Estimating the directed information to infer causal relationships in ensemble neural spike train recordings

Abstract

Advances in recording technologies have given neuroscience researchers access to large amounts of data, in particular, simultaneous, individual recordings of large groups of neurons in different parts of the brain. A variety of quantitative techniques have been utilized to analyze the spiking activities of the neurons to elucidate the functional connectivity of the recorded neurons. In the past, researchers have used correlative measures. More recently, to better capture the dynamic, complex relationships present in the data, neuroscientists have employed causal measures—most of which are variants of Granger causality—with limited success. This paper motivates the directed information, an information and control theoretic concept, as a modality-independent embodiment of Granger’s original notion of causality. Key properties include: (a) it is nonzero if and only if one process causally influences another, and (b) its specific value can be interpreted as the strength of a causal relationship. We next describe how the causally conditioned directed information between two processes given knowledge of others provides a network version of causality: it is nonzero if and only if, in the presence of the present and past of other processes, one process causally influences another. This notion is shown to be able to differentiate between true direct causal influences, common inputs, and cascade effects in more two processes. We next describe a procedure to estimate the directed information on neural spike trains using point process generalized linear models, maximum likelihood estimation and information-theoretic model order selection. We demonstrate that on a simulated network of neurons, it (a) correctly identifies all pairwise causal relationships and (b) correctly identifies network causal relationships. This procedure is then used to analyze ensemble spike train recordings in primary motor cortex of an awake monkey while performing target reaching tasks, uncovering causal relationships whose directionality are consistent with predictions made from the wave propagation of simultaneously recorded local field potentials.

Appendices

Appendix A: Proof of Lemma 1

Proof

First, prove that $\mathcal{H}(\text{Y}||\text{X})$. This proof closely follows the proof for the unconditional entropy rate in Cover and Thomas (2006). An important theorem used for the proof is the Cesaro mean theorem (Cover and Thomas 2006): For sequences of real numbers (a ₁, ⋯ a _n ) and (b ₁, ⋯ b _n ), if lim_{n → ∞} a _n  = a, and $b_n = \frac{1}{n} \sum_{i=1}^n a_n$, then lim_{n → ∞} b _n  = a.

By definition, $\text{H}(Y^n || X^n) = \frac{1}{n}\sum_{i=1}^n \text{H}(Y_i | Y^{i-1}, X^i)$. Since conditioning reduces entropy, entropy is nonnegative, and the processes are jointly stationary, we have

$$ 0 \leq \text{H}\left(Y_i | Y^{i-1}, X^i\right) \leq \text{H}(Y_1) \quad \forall \ i. $$

Observe that

$$ \text{H}\left(Y_{i} | Y^{i-1}, X^{i}\right) \leq \text{H}\left(Y_{i} | Y^{i-1}_2, X^{i}_2\right) \label{ab1} $$

(46)

$$ = \text{H}\left(Y_{i-1} | Y^{i-2}, X^{i-1}\right), \label{ab2} $$

(47)

where Eq. (46) uses the property that conditioning reduces entropy (in reverse) and Eq. (47) uses stationarity. This sequence of real numbers (once the process is defined, that is, the underlying probability distribution is specified), the entropies are deterministic numbers) $a_i \triangleq \text{H}(Y_{i} | Y^{i-1}, X^{i})$ are nonincreasing and bounded below by 0. Therefore, limit of a _n as n → ∞ exists, and thus, by employing Cesaro mean theorem, $\text{H}(\mathcal{Y} || \mathcal{X}) \triangleq \lim_{n \to \infty} \frac{1}{n}\text{H}(Y^n || X^n)$ exists.

Next, taking X ⁿ to be a deterministic sequence, and following the above, $\text{H}(\mathcal{Y} ) \triangleq \lim_{n \to \infty} \frac{1}{n}\text{H}(Y^n )$ exists. Taking the limit in Eq. (24), $\mathcal{I}(\text{X} \to \text{Y})\triangleq \lim_{n \to \infty} \frac{1}{n}\text{I}(X^n \to Y^n)$ also exists.□

Appendix B: Proof of Lemma 2

Proof

The normalized causal entropy can be rewritten as

$$ \begin{array}{lll} && {\kern-6pt} \frac{1}{n} \text{H}\left(Y^n || X^n\right) \\ && {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \!\text{H}\left(Y_i | X^i, Y^{i-1}\right) \label{bv1} \end{array} $$

(48)

$$ {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n\! \mathbb{E} \!\left[\! - \log P_{Y_i | Y^{i-1},X^i}\left(Y_i | Y^{i-1},X^i \right) \! \right] \label{bv2} $$

(49)

$$ {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \! \mathbb{E}\! \left[ \! - \log P_{Y_i | Y^{i-1}_{i-J},X^i_{i-(K-1)}} \left(Y_i | Y^{i-1}_{i-J},X^i_{i-(K-1)} \right) \! \right] \label{bv3} $$

(50)

$$ \begin{array}{lll} && {\kern6pt} = \frac{1}{n}\! \sum\limits_{i=1}^n \! \mathbb{E}\! \left[g_{JK}\left( Y^{i}_{i-J},X^i_{i-(K-1)} \right)\right] \\ && {\kern6pt} = \mathbb{E} \!\left[\! g_{JK}\left(Y^{l}_{l-J},X^l_{l-(K-1)} \right) \!\right] \label{bv4_3} \end{array} $$

(51)

where Eq. (48) follows by the definition of causally conditioned entropy, Eq. (49) follows by chain rule for entropy, Eq. (50) follows from the Markov assumption, and Eq. (51) follows from the stationarity assumption.□