Ethics statement.

All participants gave informed consent in writing and all protocols were authorized by the Institutional Review Board of the University of Pennsylvania.

Model parameters.

Under appropriate parameter choices, the WC model can give rise to oscillatory dynamics [53]. Such rhythmic activity is ubiquitous in large-scale neural systems [1] and is the dynamical behavior of interest for this investigation. While oscillation frequencies observed in neural systems can span orders of magnitude [1], local neuronal populations often exhibit gamma band (30-90Hz) rhythms as a result of feedback between coupled excitatory and inhibitory neurons [4, 85, 86]. Furthermore, gamma oscillations and synchronization between distributed brain areas are associated with the flow of information between neuronal ensembles [10, 87, 88], are modulated by stimuli [15, 16], and are thought to underlie a number of cognitive processes [6]. Because gamma oscillations are robustly observed in excitatory-inhibitory circuits, we set parameters in the phenomenological WC model such that individual brain regions oscillate in the gamma band when coupled [73] (see Table 1). We also note that it may be interesting in future work to investigate other frequency bands or multiple frequency bands simultaneously [89].

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Table 1. Parameter values for the large-scale Wilson-Cowan neural mass model and for the numerical simulations.

https://doi.org/10.1371/journal.pcbi.1008144.t001

As discussed further in Sec. SI of S1 Text, the non-specific background input P_E is the typical control parameter used to tune the behavior of an isolated WC unit. At low values of P_E, a single WC unit flows towards a low-activity steady-state (Fig. A, panel A in S1 Text), and at high values of P_E, the system reaches a stable high-activity steady-state (Fig. A, panel C in S1 Text). At intermediate values of the excitatory drive, an isolated unit—with the parameters given in Table 1—will undergo a bifurcation and exhibit rhythmic activity in the gamma frequency band (Fig. A, panel B in S1 Text). Up to a point, increasing P_E within this intermediate region leads to oscillations with increasing amplitude and frequency (Fig. A, panels D–F in S1 Text).

The situation becomes more complex when multiple WC units are coupled via the structural connectome. In this scenario, an individual region’s dynamics are determined by a combination of the constant drive P_E and the strength of delayed inputs from other parts of the network, which are modulated by the coupling C and the structural connectivity A. To account for these two influences, we consider both P_E and C as tuning parameters, and examine working points at which the combination of P_E and C generate oscillatory activity in individual brain areas. Finally, we set the signal propagation speed to a fixed value of v = 10m/s, which is in the range of empirical observations and previous large-scale modeling efforts [35].

Incorporating local perturbations into the large-scale model.

The baseline condition of the network corresponds to the situation in which all brain areas receive the same level of background drive, such that for all j ∈ {1, …, N}. To investigate how regional perturbations affect brain-wide dynamics, we examine the effects of increased excitation to a single brain area. This is modeled as a selective increase in drive to the excitatory population of the perturbed neural mass i such that P_E,i → P_E,i + ΔP_E, where ΔP_E > 0 denotes the strength of the perturbation [35] (see Fig 1E for a schematic). The dynamics of the system in the baseline state can then be compared to the situation in which unit i receives additional input (i.e., where we have and for all j ≠ i).

We note that, phenomenologically, excitation of a given brain area could occur through a number of mechanisms, including sensory input to primary sensory regions, brain stimulation, or, alternatively, via internal processes that regulate inputs to or excitability levels of specific neuronal populations. The goal of this work is to study the effects of localized excitations generally, rather than to design a detailed model of a specific type of perturbation. For this reason, we choose to study the simplest case of constant excitation.

Instantaneous phases from the Hilbert transform.

Given a real-valued signal X(t), it is possible to define instantaneous phase and amplitude variables that describe the signal using the Hilbert transform. Importantly, although the Hilbert transform can theoretically be computed for an arbitrary signal X(t), the instantaneous amplitude A(t) and phase θ(t) are only physically meaningful for relatively narrowband signals [91]. It is therefore necessary to filter a signal before taking the Hilbert transform. Here, raw time-series were bandpass filtered in a frequency range f_o ± Δf Hz using a 6th-order Butterworth filter in the forward and backward directions. In the results section, we describe how f_o and Δf are determined during the presentation of various findings that depend on computing the Hilbert phase. Filtering was carried out in MATLAB using the ‘butter’ and ‘filtfilt’ functions. After filtering the simulated activity, the Hilbert transform was applied to extract instantaneous phases for the given frequency band. The Hilbert transform was implemented using the ‘hilbert’ function in MATLAB. More details on the Hilbert transform can be found in Sec. SXIII of S1 Text.

Functional connectivity from the phase-locking value.

The outputs of the filtering and Hilbert transform processes described in the previous section are instantaneous phases θ_i(f_o, t) derived from the excitatory activity E_i(t) of each brain region i at a given central frequency f_o and time t (Fig 1F). From these phases, we can quantify the extent of phase-coherence between brain areas’ signals in a given frequency band using the phase-locking value (PLV); see Fig 1F. The PLV—here denoted symbolically as ρ_ij—between two phase time-series θ_i(t) and θ_j(t) is given by (5) where T_s is the number of sample time points over which the phase-locking is computed. If the phase difference Δθ_ij(t) = θ_i(t) − θ_j(t) is constant over a given time window, ρ_ij will be equal to 1, whereas if the phase-differences are distributed uniformly, ρ_ij will be approximately 0; in this way, ρ_ij ∈ [0, 1].

We would also like to ensure that the PLV reflects the consistency of phase relations that arise from interactions (direct or indirect), and not locking arising from the fact that two regions happen to have the same frequency, but, possibly, a different phase relation in every trial. We therefore concatenate phase time-series from different trials before computing the PLV [92], where each trial is a simulation run with different random initial conditions and noise realizations. Accordingly, a high PLV indicates that across time and trials, the activity of the corresponding regions exhibits a consistent phase relationship within a particular frequency band.

As with structural connectivity, it is useful to think of a given N × N matrix of PLV values as a network where the element (edge) ρ_ij is the phase-coherence between region (node) i and region (node) j. In contrast to the structural network, this PLV-based network represents the presence of functional associations between brain regions’ activity. Following common terminology, we will thus often refer to phase-locking as “functional connectivity” and phase-locking matrices as “functional networks”.

Long-range coupling strength and background drive tune baseline dynamical state.

We begin by computing two measures that quantify regional dynamics: (1) the time-averaged firing rate , and (2) the frequency at maximum power (peak frequency) f^peak of a given region. To obtain summary measures characterizing the state of the system as a whole, we compute network-averages of these quantities, denoted by angled-brackets. In studying as a function of and C, we observe three principal regimes (Fig 2A). When both and C are low, the system settles to a state of low average firing rate (white region); this state corresponds to a non-oscillatory, low-activity equilibrium. In contrast, when and C are both high, the average firing rate saturates at a high level (dark green region); this state corresponds to a non-oscillatory, high-activity equilibrium. Finally, at intermediate values of these parameters, the mean firing rate varies between the low and high extremes, and the regional activity is oscillatory; because we wish to consider the rhythmic nature of brain activity, this is the relevant portion of parameter space.

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Fig 2. Long-range coupling strength C and background drive modulate firing rates and oscillation frequencies at baseline.

(A) The time- and network-averaged population firing rate as a function of C and (units are arbitrary). (B) The network-averaged peak frequency of regional activity 〈f^peak〉 as a function of C and . (C) A segment of the activity of one brain area and (D) the corresponding power spectra of the same area at the working point denoted by the red dot in panels A and B (, C = 2.5). (E) A segment of the activity of one brain area and (F) the corresponding power spectra of the same area at the working point denoted by the orange dot in panels A and B (, C = 2.5). (G) A segment of the activity of one brain area and (H) the corresponding power spectra of the same area at the working point denoted by the yellow dot in panels A and B (, C = 2.5).

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Next we seek to understand how 〈f^peak〉 varies in the —C plane (Fig 2B). A clear wedge-shaped area marks parameter combinations that give rise to network-averaged peak frequencies in the gamma range. As with the firing rate, the peak frequency tends to increase (decrease) with either increasing (decreasing) background excitation or coupling strength. By comparing Fig 2B to 2A, we see that the white areas surrounding the purple wedge correspond to the regions of parameter space where the firing saturates at a fixed low or high value. In Sec. SII of S1 Text, we describe a systematic method for determining boundaries in the 2D space spanned by C and that indicate the onset or disappearance of oscillatory activity (see Fig. B in S1 Text). In what follows, we use to denote the level of background drive at which oscillations begin to emerge for a fixed coupling strength C. We refer the reader to Sec. SII of S1 Text for a detailed description of how this value is determined from the simulations. Furthermore, we often plot quantities as functions of the relative drive , such that indicates the transition point from a low-activity state to an oscillatory state at a coupling C.

To provide further intuition for how dynamics vary within this parameter space, we study example time-series and power spectra for three different baseline states (colored dots in Fig 2A and 2B). Note that these working points correspond to an intermediate coupling value of C = 2.5, but varying levels of the constant baseline input . We begin with the working point , which sits just beyond the boundary indicating the transition to sustained rhythmic activity. From the time-series, we observe that the activity is oscillating (Fig 2C), and the spectra indicates a peak frequency of ≈40Hz on a broadband background (Fig 2D). We next consider the working point . In this state, each unit receives slightly more drive, leading to higher-amplitude oscillations (Fig 2E and 2F). However, although peak spectral power increases, amplitude modulations can still be seen in the corresponding time-series (Fig 2E). Finally, we consider the working point . Here, the activity is characterized by regular, high-amplitude oscillations (Fig 2G). Furthermore, inspection of the power spectra indicates a single, narrow peak at a slightly higher frequency than the previous working point (Fig 2H).

Global phase-coherence is non-monotonically modulated by coupling strength and background drive.

Both the firing rate and the power spectra are measures that quantify the nature of individual regions’ activity. For networks, it is also imperative to define measures that capture information about the extent of dynamical order in the system as a whole. Indeed, for networks of coupled units, the system’s “state” is defined not only by the behavior of individual units, but also by how their dynamics are interrelated. Here, we are interested in the degree to which regional dynamics are coherent, which we quantify via the PLV between regions’ activities. To compute PLVs for baseline conditions, we begin by filtering the activity of each unit in the same, common frequency band. This band is determined by first finding the peak frequency of each unit at the given working point. Hence, we obtain a set of N values corresponding to the peak frequencies of all units i∈ {1, …, N} at baseline. We then filter the activity of every region in a frequency band spanning 10Hz above the maximum peak frequency in the network () and 10Hz below the minimum peak frequency in the network (). After identically filtering each unit’s activity in this common band, we extract Hilbert phases from the filtered signals. Finally, PLVs between all pairs of brain areas are computed according to Eq 5, using 50 different simulations (trials) of 5 seconds each (with noise included).

To summarize how the overall level of coherence in the network varies as a function of the background drive and coupling strength, we defined a macroscopic order parameter as the average of the PLVs over all pairs of units in the network: ρ^global = 〈ρ_ij〉. This quantity ranges between 0 and 1, where larger values indicate a more dynamically ordered state of the network. In general, we find that the background input and the coupling strength interdependently tune the level of coherence in the system (Fig 3A). At low coupling, brain areas cannot coordinate their dynamics and ρ^global remains at a relatively low value for a range of drives. In contrast, as the coupling is increased, we begin to see a qualitative change in behavior. For higher values of C, we observe that ρ^global varies non-monotonically (first increasing and then decreasing) as a function of the (relative) background drive. For a given coupling C, there appears to be a “critical” value at relatively small but non-zero where the system develops a well-defined peak in global coherence. As the drive is increased further, ρ^global begins to decrease and then eventually plateaus, albeit with some fluctuations. More specifically, at levels of background drive well beyond the state of peak coherence, ρ^global relaxes to an intermediate value between its peak and its value at the lowest background input. In this regime, the system resides in a state of partial order. Increasing the coupling has the effect of amplifying the maximum value of ρ^global (although ρ^global remains well below 1 for all couplings considered), but does not appear to significantly affect the order parameter to the right side of the peak.

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Fig 3. Long-range coupling strength C and background drive modulate network phase-coherence and relationships between structural and functional connectivity at baseline.

(A) The global order parameter ρ^global vs. , for different fixed values of C. Error bars are estimated from 100 bootstrap samples of the simulations at each coupling and background drive, and correspond to ± one standard deviation of the bootstrap disribution of ρ^global. (B) The difference between the global and local order parameters, ρ^global − ρ^local, vs. , for different fixed values of C. Error bars are estimated from 100 bootstrap samples of the simulations at each coupling and background drive, and correspond to ± one standard deviation of the bootstrap distribution of ρ^global − ρ^local. (C) Region-by-region PLV matrices for various values of at fixed C = 2.5. The boxed matrices correspond to the red, orange, and yellow working points in Fig 2 and in panels A and B of this figure. (D) The Spearman correlation r_s between structural node strength s^struc and functional node strength s^func vs. at fixed C = 2.5. Empty circles indicate that the correlation was not statistically significant at the p = 0.05 level. The arrows mark three different working points—WP1, WP2, and WP3 (which correspond to the red, orange, and yellow dots/boxes in this figure)—that will be studied in detail.

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To provide further intuition for this behavior, we focus on an intermediate coupling of C = 2.5 and examine the pairwise coherence patterns ρ_ij for several values of the background drive (Fig 3C). At the lowest (relative) baseline input (), some organization can be seen in the PLV matrix, but the system is weakly coherent overall. In this state, units exhibit relatively low amplitude oscillations, and are therefore more influenced by noise. It is thus reasonable to expect little phase-locking at low background drive. However, with only a small increase in the non-specific input (e.g., ), we observe distributed increases in coherence and a large spread of high, medium, and low coherence pairs dispersed throughout the network. Increasing the background drive slightly more (e.g., ) leads to the emergence of large, highly-coherent blocks that span the system. This working point sits near the peak of ρ^global and represents a highly ordered state of the system. As the background drive is increased further, though, phase-locking begins to decrease widely throughout the network and the coherence pattern markedly changes into a more segregated architecture. In particular, for high , we observe the emergence of smaller phase-locked clusters (Fig 3C, Row 3). To understand this shift in behavior, it is important to note that increasing increases the extent to which regional activity is independently generated in each area vs. driven by long-range network interactions. The strengthening of regional oscillations and enhanced influence of local dynamics with increasing seems to eventually hinder the ability of units to adjust their rhythms and achieve widespread coherence. Note that phase-locking is also made especially difficult by the large variance in the distribution of interareal delays imposed by the connectome’s spatial embedding, and indeed, for high background drive conditions, more strongly connected and spatially nearby units are those able to maintain stronger coherence.

In general, our observations point to complex behavior in which the macroscopic order parameter varies non-monotonically as a function of the baseline input and network coupling strength (Fig 3A and 3C). Therefore, a variety of qualitatively different regimes exist, beyond just a simple binary separation into a disordered and ordered state. To more quantitatively distinguish network states before and after the point of peak coherence, we also considered a local order parameter , which is a weighted average of ρ_ij with weights equal to the strength of structural network connections. In this way, ρ^local will be larger when more strongly connected brain areas are more phase-locked. In Fig 3B, we show ρ^local − ρ^global vs. for different values of the coupling C. Beyond a certain point, the curves for all couplings exhibit a clear upward trend where the extent of local coherence increases relative to the extent of global coherence. This behavior indicates that the macroscopic state of the system becomes increasingly constrained by structure as the background drive increases. Hence, even though the level of global coherence can be similar to the left and right of peak ρ^global, the system is in qualitatively different dynamical modes in the two regimes. Also note that for the higher couplings, ρ^local − ρ^global first decreases before consistently rising. This variation occurs because, for large enough coupling strengths, the level of global coherence is able to compete with the level of local coherence at background drives near peak ρ^global.

As a final demonstration of the complexity of the structure-function landscape across operating points, we consider the relationship between brain areas’ structural and functional connectivity strengths as a function of for a fixed coupling C = 2.5 (Fig 3D). The structural strength of node j, , is a common measure of a brain area’s importance in an anatomical network [98]. Similarly, the (baseline) functional strength of node j, , quantifies how dynamically integrated that region is to the network as a whole. From Fig 3D, we observe that shifting the system’s working point can drastically alter how—and the extent to which—structural strength and functional strength are related. Specifically, while there tends to be a weak positive correlation between s^struc and s^func at high background drives (e.g. at WP3), the correlation disappears (e.g. at WP2) and then reverses in sign (e.g. at WP1) as the background drive is lowered. Critically, these transitions occur in the absence of any change to the anatomical connectome, and are instead driven by a global change in the behavior of brain areas’ dynamics (induced by changing the level of background input). Also note that when the correlations are significant, they are intermediately-valued. Together, these results indicate that while a given structural network may only be able to support specific patterns of coordinated activity, the relationships between the two are not trivial and are modulated by dynamic properties [99, 100]. In general, functional connectivity thus reflects a complex interplay between both anatomical connectivity and the system’s dynamical state.

It is crucial to remark that the behaviors seen here are more diverse than what tends to occur in simpler phase-oscillator models, where coupling strength is the main control parameter and typically induces a monotonic increase in synchrony. A critical difference between phase-based models and the more realistic WC model considered here is that, for the latter case, unit dynamics are described and coupled by real-valued signals that represent regional activity. Hence, widespread changes in the amplitude or stability of areas’ dynamics (in addition to changes in coupling strength) can affect the macroscopic state of the network. Indeed, the preceding analyses show that global modulations in the level of diffuse, constant input to the neural populations can push the system into very different oscillatory modes, beyond just a steady progression from an incoherent to a coherent state. In what follows, we will exploit this behavior to examine how the effects of focal perturbations depend not only on which region is targeted, but also on the baseline working point of brain network dynamics as a whole.

Local excitations induce distinct modifications to power spectra.

We first examine the effects of a regional perturbation on areas’ time series and power spectra (Fig 4A–4C). In agreement with past experimental and modeling studies [16, 89, 101, 102], increased drive to the excitatory pool of region i increases the amplitude and frequency of its oscillations (Fig 4C, Left). In particular, stimulation causes an increase in the power, narrowing of the spectra (associated with an increase in periodicity of regional activity), and a shift of the peak frequency from ≈ 40Hz at baseline to ≈ 50Hz when excited. We also note the appearance of modulation sidebands in the excited spectra to the left and right of the peak frequency, which arise due to the modulation of the excited region’s time-series by the lower-frequency input it receives from other areas in the network [92]. This modulation also results in a spectral peak at ≈ 16Hz—which is the difference between the new, excited frequency and the sideband peaks, and is a marker of quasiperiodic amplitude modulation in the time-series. To more carefully quantify the effects of an excitation to region i, we consider the shift in the peak frequency of unit i, , between its excited and baseline states (Fig 4D). Calculating these differences for all choices of the stimulated brain area, we find that they range from about 6Hz to 16Hz, with an average value of Hz. These perturbation-induced shifts thus yield excited peak frequencies that are well-separated from the range of peak frequencies in the baseline state (Fig 4E).

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Fig 4. Regional excitation causes local and downstream changes to brain areas’ power spectra in different frequency bands at WP1.

(A) Schematic of a brain network depicting the stimulated site i in brightest red. The black arrows point to two other regions j and k that lie at progressively further topological distances from the perturbed area in the structural network. In this figure, regions i, j, and k correspond to brain areas 1 (R–Lateral Orbitofrontal), 4 (R–Medial Orbitofrontal), and 10 (R–Precentral), respectively. (B) Left: A segment of region i’s activity time-course in the baseline condition. Right: A segment of region i’s activity time-course when it is stimulated. (C) Power spectra of area i and two other downstream regions j and k. In all three panels, the lighter curves correspond to the baseline condition, and the darker curves correspond to the state in which i is driven with additional input. The gray vertical lines indicate the peak frequency of region i in the excited condition. (D) Histogram of the shift in peak frequency induced by stimulating unit i, plotted over all choices of the perturbed area. (E) Distribution of peak frequencies of all units in the baseline condition (light gray) and distribution of the peak frequency units acquire when directly excited (dark gray). (F) Average power spectra 〈psd〉_j≠i over all units j ≠ i at baseline (light gray) and when unit i is perturbed with additional input (dark gray). (G) Average difference of the spectra of unit j ≠ i when unit i is excited and in the baseline condition, where the average is over all units j ≠ i. For reference, the light gray vertical lines denote the minimum and maximum peak frequency across units in the baseline state, and the dark gray line indicates the peak frequency acquired by the stimulated region i. Shaded boxes denote two frequency bands of interest: (1) the baseline band (purple) consisting of the main oscillation frequencies of brain areas under baseline conditions, and (2) the excited band (green) centered around the peak frequency that the stimulated region inherits. In subsequent analyses, we assess perturbation-induced changes in the PLV between brain areas in the baseline band, (purple), and in the excited band (green).

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We next consider the power spectra of two other units j and k located at increasing topological distances from the excited region, where a shorter topological distance indicates that two areas are linked by a path of stronger structural connections [98]. (Fig 4C, Middle, Right). We observe that unit j maintains its initial frequency content, but also develops new peaks centered at the frequency of the excited region and at the difference of the excited frequency and the baseline peak. In contrast, the spectra of unit k—which is more weakly structurally connected to the stimulated site—is relatively unchanged. Hence, depending on the network structure, stimulation of region i can also cause alterations to other regions’ spectra. In general, the power modulation of a downstream area’s spectra at the peak frequency of the stimulated site decays with increasing topological distance between the dowstream area and the perturbed region (see Sec. SV in S1 Text). To summarize how the spectra of other brain areas are altered by driving region i with additional input, we compare the average power spectral density 〈psd〉_j≠i over all units j ≠ i at baseline and when unit i is stimulated (Fig 4F). At baseline, the network-averaged spectra is relatively broad and contains multiple peaks—a main one at 38Hz and a smaller peak around 34Hz. In addition, a local excitation produces complex and broadband alterations in power, as expected in a scenario of quasiperiodic entrainment between nonlinear oscillators [103]. For this example, we observe the appearance of an entirely new peak at 50Hz, but also an enhancement of the lowest baseline peak and a depression of the highest baseline peak. These changes are perhaps more apparent in Fig 4G, which shows the average difference in the spectra of unit j ≠ i between when unit i is excited and the baseline condition, where the average is over all units j ≠ i. In sum, we see that a regional enhancement of neural activity causes non-local modulations in power both at the frequency of the directly stimulated brain area, as well as at the system’s baseline oscillation frequencies. These analyses suggest that there are two relevant frequency bands to consider for subsequent analysis: (1) a relatively broad band containing the main frequencies of brain areas in the baseline state, and (2) a band centered around the peak frequency of the excited unit. In what follows, we will denote these two bands as “baseline” and “excited”, and consider changes in phase-locking, and , in each band induced by local perturbations.

Excitations of regional activity induce or alter interareal phase-locking in excited and baseline frequency bands.

We are now prepared to study how focal perturbations alter the coordination of network-wide dynamics. Specifically, we examine changes in interareal phase-locking. We separate our analysis into two frequency bands—baseline and excited—by filtering regional activity in each band, extracting Hilbert phases from the filtered signals, and then calculating the PLV for each pair of regions within each band (see Fig 1E; Materials and methods). Since spectra are relatively broad at baseline, a single baseline frequency band for the network is determined by first finding the set of peak frequencies for each unit i in the baseline state, . Next, the lower frequency for the common baseline band is set to Hz, and the upper frequency is set to Hz. A region-by-region PLV matrix corresponding to the single baseline band is then computed after identically filtering each unit’s activity in this frequency range. To examine phase-locking between units within the much narrower excited band corresponding to a given stimulated region i, we first extracted the peak frequency of region i when it is stimulated, . A PLV matrix corresponding to unit i’s excited frequency is then computed after filtering each region’s activity in the same frequency band ranging from Hz to Hz. This range was chosen to contain the majority of the excited band peak, while including as little of the original baseline band as possible. If the peak frequency of the stimulated area was not more than 3.5Hz above the largest baseline peak frequency, then we only examined PLV changes in the baseline frequency band. Our choices are motivated by the following observation: the notion of an excited frequency band is only meaningful when a perturbation introduces a new spectral peak into the system that is separated from the frequencies present in the baseline condition. Also note that, unlike in the baseline band, areas exhibit little power at the excited frequency prior to stimulation; hence, we use changes in excited-band PLV as a measure of how effectively induced activity at the excited frequency spreads in the network.

To provide intuition about how phase-locking is altered upon a local perturbation, we consider the effect of stimulating two different brain areas (left and right panels in Fig 5A and 5C). These examples show that regional stimulation induces phase-locking at the excited frequency (Fig 5C), but can also cause changes in coherence in the frequency band containing the original oscillatory activity of the system (Fig 5A). Note that the excited band effects are mostly positive (due to the fact that power in the excited band is boosted by stimulation), whereas the baseline band effects can be both positive and negative. Furthermore, the patterns induced in the excited band (Fig 5C) are distinct from the modulations that occur in the baseline band (Fig 5A), and the phase-locking changes are markedly different between perturbation of region i (left panels) and perturbation of region j ≠ i (right panels). Thus, depending on the frequency band considered and the excited area’s location within the large-scale brain network, stimulation induces different responses across the system as a whole.

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Fig 5. Phase-locking changes at WP1 are driven by local excitations of neural activity, differ between excited and baseline frequency bands, and are differentially related to structural and functional network properties.

(A) Pairwise changes in the PLV inside the baseline band Δρ^base when region i (Left) or region j ≠ i (Right) is perturbed. In this figure, regions i and j correspond to regions 4 (R–Medial Orbitofrontal) and 23 (R–Lateral Occipital), respectively. (B) Network-averaged absolute PLV changes in the baseline band caused by stimulation of different brain areas. (C) Pairwise changes in the PLV inside the excited band Δρ^exc when region i (Left) or region j ≠ i (Right) is perturbed. (D) Network-averaged absolute PLV changes in the excited band induced by stimulation of different brain areas. (E) The quantity vs. structural node strength (Left), and the quantity vs. structural node strength (Right). (F) The quantity vs. functional node strength (Left), and the quantity vs. functional node strength (Right). In panels (E) and (F), insets indicate Spearman correlation coefficients between the plotted quantities and their associated p-values).

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To summarize the global effect of regional excitation, we calculate the average absolute change in PLV induced by driving each brain area with additional input. We use the notation and to denote the network-average of the absolute PLV changes in the baseline and excited frequency bands, respectively, induced by stimulating region i. Note that since the PLV is always between 0 and 1, the maximum possible value of both quantities is 1. Furthermore, we use a phase-randomized null model (described in Sec. SXII of S1 Text) to assess whether pairwise PLV changes are significant. Prior to calculating the network-wide averages and , non-signicant changes are set to zero. We observe that the global responses exhibit a large degree of variability across different choices of the stimulated area (Fig 5B and 5D). That is, perturbation of some areas induces larger system-wide modulations of phase-locking than others. Furthermore, regions that induce the largest overall changes in PLV inside the excited frequency band are not necessarily those that cause the largest alterations of PLV in the baseline band. This observation suggests that distinct aspects of the network may be indicative of the overall effects generated at the baseline and excited frequencies.

Structural and functional connectivity are linked to different types of phase-locking modulations at WP1.

What properties of the system drive or predict the diverse, distributed responses in the baseline and excited frequency bands brought about by focal stimulation? Because the network of anatomical connections couples different brain areas and allows them to directly interact, it is reasonable to hypothesize that the organization of this network should play a role in guiding the influence of a perturbation. To test this hypothesis, we study and as functions of structural node strength (Fig 5E). Interestingly, the global PLV modulation induced in the network’s naturally-emergent frequency band is not well-predicted by the anatomical strength of the stimulated area (Fig 5E, Left). In contrast, though, we do observe a strong association (Spearman correlation r_s = 0.96, p < 0.001) between structural strength and the PLV change elicited in the excited band (Fig 5E, Right). This result indicates that more structurally connected units generate larger overall effects at the enhanced frequency of the directly stimulated area. Because the excited band response is strongly constrained by structure, we also examined whether the effects differed between two broad, anatomically-defined classes of nodes. In particular, we compared the average excited band response for stimulation of cortical vs. subcortical areas (both of which are included in the anatomical parcellation). Given this breakdown, we find that the overall effect is significantly higher upon perturbation of subcortical regions (see Fig. E in S1 Text). This result is consistent with the findings of [35, 79], and reflects the notion that subcortical nodes make strong, distributed structural connections that may support large-scale network communication [104].

Although the brain’s structural connectivity plays a crucial role, macroscale activity patterns generally reflect an interplay between connectome architecture and the network’s dynamic regime. Indeed, in Fig 3D we observed that the correlation between structural and functional strength varies in intensity and sign with working point. Importantly, the presence of a functional connection between two brain areas implies an interdependence of their dynamics—enforced by the system’s oscillatory state—that can occur even in the absence of a direct structural connection. Intuitively, we may thus expect the organization of the system’s initial functional interactions (which could be non-trivially related to structure), to be indicative of how the coherence pattern is modulated under perturbation. Given this reasoning, we study vs. (Fig 5F). Relative to structural strength, we observe a strong positive relationship (Spearman correlation r_s = 0.71, p < 0.001) between the average absolute change in baseline band coherence and functional strength. Thus, areas that are initially more coherent with other regions in the network tend to yield larger global modulations to baseline band interactions when perturbed. This should be contrasted to the results from structural node strength, for which there was not a strong relationship with absolute coherence changes at the baseline oscillation frequencies. Finally, we consider vs. (Fig 5F, Right). Though they are correlated, (r_s = −0.34; p < 0.05), the stimulation-induced responses in the excited band are much more strongly predicted by structural rather than functional strength.

The above results emphasize separate consideration of both anatomical network topology and the organization of emergent functional interactions, the latter of which is also driven by the dynamical regime of the system. In particular, for the working point considered here, structural and functional network properties relate to distinct types of perturbation-induced effects. First, phase-locking changes that arise in the excited band reflect the transmission and replication of oscillatory input from the directly excited area to and in downstream regions. If the structural connection between the stimulated site and a downstream area is strong enough, then the drive from the stimulated site will induce a new spectral component in the receiving area (see, e.g., Fig 4C); consequently, the two regions will exhibit phase-locking at the excited frequency. In addition, even two areas that are not directly linked can display a high PLV in the excited band due to strong common input from the stimulated region, or due to the propagation of the stimulated site’s signal along alternative paths in the network. In sum, because spreading of the perturbed area’s activity is highly constrained by the presence of structural connections, regions with stronger anatomical connectivity to other areas more forcefully drive downstream regions and lead to larger excited band effects. Perhaps more interesting are the modulations in coherence that occur in the baseline frequency band. These changes arise not due to a direct transmission of input, but rather via adjustments to the ongoing, mutual entrainment between units’ spontaneous rhythms. For WP1, the resulting alterations to the strength of coherent interactions are more related to the stimulated region’s initial functional connectivity rather than its anatomical strength. Intuitively, this may in part be due to the fact that perturbing a particular area tends to decouple it from other areas at the original oscillation frequencies, such that stimulating regions that are strongly coherent to begin with effectively reconfigures existing functional interactions in the baseline frequency band. Notably, the observed correlation between functional strength and baseline band coherence modulations does not uncover the deeper, precise mechanisms behind the effects. However, because the brain’s collective dynamics can reflect a complex interplay between its oscillatory state and its structural connectivity, the result highlights the importance of considering both aspects when trying to understand network-wide responses to perturbations.

Spectral changes are more restrained at the high background drive working point.

Inspection of a region’s activity time-series at baseline and when stimulated with additional input indicates noticeable differences in how perturbations alter local activity at WP3 versus at either WP1 or WP2. Specifically, when the system operates in the high-drive state, a perturbation of the same strength has a much less drastic effect on the stimulated region’s activity, inducing only relatively small changes to its amplitude and frequency (Fig 6B).

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Fig 6. Effects of local excitations on power spectra are more restricted at the high background drive working point (WP3).

(A) Schematic of a brain network depicting the stimulated site i in brightest red. The black arrows point to two other regions j and k that lie at progressively further topological distances from the perturbed area in the structural network. In this figure, regions i, j, and k correspond to brain areas 1 (R–Lateral Orbitofrontal), 4 (R–Medial Orbitofrontal), and 10 (R–Precentral), respectively. (B) Left: A segment of region i’s activity time-course in the baseline condition. Right: A segment of region i’s activity time-course when it is stimulated. (C) Power spectra of area i and two other downstream regions j and k. In all three panels, the lighter curves correspond to the baseline condition, and the darker curves correspond to the state in which i is driven with additional input. The gray vertical lines indicate the peak frequency of region i in the excited condition. (D) Histogram of the shift in peak frequency induced by exciting unit i, plotted over all choices of the perturbed area. (E) The average shift in the peak frequency of the stimulated region for WP1, WP2, and WP3 (error bars indicate the standard deviation over all choices of the excited unit). (F) Distribution of peak frequencies of all units in the baseline condition (light gray) and distribution of the peak frequency units acquire when directly excited (dark gray). (G) Average power spectra 〈psd〉_j≠i over all units j ≠ i at baseline (light gray) and when unit i is perturbed with additional input (dark gray). Because stimulation does not induce a well-separated excited frequency band, we only assess perturbation-induced changes in the PLV between brain areas for a single baseline frequency band (purple area).

https://doi.org/10.1371/journal.pcbi.1008144.g006

Consequences of regions’ enhanced baseline activity for focal stimulation are perhaps more evident from examples of areas’ spectra at baseline and under stimulation (Fig 6C). As for the lower-drive working points, the peak frequency and power of the stimulated region i shift to higher values (Fig 6C, Left). However, at WP3, the increase is modest relative to the shifts that occur at either WP1 or WP2, and no modulation sidebands arise in i’s spectra under excited conditions. As a result of the more unyielding nature of spontaneous dynamics, stimulation of unit i also has relatively little impact on the spectra of downstream regions (Fig 6C, Middle, Right), even if they are positioned topologically close to the perturbed site. To more generally quantify the effects of regional stimulation on areas’ power spectra, we examine the distribution of the shifts in peak frequency that occur due to perturbation of each unit. The largest of these shifts is only about 3Hz (Fig 6D). Hence, relative to WP1 and WP2, the average shift in peak frequency is greatly reduced at the high-drive working point (see Fig 6E). Furthermore, unlike the situation in the low-drive state, the distributions of peak frequencies at baseline and under focal stimulation begin to overlap at WP3 (Fig 6F), precluding the notion of separate baseline and excited frequency bands. For this reason, in our subsequent analyses we only consider phase-locking changes inside a single frequency band (Fig 6G). While we refer to this as the “baseline band”, we note that it still contains the peak frequency of the directly excited unit, since its frequency shift is so small.

The results presented in this section indicate that regional dynamics are more robust to perturbations at the high-drive working point. This can in part be understood by considering the effects of the background drive , which is the parameter tuned to move from WP1 → WP2 → WP3. In particular, the increased baseline input level at WP3 means that each unit, if disconnected from the network, would operate closer to the bifurcation separating the quiescent and oscillatory state than would be the case at WP1 or WP2. As a result, stronger oscillations emerge at WP3 when the network coupling is introduced, reflecting the increased influence of recurrent dynamics. The high-amplitude rhythms that arise in the high-drive state are more difficult to disrupt, leading to minimal changes in the power spectra under local perturbations. For the same reasons, it is also more difficult for a local change in activity to propagate and influence the dynamics of remote areas. In contrast, when the system operates at either WP1 or WP2, the baseline oscillations at each brain area are weaker. This lower-amplitude activity is easier to override, yielding the system more plastic and susceptible to local perturbations. This flexibility at WP1 and WP2 is reflected by clear modifications to regional spectra upon local stimulation and the signatures of the stimulation effect in downstream regions (Fig 4 and Fig. C in S1 Text).

Focal perturbations yield a distinct and more homogenous set of phase-coherence modulations at the high-drive working point.

While the rigidity of baseline rhythms at WP3 prevents the emergence of a well-defined excited frequency band, we can still assess the effects of selective perturbations on interareal phase-locking in the single, baseline frequency band. We carry out such an analysis in this section, focusing on contrasting the results obtained at WP3 to those obtained previously at WP1 (and WP2). To begin, we show examples of the pairwise changes in PLV induced by stimulation of two different brain areas i and j (Fig 7A), and the average absolute changes in phase-locking driven by perturbation of each brain area in the network (Fig 7B). As with WP1, the statistical significance of PLV changes is first assessed with a phase-randomized null model (see Sec. SXII of S1 Text) before computing network averages. Despite differences in the induced average responses across regions, visual comparison of the distribution at WP3 (Fig 7B) and at WP1 (Fig 5B) suggests that there may be less variation across regions when the system operates in the high-drive state. Indeed, although the mean of the average absolute changes across all brain areas is approximately the same at WP1 and WP3, when we compare the coefficient of variation (CoV) of the two distributions of , we find that the CoV for WP3 is 0.25, while for WP1 it is 0.45. This indicates that global coherence modulations elicited by regional stimulation are more homogenous across different choices of the stimulated brain area when the system operates in the high background drive regime. Because we are particularly interested in how perturbations may differentially affect network dynamics depending on baseline state, we also investigate if the average absolute PLV modulations are correlated between WP1 and WP3. This relationship is not significant (Fig. G in S1 Text), indicating that the rank-ordering of is quite different between the low and high drive regimes.

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Fig 7. Phase-locking modulations induced by regional stimulation at WP3 and their associations with network properties.

(A) Pairwise changes in the PLV inside the baseline band Δρ^base when region i (Left) or region j ≠ i (Right) is perturbed. In this figure, regions i and j correspond to regions 10 (R–Precentral) and 15 (R–Isthmus), respectively. (B) Network-averaged absolute PLV changes in the baseline band induced by stimulation of different brain areas. (C) The quantity vs. structural node strength (Left) and vs. functional node strength (Right). Insets indicate Spearman correlation coefficients between the plotted quantities and their associated p-values).

https://doi.org/10.1371/journal.pcbi.1008144.g007

We next considered the associations between structural or functional properties of network nodes and the global phase-locking changes induced by regional stimulation (Fig 7C). At WP3, we find a strong positive correlation between and structural strength (r_s = 0.82, p < 0.001; Fig 7C, Left). Thus, larger coherence modulations at the high-drive working point are dependent on the presence of strong anatomical connections emanating from the stimulated region. Note that functional strength also exhibits a positive correlation with (r_s = 0.34, p < 0.05; Fig 7C, Right), which we might expect given the weak but positive correlation between s^struc and s^func at this working point (Fig 3D). However, structural connectivity is undoubtedly a more robust predictor of these effects.

A supplementary analysis (see Fig. H in S1 Text) indicates that the association between and is (largely) explained by a relationship between structural strength and the bulk decreases in coherence induced by perturbations. Hence, units with stronger anatomical connectivity to the network as a whole generate larger global breakdowns in coherence when stimulated with constant, additional input. To gain intuition for this result, recall that in the high-drive regime, brain areas exhibit relatively strong and inflexible oscillations at baseline. Furthermore, though not enough to bring about an entirely new excited band, stimulation still slightly accelerates the frequency of the perturbed unit. As a consequence, we may expect stimulation of regions with stronger structural connections—which have more direct influence on other areas—to drive more disruptions in the network’s ongoing dynamics. That is, a given node will tend to be driven out of coherence with other units when stimulated (since it acquires a slightly increased frequency), and high-strength nodes in particular will also be able to decouple other areas from their baseline functional assemblies, in turn causing larger dissociations of functional connectivity.

To conclude this section, we highlight the observation that depending on the oscillatory regime, different aspects of the system best predict global responses to focal stimulation. In the low-drive state, emergent coordination between units activities at baseline is more strongly related to coherence modulations at regions’ spontaneous frequencies, whereas structure is more strongly associated with the effects in the high-drive regime.