Inferring evoked brain connectivity through adaptive perturbation

Abstract

Inference of functional networks—representing the statistical associations between time series recorded from multiple sensors—has found important applications in neuroscience. However, networksexhibiting time-locked activity between physically independent elements can bias functional connectivity estimates employing passive measurements. Here, a perturbative and adaptive method of inferring network connectivity based on measurement and stimulation—so called “evoked network connectivity” is introduced. This procedure, employing a recursive Bayesian update scheme, allows principled network stimulation given a current network estimate inferred from all previous stimulations and recordings. The method decouples stimulus and detector design from network inference and can be suitably applied to a wide range of clinical and basic neuroscience related problems. The proposed method demonstrates improved accuracy compared to network inference based on passive observation of node dynamics and an increased rate of convergence relative to network estimation employing a naïve stimulation strategy.

Appendix: Posterior update equation

$$ \begin{array}{rll} &&{\kern-6pt} {\rm P\left({\bf e}^{(k)} | {\bf d}^{(k)}, {\bf s}^{(k)}, H_k\right)} \\ &&{\kern6pt} \propto \rm P\left( {\bf d}^{(k)} | {\bf e}^{(k)}, {\bf s}^{(k)}, H_k \right) P\left( {\bf e}^{(k)} | {\bf s}^{(k)}, H_k \right) \ , \\ &&{\kern6pt} = \rm P\left( {\bf d}^{(k)} | {\bf e}^{(k)}, {\bf s}^{(k)} \right) \ P\left( {\bf e}^{(k)} | {\bf s}^{(k)}, H_k \right) \ . \end{array} $$

(22)

The prior probability mass function of the edges, $\rm P\left( {\bf e}^{(k)} | {\bf s}^{(k)}, H_k \right)$ can be further decomposed by applying the Chapman–Kolmogorov equation.

$$ \begin{array}{rll} &&{\kern-6pt} {\rm P\left( {\bf e}^{(k)} | {\bf s}^{(k)}, H_k \right)}\\ &&{\kern6pt} = {\rm \sum\limits_{{\bf c} \in C} P\left( {\bf e}^{(k)} | {\bf e}^{(k-1)} = {\bf c}, {\bf s}^{(k)}, H_k \right)} \\ &&{\kern18pt} \times {\rm P\left( {\bf e}^{(k-1)} = {\bf c} | {\bf d}^{(k-1)}, {\bf s}^{(k-1)}, H_{k-1} \right)} \ , \\ &&{\kern6pt} = \rm {\sum\limits_{{\bf c} \in C} P\left( {\bf e}^{(k)} | {\bf e}^{(k-1)} = {\bf c} \right)} \\ &&{\kern18pt} \times {\rm P\left( {\bf e}^{(k-1)} = {\bf c} | {\bf d}^{(k-1)}, {\bf s}^{(k-1)}, H_{k-1} \right)} \ , \end{array} $$

(23)

where C is the set of $\rm 2^{N_{edge}}$ possible networks. Here it is assumed that e ^(k) is conditionally independent of s ^(k) and H_k given e ^{(k − 1)}. Heuristically, this assumption implies that given knowledge of the network connections at stimulus index k − 1, knowledge of the stimulus at the stimulation index k does not provide information about e ^(k). This is a reasonable assumption since the stimulus at index k is determined from the posterior of e ^{(k − 1)}.

Without a restriction upon C the sum in Eq. (23) is computationally intractable. As discussed in Section 3, this restriction is attained by assuming that the edges are independent of each other. With this assumption the probability mass functions factor and the prior probability mass function can be written,

$$ \begin{array}{rll} &&{\kern-6pt} {\rm {P,\left( {\bf e}^{(k)} | {\bf s}^{(k)}, H_k \right) }}\\ &&{\kern6pt} = {\rm \prod\limits_{j=1}^{N_{edge}} P\left( ({\bf e}^{(k)} )_j | ({\bf s}^{(k)})_j, H_k \right)} \\ &&{\kern6pt} = \rm \sum\limits_{{\bf c} \in C} \prod\limits_{j=1}^{N_{edge}} P\left( ({\bf e}^{(k)})_j | ({\bf e}^{(k-1)})_j = ({\bf c})_j \right) \\ &&{\kern18pt} \times\, {\rm P\left( ({\bf e}^{(k-1)})_j = ({\bf c})_j | ({\bf d}^{(k-1)})_j, ({\bf s}^{(k-1)})_j, H_{k-1} \right)} \ , \\ &&{\kern6pt} = {\rm \prod\limits_{j=1}^{N_{edge}} \sum\limits_{c_j = 0}^1 P\left( ({\bf e}^{(k)})_j | ({\bf e}^{(k-1)})_j = c_j \right)}\\ &&{\kern18pt} \times {\rm P\left( ({\bf e}^{(k-1)})_j = c_j | ({\bf d}^{(k-1)})_j, ({\bf s}^{(k-1)})_j, H_{k-1} \right)} \ . \end{array} $$

(24)

Combine Eq. (24) with Eq. (22), along with the edge independence assumption to obtain:

$$ \begin{array}{rll} && {\kern-6pt}{\rm \prod\limits_{j=1}^{N_{edge}} P\left( ({\bf e}^{(k)})_j | {\bf d}^{(k)}, {\bf s}^{(k)}, H_k \right)} \\ &&{\kern6pt}\propto {\rm \prod\limits_{j=1}^{N_{edge}}\! P\!\left( ({\bf d}^{(k)})_j | {\bf e}^{(k)}, {\bf s}^{(k)} \right)} \!\times\! {\rm \sum\limits_{c_j = 0}^1 P\!\left( ({\bf e}^{(k)})_j | ({\bf e}^{(k-1)})_j = c_j \right)} \\ &&{\kern18pt}\times {\rm P\left( ({\bf e}^{(k-1)})_j = c_j | ({\bf d}^{(k-1)})_j, ({\bf s}^{(k-1)})_j, H_{k-1} \right)} \ . \end{array} $$

(25)

Thus, recursive update can be computed on an edge by edge basis,

$$ \begin{array}{rll} &&{\kern-6pt}{\rm P\left( ({\bf e}^{(k)})_j | {\bf d}^{(k)}, {\bf s}^{(k)}, H_k \right)} \\ &&{\kern6pt} \propto {\rm P\left( ({\bf d}^{(k)})_j | {\bf e}^{(k)}, {\bf s}^{(k)} \right) } \times {\rm \sum\limits_{c_j = 0}^1 P\left( ({\bf e}^{(k)})_j | ({\bf e}^{(k-1)})_j = c_j \right)} \\ &&{\kern6pt} \times {\rm P\left( ({\bf e}^{(k-1)})_j = c_j | ({\bf d}^{(k-1)})_j, ({\bf s}^{(k-1)})_j, H_{k-1} \right)} \ , \end{array} $$

(26)

for j = 1 , ... , N_edge; greatly simplifying computations. This factoring allows for tractable computational inference, reducing both memory requirements and computation time.