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Functional connectivity is among the widely used metrics to assess the network-level attributes of brain function. While most existing analysis frameworks assume static functional connectivity during the course of an experiment, to capture neural dynamics over short time scales, a time-varying notion of functional connectivity is required. By revealing how neural networks reconfigure in response to changing external stimuli, internal states, and task demands, time-varying functional connectivity can be leveraged to study flexible cognition, such as working memory, attention, and decision-making. A major challenge in estimating time-varying functional connectivity from high-dimensional neural is the associated computational complexity. Existing methods trade off accuracy for computational efficiency, especially in applications that require real-time or near real-time processing. Here, we build on existing work using covariance-domain state-space models and introduce a framework based on variational inference that allows low-complexity estimation of time-varying functional connectivity and construction of confidence intervals. We validate the performance of the proposed method using simulation studies. Our results reveal significant gains in computational complexity compared to existing methods, while maintaining high accuracy.