Bayesian estimation for large scale multivariate Ornstein-Uhlenbeck model of brain connectivity

05/25/2018 ∙ by Andrea Insabato, et al. ∙ Universitat Pompeu Fabra Columbia University 0

Estimation of reliable whole-brain connectivity is a crucial step towards the use of connectivity information in quantitative approaches to the study of neuropsychiatric disorders. When estimating brain connectivity a challenge is imposed by the paucity of time samples and the large dimensionality of the measurements. Bayesian estimation methods for network models offer a number of advantages in this context but are not commonly employed. Here we compare three different estimation methods for the multivariate Ornstein-Uhlenbeck model, that has recently gained some popularity for characterizing whole-brain connectivity. We first show that a Bayesian estimation of model parameters assuming uniform priors is equivalent to an application of the method of moments. Then, using synthetic data, we show that the Bayesian estimate scales poorly with number of nodes in the network as compared to an iterative Lyapunov optimization. In particular when the network size is in the order of that used for whole-brain studies (about 100 nodes) the Bayesian method needs about eight times more time samples than Lyapunov method in order to achieve similar estimation accuracy. We also show that the higher estimation accuracy of Lyapunov method is reflected in a much better classification of individuals based on the estimated connectivity from a real dataset of BOLD fMRI. Finally we show that the poor accuracy of Bayesian method is due to numerical errors, when the imaginary part of the connectivity estimate gets large compared to its real part.

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Introduction

Estimation of connectivity from time series data is a common goal for many disciplines, such as economics, system biology or neuroscience. In neuroscience there has been increasing interest over the last years for connectivity measures at the whole-brain scale as defined by recording techniques like fMRI or EEG. Examples of connectivity measures include statistical dependence measures such as Pearson’s correlation or mutual information and network models such as vector autoregressive models, dynamic causal model

(Friston, 2011) or multivariate Ornstein-Uhlenbeck (Gilson et al., 2016).

The estimated connectivity can then be used to study individual traits (Finn et al., 2015; Pallares et al., 2017), cognitive states (Gonzalez-Castillo et al., 2015; Pallares et al., 2017) or clinical conditions (Rahim et al., 2017). In particular connectivity measures open new perspectives as biomarkers for neuropsychiatric diseases (Arbabshirani et al., 2017; Greicius, 2008; Meng et al., 2017).

Estimating brain connectivity is in general challenging due to the scarcity of time samples (in the order of hundreds for fMRI) and the large dimensionality of the measurements (typical brain parcellation about 100 regions or more). Bayesian estimation could bring some advantages such as the ability to naturally perform connectivity detection, by making statistical tests exploiting the posterior probability distribution of the connectivity. In addition the inclusion of a prior over model parameters can act as a regularization term and in turn improve estimation accuracy when only few samples are available. Finally the Bayesian framework naturally allows to extend models to include modelling of population variability through hierarchical models and different observational models

(Linderman et al., 2016).

Here we study a Bayesian estimation method for the multivariate Ornstein-Uhlenbeck model (mOU) and compare it with other available estimation methods: the moments method and a Lyapunov optimization (Gilson et al., 2016).

1 Model definition

In this work we consider a multivariate Ornstein-Uhlenbeck (mOU) process. The process instantiates a network model where the dynamics of each node is governed by an OU equation with additional input from all connected nodes:

(1)

where is the activity of the th node, is the time constant of the node, is the connectivity matrix where each entry represent the link strength from node to node (the diagonal of the is set to 0 to avoid self connections),

is the noise variance of each node that scales the Gaussian noise

. All can be collected in the diagonal of noise covariance matrix . Here we assume uncorrelated noise, hence is diagonal, but the model can be easyly generalized to correlated noise. We also assume the time constant to be omogeneous between nodes. Here we are not consider any additional node specific term (usually called drift). However we note that a drift term could be introduced into the equation and its value estimated for decoding purposes, e.g. in scenarios where external stimuli are relevant.

2 Model estimation

In this section, following Gilson et al. (2016) and Singh et al. (2017) we first show how to calculate model covariance knowing the parameters then we outline the basic steps to estimate mOU parameters using three different procedures: moments method, Lyapunov optimization and Bayesian posterior mean.

2.1 Forward step

The Jacobian of the system is: , where is Kronecker’s delta. Deriving the covariance of the system we obtain the Lyapunov equation:

(2)

where is the diagonal matrix . From this equation the covariance can be calculated using the Bartels-Stewart algorithm (Bartels and Stewart, 1972).

Similarly the derivation of the time-lagged covariance for a given lag yealds:

(3)

where is the the matrix exponential of .

Hence knowing , can be calculated.

These equations allow to compute the farward step: calculate (time-lagged) covariance matrix knowing the parameters of the system .

2.2 Moments method

As shown by Gilson et al. (2016), the moments method can be applied in order to estimate the parameters (called direct estimation in Gilson et al. (2016)). We can substitute the theoretical covariances in eq. (3) for their empirical counterparts , and solve for to get:

(4)

where is the matrix logarithm of .

Then with the estimate , an estimate of can then be calculated from Lyapunov equation (2):

(5)

2.3 Lyapunov optimization

Gilson et al. (2016) developed a Lyapunov optimization for the parameters of mOU. This optimization procedure minimizes the Lyapunov function:

(6)

By differentiating this function the parameters update can be obtained as:

(7)
(8)

The reader is referenced to Gilson et al. (2016) for further details.

2.4 Bayesian approach

Here we follow Singh et al. (2017) in the definition of a Bayesian estimate.

The probability of the state of the system at time given its state at time is given by:

(9)

where and .

The stationary distribution is: . We recall that system’s covariance matrix is related to by Lyapunov equation: .

Given a dataset where x is sampled at regular intervals , and is the times matrix collecting all the observations for each of the nodes, the likelihood function is given by:

(10)

Then the posterior distribution of parameters is given by Bayes theorem:

(11)

Assuming a uniform prior over the paramters and substituting the explicit pdfs the log posterior is:

(12)

where and .

The first term of (12) can be expanded to show that the log posterior is normal in . Finally, defining and , the posterior mean for can be written as: ( can also be estimated with the same method but we don’t pursue it here since it is not related to connectivity estimation).

It follows than that:

(13)

Sigma can then be estimated again from the Lyapunov equation:

(14)

By noting that and are proportional to the empirical covariance matrix and the transposed empirical lagged covariance matrix it can be easily seen that the Bayesian and the moments methods yield the same solution.

3 Estimation accuracy for large scale systems

Here we study the accuracy of the estimation methods presented above in the context of systems where the number of variables is large. We first study the estimation accuracy of synthetic data, where the ground truth is known and then move to the estimation with empirical data.

3.1 Synthetic network with random connectivity

Synthetic data are generated by simulating a mOU process. To this aim we used Euler integration with time step of 0.05, which is small compared to the time constant . The resulting simulated time series are then downsampled to 1 second (to have similar time resolution as typical fMRI recordings). For the simulations we fixed second and to a diagonal matrix , where

is sampled from a uniform distribution between 0 and 1.

Connectivity matrix was constructed as the Hadamard (element-wise) product of a binary adjacency matrix and a log-normal weight matrix.

(15)

where and

Thus the underlying network is Erdős-Rényi while the strength of each link is sampled from a log-normal distribution. While this corresponds to assuming a very simple model we show below that our results hold also for more structured networks.

When varying the number of nodes C’ gets normalized in order to avoid explosion of activity.

(16)

In general, given a fixed amount of samples, the more parsimonious (i.e. less parameters) a model the higher the precision in the estimate of its parameters. The model under consideration has parameters (including and ), where is the number of nodes in the network.

We show here how the size of the network and the number of time samples influences the accuracy of the estimation. For a given size we draw a connectivity matrix and simulate the model for 500 seconds. Then we estimate the connectivity from the simulated time series using Lyapunov and Bayesian methods. Finally, we calculate Pearson’s correlation coefficient between the true connectivity (used to generate the time series) and the estimated connectivity, as a measure of the accuracy of estimation. We repeat these steps 100 times for each value of . As expected, the largest effect is on the estimation of as illustrated in fig. 1. The accuracy for the estimation of presents a small decrease for Bayesian method and no notable modulation for Lyapunov method. The accuracy in the estimate of decreases as a function of network size for both estimation methods. However Lyapunov estimation shows a slower decrease of accuracy and allows more reliable estimates.

To complement this analysis we also show the effect of the amount of time samples for a given network size. For a each number of time samples we draw a connectivity matrix and simulate the model with 50 nodes. Then we estimate the connectivity using Lyapunov and Bayesian methods and evaluate estimation accuracy as above. As can be observed in fig. 1 Lyapunov estimation needs about four times fewer samples to achieve similar accuracy as Bayesian method for the estimation of . The estimation of shows a small increase in accuracy for both estimation methods (notice that the accuracy is already very high for ).

Figure 1: Estimation accuracy with a random generative connectivity matrix. Accuracy for (left column) and (rigth column), varying the number of nodes in the network (upper row, withfixed ) or the number of time samples (bottom row, with fixed ). The estimation is performed with Lyapunov (blu curves) or Bayesian methods (orange curves).

3.2 Synthetic network with connectivity estimated from empirical data

Typical estimate of brain structural and functional connectivity show significantly more structure than Erdős-Réniy networks. Here we show that the above results also hold for a real brain topology obtained from anatomical contraints. In order to obtain a generative model similar to experimental data we estimate the connectivity matrix from the BOLD time series of an fMRI recording session, using Lyapunov optimization and a structural connectivity template (from diffusion tensor imaging) as a mask for the estimation. The BOLD time series used was taken from

Zuo et al. (2014) session 1 of subject with id 25427 preprocess with AAL parcellation ( nodes) as described in Pallares et al. (2017). The structural connectivity corresponds to a generic template (obtained from other subjects) for the AAL parcellation (Tzourio-Mazoyer et al., 2002), which enforced a non-random topology (see left panel in fig. 2). We then use this connectivity matrix to generate simulated time series using the mOU as above.

In fig. 2 right we show the influence of the amount of time samples on the estimation accuracy. For a given number of time samples we simulate the model using the connectivity matrix estimated from empirical data (fig. 2). Then we estimate the connectivity from the simulated time series using Lyapunov and Bayesian methods and evaluate estimation accuracy as above.

It can be observed that the accuracy increases as a function of the number of time samples but Lyapunov method has always a higher accuracy and also a faster increase compared to Bayesian method. We note that, as shown in fig. 1 Bayesian method suffers more from high dimensionality of the system; as a result here () Bayesian needs 8 times more time sample to achieve a similar accuracy as Lyapunov method.

Figure 2: Left: Connectivity matrix estimated from BOLD fMRI data. Right: Estimation accuracy of for time series simulated using the connectivity matrix in left panel.

4 Higher estimation accuracy is reflected in better subjects identification

There is an increasing interest in measure of brain connectivity for identifying subjects (Finn et al., 2015; Pallares et al., 2017), cognitive states (Gonzalez-Castillo et al., 2015; Pallares et al., 2017) or clinical conditions (Rahim et al., 2017). From the previous results we expect the Lyapunov method to extract more informative parameters value than Bayesian, which should be in turn advantageous for classification.

In the following we show how the Lyapunov and Bayesian estimates of connectivity perform in a classification task of subjects identity. For this we used a dataset where 30 subject underwent 10 fMRI resting state scanning sessions Zuo et al. (2014). For each subject and session we estimate the connectivity with either the Bayesian or Lypunov method. Here we use the generic structural connectivity template (Tzourio-Mazoyer et al., 2002) to mask the estimated value of

where there is no connection in the structural connectivity. We then classify the identity of the subjects using the values of estimated connectivity. To this aim our data for classification is composed of 300 points (one for each subject and session) in 4056 dimensions (all non-zero values in the structural connectivity matrix). We randomly split the data using 80% as training set and the rest for testing. We train a multinomial logistic regression classifier on the training set and evaluat its performance on the test set. This procedure is repeated 100 times. For a more complete study of subject and behavior classification see

Pallares et al. (2017).

Fig. 3 shows the distribution of classifier accuracy as violin plots. As can be observed the higher reliability of lyapunov method is reflected in higher classification accuracy.

Figure 3: Distribution of classification accuracy for 100 test-sets comprising 20% of randomly chosen samples for the classification of subjects identity based on the value of links in the connectivity matrix estimated either with Lyapunov or with Bayesian method.

5 Responsibility of estimation error for Bayesian method

The Bayesian method is based on the matrix logarithm of , where precision matrix and lagged-covariance are both estimated from the time series. The large estimation error shown above is presumably due to numerical errors. In order to understand which part of this estimation has the largest responsibility in the estimation error, in fig. 4 we show the similarity (measured by Pearson’s correlation) between the estimated and theoretical counterparts of the different steps in the estimation of connectivity. As above we simulate the mOU for varying network size and compare the estimate with the generative connectivity and the estimates , , and with their theoretical counterparts calculated using the parameters of the generative model.

It can be observed in left panel that the estimation of lagged-covariance suffers much more than covariance and precision from the increase of network size. This gets also reflected in the similarity of with its theoretical counterpart. However the largest responsibility for estimation error comes from the matrix logarithm (blue line in fig. 4 left). Indeed when number of nodes increases the imaginary part of becomes larger compared to its real part as shown in fig. 4 right.

Figure 4: Left: Estimation accuracy for intermediate steps in the calculation of Bayesian estimate as a function of the number of nodes in the network. The estimate is compared with the generative connectivity , while all other estimates are compared to their theoretical equivalents calculated from the parameters. Right: Ratio of the norms of the imaginary and real part of connectivity

as a function of the number of nodes in the network. Errorbars show standard deviation over 100 simulations.

6 Discussion

Here we show that Bayesian estimate for mOU model is equivalent to an estimation based on moments method. We also show that Bayesian method is more sensible to network size and number of time samples than Lyapunov method. A number of improvement could be done for the Bayesian estimation (and will form the basis for future work). First, a different prior could be used for the Bayesian estimate. Here we used a uniform prior, so a natural extension could be a Gaussian prior that would act as a regularization term, thereby probably increasing the accuracy for small number of time samples. We note that the Lyapunov estimation used here could be easily extended to also include a regularization term.

Another option whould be the use of an empirical prior, for example derived from structural connectivity or covariance matrix of BOLD time series, possibly in conjunction with an expectation-maximization algorithm. Indeed, while we used the structural connectivity matrix to mask estimated values, these are not excluded from the estimation procedure. On the contrary, the iterative Lyapunov optimization clips to zero those links that are zero in the structural connectivity, thereby allowing other links to vary more. Hence the structural connectivity acts like a sort of empirical prior in the Lyapunov estimation procedure.

Finally another natural extension of the Bayesian estimation would be to perform link detection to filter out links that are estimated non-zero merely as an effect of noise. This would produce a sparse estimate for the connectivity and presumably increase the estimation accuracy.

Acknowledgments

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