Model description

Fig 1A illustrates the architecture of our model—containing both geometrically constrained and fast non-local connectomic mechanisms of cortico-cortical interaction—which we describe below. The cortical sheet of a cerebral hemisphere is represented as a continuous two-dimensional surface, denoted . Cortical populations are identified by their spatial location on , and their neural activity over time is represented by the field variable . The spatiotemporal evolution of over time is governed by the non-local partial differential equation:

(1)

(2)

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Fig 1. Schematic of our mathematical model of macroscale cortical activity with local geometric (blue) and non-local connectomic (green) mechanisms of cortico-cortical interaction.

The model equations are defined in Eq. (1). A. The two-dimensional continuous surface represents the cortical sheet of a cerebral hemisphere, on which neural populations are represented as points. The mean population-level activity at each point, on millimeter lengthscales and above, is described by variable . This model includes both local non-local mechanisms of interaction between points on . Local geometric propagation of activity takes the form of traveling waves along , while non-local connectomic propagation takes the form of rapid, non-local interactions between spatially distant pairs of points on (an example of which is illustrated as and ). B. Geometric propagation (traveling waves) is derived from a spatially homogeneous and isotropic connectivity rule [33–35]. C. Connectomic propagation between specific pairs of points is facilitated by fast-conducting, non-local projections (FNPs), which are axonal fibers that propagate activity faster than traveling waves along , and is considered to underpin macroscale cortical function by facilitating rapid inter-communication between spatially distributed neural systems. FNPs introduce spatial inhomogeneities and anisotropies to the otherwise geometric connectivity.

https://doi.org/10.1371/journal.pcbi.1014673.g001

In Eq. (1), is a second-order linear differential operator in time, and the term captures local spatial relations on for different points . Both terms together capture local propagation as traveling waves along , which arises from a geometric (spatially homogeneous and isotropic) connectivity rule (illustrated in Fig 1B) [33–35]. The term captures non-local propagation between specific points on facilitated by FNPs (forming ‘shortcuts’ through , as illustrated in Fig 1C). Each FNP is a unidirectional projection from an assigned source point (denoted ‘’ in Fig 1A) to a target point (denoted ‘’ in Fig 1A), through which activity propagates more rapidly than by a traveling wave along . The term is the external input that drives the model’s activity. In the absence of input, , the initial conditions in Eq. (2) define zero activity at all points and for all time t. Further details on the above terms, and a derivation of Eq. (1) from physiological principles, are in S1 Text, Sec A.

A notable property of the non-local term is that, for any given input f, the total activity in the system integrated over space and time is preserved after the addition of any FNP to the structural connectivity. This property prevents non-oscillatory instabilities in the dynamics, which can occur when activity is regenerated from FNPs at a faster rate than activity globally dissipates [45,46], and therefore allows multiple FNPs to be added with arbitrary connectivity strengths. In alternative models, total activity is not preserved (e.g., the neural field model by Jirsa and colleagues [45–47]), such that the connectivity strengths of each FNP must be progressively reduced to maintain linear stability.

The model defined by Eq. (1) will be used to simulate cortical dynamics for different external inputs , so that the perturbatory influence of FNPs on the dynamics can be evaluated separately for each input setting. For example, to simulate stimulus-evoked dynamics in response to an impulse stimulus, f is set as an impulse-like input that is Gaussian in both space and time: [48,49]:

(3)

where are the position and onset time of the stimulus, respectively, and are the spatial and temporal width of the stimulus, respectively. In contrast, to simulate noise-driven dynamics—that is, temporally unstructured inputs—we set f as a continuous space–time stochastic process:

(4)

where on is a space-time white noise process [29,50], and on is a smooth deterministic function which controls the variance of the process at different points on .

Model implementation

To numerically solve the model defined by Eq. (1), we developed a finite-difference scheme used for non-local partial differential equations, in which derivatives in time and space are approximated by centered finite differences and definite integrals are approximated using a quadrature rule [51]. For our investigations, is taken to be a square with periodic boundary conditions, making its geometry equivalent to a flat torus. A torus has a closed surface, like a more physically realistic sphere [52], but permits more efficient numerical simulations on a square grid. The side length of the square was set to L = 0.4 m, so that the area of is approximately that of a human cortical hemisphere [35]. We then uniformly partitioned a grid of points from with grid spacing , and time points from with timestep . With this time and space partitioning, we used a finite-difference scheme to numerically solve Eq. (1). Further details and the derivation of this scheme, including the chosen grid spacing and timestep for our investigations, can be found in S1 Text, Sec B.

The parameters of Eq. (1) (detailed in Eqs (S18), (S19), (S20) in S1 Text, Sec A) and their values used in our investigations are given in Table 1. For clarity, we divide the parameters into three groups: (i) parameters that govern geometric propagation through traveling waves: r, , ; (ii) parameters that govern interaction via FNPs: , ; and (iii) parameters that govern the spatiotemporal properties of the external input f: . The value of parameter r, the characteristic lengthscale of the geometric connectivity rule, and the value of , the traveling wave speed divided by r, were both taken from the two-dimensional neural field model developed by Robinson et al. for the human cortex [35,53]. The value of , which is the effective rate at which activity at a given position is regenerated, is fixed in all experiments conducted here, and was taken from the excitatory-only neural field model developed by Jirsa and Haken [34].

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Table 1. Nominal model parameter values for numerical experiments presented here.

https://doi.org/10.1371/journal.pcbi.1014673.t001

The formulation for the non-local term is given by

(5)

As detailed in S1 Text, Sec A, are positions of the source and target populations of the mth FNP, and physiologically represents the effective increase in activity at the target population per unit activity that the FNP exhibits at its source. We set , which is the connectivity strength of the mth FNP, to equate the strength of local and non-local propagation from the source of the FNP. Accordingly, we set for each m, so that when the model is numerically treated, the connectivity strength of the FNP () in Eq. (S24) in S1 Text, Sec B) is equal to the strength of coupling between geometrically nearest neighbors on the grid from the local geometry () in Eq. (S23) in S1 Text, Sec B). We set the value of , which represents the conduction time of the mth FNP, so that it is always smaller than the conduction time of a traveling wave between the FNP source and target, but is also small enough to avoid oscillatory instabilities independently of the chosen connectome configuration [45,46]. However, the conduction speeds of FNPs vary widely from factors other than their length [54], and the critical values of the values separating stability from instability depends on the spatial configurations of the FNPs [45]. For this reason, we fixed for each m. The values of (in Eq. (3)), the spatial and temporal widths of the impulse stimulus, respectively, were set to sufficiently small values such that the input mimics a delta impulse function without causing spurious oscillations from the numerical scheme.

The majority of this paper characterizes the effect of a single FNP as a perturbation to a background of geometrically constrained connectivity, in shaping the response to an impulsive stimulus of a given spatial profile and for spontaneous dynamics. We then investigate the results of a network of FNPs embedded in the domain , considering random (unconstrained) connectivity as well as a range of idealized constraints on connectome wiring inspired by network structures observed in real structural connectome data: (i) distance-dependent wiring of FNPs via an exponential distance rule (indexed by a given lengthscale ) [55]; (ii) a hub-specificity constraint (that preferences connections to involve specified hub locations, indexed by parameter ) [36]; and (iii) rich-club specificity (that preferences hub–hub connectivity, indexed by parameter ) [56]. Details of each algorithm for generating a given arrangement of FNPs are described in S1 Text, Sec C.

Model demonstration

We first introduce the behavior of the model, through a simple demonstration of how the incorporation of FNPs to the model’s structural connectivity perturbs its stimulus-evoked dynamics (i.e., evoked response to an impulse stimulus, cf. Eq. (3)). Two configurations of structural connectivity were considered. The geometric model, depicted in Fig 2A, is governed by the spatially homogeneous and isotropic structural connectivity rule of neural field models, thus completely constrained by the geometry of and absent of any FNPs [35]. The hybrid model, depicted in Fig 2B, additionally incorporates 50 FNPs whose source and target positions were each sampled uniformly at random from . We then computed the response of both models to an impulse stimulus at position , shown with a red dot in Fig 2A and B. The stimulus-evoked responses of the geometric and hybrid models are plotted in Fig 2C and D, respectively, as heatmaps of their spatial activity profiles , at each of six time points following stimulus onset. Consistent with the connectivity rule of neural field models, the stimulus-evoked response of the geometric model is a traveling wave emanating from . By contrast, the response of the hybrid model (containing FNPs) also comprises the activation of specific non-local points due to non-local activity propagation via FNPs. These non-local activations lead to additional traveling wave fronts under the geometric connectivity, yielding dynamics comprising a superposition of multiple traveling waves. In summary, FNPs yield more complex stimulus-evoked dynamics by non-locally propagating activity between spatially distant populations at speeds faster than the traveling waves constrained by cortical geometry.

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Fig 2. FNPs shape stimulus-evoked dynamics by enabling rapid, non-local propagation of activity between neural populations.

We used the model defined in Eq. (1) to examine the influence of FNPs on stimulus-evoked dynamics, by comparing the model’s stimulus-evoked dynamics under two structural connectivity configurations, in response to the same impulse stimulus. A. The geometric model is governed by a spatially homogeneous and isotropic connectivity rule, and is purely constrained by the geometry of (shown as the square). B. The hybrid model’s structural connectivity additionally contains 50 FNPs, depicted as arrows from their source to target. The impulse stimulus applied to both models was positioned at the point , annotated with a red dot. C. The geometric model’s stimulus-evoked response is illustrated as heatmaps of activity profile at time points, t = 0,2,4,6,8,10 ms, measured relative to stimulus onset. D. The hybrid model’s stimulus-evoked response is illustrated in a similar manner as in C. Specific FNPs of the hybrid connectivity (depicted in B) facilitate rapid activity propagation from their source to target at different times and are annotated manually in each heatmap. Note that we have imposed an upper limit of on the color scale for all heatmaps in C and D, to facilitate clear visibility across all time points.

https://doi.org/10.1371/journal.pcbi.1014673.g002

Influence of a single FNP on cortical dynamics resolved across varying timescales

We aim to understand how a single FNP’s perturbation on the model’s stimulus-evoked dynamics evolves over different timescales. We then use this distinction of perturbation by timescale to shed light on how the influence of FNPs on measured macroscale cortical dynamics could differ between activity resolved over fast sub-second timescales by modalities such as voltage-sensitive dye (VSD) imaging, EEG or calcium imaging [15–18,40,41]; and activity resolved over slower, second timescales by modalities such as fMRI [29–31].

We compared the stimulus-evoked response of the geometric model depicted in Fig 3A(i), and a hybrid model containing a single additional FNP from , depicted in Fig 3A(ii). The length of this FNP () is comparable to the longest intra-hemispheric FNPs measured in human cortex [57]. The impulse stimulus applied to both model was set to position , shown as a red dot in Fig 3A (i) and (ii), which was selected to ensure that the FNP would propagate activity from to immediately at stimulus onset. The stimulus-evoked responses of both the geometric and hybrid model to this impulse stimulus are shown in Fig 3A(iii) and (iv). As expected, the FNP perturbs the stimulus-evoked dynamics by non-locally activating , which results in an additional traveling wave emanating from . However, from the point that the two traveling waves begin to superimpose, from onwards, the geometric and hybrid model’s response becomes progressively similar. At the later time points shown in Fig 3A, t = 30, 50 ms, we see that the two models exhibit very similar spatial patterns in their responses, with the hybrid model’s spatial activity pattern displaying a very small perturbation, near and , due to the perpetual propagation of activity between these two points via the FNP. To quantify the timescale dependency of the FNP’s perturbation on the dynamics, we measured the cosine dissimilarity metric (or cosine distance) between the spatial activity patterns of the responses of the two models over time t, denoted as C(t) (see Eq. (S30) in S1 Text, Sec C for mathematical formulation). A plot of C(t) in Fig 3B shows that the FNP maximally perturbs the dynamics at time , which closely corresponds to the time at which the two waves from and meet. We thus find that a single FNP has a strongly timescale-dependent perturbation on the model’s dynamics, being substantial over short, millisecond timescales () but minimal on longer timescales (> 30 ms).

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Fig 3. The addition of a single fast-conducting non-local projection (FNP) predominantly shapes stimulus-evoked dynamics on short timescales .

A. We compared the stimulus-evoked dynamics of: i. the geometric model, and ii. a hybrid model whose structural connectivity additionally contains a single FNP from to . An impulse stimulus was applied to both models at position , shown by the red dot. iii. and iv. show heatmaps of the stimulus-response dynamics of the geometric and hybrid models for t = 0, 5, 10, 20, 30, 50 ms. Note that the first snapshot, labeled , was taken at a time slightly earlier than stimulus onset (), to indicate that is activated after in the hybrid dynamics. The color scale for all heatmaps is shown on the right of the figure. B. The cosine dissimilarity (or cosine distance) between the spatial activity patterns of the geometric and hybrid model’s stimulus-evoked dynamics over time, denoted C(t) (Eq. (S30) in S1 Text, Sec C), quantifies the FNP’s perturbation on the model’s stimulus-evoked dynamics over time. C. We examined the difference between the geometric and hybrid model’s stimulus-evoked response at two locations: i. , and ii. . Insets in the top right corner of each plot show the positions of and . The solid and dashed red lines in both plots represent the time at which the perturbation of the FNP on the dynamics commences and ends , respectively.

https://doi.org/10.1371/journal.pcbi.1014673.g003

To understand why the stimulus-evoked dynamics are most strongly influenced by a stimulus-proximate FNP over short, millisecond timescales (), we observed the evoked response of the geometric and hybrid models at two specific points (in Fig 3C(i)) and (in Fig 3C(ii)), which are positioned symmetrically on the interval from to . At both and , the FNP’s perturbation on the response commences at time (illustrated by the solid red line), through a reduction in activity at and an increase in activity at . The increase in activity at arises from the additional traveling wave emanating from , which arrives at before the wave emanating from . Contrastingly, the decrease in activity at arises from the total activity preservation property of the model, where the non-local propagation of activity from to reduces the amplitude of the traveling wave emanating from . However, the perturbation induced by the FNP is mostly ‘canceled’ once the waves emanating from and begin to interfere. For and , this cancellation occurs at (illustrated by the dotted red line in Fig 3C(i) and (ii)), where the negative perturbation at is compensated for by the superposition of activity arriving from the secondary wave emanating from , and the positive perturbation at is canceled by a weaker incident primary traveling wave from from . This experiment indicates that, at any given point on , the FNP can only substantially perturb the stimulus-evoked response over a specific time interval: from the time at which the first wavefront (from either or ) arrives at the point, to the time at which the second wavefront (from either or , respectively) arrives. By generalizing this result across all points on , we therefore expect that the perturbation C(t) in Fig 3B would be maximal at a time soon after , when the two waves in the hybrid dynamics superimpose; but would become minimal from time , when the traveling wave from reaches every point on . These simple calculations capture the simulation results of Fig 3B well and explain why an FNP strongly affects spatial activity patterns only over a relatively short time interval on short (millisecond) timescales.

The findings above suggest that the perturbation of the FNP on the model’s stimulus-evoked dynamics would reduce in magnitude as the resolved dynamics becomes increasingly restricted to longer timescales, such as the order-of-seconds timescales resolved by fMRI [58]. We explicitly tested this prediction by examining the perturbation of the FNP on the model’s BOLD response to the impulse stimulus, i.e., the BOLD signal observation of the stimulus-evoked response. At each , the BOLD response was approximated as , the stimulus-evoked response’s zero temporal frequency component [58], or equivalently, the time-integrated response as depicted schematically in Fig 4A (see Eq. (S28) in S1 Text, Sec B for details on how is computed numerically). This linear approximation closely aligns with, and hence can be generalized to BOLD responses computed with more complex nonlinear hemodynamic forward models [59] preserve more low-frequency dynamics, including physiologically realistic [60] and phenomenological [61] constructions.

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Fig 4. When neural dynamics are resolved only over long timescales (such as those accessible using fMRI), a single FNP has a much smaller impact on measured neural dynamics.

A. The modeled BOLD response (BOLD signal observation of the stimulus-evoked response), denoted z, is computed as the zero-frequency temporal Fourier component of the model’s stimulus-evoked response. i. To demonstrate the contrast between the stimulus-evoked response and the BOLD response, we apply an impulse stimulus (positioned at the red dot) to the geometric model. ii. The stimulus-evoked response of the geometric model, , is then integrated over time to approximate the slow BOLD response, shown by the heatmap of . B. We modeled the evoked BOLD response of i. the geometric model, and ii. the hybrid model containing the FNP , when the stimulus is positioned at (indicated by a red dot). iii. and iv. plot the BOLD response as a heatmap for the geometric and hybrid models, respectively. C. The cosine dissimilarity between the BOLD responses of the geometric and hybrid models, denoted (Eq. (S31) in S1 Text, Sec C), quantifies the perturbation of the FNP on the evoked BOLD response. is illustrated as the horizontal gray line, C(t) from Fig 3B is illustrated as the black curve, and is illustrated as the dotted black line.

https://doi.org/10.1371/journal.pcbi.1014673.g004

Fig 4B(i) and (ii) illustrate again the geometric and hybrid model respectively from Fig 3A(i) and (ii), and the impulse stimulus positioned at . Fig 4B(iii) and (iv) are heatmaps of the evoked BOLD response of both models to this stimulus, i.e., the time-integrated spatial activity pattern of the stimulus-evoked responses in Fig 3A(iii) and (iv). The FNP’s perturbation on the modeled BOLD response is concentrated near and , increasing the response near and decreasing it near , which is consistent with the same FNP’s perturbation on the model’s stimulus-evoked dynamics on longer timescales (> 30 ms) in our experiments above (Fig 3A(iii) and (iv)). To quantify this perturbation, we measured , the cosine dissimilarity between the BOLD responses of the geometric and hybrid model (see Eq. (S31) in S1 Text, Sec C for mathematical detail). A comparison between and C(t) in Fig 4C shows that is larger than the limiting value of C(t) as , meaning that the FNP’s strong perturbations over short timescales () are partially captured by the BOLD response. However, is less than a fifth of , the maximum attained value of C(t) at time , indicating the short-timescale perturbation captured by the BOLD response is still substantially lower than what would be measured if the dynamics were properly time-resolved. Hence, restricting our observations of the activity dynamics to the longer, order-of-seconds timescales captured by fMRI indeed limits our ability to access the shorter timescales over which FNPs most prominently shape the stimulus-evoked dynamics. These findings provide a potential explanation for the discrepant empirical findings on the influence of FNPs on cortical dynamics across differing timescales: An FNP facilitates rapid non-local activity propagation on the sub-second timescales resolved by VSD imaging, EEG and calcium imaging [15–18,40,41], but has a weaker influence on the longer timescales captured by fMRI, in which the idealized geometric connectivity rule of neural field models becomes a more valid approximation [30,31].

Influence of a single FNP on cortical dynamics driven by inputs of varying spatial precision

We next investigated how the perturbation of a single FNP on the model dynamics depends on the spatial precision of the driving input. We particularly explored perturbations on noise-driven dynamics (dynamics driven by noisy or temporally unstructured input), so that we could directly measure the FNP-induced perturbations to spontaneous activity, which is a commonly studied form of noise-driven dynamics where the input is spatially uniform [62,63].

The noise-driven linear spatiotemporal dynamics of cortical activity can be decomposed as a superposition of linear evoked responses to spatially uncorrelated impulse stimuli across the cortical surface [58,64,65]. Hence, the perturbation of an FNP on noise-driven dynamics can be interpreted as its perturbation on stimulus-evoked dynamics, aggregated across all stimulus positions on the cortical surface, and weighted in accordance to the variance of the noisy input at each position. The previous experiments in Fig 3 characterized the perturbation to geometric stimulus-evoked dynamics caused by a single FNP when the stimulus is incident at the FNP source location. However, a sweep of all possible stimulus positions on shows that the extent of this perturbation (quantified by both and ) decreases with its distance from the source of the FNP (S1 Fig), indicating that the FNP’s non-local propagation is highly specific to activity that is proximate to its source location. Under our aforementioned interpretation, this result in the stimulus-evoked setting therefore suggests that a single FNP’s perturbation to noise-driven dynamics would depend not only on the configuration of the FNPs, but also on the spatial precision of the driving noisy input. In particular, we predict that the FNP’s perturbation on noise-driven dynamics increases as the input becomes more spatially localized at the FNP source, and decreases as the input becomes more spatially uniform in variance per spontaneous dynamics.

To test and quantify this prediction, we investigated how the FNP perturbs the dynamics of a range of noise-driven activity—from activity driven by noisy input spatially localized at , to spontaneous activity (activity driven by spatially uniform noisy input)—and compared the magnitude of the perturbation of the FNP across this variation of input precision. As illustrated in Fig 5A, we formulated the noisy input f^stim in Eq. (4) by setting , the input variance, as a Gaussian with fixed center (Fig 5A(i)). By varying the spatial width , we could assign different spatial profiles to the variance of the noisy inputs, ranging from spatially localized input at (), to spatially uniform input (uniformly distributed in the limit ). For any choice of , we then summarized the statistics of the resultant noise-driven activity via the spatial correlation function from reference point , denoted (Fig 5A(ii)), which measures the Pearson correlation between the activities at and for each (see S1 Text, Sec C for details on computation). The reference point was natural to focus on, as it allows us to understand how the FNP is perturbing the dynamics by propagating activity from (to ).

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Fig 5. The influence of an FNP on noise-driven dynamics is strongest for spatially precise input drives centered on the FNP source.

A. We investigated the influence of a single FNP on noise-driven dynamics for noisy inputs for varying levels of spatial precision. i. We formulated the noisy input in Eq. (4) by setting , a Gaussian function with center and spatial width (indicated by the radius of the red circle). ii. We summarized the spatial correlation structure of the noise-driven model dynamics via the spatial correlation function to the reference point (with other points ), which we denote as . B: A heatmap of , the spatial correlation function with reference point of the hybrid model containing the FNP, for four different spatial widths of noisy input: i. ; ii. ; iii. ; and iv. spatially uniform input (). The values are indicated by the radius of an annotated red circle for i., ii., and iii. For each heatmap, the maximum value of the color scale is the maximum attained value of . C: Scatter plots of the elements of the functions and (the spatial correlation function with reference point of the geometric model) for the four values. In all scatter plots, the green dot corresponds to the point , the gray line is the linear fit between the elements, and r is the Pearson correlation.

https://doi.org/10.1371/journal.pcbi.1014673.g005

Fig 5B plots heatmaps of , the spatial correlation function of the hybrid model, for , and (corresponding to space-time white noise). We first note for all the expected decay of spatial correlation as a function of distance from , mediated by local geometric propagations of activity. The FNP also expectedly perturbs the geometry-mediated noise-driven dynamics by mediating long-range correlations between and , the FNP target, due to non-local propagation between the two points. To quantify this perturbation, we compared with , the spatial correlation function of the geometric model with reference point , for the same selections of (shown in S2 Fig). Comparisons of the elements of and for each are plotted as scatters in Fig 5C(i)–(iv) for each value. These plots show that the FNP’s perturbation on noise-driven dynamics is observable in all cases, with the largest deviation in the linear fit between and occurring at , denoted by the green dot. However, the Pearson correlation between the elements of and , denoted r, increases with . The noise-driven dynamics of the geometric and hybrid models therefore becomes increasingly similar in correlation structure as the noisy input becomes more spatially uniform, characteristic of spontaneous activity. This finding is in line with our expectation and suggests that differences in the spatial precisions of input may also contribute to conflicting empirical findings about the influence of FNPs on cortical dynamics: while an FNP exerts its strongest influence when the input is spatially localized at its source [15–18], it has a weaker influence on spontaneous dynamics driven by spatially uniform inputs, which can be captured by a simpler geometric connectivity rule [29–31].

Influence of a complex network of FNPs on cortical dynamics

The results above demonstrate a single added FNP perturbs the model dynamics most strongly when the dynamics are observed on relatively short timescales () and when the driving input is spatially localized near the source of the FNP. We now extend to a more realistic setting in which a connectome of multiple FNPs are distributed across the cortical sheet, facilitating rapid and non-local interactions across a complex distributed network. This setting allows us to explore specific network properties of FNPs that violate the spatially homogeneous and isotropic connectivity rule commonly assumed by neural field models [36–39].

The perturbation of a single FNP on stimulus-evoked dynamics being strongest over timescales (in Fig 3) could be explained by a reduction in perturbation over longer timescales (> 30 ms) following the superposition of traveling waves from the source and target of the FNP. However, when multiple FNPs are present, the superposition of waves from the source and target of one FNP can be followed by a subsequent perturbation by an FNP at another location, as demonstrated in Fig 2 (using an example connectome of 50 random FNPs). Motivated by this possibility, here we aimed to quantitatively test whether the timescale-dependent perturbation of a single FNP, as demonstrated above, can be extended to a connectome of multiple FNPs.

For a fixed number of FNPs N, we generated an ensemble of 200 connectomes, each with N randomly positioned FNPs with source and target positions sampled uniformly on . An example of a generated connectome for each of N = 10, 20, 50, 100 FNPs is illustrated in Fig 6A. As above, we measured the perturbation of each connectome on the stimulus-evoked dynamics using C(t) (cf. Eq. (S30), S1 Text, Sec C), in a stimulus-response setting using an impulse stimulus positioned at the source location of the first randomly sampled FNP. This stimulus position was selected to ensure that the perturbation of all sampled connectomes begins from the same time (the time of stimulus onset), so that the C(t) curves align and can be compared straightforwardly.

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Fig 6. A connectome of multiple randomly positioned FNPs perturbs geometric dynamics more strongly with the number of FNPs N over timescales .

A. For each of four different numbers of FNPs: N = 10, 20, 50, 100, we generated an ensemble of 200 connectomes, with source and target locations sampled uniformly at random from . B. Realizations of C(t) for each connectome in each of the four ensembles, with each ensemble’s average shown as a thicker curve (computed as the ensemble average of C(t) for a fixed t).

https://doi.org/10.1371/journal.pcbi.1014673.g006

The perturbation curves C(t) across ensembles of connectomes for each of N = 10, 20, 50, 100 are shown in Fig 6B. For all t, we find that C(t) tends to increase with the number of FNPs N. This finding is consistent with our expectation, since increasing N increases the expected number of rapid non-local activations during the response. C(t) for all N also tends to increase to a maximum value at a time between 10 ms and 20 ms. Note that this time to maximum is larger than that attained with a single FNP (, see Fig 3B), reflecting the different times from stimulus onset at which each FNP perturbs the dynamics. However, after reaching the maximum, C(t) for all N tends to reduce to a smaller limiting value over longer timescales (> 30 ms). This subsequent behavior of C(t) for each N therefore demonstrates the timescale-dependent perturbation of a connectome of multiple randomly positioned FNPs on cortical dynamics, with the most salient perturbation occurring over shorter timescales (.

In our investigations of noise-driven dynamics (in Fig 5), a single FNP was shown to contribute a relatively small perturbation to geometrically mediated spontaneous dynamics (driven by a noisy input of spatially uniform variance). This was explained by the inverse relationship between the perturbation ( or ) and the distance between the stimulus position and the source of the FNP (S1 Fig). However, connectomic data indicates that FNPs tend to organize into non-random topological configurations, leading to spatially inhomogeneous and anisotropic connectivity patterns that violate the average connectivity rule typically used in neural field models [39]. A well-described characteristic of cortical structural connectivity is the non-uniform distribution of FNPs, which tends to concentrate connectivity on a subset of highly connected cortical regions called ‘hubs’ [4,36–38]. Hub–hub connectivity makes the existence of chains of inter-linked FNPs (i.e., where the target of one FNP is proximate to the source of another FNP) more likely and thus facilitating distributed, multi-FNP interactions. Such effects would contribute differently in hub areas (where multi-FNP chains are more likely than in random networks) than in non-hub areas (where multi-FNP chains are less likely than in random networks). Accordingly, we wanted to investigate how the concentration of connectivity on specific spatial locations may lead to higher-order network effects (beyond that of a single FNP) that result in different behavior to that observed for single FNPs in isolation.

Relative to random (unconstrained) connectivity, we explored the influence on spontaneous dynamics (quantified as ) of three key connectomic constraints: the exponential distance rule (indexed by parameter ), hub specificity (indexed by parameter ), and rich-club specificity (indexed by parameter ). For a given , or , connectomes were generated with a rejection sampling algorithm (see S1 Text, Sec C for details on each algorithm). We explain each constraint in turn.

The exponential distance rule (EDR) captures the empirical finding that the probability of a FNP existing between two given cortical populations decays exponentially as a function of the length of the projection [2,55,66]. To generate a connectome with a given EDR decay rate, we used a decay parameter to control the rate at which the probability of an FNP existing between two given points on decays exponentially with its (Euclidean) separation distance (accounting for periodic boundary conditions) (see Eq. (S32) in S1 Text, Sec C for the formulation of this probability). Examples of three different connectomes with 10 FNPs and varying EDR decay rates, generated from , are shown in Fig 7A(i), demonstrating the increasing penalization of long-range connections with increasing .

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Fig 7. The influence of a connectome of FNPs on spontaneous dynamics is either unaffected or reduced by the incorporation of distance-dependent or positional specificity-based connectomic constraints.

A. We examine the contribution of three connectivity constraints using a parameter that quantify the extent of the constraint: . i. An exponential distance rule (EDR) constraint is captured by the parameter , which controls the rate at which connection probability decays with separation distance. ii. A hub specificity constraint is captured by the parameter , which controls the probability that a given FNP is incident with a hub area (colored orange). iii. A rich-club specificity constraint is captured by the parameter , which controls the probability that a given FNP connects a pair of hubs. B. The distribution of over an ensemble of 200 connectomes containing N FNPs each and 200 uniformly sampled stimulus positions, for increasing values of: i. ; ii. ; and iii. . Each distribution is represented with error bars of one standard deviation from the mean. C. We compared the distribution of for different stimulus positions, across three sampled connectomes: i. A random (unconstrained) connectome; ii. A maximally hub constrained connectome (); and iii. A maximally rich-club constrained connectome (); each containing N = 100 FNPs iv., v., vi. Heatmaps of by stimulus position of the sampled random, hub-constrained, and rich-club-constrained connectomes shown in i., ii., and iii., respectively. The stimulus positions were a grid of equally spaced points on , . Source positions of all FNPs are indicated by black dots. D. Scatter plots of the sampled values of and the distance between the sampled stimulus and center of the nearest hub, for an ensemble of 200 maximally hub constrained () connectomes and 200 maximally rich-club constrained () connectomes. The mean of both distributions is annotated using a dotted horizontal line, and the average value of over an ensemble of 200 random connectomes (with no hubs). Exponential fits to the data points from the maximally hub and rich-club constrained ensemble are annotated by the solid curves.

https://doi.org/10.1371/journal.pcbi.1014673.g007

Hub specificity captures the empirical observation that specific hubs of the cortex tend to be particularly strongly connected and are thought to play a major role in mediating distributed information transfer across the cortex [36,37,67]. To generate a connectome with a given level of hub specificity, we used a scaling parameter to assign the probability that each FNP connects a population inside a hub to (or from) a population outside the same hub; ranging from the probability attained by chance from uniform sampling () to certainty () (see Eq. (S33) in S1 Text, Sec C for the formulation of this probability). For simplicity, we modeled the case of four hub areas (which also accords with some early work on hub identification in the human cortex [68,69]) which were chosen as square areas on the domain . As depicted in Fig 7A(ii), hub areas were allocated with centers to maximize spacing between hubs, and side length since the cortex of each cerebral hemisphere was parcellated into 34 regions [68,69]. Examples of three different connectomes with 10 FNPs for are illustrated in Fig 7A(ii). At we see spatially uniform random FNP connectivity unaffected by the assignment of hub areas, whereas at we see that all FNPs connect with a hub. We remark that, given our continuous spatial formulation, a hub here is treated as a spatial area of the cortical sheet with preferential connectivity; if this continuum were to be parcellated into discrete nodes, the node(s) corresponding to such high-connectivity areas would become highly connected hub nodes in the resulting network (as would be analyzed in network neuroscience).

Rich-club specificity captures the empirical phenomenon of dense inter-hub connectivity, resulting in a rich-club core of the connectome [4,38,56]. To generate a connectome with a given level of rich-club specificity, we used a parameter to assign the probability that each FNP connects two distinct hubs; ranging from the probability attained by chance from spatially uniform random connectivity () to certainty () (see Eq. (S34) in S1 Text, Sec C for the formulation of this probability). Example realizations of connectomes containing 10 FNPs for are plotted in Fig 7A(iii).

We investigated how incorporating each of these three connectomic constraints (EDR, ; hub specificity, ; or rich-club specificity, ) affects the perturbation to the model’s spontaneous dynamics. We generated an ensemble of 200 connectomes each containing N = 10, 20, 50, 100 FNPs, now repeating for different values of (for the EDR constraint), (for the hub specificity constraint), or (for the rich-club specificity constraint). We then quantified the average perturbation of each ensemble on the modeled spontaneous dynamics. Specifically, we measured the average perturbation of each connectome in the ensemble on stimulus-evoked dynamics, for a uniformly sampled stimulus position on . This quantification approach, which draws on the relationship between noise-driven activity and stimulus-evoked dynamics demonstrated in the previous subsection, is computationally faster than the conventional approach of computing correlation structures of spontaneous dynamics (see S3 Fig for similar results of the experiments using correlation structures for a smaller ensemble sample size). Accordingly, for each sampled connectome we sampled an impulse stimulus uniformly at random from , and measured (cf. Eq. (S30), S1 Text, Sec C), the maximum perturbation that the connectome induces on the evoked response to this stimulus over time if the dynamics was resolved to sufficiently fine timescales.

Distributions of as a function of , , and are plotted in Fig 7B. For a fixed , or , we see an increase in the average with N (the number of FNPs), matching the increase in perturbation with N across timescales shown above for the random connectome (Fig 6B). The decreasing in for increasing —i.e., as progressively shorter FNPs are favored—matches the expectation that fast-conducting perturbations induced by FNPs increase when the FNPs propagate activity over longer distances. Indeed, in the limiting case of , the average length of each sampled FNP approaches zero, and in turn approaches zero.

Interestingly, the average did not increase with , nor with , remaining relatively constant for small N = 10, 20, and decreasing for larger N = 50, 100 (Fig 7B(ii) and (iii)). To better understand this result, we investigated how and affected the dependence of on the position of the stimulus. To this end, we sampled and studied three connectomes: a random (unconstrained) connectome (Fig 7C(i)); a maximally hub constrained connectome, , in which every FNP connects a hub (Fig 7C(ii)); and a maximally rich-club constrained connectome, , in which every FNP connects a pair of hubs (Fig 7C(iii)). We constructed all three connectomes with N = 100 FNPs, since was most sensitive to and in Fig 7B(ii) and (iii) for this value of N. Fig 7C(iv)–(vi) show heatmaps of by stimulus position for the random, hub-constrained (), and rich-club constrained () connectomes, respectively. Compared to random connectivity, the hub constrained and rich-club constrained connectomes exhibit a more spatially heterogeneous distribution of , with higher values for stimuli targeted on or near hub areas, and lower values as the stimulus becomes more distant from the hubs. This spatial non-uniformity can be explained by the fact that a single FNP perturbs stimulus-evoked dynamics most strongly when the stimulus is positioned at the FNP source location (S1 Fig), with the non-random connectomes displaying a higher density of FNP sources (indicated by the black dots) at hub areas. Fig 7D demonstrates more concretely that the spatial nonuniformity of attained from hub specificity and rich-club specificity persists across an ensemble of 200 maximally hub constrained connectomes () and 200 maximally rich-club constrained () connectomes. Consistent with the concentration of high near hub locations, the plot explicitly demonstrates the decreasing relationship between and the distance between the stimulus and the nearest hub for the hub constrained and rich-club constrained models (shown by solid curves). However, when compared to a random ensemble of connectomes (dashed horizontal lines in Fig 7D) the non-random connectomes tend to have larger values of for stimulus positions proximate to a hub (stimulus–hub distances <0.03 m), and smaller values of for stimulus positions distant from a hub (stimulus–hub distances >0.03 m). Aggregating across all stimulus locations results in the hub and rich-club constrained ensembles having a lower average than the random ensemble, indicating that the decrease in for stimuli distant from hub locations outweighs the increase in for stimuli proximate to hubs.

The results from Fig 7C and D in the stimulus-evoked setting suggest that the hub and rich-club constraints result in a corresponding spatial specificity of the connectomic perturbation to spontaneous dynamics, which becomes concentrated on hubs. However these spatially localized perturbations are largely ‘washed out’ over larger lengthscales characteristic of the entire cortical surface, which is likely attributable to the fact that the hubs, where the perturbations increase, form a small proportion of the cortical surface ( in our experiments). Taken together, while the hub and rich-club constraints can amplify the perturbation of FNPs on evoked responses to (specifically) hub stimulation, it has a comparatively weaker effect on the perturbation to spontaneous dynamics. This suggests that the impact of hub and rich-club connectivity on model dynamics depends on the spatial precision of the input driving the dynamics.

In summary, a connectome of multiple FNPs, that facilitates rapid and distributed communication across the cortex, perturbs the geometric dynamics more strongly with an increased number of FNPs but—as with prior results using a single FNP—these perturbations remain most prominent on timescales . Furthermore, relative to randomly embedded networks, simple models of spatially and topologically constrained connectomes (including hub and rich-club specificity) tend to concentrate the perturbative impact of FNPs on spontaneous dynamics at specific hub regions, but did not tend to elevate the aggregate perturbation over the entire cortical surface (and actually tended to reduce the spatially aggregated perturbation when the connectome contained sufficiently large numbers of FNPs, N = 50, 100). Our findings provide a plausible and testable account for why the spatially homogeneous and isotropic average connectivity rule of neural field models can approximate resting-state fMRI dynamics (spontaneous dynamics over timescales ), despite the presence of a spatially embedded and topologically complex network of FNPs.