating the pairwise functional connectivity between brain regions.
Fig 2.
Nonlinear brain dynamics and variability.
(a) Excitatory-inhibitory phase space plots depicting behavior for a single Wilson-Cowan oscillator in the presence of no external current input (left; P = 0; low-fixed point), moderate external current input (middle; P = 1.25; limit cycle), and high external current input (right; P = 2.5; high fixed point). All simulations are started with initial conditions E = 0.1, I = 0.1 and nullclines are plotted in green. (b) The corresponding firing rate of the excitatory population plotted as a function of time for the simulations depicted in (a). (c) Box plots showing the value of global coupling parameter at which the system transitions from the low fixed-point state to the oscillatory regime for models derived from three different structural scans from each of eight subjects. (d) Within and between subject reproducibility (left) and variability (right) for the data shown in (c). Reproducibility is measured by the intraclass correlation coefficient (ICC) and is high within subjects indicating a high level of reproducibility between scans of a single subject, but low between subjects indicating a low reproducibility between scans of different subjects. This is additionally reflected in the low within subject variability, measured as the average variance, and high between subject variability.
Fig 3.
Linear regional controllability.
(a) Average controllability plotted as a function of regional degree for each of the 83 brain regions. (b) Average controllability plotted as a function of regional degree. Controllability predictions were performed for each of the three scans for each subject and the data points reflect controllability and degree values averaged across scans.
Fig 4.
(a) Stimulation of a single region pushes the region from fluctuations around its low fixed point to the oscillatory state. (b) Example brain regions identified as having low average controllabllity (pars opercularis, blue), medium average controllability (post central, green) and high average controllability (isthmus cingulate, orange) (c) Simulation of example regions in panel (b) differentially move the system into new functional states. Stimulation applied to regions of high average controllability imparts more change in the functional brain state than stimulation applied to regions of low average controllability.
Fig 5.
Functional effect of stimulation.
(a-b) The functional effect of regional stimulation plotted as a function of the average (a) and modal (b) controllability for each of the 83 brain regions. (c-d) The structural effect of regional stimulation plotted as a function of the average (c) and modal (d) controllability for each of the 83 brain regions. Controllability predictions, simulations, and calculation of the functional and structural effects were performed for each of the three scans for each subject and the data points reflect values averaged over scans.
Fig 6.
(a-b) The absolute change in functional connectivity and resulting fractional activation shown for a threshold value of 0.6. (a) Example of stimulation applied to a region of low average controllability resulting in a focal effect on the resulting functional connectivity matrix. (b) Example of stimulation applied to a region of high average controllability resulting in a global effect on the resulting functional connectivity matrix. (c) The relationship between functional effect and fractional activation due to regional stimulation. We observe a high positive correlation between functional effect and fractional activation (Spearman’s ρ = .992, p ≪.001), indicating that a high functional effect corresponds to a global impact of stimulation while a low functional effect corresponds to a focal impact of stimulation. (d) The relationship between structural effect and fractional activation due to regional stimulation. Calculations of the functional and structural effects and fractional activation were performed for each of the three scans for each subject and the data points reflect values averaged over scans.
Fig 7.
(a) Structural and functional effect values for stimulation of individual brain regions sorted into 9 cognitive systems. Colored ellipses are centered on the mean structural and functional effects for a given cognitive system and the major and minor axis of the ellipse represent the standard error of the mean for the associated system. (b) Same as (a) but data is further course grained into 4 cognitive system types as in [36]. The colored regions indicate the convex hull surrounding the data points associated with the given system. Simulations, and calculations were performed for each of the three scans for each subject and the data points reflect values averaged over both scans and individuals. (c) Average density of connections within and between the four cognitive system types. Colors correspond to system assignments in (b) and dark shades represent the average density of connections between regions within a single cognitive system while light shades represent the average density of connections between regions within that system and regions outside of the system.
Table 1.
Average density of connections between and within subnetworks of four cognitive systems.