bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Modeling Brain Connectivity Dynamics in Functional Magnetic Resonance Imaging via Particle Filtering

Pierfrancesco Ambrosi, Mauro Costagli, Ercan E Kuruoglu,˘ Laura Biagi, Guido Buonincontri, Michela Tosetti

Abstract—Interest in the studying of functional connections in anatomical link between brain areas. Functional connectivity the brain has grown considerably in the last decades, as many regards connections as statistical codependencies, and conse- studies have pointed out that these interactions can play a role as quently it is a non-directional and model-free description of the markers of neurological diseases. Most studies in this field treat the brain network as a system of connections stationary in time, brain network. On the contrary, effective connectivity defines but dynamic features of brain connectivity can provide useful the temporal relationship and causal influences in the brain in information, both on physiology and pathological conditions a given network model [2]. of the brain. In this paper, we propose the application of a Functional Magnetic Resonance Imaging (fMRI) is fre- computational methodology, named Particle Filter (PF), to study quently employed in brain connectivity studies, given its non- non-stationarities in brain connectivity in functional Magnetic Resonance Imaging (fMRI). The PF algorithm estimates time- invasiveness and satisfactory spatiotemporal resolution, both varying hidden parameters of a first-order linear time-varying in physiology and pathology (e.g. Alzheimer’s disease [3]–[5], Vector Autoregressive model (VAR) through a Sequential Monte schizophrenia [6] and Major Depression Disorder [7]). From Carlo strategy. On simulated time series, the PF approach brain connectivity studies it emerged that brain dynamics, effectively detected and enabled to follow time-varying hidden in particular effective connectivity, may provide a biological parameters and it captured causal relationships among signals. The method was also applied to real fMRI data, acquired in marker for specific brain disease and a tool for monitoring presence of periodic tactile or visual stimulations, in different responses to treatments of these pathologies [8]–[11]. sessions. On these data, the PF estimates were consistent with Granger Causality (GC) and Dynamic Causal Modeling current knowledge on brain functioning. Most importantly, the (DCM) are methods to investigate effective connectivity. GC approach enabled to detect statistically significant modulations is present when knowledge on temporal evolution of the signal in the cause-effect relationship between brain areas, which cor- related with the underlying visual stimulation pattern presented in a certain brain region A improves the predictability of during the acquisition. another brain region B [12], [13]. This approach is based on Index Terms—brain connectivity, fMRI, Particle Filter, sequen- the evaluation of a linear codependence among time series, tial Monte Carlo, VAR model and it is therefore limited to a stationary framework or needs a sliding-window approach to address time-varying coupling I.INTRODUCTION between regions, which has limitations [14]. Differently, in DCM the predicted relationship between neural activity and The understanding of brain functioning is linked to the study observed fMRI signal needs to be specified in a pre-determined of the dynamic interaction among anatomically segregated model, hence requiring previous knowledge about the timing brain areas. These interactions are labeled functional and and effect on signals of the connectivity modulation [15]. effective connectivity and refer to distinct ways of consider- The Sequential Monte Carlo (SMC) methodology [16] is ing connections among brain region. While complementary crucially different from these two strategies. SMC approaches to structural connectivity, which describes anatomical con- estimate the hidden states of a dynamic system with only nections between brain regions [1], they concern functional partial and noisy observations, without further assumptions connections that are not necessarily achieved through a direct on the presence of variations in connectivity. A specific SMC methodology called Particle Filter (PF) employs discrete P. Ambrosi is with the Department of Neuroscience, Psychology, sampling to approximate probability density functions and it Pharmacology and Child Health, University of Florence, Florence, Italy updates the posteriors with the arrival of new samples. ([email protected]). M. Costagli is with the Department of Neu- The SMC algorithm proposed here was recently developed rosciences, Rehabilitation, Ophthalmology, Genetics, and Maternal-and- Child Sciences, University of Genoa, Genova, Italy and the Labora- by Ancherbak et al. [17], originally for time-varying gene tory of Medical Physics and Magnetic Resonance, IRCCS Stella Maris, network modeling. We adapted it for the study of brain Pisa, Italy ([email protected]). L. Biagi, G. Buonincontri and M. connectivity using fMRI data and the feasibility and behaviour Tosetti are with the Imago 7 Research Center, Pisa, Italy and the Laboratory of Medical Physics and Magnetic Resonance, IRCCS Stella of the proposed approach has been studied on synthetic data Maris, Pisa, Italy ([email protected], [email protected], mimicking fMRI time-series. When applied to real fMRI [email protected]). E.E. Kuruoglu is on leave from Istituto di datasets, results were compared to correlation between delayed Scienza e Tecnologie dell’Informazione, CNR, Pisa, Italy. He is now with the Tsinghua-Berkeley Shenzhen Institute, Data Science and Information time series, considered as a proxy measure for stationary Technology Center, Shenzhen, China ([email protected]). effective connectivity. Two different experimental paradigms bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. were tested, one during periodic tactile stimulation and one probability density function of the parameters of interest a(t) during visual periodic stimulation. Two different experimental can be estimated via Bayes theorem as follows: paradigms were tested: the first one, whose preliminary results p(x(t)|a(t))p(a(t)|x(1,...,t − 1)) were presented in abstract form [18], involved tactile stimula- p(a(t)|x(1,...,t)) = p(x(t)|x(1,...,t − 1)) tion during an fMRI acquisition with temporal resolution of 2s. (5) The second experiment employed a periodic visual stimulation and with the assumption of Gaussian noise we have with significantly improved time resolution of 0.8s. 1  (x (t) − xˆ (t))2  p(x (t)|a (t)) = − i i i i 2 R/2 exp 2 (6) II.METHODOLOGY (2πση) 2ση

A. Model and algorithm where xˆi(t) are the data estimated through equation (1) at time Particle filter [17], [19]–[22] is a sequential Monte Carlo t for the i-th ROI and ai(t)= {ai1, . . . , aiR} is the vector of methodology based on the Bayes theorem on conditional hidden variables associated with the i-th ROI at time t, that probability. Particle filters estimate the probability distributions is, the i-th row of matrix at. of hidden variables of interest, modeled according to a hypoth- In most applications, equation (5) cannot be solved ana- esized state-space equation. The probability density function lytically [26], but it can be computed through the Sequential (pdf ) of the hidden variables is allowed to be time-varying and Monte Carlo sampling scheme, which consists in representing is therefore sequentially updated when new data become avail- the pdf p(a(t)|x(1,...,t)) as a discrete set of N weighted able. Such probability distribution is estimated from the data, samples called particles: modeled according to a hypothesized observation equation. In N X (n) (n) brain connectivity studies based on fMRI data, the relationship p(ai(t)|x(1,...,t)) ≈ wt δ(ai(t) − ai (t)) (7) among the time-series of R different brain Regions of Interest n=1 (ROIs) x = {x (t),...,x (t)} can be modeled as a first t 1 R where w(n) is the weight associated to the n-th particle order linear Vector Autoregressive (VAR) model [12], [21], t vector a(n)(t) for the i-th row of matrix a(t) at time t. [23]–[25] as: i The Sequential Importance Sampling (SIS) [26] methodology provides a strategy to compute the weights. It has been shown R X [20] that the weights can be sequentially updated as follows: xi(t)= aij(t)xj(t − 1) + ηi(t) i = 1, ··· ,R (1) (n) (n) (n) j wt ∝ wt−1p(x(t)|ai(t) ) (8) or in matrix notation: where the proportionality takes into account normalization factors. With this approach, at each time instant t we have x(t)= a(t)x(t − 1) + η(t) (2) (n) (n) a sample set {ai(t) , wt } for n = 1,...,N and for where i = 1,...,R which can be used to estimate the pdf of the parameters and to infer information about the network. a (t) a (t) . . . a (t) 11 12 1R However, after some iterations, most of the particles will have a21(t) a22(t) . . . a2R(t) a(t)=   (3) a very low statistical weight, resulting in a lower exploration  . . .. .   . . . .  efficiency of the algorithm. To overcome this typical problem aR1(t) aR2(t) . . . aRR(t) of sequential Monte Carlo methodologies, a step called Re- sampling is performed. The number of effective particles was which is employed as the observation equation describing defined in [27] as the relationship between the observations x(t) at time t and 1 those at time t − 1 (that is, x(t − 1)); η(t) is the vector of Neff = PN (n) 2 observation noise; the matrix of hidden parameters of interest n=1(wt ) a(t) represents the causal influence exerted between different If Neff is below a certain arbitrary threshold the Resampling areas. In particular, it can be assumed that elements of a(t) is performed: particles with weight below a certain threshold are allowed to be time-varying: are substituted by copies of particles with sufficiently high weights and to each of the new particles set is assigned the aij(t)= aij(t − 1) + νij(t) (4) same weight 1/N. This results in a more effective exploration where aij(t) is the ij-th element of the matrix a(t), describing of the solution space, because only statistically relevant parti- the influence of the j-th region over the i-th region, and νij(t) cles remain after this step. is the process noise (innovation) term. To sum up, the resulting algorithm can be schematically The PF algorithm evolves from an initial probability distri- expressed as in Table I. In our implementation the procedure bution for aij(t − 1), which we chose to be uniform at t = 1, is repeated Nr = 100 times, all independent from each other, and through equation (4) it generates new possible values for to provide a better exploration of the solution space, and aij(t); then, with equation (2), the PF algorithm generates resampling was performed when Neff < 30% of the total predicted values of the observations at time t. The desired number of particles. The final outputs of the algorithm are bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Table I. • Motor Task: Time-series consisting of 240 time points Schematic description of the PF algorithm. with a temporal resolution of 2s were acquired on two subjects. During acquisition, the subjects’ thumb- and

Input the BOLD fMRI xt index-fingertips were stimulated via a pneumatic device data series (Linari Engineering, Pisa, Italy). The subjects’ task was to move the finger whenever it was stimulated. Four Set the number of parti- N (=2000 in our case) ROIs were studied covering primary somatosensory (S1), cles primary motor (M1), supplementary motor (SM) and parietal (P) cortices. All ROIs consisted in four voxels Set starting point for coef- at=0 = 0 ficients values and were manually drawn on each subject on one slice only, to avoid potential slice timing confounds (Fig. 1). The resultant time-series of each ROI were obtained by Start PF for t = 1 : T averaging the four time-series of individual voxels. • Visual Task: updating step generate N particles from previous Time-series consisting of either 300 or 600 coefficients’ values through time points with a temporal resolution of 0.8s were aij (t) = aij (t − 1) + νij (t) acquired on four trials. Subjects underwent a periodic visual stimulation alternating between black and white estimation step predict the values of the observa- dots moving along spiral trajectories over a gray back- tions at time t from values at time ground (stimulation ON) and presentation of the gray t − 1 with x(t) = a(t)x(t − 1) + η(t) background alone (stimulation OFF). Four ROIs were compute the likelihood between studied covering the Lateral Geniculate Nucleus (LGN), predicted values and observed val- the Middle temporal cortex (MT), the Primary Visual area ues with (6) (V1) and one control ROI in the tempo-parietal cortex normalize the weights and resam- (CTRL). Four voxels wide ROIs were manually drawn ple on each subject on one slice only. End PF end for on t

the at computed as the average of the Nr repetitions. The algorithm was implemented in MATLAB (Mathworks, Natick, MA, U.S.A.) R2017b. B. Synthetic data To validate the proposed approach, two different synthetic networks were used. • One network with R = 6 nodes, each with T = 100 time points, stationary coefficients generated with the MATLAB function varm() with a Signal-to-Noise Ratio (SNR) set to either ∞ (ση = 0, ideal case) or 6dB. • Another network with R = 2 and T = 250 was used to assess the PF capability to capture time-varying hidden parameters. In this case, aij coefficients were zero except for coefficient a21, whose value switched from 1 to −1 with a period of 125 time points. The SNR was 10dB. The first synthetic dataset allowed us to verify the reliability of the results of the PF and to decide the optimal values Fig. 1: ROIs drawn on one representative subject, representing of the parameters; the second one allowed us to address the primary somatosensory (S1), primary motor (M1), supplemen- capability of the PF to track variations through time of the AR tary motor (SM) and parietal (P) cortices. coefficients. The optimal order of the autoregressive model describing C. Real fMRI data the time series was 1, as estimated by the Schwartz criterion The proposed approach was also retrospectively applied to [12]. real fMRI data acquired on healthy volunteers in two different The particle filter was applied not only to the original time experimental set-ups, acquired at isotropic spatial resolution of series but also to the same fMRI data shuffled in time. This (1.5 mm)3 on a 7T MRI system: test allowed to address the effective dependency of the results bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. from the temporal order of the data, i.e. to address causal dependency through time in the data. Subsequently, the aij coefficients estimated by particle filtering were compared to the delayed correlation (DC) cij between signals xi(t) and xj(t − 1) which reflect the time-invariant causal influence exerted by network node (ROI) j over node i. Furthermore, the results of the PF were also compared to coefficients estimated in a stationary framework fitting a four-node first order linear AR model to the data via maximum likelihood.

Fig. 3: Scatter plots that relate PF estimates (x axis) and true values (y axis) of the autoregressive model for a 6-node network with 100 time samples, in the absence of noise (left) and with SNR = 6dB (right). The lines are the results of a linear fit of the data: in the noiseless case the slope m and the offset q were 1.39 and −1.62·10−2, respectively; in the noisy case, m = 1.62 and q = 8 · 10−3.

Fig. 2: The plot on top shows an example of PF estimates of causal dependency between time-series. Plot in the center shows the actual stimulation pattern as a square wave, where the presence and the absence of stimulation are represented by 1 and 0 respectively. The bottom plot shows a time shifted stimulation pattern, which has a half-period offset from the actual stimulation pattern. The t-test was run comparing connectivity values in correspondence of ones (stimulation ON) and zeros (stimulation OFF) in both cases. The second test allowed to exclude spurious changes in connectivity values Fig. 4: Time courses of the hidden parameters a in the not due to the stimulation. ij case of a 2-node network with non-stationary coefficient a21 alternating between 1 and −1. Red lines represent the true With a t-test between coefficient values in the presence values, while blue lines represent the estimates obtained by and in the absence of stimulation we searched statistically PF. significant (p-value < 0.05) changes in connectivity following the underlying stimulation pattern. Also, as a control analysis, a second t-test was performed, simulating a different stimu- other coefficients are correctly estimated to be close to the lation pattern from the actual one, which allowed to exclude nominal null value. variations deriving from spurious fluctuations of the results, as explained in Fig. 2. B. Real fMRI Data 1) Motor Network: Red lines in Fig. 5 represent aver- III.EXPERIMENTAL RESULTS AND DISCUSSIONS age values of the aij coefficients obtained on fMRI time series and the blue histogram represents the corresponding A. Synthetic Data distribution of mean values of causal interactions for the Scatter plots in Fig. 3 demonstrate that the causality coef- permuted time series. Since temporal permutation suppresses ficients estimated by PF in a stationary network satisfactorily the causal dependency between subsequent values, it was correlate with the true coefficients, both in the noiseless expected to observe zero-mean gaussianly distributed results, synthetic dataset (Pearson’s ρ = 0.96) and in the noisy that is, no causal dependency. On the other hand, coefficients scenario with SNR = 6dB (Pearsons’ ρ = 0.59). The case of representing the causal relationship between two interacting a network with one time-varying coefficient is shown in Fig. brain areas should have values lying far away from the null 4. The PF tracks the changes of the non-stationary coefficient distribution. This is what Fig. 5 shows, demonstrating that the a21, although the estimated values do not immediately follow particle filter effectively discriminates between unrelated and the abrupt changes between 1 and −1 and viceversa. All the causally related time series. bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Fig. 7: Plots in blue color show the PF-estimated time courses of three representative hidden parameters aij in the case of a 4-node motor network, estimated in real fMRI data in Fig. 5: Mean results of causal interactions between real fMRI one subject. Top panel depicts the coefficient describing the data computed through the particle filter: in blue, the histogram negligible causal effect exerted by area S1 over P. Central and of mean values obtained on time-series randomly permuted bottom panels represent the causal effect exerted by the SM in time, while the red line shows the corresponding mean area over M1 and viceversa respectively. value obtained from non-permuted time-series. Many of these values lie well outside of the null distribution, therefore their delayed correlation were those which represent the causal value reflects the effective mean causal interaction and it is influence exerted by areas M1 and S1, in agreement with not produced by chance. current knowledge of brain functioning during a sensory-motor task. Fig. 7 exemplifies the temporal evolution of brain connec- tivity through three representative aij coefficients in Subject 2. The top panel displays one representative coefficient involving the control ROI P, which is approximately 0. The two bottom panels demonstrate the expected reciprocal influence between M1 and SM areas. It is worth noting that the influences in the two directions, i.e. M1 over SM and vice versa, have different values, as a consequence of the adopted model that allows non-symmetric matrix of coefficients. 2) Visual Network: The PF detected causal dependencies between brain areas of the visual network which also cor- related with the delayed correlation (Pearson’s ρ = 0.70 and p < 0.001, as shown in Fig. 8a). Interestingly, all four fMRI datasets showed statistically significant dynamic changes (p < 0.05) with a pattern following the underlying stimulation in the effective connectivity coefficient regarding the influence of MT on V1, which are known to take part in the Fig. 6: Scatter plot showing the relationship between mean processing of visual stimuli. As an example, Fig. 9 represents PF estimates (horizontal axis) and delayed correlation (vertical the statistically significant differences in the causal influence axis) on the two sensory-motor experiments. On both results of MT on V1 in the presence or absence of the visual stimulus, taken altogether the Pearson’s correlation coefficient ρ is 0.74, observed consistently in all four fMRI datasets. Blue crosses which corresponds to a statistically significant correlation with show the quality parameter Q = T/σ, where T is the temporal p < 0.001. Slope and offset of the linear fit were 0.83 and 0.24 length of the time series and σ is the standard deviation of respectively. the data, taken as a measure of noise. Q values indicate a possible explanation for inter-dataset variability: the particle filter performance are enhanced by the timeseries length (T) The PF captured causal interactions between brain areas, and reduced by noise (estimated as 1/σ). Not all connectivity which significantly correlated with a proxy measure of effec- coefficients share this same behavior, therefore other factors tive connectivity, that is, delayed correlation (DC) (p < 0.001, might influence the results. Fig. 10 represents one example of Pearson’s correlation coefficient ρ = 0.74, Fig. 6). In particular, absence of detected causal relationship between area MT and in both subjects, the highest aij coefficients in both PF and the control region CTRL, and one example of non-symmetrical bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Fig. 9: Comparison between presence (green bar) and absence (a) (red bar) of visual stimulation for the mean aij coefficient representing the causal influence of MT on V1 in the four different datasets. Bars indicate the mean values and the stan- dard deviation of the mean. Higher values of the coefficients are obtained in data sets with better quality parameter Q, as shown by blue crosses.

(b) Fig. 8: (a) Scatter plot between mean PF estimates (horizontal Fig. 10: Plots in blue color show the PF-estimated time courses axis) and delayed correlation (vertical axis) on the four visual of three representative hidden parameters a in the case of experiments fMRI data. A p < 0.001 was obtained with a ij the 4-node visual network, estimated in real fMRI data in Pearson’s correlation coefficient ρ = 0.70. The black line is one subject. Top panel depicts the coefficient describing the the result of a linear fit. Slope and offset of the linear fit were negligible causal effect exerted by area MT over the control 1.10 and 0.25 respectively. (b) Scatter plot between average area. Central and bottom panels represent the non-symmetrical PF estimates (horizontal axis) and stationary AR coefficients causal effect exerted by the V1 area over MT and viceversa (vertical axis). The resulting Pearson’s correlation coefficient respectively. ρ was 0.94, with a p < 0.001. The black line is the result of a linear fit, with slope and offset 1.2 and -0.004 respectively. results were not consistent between the four datasets. Fig. 11 shows the relationship between coefficients obtained causality exerted by area V1 over MT and viceversa. by PF and DC with explicit reference to the causal influence Fig. 8b shows the comparison between results of the PF they are referred to. We searched for the appropriate clustering and stationary AR coefficients (Pearson’s ρ = 0.94 and of the data in Fig. 11. The number of clusters was chosen p < 0.001). An excellent agreement was found between the as the knee of the number-of-clusters vs distance-to-every- two methods. In addition, changes through time of the AR centroid curve. The optimal number was 3 and each cluster’s coefficient were searched through a sliding-window approach. centroid was plotted as a black cross in Fig. 11. All coefficients While in this way some coefficients were found to vary, the representing the causal interactions involving the CTRL region bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

In a different study, Ahmad et al. [32] adopted a symmetric, linear, first-order, time-varying Autoregressive (TVAR) model and used a Rao-Blackwellized PF to estimate the temporal relationships among fMRI time-series representing four brain regions during resting state. The assumption of symmetry in coupling coefficients, that is aij = aji, reduced model complexity, but did not permit to infer neither the directionality of the network nor any possibly asymmetric cause/effect inter- action between brain areas. Therefore their approach cannot be used to investigate effective connectivity. Also, the results were not benchmarked with the outcome results from different analyses and the resting-state paradigm did not allow any analysis on the temporal evolution of the results. In our implementation on fMRI data, the time-averaged PF estimates were in agreement with a proxy measure of causality, that is, delayed correlation. Part of the mismatch between the proposed method and delayed correlation could be explained by the fact that the PF algorithm studies the network as a whole and produces estimates of aij coefficients that update at every time instant, while delayed correlation is a measure of pair- wise causality that does not take into account possible non- stationarities and spurious cause-effect relationships mediated by other nodes of the network. In the first experimental setup involving the motor network, statistically significant changes in connectivity were not found, Fig. 11: Fig. 8a with ROI-based encoding, explained in the but the poor temporal resolution of the data (2s) may have legend above the figure: same color refer to the same causing prevented the detection of these changes. ROI and same symbols to same caused ROI. Black crosses are On the contrary, in our study of the visual network sta- clusters centroids found through the Matlab function kmeans(), tistically significant changes in connectivity were found in called A, B and C as in figure. The horizontal red line and four different datasets with a pattern following the underlying vertical green line intersect in the point equally distant from stimulation, without requiring any previous knowledge on the centroids A and B. actual stimulation paradigm during the estimation process. These variations interested the influence of MT on V1 which is consistent with our understanding of cortical processing in the (black symbols) belong to cluster A, that is, small values of early visual cortex [24]. Good agreement was found between DC and mean PF estimates, except for the self-causality terms, PF results and an AR coefficient evaluation method, even if the which all lay in the vicinity of the C centroid. Every non- latter failed in detecting variations through time in a sliding- diagonal coefficient involving ROIs LGN, MT and V1 belong window fashion. This result suggests that this second method to the cluster B. In particular, in Fig. 11, the horizontal red might require more data, i.e. longer exams, to achieve the line and vertical green line intersect in the point equally distant sensitivity obtained with the PF. Furthermore, while the PF to the centroids A and B. All values representing the causal is ”blind,” sliding window analysis require the inclusion of influence of MT on V1 (which vary following the stimulation information regarding the stimulation timing, which in some pattern, represented by highlighted red diamonds in the figure) cases may not be available. and that of V1 on MT (green stars) lay right to the green line Mean causal interaction values were found to vary between and above the red line. different datasets. As Fig. 9 shows, in some cases these differ- ences can be explained through the simultaneous contribution IV. DISCUSSION of noise and temporal length of the data, i.e. the quality Other implementations of SMC algorithms were previously parameter Q = T/σ. The brain haemodynamic responses may proposed to investigate brain connectivity in fMRI data. Mur- also be involved, which vary not only among subjects but also ray and Storkey [28] proposed a forward-backward Particle between different areas in the same subject [33], which was Filter using, as observation equation, a stochastic extension not taken into account in this study. of the balloon model, which was proposed to describe the haemodynamics that follow brain activity [29]. In their study, V. CONCLUSIONS the hidden parameters of the model resulted approximately We used Particle Filter to test and identify the time- constant, probably as a consequence to the complexity of the varying brain connectivity as evidenced in fMRI images. Our model itself [30], [31]. experiments confirmed the hypothesis of time-varying brain bioRxiv preprint doi: https://doi.org/10.1101/2021.01.19.427249; this version posted January 20, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

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