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 The copyright holder for this preprint (which was not certified by peer review) is the author/funder.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)ja(t))p(a(t)jx(1;:::;t − 1)) were presented in abstract form [18], involved tactile stimula- p(a(t)jx(1;:::;t)) = p(x(t)jx(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) − x^ (t))2 p(x (t)ja (t)) = − i i i i 2 R=2 exp 2 (6) II. METHODOLOGY (2πση) 2ση A. Model and algorithm where x^i(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)= fai1; : : : ; aiRg 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)jx(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)jx(1;:::;t)) ≈ wt δ(ai(t) − ai (t)) (7) among the time-series of R different brain Regions of Interest n=1 (ROIs) x = fx (t);:::;x (t)g 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)jai(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 fai(t) ; wt g 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. 2a (t) a (t) : : : a (t)3 11 12 1R However, after some iterations, most of the particles will have 6a21(t) a22(t) : : : a2R(t)7 a(t)= 6 7 (3) a very low statistical weight, resulting in a lower exploration 6 . .. 7 4 . 5 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.