Measuring functional connectivity from fMRI recordings is important in understanding processing in cortical networks. However, because the brain's connection pattern is complex, currently used methods are prone to producing false functional connections. We introduce differential covariance analysis, a new method that uses derivatives of the signal for estimating functional connectivity. We generated neural activities from dynamical causal modeling and a neural network of Hodgkin-Huxley neurons and then converted them to hemodynamic signals using the forward balloon model. The simulated fMRI signals, together with the ground-truth connectivity pattern, were used to benchmark our method with other commonly used methods. Differential covariance achieved better results in complex network simulations. This new method opens an alternative way to estimate functional connectivity.

With our ability to record more neurons simultaneously, making sense of these data is a challenge. Functional connectivity is one popular way to study the relationship of multiple neural signals. Correlation-based methods are a set of currently well-used techniques for functional connectivity estimation. However, due to explaining away and unobserved common inputs (Stevenson, Rebesco, Miller, & Körding, 2008 ), they produce spurious connections. The general linear model (GLM), which models spike trains as Poisson processes (Okatan, Wilson, & Brown, 2005 ; Truccolo, Eden, Fellows, Donoghue, & Brown, 2005 ; Pillow et al., 2008 ), avoids these confounds. We develop here a new class of methods by using differential signals based on simulated intracellular voltage recordings. It is equivalent to a regularized AR(2) model. We also expand the method to simulated local field potential recordings and calcium imaging. In all of our simulated data, the differential covariance-based methods achieved performance better than or similar to the GLM method and required fewer data samples. This new class of methods provides alternative ways to analyze neural signals.

Entropy rate quantifies the change of information of a stochastic process (Cover & Thomas, 2006 ). For decades, the temporal dynamics of spike trains generated by neurons has been studied as a stochastic process (Barbieri, Quirk, Frank, Wilson, & Brown, 2001 ; Brown, Frank, Tang, Quirk, & Wilson, 1998 ; Kass & Ventura, 2001 ; Metzner, Koch, Wessel, & Gabbiani, 1998 ; Zhang, Ginzburg, McNaughton, & Sejnowski, 1998 ). We propose here to estimate the entropy rate of a spike train from an inhomogeneous hidden Markov model of the spike intervals. The model is constructed by building a context tree structure to lay out the conditional probabilities of various subsequences of the spike train. For each state in the Markov chain, we assume a gamma distribution over the spike intervals, although any appropriate distribution may be employed as circumstances dictate. The entropy and confidence intervals for the entropy are calculated from bootstrapping samples taken from a large raw data sequence. The estimator was first tested on synthetic data generated by multiple-order Markov chains, and it always converged to the theoretical Shannon entropy rate (except in the case of a sixth-order model, where the calculations were terminated before convergence was reached). We also applied the method to experimental data and compare its performance with that of several other methods of entropy estimation.