Anticorrelations among brain areas observed in fMRI acquisitions under resting state are not endowed with a well-defined set of characters. Some evidence points to a possible physiological role for them, and simulation models showed that it is appropriate to explore such an issue. A large-scale brain representation was considered, implementing an agent-based brain-inspired model (ABBM) incorporating the SER (susceptible-excited-refractory) cyclic mechanism of state change. The experimental data used for validation included 30 selected functional images of healthy controls from the 1000 Functional Connectomes Classic collection. To study how different fractions of positive and negative connectivities could modulate the model efficiency, the correlation coefficient was systematically used to check the goodness-of-fit of empirical data by simulations under different combinations of parameters. The results show that a small fraction of positive connectivity is necessary to match at best the empirical data. Similarly, a goodness-of-fit improvement was observed upon addition of negative links to an initial pattern of only-positive connections, indicating a significant information intrinsic to negative links. As a general conclusion, anticorrelations showed that it is crucial to improve the performance of our simulation and, since these cannot be assimilated to noise, should be always considered in order to refine any brain functional model.

The not-well-defined nature of negative correlations stimulated several authors to study the persistence of significant negative correlations by means of fMRI-specific correction methods and to propose a possible physiological role for them [

Different models have been proposed [

In order to reproduce the brain resting state from fMRI acquisitions, the long-range myelinated fiber connections by diffusion imaging, or the folded cortical surface by high resolution imaging [

An alternative approach to the large-scale brain modeling is to simulate the brain activity using the functional connectivity map itself as a background. In such a context, Joyce et al. [

Here we develop a model using an ABM model and a biologically plausible SER model, which should account for both positive and negative interactions between large-scale brain areas. Different levels of functional connectivity in the background modulate the goodness-of-fit of simulations, and we focus, in particular, on the fraction of negative links to test their role in the organization of structured networks.

The sample is composed of 30 selected functional images of healthy controls from the Beijing Zang dataset (180 subject) in the 1000 Functional Connectomes Classic collection (^{2}; resolution, 64 × 64; flip angle, 90°). For the anatomical images, a T1-weighted sagittal three-dimensional magnetization prepared rapid gradient echo (MPRAGE) sequence was acquired, covering the entire brain: 128 slices, TR = 2530 ms, TE = 3.39 ms, slice thickness = 1.33 mm, flip angle = 7°, inversion time = 1100 ms, FOV = 256 × 256 mm, and in-plane resolution = 256 × 192.

The first 10 scans of each subject were removed, and the remaining functional images were analyzed according to the procedures fully described elsewhere [

The images from each subject were divided into 105 ROIs without brainstem and cerebellum (see Figure

For each subject, the activation time series of 105 ROIs extracted from 240 functional images (see Data Collection) were coupled and correlated in all possible combinations, producing an individual connectivity matrix. Then, a global average concerning the whole group of subjects is obtained by averaging the 30 individual matrices, as schematized in Figure

For both positive and negative interactions, in the above average matrix, a series of 20 binary and thresholded matrices are constructed, taking fractions of the highest absolute correlation values in the range from 0% to 100% at 5% steps: this represents the

A further set of binary and thresholded matrices is calculated in order to distinguish the most significant correlation value for each sign: 15 matrices from the 0%–70% cost (maximum fraction of positive links), containing only positive values, and 7 matrices from the 0%–30% cost (maximum fraction of negative links), containing only negative values. Thus, we have different amounts of positive and negative correlations for the same fraction of total links. We call this type of threshold

Finally, all the combinations of positive and negative matrices for different thresholds are joined, producing

Brain parcellation. Location of the brain regions considered in the extraction of the BOLD signal and visible in a sagittal brain representation. For the complete list of the 105 regions considered in this work, taken from FSL Harvard-Oxford maximum likelihood cortical and subcortical atlas, see the Appendix.

Working out the connectivity matrices. (a) Refers to point (1) of the procedure detailed in the text. The fractions in (b) concern the highest absolute correlation values of the threshold in the corresponding matrices (see point (2) in the text for details).

An agent-based approach was used in a large-scale brain network simulation able to account for the independent behavior of each brain region as well as for the interactions between different regions. Each node in the network represents, according to the susceptible-excited-refractory (SER) formalism [

The interactions among the nodes (agents) characterized by the (SER) states are defined through positive and negative links in a binary and thresholded matrix derived from empirical data and simulated through an agent-based brain-inspired model (ABBM) of the type suggested by Joyce [

In particular, each node is characterized by three variables (_{s}, _{p}, and _{n}) and two parameters (_{p} and _{n}) (see Figure

_{s} = 1 if the node is in the S (susceptible) state, namely, prone to change (otherwise, _{s} = 0).

_{p} and _{n} are calculated from the average contribution of positive and negative neighbors, respectively; each neighbor contributes to the average if in the active (on) state.

_{n} and _{p} are threshold parameters above which the average of negative and positive neighbors (_{p} and _{n}) are set to 1 (otherwise, are set to 0).

State balance of an agent (_{p} = 0.5 threshold. This is also the case for the both active (1) and negatively linked neighbors, since _{n} = 0.5 also.

Taking into account the previous variables, we characterized an agent by three binary variables (_{s}, _{p}, and _{n}), namely, by one of 2^{3} possible combinations (111, 110, 101, 011, 100, 001, 010, 000). Simulations were carried out concurrently for all agents and for each step, and in contrast with Morris and Lecar [

Transition rules adopted in the model.

_{s} |
_{p} |
_{n} |
State transition |
---|---|---|---|

0 | 0 | 0 | E → R; R → S |

0 | 1 | 1 | E → R; R → S |

0 | 0 | 1 | E → R; R → S |

0 | 1 | 0 | E → R; R → S |

1 | 0 | 0 | S → E; S → S |

1 | 1 | 1 | S → E; S → S |

1 | 0 | 1 | S → S |

1 | 1 | 0 | S → E |

The fourth column reports the type of transition at a given step (

Various combinations of the _{p}, _{n} (connectivity dependent) couples of parameters have been checked in the above-described model in order to simulate at best the whole empirical, positive connectivity matrix by a given fraction of positive and negative links. In particular, if negative links are associated with noise, the simulation quality should decrease when their fractional amount increases and, inversely, increase in the opposite, symmetrical condition.

Simulations were repeated 100 times for each different combination of parameters, assigning to nodes a random series of 0 and 1 and a random SER state. Notice that in the case of the _{p}, _{n} couple, the same value for each member of the couple was used. Each simulation included 200 time steps and produced a matrix of 105 columns (brain regions) and 200 rows (total time steps); see Figure

Example of a simulated time series. The time series corresponds to the condition included in Figure _{p} = _{n} = 0.1, and absolute-values-proportional-threshold = 100%. The spots indicate an excited state (E) for each of the 105 brain regions in each step of the simulation.

The whole procedure included three series of simulations: The first two series aimed to optimize the parameter values; in the third series, the importance of different fractions of negative and positive connectivities in the reproduction of the positive connectivity itself was estimated. In particular, the following should be noted:

In the first series of simulations, each of the 20 matrices characterized by an absolute-values-proportional-threshold (from 0% to 100% of absolute value threshold with 5% steps) was used as a background, as well as large variations of the other parameters (_{p}/_{n} from 0.1 to 1, step 0.1).

The second series of simulations aimed to improve the parameter precision within the range identified in the previous set of simulations.

Finally, the third series of simulations was carried out upon considering, within the 105 matrices characterized by any possible combination of 15 positive and 7 negative signed-values-proportional-thresholds, the one showing the best simulation performance, namely, the best reproduction of the original connectivity pattern.

The significance of the fitting performance was assessed as follows: in order to check the effect of positive and negative connectivities, 15 and 7 different fractions of positive and negative links, respectively, were used and subjected to a Friedman test. Then, a post hoc analysis using the ranks of the goodness-of-fit was performed by the Tukey-Kramer test.

In the first exploratory phase of the model validation, the goodness-of-fit between empirical data and simulations, as monitored by the Pearson (_{p}, _{n}) parameters, namely, 0.25–0.50–0.75 and from 0.1 to 1 at 0.1 steps, respectively.

In Figure _{p} and _{n} values associated with the goodness-of-fit peaks show a trend increasing with both _{p} and _{n}, namely, _{p} and _{n} = 0.1, under the condition of low excitability (

Fitting empirical data by the ABM model: dependence upon model’s parameters. (a) Connectivity-dependent parameters (_{p} and _{n}) on the x-axis. Blue, green, and red lines indicate, respectively, _{p} and _{n}. Notice that a peak of the goodness-of-fit appears at _{p}, _{n} = 0.1, in the lower range only of the network density. In all cases, the Pearson correlation (

The above considerations suggest to focus on the lower range of parameters, namely, _{p} and _{n} from 0.025 to 0.1 (step = 0.025). Thus, the matching between simulation and empirical data could be improved by reaching the maximum value of 0.50 at the following connectivity-independent parameter values:

As shown in Figure _{p} = _{n} = 0.1 and using a small connectivity density (15%). At increasing _{p} and _{n} values, the trend changes gradually until at _{p} = _{n} = 0.1 an absolute minimum in the lower range of connectivity density can be observed, as well as a maximum in the higher range of connectivity density. Notice that

This behavior can be ascribed to the different amounts of positive and negative links using the absolute-values-proportional-threshold: The number of negative links is lower (almost nonsignificant for the lower level of general connectivity cost), and a more conservative threshold _{n} would further decrease the associated information. Thus, with a more labile threshold of _{n}, more information from the negative connectivities can be extracted, which increases their modulation role. Due to the unbalanced distribution of positive and negative links, however, the simulation reaches a maximum value of goodness-of-fit only in the higher range of connectivity density (where a significant amount of negative connectivity is also increasing). At the same time, a lower threshold _{p} can introduce random positive connections, decreasing the goodness-of-fit in the lower range of the connectivity density.

In this phase, the task is to define the dependence of the fitting procedure on the relative amounts of positive and negative links, using the parameter values identified in the previous steps, namely, _{p} = _{n} = 0.1. In Figure

Fitting empirical data by combinations of positive and negative cost. The false-color scale visualizes the Pearson correlation between experiments and simulations obtained using the fractions of negative and positive links indicated in the horizontal and vertical axes, respectively.

A nonparametric statistical analysis (Friedman test) reported in Figure ^{2} = 97.3, df = 1) of positive links on the fitting performance of the model. The effect of negative links, however, is not significant (^{2} = 4.9, df = 6). The significant post hoc difference in the positive links is apparent in the range from 5% to 30% of positive network density (Figure ^{2} = 37.1, df = 6) emerges. This indicates a possible interaction between different amounts of positive and negative links, so that only in the range of 5%–30% positive

Post hoc analysis. Mean differences of the goodness-of-fit using an increasing amount of positive and negative links. (a) Goodness-of-fit as a function of positive links. (b) Goodness-of-fit as a function of negative links. (c) Goodness-of-fit as a function of negative links in the range of 5%–30% positive cost; a significant difference between the first mean value in blue (no negative links) is reached for the highest value (in red) of negative cost: 25%–30%.

Given the noticeable level of individual variability in brain functional connectivity, the model has been individually applied on a small sample of subjects. For each of eight randomly chosen subjects, the simulations were repeated in the positive cost range indicated as significant by our previous work (positive cost: 5%–30%), and keeping the same values of the _{p}/_{n} parameters. The results, shown in Figure

Modeling individual patterns. The goodness-of-fit values as a function of increasing amount of negative links (average of the fraction of positive links between 5% and 30%) concern 8 randomly chosen subjects. For the average values of the whole group of subjects, see Figure

In this work, we propose a simple agent-based model able to simulate brain functional connectivity. Our results stress once again on how a set of simple rules between interacting agents can show a complex dynamics [

Our simulations exploit the appealing features of an ABBM-based strategy already used for the same purpose among several possible alternatives [

Different trends were found by our simulations depending upon the relative amount of positive and negative connectivities: In the former case (positive connectivities), the goodness-of-fit shows a peak at lower

As for positive connectivities, the statistical analysis showed clear differences between the random model (no connections among nodes, and all brain regions showing random oscillations) in the range between 5% and 30%. This result is in line with previous findings pointing to a small-world topology in that range [

The results gathered by our model on single subjects are in agreement with those on the average matrix, indicating a good reproduction of individual variability. As a more general validation of our study, the same analysis carried out over another set of 30 randomly chosen individuals from the same database (Beijing Zang dataset, the 1000 Functional Connectomes Classic collection) produced pretty similar results (not shown).

An objective interpretation of our observations should take into account several factors: (1) More positive than negative modulations could be favoured by our model; (2) the anticorrelations have a more variable dynamics, more dependent on experimental conditions. From this point of view, such interactions are characteristic of the resting state itself and have a more local than global meaning; (3) our preprocessing method (aCompCorr [

About the last issue, however, there is no univocal consensus, and alternative methods have been proposed [

A direct comparison of aCompCorr with GSR [

All in all, the target of the present work was not to develop an alternative to the already used large-scale brain models but to underpin the importance of different connectivity types for the brain system. To this aim, we introduced a simple model able to fit empirical data, provided a method to identify the random (or noisy) functional connections, and found some evidence about the importance of anticorrelations for the optimal characterization of connectivity patterns.

It seems fair to conclude that anticorrelations (1) should be distinguished from noise and (2) may improve the characterization of positive connectivity and contribute to the refinement of the global brain functional system in fMRI acquisitions.

FP r (frontal pole right)

FP l (frontal pole left)

IC r (insular cortex right)

IC l (insular cortex left)

SFG r (superior frontal gyrus right)

SFG l (superior frontal gyrus left)

MidFG r (middle frontal gyrus right)

MidFG l (middle frontal gyrus left)

IFG tri r (inferior frontal gyrus, pars triangularis right)

IFG tri l (inferior frontal gyrus, pars triangularis left)

IFG oper r (inferior frontal gyrus, pars opercularis right)

IFG oper l (inferior frontal gyrus, pars opercularis left)

PreCG r (precentral gyrus right)

PreCG l (precentral gyrus left)

TP r (temporal pole right)

TP l (temporal pole left)

aSTG r (superior temporal gyrus, anterior division right)

aSTG l (superior temporal gyrus, anterior division left)

pSTG r (superior temporal gyrus, posterior division right)

pSTG l (superior temporal gyrus, posterior division left)

aMTG r (middle temporal gyrus, anterior division right)

aMTG l (middle temporal gyrus, anterior division left)

pMTG r (middle temporal gyrus, posterior division right)

pMTG l (middle temporal gyrus, posterior division left)

toMTG r (middle temporal gyrus, temporooccipital part right)

toMTG l (middle temporal gyrus, temporooccipital part left)

aITG r (inferior temporal gyrus, anterior division right)

aITG l (inferior temporal gyrus, anterior division left)

pITG r (inferior temporal gyrus, posterior division right)

pITG l (inferior temporal gyrus, posterior division left)

toITG r (inferior temporal gyrus, temporooccipital part right)

toITG l (inferior temporal gyrus, temporooccipital part left)

PostCG r (postcentral gyrus right)

PostCG l (postcentral gyrus left)

SPL r (superior parietal lobule right)

SPL l (superior parietal lobule left)

aSMG r (supramarginal gyrus, anterior division right)

aSMG l (supramarginal gyrus, anterior division left)

pSMG r (supramarginal gyrus, posterior division right)

pSMG l (supramarginal gyrus, posterior division left)

AG r (angular gyrus right)

AG l (angular gyrus left)

sLOC r (lateral occipital cortex, superior division right)

sLOC l (lateral occipital cortex, superior division left)

iLOC r (lateral occipital cortex, inferior division right)

iLOC l (lateral occipital cortex, inferior division left)

ICC r (intracalcarine cortex right)

ICC l (intracalcarine cortex left)

MedFC (frontal medial cortex)

SMA r (juxtapositional lobule cortex—formerly supplementary motor cortex right)

SMA L (juxtapositional lobule cortex—formerly supplementary motor cortex left)

SubCalC (subcallosal cortex)

PaCiG r (paracingulate gyrus right)

PaCiG l (paracingulate gyrus left)

AC (cingulate gyrus, anterior division)

PC (cingulate gyrus, posterior division)

Precuneus (precuneus cortex)

Cuneal r (cuneal cortex right)

Cuneal l (cuneal cortex left)

FOrb r (frontal orbital cortex right)

FOrb l (frontal orbital cortex left)

aPaHC r (parahippocampal gyrus, anterior division right)

aPaHC l (parahippocampal gyrus, anterior division left)

pPaHC r (parahippocampal gyrus, posterior division right)

pPaHC l (parahippocampal gyrus, posterior division left)

LG r (lingual gyrus right)

LG l (lingual gyrus left)

aTFusC r (temporal fusiform cortex, anterior division right)

aTFusC l (temporal fusiform cortex, anterior division left)

pTFusC r (temporal fusiform cortex, posterior division right)

pTFusC l (temporal fusiform cortex, posterior division left)

TOFusC r (temporal occipital fusiform cortex right)

TOFusC l (temporal occipital fusiform cortex left)

OFusG r (occipital fusiform gyrus right)

OFusG l (occipital fusiform gyrus left)

FO r (frontal operculum cortex right)

FO l (frontal operculum cortex left)

CO r (central opercular cortex right)

CO l (central opercular cortex left)

PO r (parietal operculum cortex right)

PO l (parietal operculum cortex left)

PP r (planum polare right)

PP l (planum polare left)

HG r (Heschl’s gyrus right)

HG l (Heschl’s gyrus left)

PT r (planum temporale right)

PT l (planum temporale left)

SCC r (supracalcarine cortex right)

SCC l (supracalcarine cortex left)

OP r (occipital pole right)

OP l (occipital pole left)

Thalamus r

Thalamus l

Caudate r

Caudate l

Putamen r

Putamen l

Pallidum r

Pallidum l

Hippocampus r

Hippocampus l

Amygdala r

Amygdala l

Accumbens r

Accumbens l

The authors declare that there is no conflict of interest regarding the publication of this paper.