The variability of cortical activity in response to repeated presentations of a stimulus has been an area of controversy in the ongoing debate regarding the evidence for fine temporal structure in nervous system activity. We present a new statistical technique for assessing the significance of observed variability in the neural spike counts with respect to a minimal Poisson hypothesis, which avoids the conventional but troubling assumption that the spiking process is identically distributed across trials. We apply the method to recordings of inferotemporal cortical neurons of primates presented with complex visual stimuli. On this data, the minimal Poisson hypothesis is rejected: the neuronal responses are too reliable to be fit by a typical firing-rate model, even allowing for sudden, time-varying, and trial-dependent rate changes after stimulus onset. The statistical evidence favors a tightly regulated stimulus response in these neurons, close to stimulus onset, although not further away.

Let ${\cal X}$ be a finite space and let $\pi$ be an underlying probability on ${\cal X}$. For any real-valued function $f$ defined on ${\cal X}$, we are interested in calculating the expectation of $f$ under $\pi$. Let $X_0,X_1,\ldots,X_n,\ldots$ be a Markov chain generated by some transition matrix $P$ with invariant distribution $\pi$. The time average, $\frac{1}{n}\sum_{k=0}^{n-1} f(X_k)$, is a reasonable approximation to the expectation, $E_\pi[f(X)]$. Which matrix $P$ minimizes the asymptotic variance of $\frac{1}{n}\sum_{k=0}^{n-1} f(X_k)$? The answer depends on $f$. Rather than a worst-case analysis, we will identify the set of $P$'s that minimize the average asymptotic variance, averaged with respect to a uniform distribution on $f$.

Cryo-electron microscopy (cryo-EM) has recently emerged as a powerful tool for obtaining three-dimensional (3D) structures of biological macromolecules in native states. A minimum cryo-EM image data set for deriving a meaningful reconstruction is comprised of thousands of randomly orientated projections of identical particles photographed with a small number of electrons. The computation of 3D structure from 2D projections requires clustering, which aims to enhance the signal to noise ratio in each view by grouping similarly oriented images. Nevertheless, the prevailing clustering techniques are often compromised by three characteristics of cryo-EM data: high noise content, high dimensionality and large number of clusters. Moreover, since clustering requires registering images of similar orientation into the same pixel coordinates by 2D alignment, it is desired that the clustering algorithm can label misaligned images as outliers. Herein, we introduce a clustering algorithm $\gamma$-SUP to model the data with a $q$-Gaussian mixture and adopt the minimum $\gamma$-divergence for estimation, and then use a self-updating procedure to obtain the numerical solution. We apply $\gamma$-SUP to the cryo-EM images of two benchmark macromolecules, RNA polymerase II and ribosome. In the former case, simulated images were chosen to decouple clustering from alignment to demonstrate $\gamma$-SUP is more robust to misalignment outliers than the existing clustering methods used in the cryo-EM community. In the latter case, the clustering of real cryo-EM data by our $\gamma$-SUP method eliminates noise in many views to reveal true structure features of ribosome at the projection level.