Functional organization of cortical dynamics, in the model and in the data, can be operationalized by constructing cortical GBC maps, for comparison to our prior empirical findings (Preller et al., 2018). The change in GBC induced by LSD provides a topographic map across cortical regions of the mean effect on FC, which we have previously compared to the topography of the HTR2A gene expression map (Preller et al., 2018). The first step in computing a GBC map is to construct an FC matrix, defined as the matrix of Pearson correlation coefficients between pairs of brain regions’ BOLD signal time traces. We apply the Fisher r-to-Z transformation to off-diagonal elements of this matrix such that values are approximately normally distributed. Off-diagonal elements are then averaged along either rows or columns (as the FC matrix is symmetric), yielding a map of GBC which consists of a single scalar value for each region.

As reported in Preller et al., 2018, GBC maps were constructed from resting-state BOLD fMRI scans for 24 study participants. Subjects served as their own controls. GBC maps were constructed for each subject in both drug conditions (placebo and LSD). The group-averaged contrast map of the change in GBC for LSD vs. placebo drug conditions – henceforth referred to as the ΔGBC map – specified the target model output (Figure 1D). In other words, we sought to recapitulate in our model the pattern of functional alterations indicated by changes in cortical GBC topography.

Empirically, the group-averaged ΔGBC map reveals widespread LSD-induced hyper-connectivity of sensory-somatomotor regions, and hypo-connectivity of association regions (Preller et al., 2018). In the models, we generate a GBC map for both the unperturbed and neuromodulated models, corresponding to models for the placebo and LSD conditions, respectively (described in more detail below). From these maps, we compute a model ΔGBC map which is then quantitatively compared to the empirical ΔGBC map (Figure 1D).

The operating point of the unperturbed model was first calibrated such that it best captured empirical FC in the placebo condition. This was achieved by performing a one-dimensional grid search over the value of a global long-range coupling parameter, which uniformly scales the strengths of all long-range connections between nodes (Deco et al., 2014). We determined the value of this coupling parameter that yielded the greatest Spearman rank correlation between the off-diagonal elements of the simulated and empirical FC matrices. The coupling parameter was subsequently fixed at this value. The model FC matrix was then used to compute a baseline model GBC map – our computational analog of the empirical GBC map in the placebo condition.

Next we performed a two-dimensional grid search over the two free model parameters which set the strength of gain modulation on excitatory and inhibitory neurons (δE and δI), such that a modulation of gain by 1% corresponds to δ=0.01 (Figure 2A). For each combination of parameters (i.e., at each point on the parameter space grid), we updated the simulated gain functions and re-computed the system’s stable fixed point. This now-perturbed system is used to generate a new FC matrix, from which we again compute a model GBC map – this time, our computational analog of the empirical GBC map in the LSD condition. For each combination of free parameters, we then computed a model ΔGBC map by calculating the difference between the perturbed and unperturbed model GBC maps. To evaluate the quality of the model for each set of parameters, we evaluated the statistical similarity of each model ΔGBC map to the empirical ΔGBC map. We quantified similarity by computing the ‘loading’ of the model ΔGBC map onto the empirical ΔGBC map, which we define as the Cartesian dot product between the two ΔGBC maps (expressed as vectors), divided by the squared norm of the empirical ΔGBC map (the natural scale in the quantity). This metric was chosen specifically for its sensitivity to both the topography and magnitude of the model perturbation.

We found that the model ΔGBC map loaded most strongly onto the empirical ΔGBC map across a quasi-degenerate range of parameter values (Figure 2A, off-diagonal yellow band). The location of this regime suggests that neural gain is preferentially modulated on excitatory pyramidal neurons, which nonetheless remain in a specific ratio with inhibitory interneurons. To investigate this further, we next characterized the relationship between changes in model excitatory-to-inhibitory (E/I) balance and model-empirical loading. For each combination of gain modulation parameters (i.e., at each location on the two-dimensional grid in parameter space), we computed the model’s E/I firing rate ratio, relative to its unperturbed value, where we define E/I ratio as the ratio of the mean excitatory firing rate (computed across nodes) to the mean inhibitory firing rate. We find that model-empirical loadings, when plotted as a function of fractional change in E/I ratio, collapse to form an approximately one-dimensional curve which peaks at positive shifts in E/I ratio (Figure 2B). This one-dimensional collapse suggests that gain modulation-induced changes in E/I balance are the primary mechanism through which GBC topography is altered under LSD. Moreover, the fact that the peak occurs for positive shifts in E/I balance suggests that LSD has a net-excitatory effect on cortical activity.

The model generates a strong functional perturbation that is aligned with the data, but does it also capture fine-grained functional network-specific and parcel-level effects? Empirically, LSD exhibits bidirectional effects across sensory and association brain networks, inducing hyper-connectivity of constituent brain regions in sensory and somatomotor networks, while inducing hypo-connectivity of regions in associative networks (Preller et al., 2018). To investigate network effects in the model, we set the gain modulation parameters to the values that yielded the greatest model-empirical loading (indicated by the black star in Figure 2A). We then characterized the distribution of model ΔGBC values across parcels within each functional network defined in the Cole-Anticevic Brain Network Parcellation (CAB-NP) (Figure 2C; Ji et al., 2019). We found that empirical network specificity of GBC changes was recapitulated in the model (Figure 2D–E). This finding indicates that the strong model-empirical loading is not driven by select functional networks or by a small subset of all cortical parcels. In fact, at the level of parcels we find a remarkably strong spatial correspondence between ΔGBC topographies in the model and data (rs=0.73; p < 10⁻⁵; Spearman rank correlation) (Figure 2F). Note that brain map values, here and in subsequent figures, are plotted on a flattened (i.e., unfolded) representation of the cortical surface, to better illustrate maps’ contiguous spatial topographies. This statistical association was significantly stronger than the association between the HTR2A gene expression map and the empirical ΔGBC map (p = 0.011; test for difference between dependent correlations) (Figure 2H). Heterogeneous physiological responses to LSD thereby play a critical role in shaping the cortical topography of LSD-induced GBC changes.

Global signal regression (GSR) remains a debated pre-processing step that is frequently included in fMRI analyses (Murphy and Fox, 2017). Preller et al., 2018 extensively characterized these data, with and without GSR applied. There, a statistically significant brain-behavior relationship between the neural and experiential effects of LSD was observed when – and only when – GSR was performed. Moreover, GSR was found to substantially alter the topography of the empirical ΔGBC map: when GSR was not used, the spatial correspondence of the ΔGBC map with the HTR2A expression map is dramatically attenuated (Preller et al., 2018). Here, we find that when GSR is not performed, maximal model-empirical loading and Spearman rank correlation between the model and empirical ΔGBC maps are both reduced significantly (Figure 2—figure supplement 1). This finding further supports the notion that GSR attenuates non-neural components or artifacts which otherwise obscure meaningful neuronal effects in pharmacological fMRI signals.

In addition to stimulating 5-HT_2A receptors, LSD exerts predominately agonistic activity at (5-HT)-2C, -1A/B, -6, and -7 receptors, as well as dopamine D1 and D2 receptors (Nichols, 2004; Passie et al., 2008). Molecular signaling across these disparate receptor subtypes would impact neuronal physiology differently; however, these neuromodulatory systems are known to be important regulators of cortical gain (Thurley et al., 2008; Ferguson and Cardin, 2020). We therefore asked in our model whether modulation of gain via these other receptor subtypes also explains the spatial topography of cortical GBC changes. To test this, we repeated our grid search over model parameters, except that instead of modulating gain in proportion to HTR2A expression levels, we modulated gain in proportion to the expression level of, in turn: serotonergic genes HTR1A, HTR2C, and HTR7, and dopaminergic genes DRD1 and DRD2 (excluding 5-HT_1B and 5-HT₆ encoding genes for lack of reliable transcriptomic data in the AHBA). The result of this process is illustrated in Figure 3A. Of the receptor-encoding genes that we tested, using the HTR2A map in the model yields the maximal model-empirical loading and Spearman rank correlation between empirical and simulated ΔGBC maps (Figure 3B). These findings highlight the importance of the spatial distribution of drug targets in mediating their large-scale functional effects.

We examined whether a strong model-empirical loading could be simply explained by general statistical properties of the HTR2A map, rather than its specific topographic pattern. To quantify the significance of the HTR2A map’s topography, we sought to establish statistical expectations for model-empirical loading under an appropriate null hypothesis. A widely adopted approach to quantifying significance for a complex, model-generated statistic (such as model-empirical loading) is to perform a non-parametric permutation test: by randomly permuting the brain map and re-generating a quantity of interest, one can construct a null distribution of expected outcomes due to random chance (Deco et al., 2018). When permutation tests of this type are applied to brain maps, the implicit null hypothesis is that any brain map with the same distribution of values but with a random topography is statistically likely to have produced a comparable or more extreme effect.

One important problem with this approach is that spatial autocorrelation – a characteristic statistical property of brain maps – is necessarily destroyed by random spatial permutations. Spatial autocorrelation is fundamentally important for two reasons: (i) it violates the assumption that samples are independent or exchangeable, an assumption which underlies many common statistical tests (including the permutation test); and (ii) this violation dramatically inflates p-values in studies of brain maps (Burt et al., 2020). In light of this limitation, here we leverage a recent method to generate autocorrelation-preserving surrogate brain maps (Burt et al., 2020). By construction, these surrogate brain maps have randomized topographies but a fixed spatial autocorrelation (Figure 3C). Building a null distribution from these autocorrelation-preserving brain maps facilitates a test that controls for this important property.

To perform such a test, we repeatedly redefined gain functions in the model, each time by modulating gain in proportion to regional values of a random spatial autocorrelation-preserving surrogate brain map. Each surrogate map was constructed to have the same spatial autocorrelation structure as the HTR2A map. For each surrogate map, we regenerated a model ΔGBC map, from which we computed a model-empirical loading. The resulting null distribution of model-empirical loadings was then used to evaluate the significance of the HTR2A map’s specific topography (Figure 3D). We found that modulating gain in proportion to HTR2A expression levels produces a statistically significant loading (p = 0.008). The null distribution further reveals that none of the alternative gene expression maps reach significance of p < 0.05. These findings further implicate the 5-HT_2A receptor in LSD’s mechanism of action, while simultaneously acting as a statistical control for the model complexity.

Our model captures cortical GBC alterations at the group level, but is it expressive enough to capture individual differences? To characterize individual variation – first, in the empirical data – we performed a principal components analysis (PCA) on subjects’ individual ΔGBC maps. We focused on the leading principal component (PC1) of ΔGBC variation (Figure 4A). By definition, PC1 corresponds to the spatial map that (linearly) captures maximal variance in ΔGBC across subjects. We found that PC1 correlates strongly with the group-averaged ΔGBC map (rs=0.72; p < 10⁻⁵; Spearman rank correlation). This indicates that individual differences are primarily driven by the strength with which subjects exhibit the group-averaged pattern.

To explore individual variation in the model, for each subject, we repeated the two-dimensional grid search over the two gain-modulating parameters. Model-empirical loading for each subject was computed using that subject’s ΔGBC map as the target model output. For each subject, we selected the combination of gain modulation parameters that maximized model-empirical loading. We then used those parameters to construct subjects’ model ΔGBC maps (Figure 4—figure supplement 1). We performed PCA on these subject-specific maps to derive PC1 of ΔGBC variance in the model (Figure 4B). We found that, as in the empirical data, PC1 is topographically aligned with the group-level ΔGBC map (rs=0.87; p < 10⁻⁵; Spearman rank correlation). Moreover, model PC1 and empirical PC1 exhibited similar spatial topographies (rs=0.63; p < 10⁻⁵; Spearman rank correlation). These findings reveal a convergence between the dominant modes of neural variation in the data and in the model.

The model PC variance spectrum shows that variation across subjects’ simulated ΔGBC maps is almost entirely captured by the two leading model PCs (Figure 4C). That model variation lies within a two-dimensional subspace is expected due to our fitting of two free parameters to ΔGBC. We also find topographic alignment between model PC2 and empirical PC2 (rs=0.51; p < 10⁻⁵; Spearman rank correlation) (Figure 4D). However, the empirical PC variance spectrum reveals that the data exhibit approximately five significant modes of variation (Figure 4C). The data therefore exhibit three more modes than the model. A linear subspace analysis revealed that 67% of model variance fell within the subspace defined by the data’s leading five PCs (see Materials and methods). In other words, we found that the majority of model variation lies within a low-dimensional subspace defined by the principal modes of empirical variation.

With only two significant modes of variation, is the model expressive enough to capture individual differences in LSD-induced alterations of consciousness? As reported by Preller et al., 2018, changes in conscious experience were determined psychometrically using the five-dimensional altered states of consciousness (5D-ASC) questionnaire (Dittrich et al., 2010), which was administered to study participants 180 min after drug infusion. The short version of the 5D-ASC questionnaire quantifies changes in conscious experience along five distinct experiential dimensions: experiences of disembodiment, elemental imagery, changed meaning of percepts, blissful state, and spiritual experience – each of which is profoundly altered under LSD (Figure 4E). We performed a neurobehavioral linear regression analysis to find patterns of altered GBC that were predictive of experiential effects. Specifically, for each (parcel, experiential dimension) pair, we performed a linear regression across subjects, such that each sample in the regression between ΔGBC in parcel i (the predictor variable) and change along experiential dimension j (the target variable) corresponds to a (ΔGBC_i, ΔScore_j) pair for one subject.

For each experiential dimension, we performed one regression per parcel and aggregated the linear regression coefficients across all parcels to create a spatial brain map (Figure 4F). We henceforth refer to these maps as ‘experiential regression maps’. By construction, experiential regression maps reflect ΔGBC patterns that predict individuals’ change in experience. For instance, if a subject’s ΔGBC map strongly resembles the Meaning regression map (Figure 4F), then we expect that subject to have reported a relatively strong change in the meaning of percepts. How consistent are these experiential regression maps across experiential dimensions? We found evidence for a single dominant experiential regression map pattern (Figure 4G).

Returning to our original question – do modes of model variation have experiential relevance? – we compared PC1 maps (which characterize individual variation) to experiential regression maps (which are defined by their experiential relevance) (Figure 4G). We find notable correlations of model PC1 with the Meaning (rs=0.38; p < 10⁻⁵; Spearman rank correlation) and Disembodiment (rs=0.61; p < 10⁻⁵; Spearman rank correlation) experiential regression maps. Most strikingly, however, we find that the model PC1 map strongly resembles the dominant experiential regression map pattern (rs=0.65; p < 10⁻⁵; Spearman rank correlation), which is defined as the first PC of the experiential regression maps. This indicates that the model captures patterns of functional individual variation that are linked to individual differences in pharmacologically induced changes in experience.

Group-averaged left-hemispheric SC matrices were constructed from diffusion MRI data using probabilistic tractography for 339 unrelated subjects from the HCP 900-subject data release (Van Essen et al., 2013). SC matrices were parcellated into 180 areas using the HCP’s MMP1.0 (Glasser et al., 2016). Diagonal elements of the SC matrix were set identically to zero, as the dynamical model (described below) explicitly includes self-coupling terms. Moreover, the SC matrix was row-wise normalized such that the total long-range inputs to each node were normalized. This normalization procedure instantiates the assumption that each local microcircuit receives a balance of local and long-range inputs. The CAB-NP was used to determine parcels’ functional network assignments (Ji et al., 2019). Pairwise inter-parcel geodesic distance matrices, which were used for constructing spatial autocorrelation-preserving surrogate maps (Burt et al., 2020), were computed by averaging over surface-based distances between grayordinate vertices in each pair of parcels i and j, where surface-based distances were computed across the left-hemisphere midthickness surface in the HCP atlas.

A complete description of data collection and pre-processing procedures was reported in Preller et al., 2018. Briefly, fMRI data derived from a double-blind, placebo-controlled, and within-subject study design, ensuring that each subject served as their own control. Twenty-five healthy human participants received either (i) placebo or (ii) LSD (100 µg po). Resting-state scans were collected 75 min following drug administration. fMRI data were obtained and processed following HCP-compliant acquisition standards (Glasser et al., 2013). Behavioral measures were derived from the short version of the 5D-ASC questionnaire (Dittrich et al., 2010), which includes 45 items that comprise five different scales: spiritual experience, blissful state, disembodiment, elementary imagery, and changed meaning of percepts. The 5D-ASC questionnaire was administered 180 min following drug infusion. One subject was excluded from analysis due to failure in registration caused by an improper head position. Thus, 24 study participants were analyzed in this study.

The AHBA is a publicly available transcriptional atlas of microarray data containing samples from hundreds of neuroanatomical structures in six normal post-mortem human brains. Microarray expression data and all accompanying metadata were downloaded from the AHBA (http://human.brain-map.org) (Hawrylycz et al., 2012; Hawrylycz et al., 2015). An exhaustive description of our pre-processing procedure was reported in Burt et al., 2018. Briefly, cortical microarray data in volumetric space were mapped onto subjects’ native two-dimensional cortical surfaces by minimizing three-dimensional Euclidean distance between microarray samples and grayordinate vertex coordinates. For parcels that were not directly sampled, we performed a surface-based interpolation of sample expression values using a Voronoi diagram approach to create a dense gene expression map (at the level of grayordinates); this interpolated map was then parcellated by averaging across grayordinate vertices in each parcel. One representative microarray probe was selected for each unique gene, and gene expression profiles for selected gene probes were z-scored. Finally, group-averaged expression profiles were computed for genes whose expression profiles were reliable in at least four of the six subjects. These steps yielded group-averaged gene expression values across 180 left-hemispheric cortical areas.

Single-nucleus transcriptomic data that quantify the expression of HTR2A across distinct cell types in human cortex was obtained from the Allen Brain Institute’s Human Multiple Cortical Areas SMART-seq database (https://portal.brain-map.org/atlases-and-data/rnaseq/human-multiple-cortical-areas-smart-seq). We downloaded the CSV file containing ‘Gene expression by cluster, trimmed means’ and aggregated columns that included either ‘Exc’ or ‘Inh’.

We adapted the biophysically based large-scale computational model described in Deco et al., 2014. This model reduces the complexity and the number of local microcircuit parameters in a spiking neural network model using a dynamical mean-field approach (Wong and Wang, 2006). By leveraging a statistical description of ensemble neural activity and exploiting the long time constants of NMDA receptors, the model reduces a high-dimensional spiking neural network to a computationally tractable two-dimensional dynamical system. Each node in the model comprises recurrently coupled excitatory (E) and inhibitory (I) populations, the dynamics of which are described via coupled non-linear stochastic differential equations. Time-varying activity of the excitatory and inhibitory synaptic currents in brain region i, denoted respectively by IiE and IiI, is defined as(1)IiE(t)=WEIb+wEESiE(t)+GJΣjCijSjE(t)−wEISiI(t)IiI(t)=WIIb+wIESiE(t)−SiI(t)where Ib is constant background input current; S is the synaptic gating variable, that is, the fraction of open channels; G is a global conductance parameter which scales the strengths of long-range connections; J is the effective NMDA conductance; C is the SC matrix; and remaining parameters are constants. Synaptic current Ip for population p∈{E,I} is transformed into firing rate rp via the transfer function(2)rip⁢(t)=ϕ⁢(Iip⁢(t))=ap⁢Iip⁢(t)-bp1-e-dp⁢(ap⁢Iip⁢(t)-bp),where a, b, and d are cell-type specific parameters governing the form of the input-output relation. In particular, note that parameter a, which regulates the slope of the transfer function, corresponds to the neural gain parameter. Note that while this function is unbounded and therefore does not saturate, we confirmed that the node-averaged firing rates in the gain-modulated model (which do not exceed ∼15 Hz throughout the parameter sweep in Figure 2A) remain in a neurobiologically plausible firing rate regime where this approximation of the F-I curve does not break down.

Finally, the time evolution for synaptic gating variable S is given by(3)dSiE(t)dt=−SiE(t)τE+(1−SiE(t))γriE(t)+σνi(t)dSiI(t)dt=−SiI(t)τI+riI(t)+σνi(t)where τ is a synaptic time constant, γ is a kinetic parameter, and σ is the standard deviation of the stochastic Gaussian input noise ν (Wong and Wang, 2006; Deco et al., 2014). Following Deco et al., 2014, simulated synaptic covariance matrices were approximated analytically by linearizing Equations 1 to 3 around the dynamical system’s stable fixed point.

Neural gain modulation was implemented in the model by introducing a parametric rescaling of gain parameter a for each cell type, in proportion to regional gene expression levels of HTR2A:(4)aiE=hi(1+δE)aEaiI=hi(1+δI)aIwhere h_i denotes the expression level of HTR2A in brain region i, δE and δI regulate the strength of the perturbation applied to excitatory and inhibitory cell populations, respectively. Thus, two-dimensional parameter sweeps reported in this study, for instance in Figure 2A, correspond to sweeps over free parameters δE and δI. Where brain maps (gene expression maps as well as surrogate maps) were used to modulate gain, they were first linearized using the error function (following Demirtaş et al., 2019).

Following Deco et al., 2014, we implemented feedback inhibition control (FIC) in the model such that the firing rates riE were constrained to be 3 Hz prior to neural gain modulation. This is accomplished by tuning the weight wE⁢I, which regulates the strength of local inhibitory-to-excitatory feedback, within each simulated brain region. Crucially, we note that FIC was calculated prior to but not following gain modulation, under the assumption that FIC represents neurobiological self-regulatory processes which balance neural activity in the brain on timescales much longer than the characteristic timescales of transient pharmacological manipulations.

To facilitate model comparisons with empirical fMRI data, excitatory synaptic activity within each brain region (given by SE in Equation 3) was transformed to a BOLD signal using the Balloon-Windkessel model (Friston et al., 2003). In this model, the hemodynamic response obeys the following coupled system of equations: (5)d⁢xi⁢(t)d⁢t=SiE⁢(t)-k⁢xi-γ⁢(fi⁢(t)-1)(6)d⁢fi⁢(t)d⁢t=xi⁢(t)(7)τ⁢d⁢vi⁢(t)d⁢t=fi⁢(t)-vi1α⁢(t)(8)τ⁢d⁢qi⁢(t)d⁢t=fi⁢(t)ρ⁢[1-(1-ρ)1fi⁢(t)]-qi⁢(vi1-αα⁢(t))where x denotes the vasodilatory signal, f the blood inflow, v the blood volume, and q the deoxyhemoglobin content; and parameters ρ, τ, κ, γ, and α are the resting oxygen extraction fraction, hemodynamic transit time, rate of signal decay, rate of flow-dependent elimination, and the Grubb’s exponent, respectively (Friston et al., 2003). In turn, the corresponding BOLD signal y is calculated as(9)yi⁢(t)=V0⁢[k1⁢(1-qi⁢(t))+k2⁢(1-qi⁢(t)vi⁢(t))+k3⁢(1-vi⁢(t))]where V₀ is the resting blood volume fraction. The three dimensionless magnetic field strength-dependent parameter values k₁, k₂, and k₃ were derived for a magnetic field strength of 3 T using Appendix A of Heinzle et al., 2016; all other hemodynamic parameter values were taken from Obata et al., 2004. Simulated BOLD covariance matrices were derived by linearizing these equations and then algebraically transforming the linearized covariance matrix, using a procedure which we previously reported in Demirtaş et al., 2019. Specifically, we semi-analytically derive the covariance matrix P of the dynamical system by numerically solving the Lyapunov equation:(10)A⁢P+P⁢AT+QN=0where A is the Jacobian matrix and QN is the noise covariance matrix. Note that this expression is solved for the full six-dimensional dynamical system which includes two synaptic variables, SE and SI, as well as four hemodynamic state variables x, f, v, and q. The linearized BOLD covariance matrix is then given by(11)PBOLD=K⁢P⁢K†where K is a matrix of partial derivatives of the BOLD signal with respect to the six dynamical variables (Demirtaş et al., 2019).

Values for all static parameters in the model are provided in Table 1.

Tuning the model parameters consisted of two key sequential steps. First, the dynamical operating point of the model was calibrated such that (prior to gain modulation) the model optimally captured empirical FC derived from the placebo-condition fMRI scan. To this end, we performed a one-dimensional grid search over the global coupling parameter G, which linearly scales all long-range interactions between nodes (Equation 1). The grid search was over the range [0.01, 0.85] with a step size of 0.01. For each value, we solved for the stable fixed point of the system and then computed the linearized BOLD covariance matrix around that point. From the BOLD covariance matrix, we computed the FC matrix using the transformation from covariance to Pearson correlation coefficient. We computed the Spearman rank correlation between the N=180⁢(180-1)/2 upper-triangular elements of the model FC matrix and the placebo-condition empirical FC matrix. At the group level (i.e., using the group-averaged empirical FC), this step yielded an optimal global coupling value of G=0.85, which resulted in a Spearman rank correlation of rs=0.45. This value of G was used for all group-averaged analyses in this study. We note that if the upper end of the range was extended beyond G=0.85, the maximal fit to the group-level data occurred at G=0.89. However, at this value the fit increases only marginally (δ⁢rs<10-3), while the system is then positioned so close to a critical bifurcation that even modest modulations of gain will dynamically destabilize the system. This class of computational models is known to produce optimal fits near the edge of stability (see, e.g., Deco et al., 2013).

After calibrating the system, next we performed a two-dimensional grid search over model parameters δ⁢E and δ⁢I, which determine the strength of excitatory and inhibitory gain modulation, respectively. For each combination of parameters, we updated the neural gain equations (Equation 5) in the model. We solved for the new fixed point and computed the linearized BOLD covariance matrix around that point. FC and GBC were then computed as described above. The difference between model GBC maps (expressed as Fisher Z-values) – perturbed minus baseline – yielded a model ΔGBC map. We then computed the model-empirical loading between the model ΔGBC map and the empirical ΔGBC map (LSD minus placebo). As illustrated in the heatmaps, the grid search for the gain modulation parameters was over the range [0, 0.03] with a step size of 0.003. Gain parameters that maximized model-empirical loading were then used for all analyses involving a model ΔGBC map, either at the group or subject level.

Empirical GBC was computed using in-house Matlab tools for all grayordinates in the brain using an HCP-harmonized version of the FreeSurfer software (Fischl et al., 2002; Cole et al., 2011; Anticevic et al., 2013; Anticevic et al., 2014), as reported in Preller et al., 2018. Briefly, for each subject, we computed the fMRI time-series correlation between pairs of grayordinates, thereby constructing an FC matrix. Off-diagonal FC matrix elements were Fisher Z-transformed and then averaged across rows to compute GBC values. Maps were then parcellated using the HCP’s MMP1.0 (Glasser et al., 2016). This yielded a parcellated GBC map for each subject where each parcel’s value represents the mean FC of that region to all other regions. Parcellated GBC maps were averaged across subjects to construct the group-averaged GBC map. Similarly, to compute model GBC maps, we used the model to generate a parcellated BOLD FC matrix. Off-diagonal FC matrix elements were Fisher Z-transformed and then averaged across rows to compute GBC values. All illustrated GBC map values (empirical and simulated) are Fisher Z-values.

GSR was performed on empirical fMRI-derived time-series data using widely adopted approaches (Anticevic et al., 2013; Cole et al., 2011), as described in Preller et al., 2018. GSR was also performed in the model, as GSR not only removes artifactual signal components but neuronal signal as well. In the model, however, we performed GSR directly on the analytically approximated BOLD covariance matrices using the following approach. Let y_i be the BOLD signal in brain region i, and let x be the GS – that is, the mean gray-matter BOLD signal across regions. Note that x and y are both implicit functions of time. We construct a linear regression model of the form (12)yi^=αi+βi⁢x,where the residual signal following GSR is given by(13)ϵi≡yi-yi^.

What we seek is an expression for the covariance between GS-regressed BOLD signal residuals in regions i and j, that is,(14)Cov⁢(ϵi,ϵj)=⟨(ϵi-ϵi¯)⁢(ϵj-ϵj¯)⟩,where ϵ¯ denotes the temporal average of ϵ. Substituting Equations 12 and 13, this expression can be rewritten as(15)Cov(ϵi,ϵj)=⟨(yi−yi¯),(yj−yj¯)⟩−⟨(yi−yi¯),βj(xj−xj¯)⟩−⟨βi(xi−xi¯)(yj−yj¯)⟩+⟨βi(xi−xi¯)βj(xj−xj¯)⟩.

By manipulating Equation 12, β can be expressed in terms of covariances:(16)β=Cov⁢(x,y^)Cov⁢(x,x)=⟨(xi-xi¯)⁢(yj^-yj^¯)⟩⟨(xi-xi¯)⁢(xj-xj¯)⟩.

Substituting this expression into Equation 15 and simplifying, we obtain(17)Cov⁢(ϵi,ϵj)=⟨(yi-yi¯)⁢(yj-yj¯)⟩-⟨(yi-yi¯)⁢(x-x¯)⟩⁢⟨(yj-yj¯)⁢(x-x¯)⟩⟨(x-x¯)⟩2.

From the definition of GS x=∑iyi, Equation 17 can be rewritten as(18)Cov⁢(ϵi,ϵj)=⟨(yi-yi¯)⁢(yj-yj¯)⟩-⟨(yi-yi¯)⁢(∑kyk-∑kyk¯)⟩⁢⟨(yj-yj¯)⁢(∑lyl-∑lyl¯)⟩⟨(∑kyk-∑kyk¯)⟩2.

Collecting the summations and noting that(19)Cov⁢(yi,yj)=⟨(yi-yi¯)⁢(yj-yj¯)⟩,we obtain(20)Cov⁢(ϵi,ϵj)=Cov⁢(yi,yj)-∑kCov⁢(yi,yk)⁢∑lCov⁢(yj,yl)∑m,nCov⁢(ym,yn).

Equation 20 analytically relates the covariance matrix of the raw BOLD signal (right-hand side) to the covariance matrix of GS-regressed residuals (left-hand side). We used this expression to perform GSR on model BOLD covariance matrices. FC matrices (i.e., Pearson correlation matrices) were then computed from GS-regressed BOLD covariance matrices.

Spatial autocorrelation-preserving surrogate maps were constructed following Burt et al., 2020 using the Python-based BrainSMASH toolbox (https://brainsmash.readthedocs.io/). This approach operationalizes spatial autocorrelation in a brain map by computing a variogram, which quantifies variance in the brain map as a function of pairwise distance between areas. To construct each surrogate map, the empirical brain map is iteratively shuffled (to randomize topography), smoothed (to reintroduce spatial autocorrelation), and rescaled (to recover empirical spatial autocorrelation). After performing this procedure, we resampled our surrogate map values from the empirical brain map such that the distribution of values does not change: each surrogate map can then be conceptualized as one random realization of an (approximately) spatial autocorrelation-preserving permutation of the original brain map.

For each surrogate map h→, the surrogate map values {hi} were substituted into Equation 5 such that the modulation of gain in each region was scaled in proportion to the values of the surrogate map, rather than the empirical HTR2A map. This ‘surrogate model’ was then used to re-compute the statistic under consideration. Repeating for and aggregating across all surrogate maps yields a null distribution for the expected value of the statistic under the null hypothesis that the outcome of the statistical test is an artifact of spatial autocorrelation, and therefore not unique to a specific spatial topography. Specifically, where we report p-values that derive from a spatial autocorrelation-preserving surrogate map test, we are reporting the proportion of samples in the null distribution which were more extreme than the test statistic.

To identify characteristic patterns of changes in GBC that predict subject-level changes in 5D-ASC scores, we constructed spatial brain maps of linear regression coefficients (i.e., beta coefficient maps or simply ‘beta maps’). To calculate the beta map value in the i th cortical parcel for experiential dimension d, we solved for βid in the linear regression defined by(21)δ⁢sd→=αid+βid⋅Δ⁢GBCi→where δ⁢sd is the change in score δ⁢s along experiential dimension d; α and β are constant and linear regression coefficients, respectively; and Δ⁢GBCi is the change in GBC for parcel i; and vectors include 24 elements (one per subject).

To define low-dimensional linear subspaces within the 180-dimensional neural state space, we used PCA to determine the dominant spatial modes of variation across different collections of brain maps. We let 𝐗 be the M×N matrix comprised of M brain maps, each represented as an N-dimensional vector, where columns of 𝐗 have been mean-subtracted. The spatial covariance matrix is then given by the N×N symmetric matrix 𝐂, where(22)𝐂=1M-1⁢𝐗T⁢𝐗.

The symmetric matrix 𝐂 can in general be decomposed according to(23)𝐂=𝐏⁢𝚲⁢𝐏T,where 𝐏 is an N×N orthogonal matrix whose i th column pi→ is an eigenvector of 𝐂, and 𝚲 is an N×N diagonal matrix whose i th diagonal element λi is the eigenvalue associated with eigenvector p_i. Eigenvector pi→ is a PC of 𝐗, and the normalized eigenvalue λi/Tr⁢(𝚲) is the fraction of total variance in 𝐗 that occurs along pi→. We will assume without loss of generality that eigenvectors are ordered such that λi+1≤λi. In other words, we assume that eigenvectors are arranged such that eigenvalues are placed in descending order. Finally, note that if M<N, then N-M+1 eigenvalues are identically zero (i.e., 𝐂 will not be full rank).

To determine which PCs were statistically significant, we performed simple permutation testing. Each row in 𝐗 (i.e., each brain map) was randomly permuted to obtain a new matrix 𝐗′. The procedure described above was then performed on 𝐗′ and the fraction of variance captured per PC (i.e., λ′→/Tr⁢(𝚲′)) was recorded. To construct the gray dashed lines in Figure 4, we performed this procedure N=1000 times and then computed the fifth percentile of the resulting null distribution (for each individual PC).

To investigate shared dimensions of variability in the model and in the data, we first performed PCA on both the model and empirical subject-specific ΔGBC maps. Permutation testing revealed that the leading two model PCs and the leading five empirical PCs were statistically significant. We then derived the covariance matrix 𝐂 for the 24×180 data matrix 𝐗, comprised of subjects’ model ΔGBC maps, per Equation 22. Total model variance V was determined by computing the trace of 𝐂. Let 𝐏 denote the 180×5 matrix whose columns correspond to the leading five empirical PCs; then we performed a linear transformation of 𝐂 into the five-dimensional subspace defined by the columns of 𝐏 according to(24)𝐂′=𝐏T⁢𝐂𝐏.

Finally, we determined the total variance V′ contained in the subspace by computing the trace of 𝐂′. Then the fraction of model variance that falls within the data’s five-dimensional subspace (67%) is given by V′/V.

A Python-based implementation of the model is provided as an open-access software package, BRAINTRIPS (BRainwide Activity Induced by Neuromodulation via TRanscriptomics-Informed Pharmacological Simulation): https://github.com/murraylab/braintrips (Burt, 2021; copy archived at swh:1:rev:2f4c9bb63ec01c40e4374e277d03c7ff28e92713). All processed neuroimaging data needed to reproduce main findings in this study are made publicly available on BALSA (https://balsa.wustl.edu/study/show/5XMxv).