How Dendrites Affect Online Recognition Memory.

In order to record the stream of autobiographical information that defines our unique personal history, our brains must form durable memories from single brief exposures to the patterned stimuli that impinge on them continuously throughout life. However, little is known about the computational strategies or neural mechanisms that underlie the brain's ability to perform this type of "online" learning. Based on increasing evidence that dendrites act as both signaling and learning units in the brain, we developed an analytical model that relates online recognition memory capacity to roughly a dozen dendritic, network, pattern, and task-related parameters. We used the model to determine what dendrite size maximizes storage capacity under varying assumptions about pattern density and noise level. We show that over a several-fold range of both of these parameters, and over multiple orders-of-magnitude of memory size, capacity is maximized when dendrites contain a few hundred synapses-roughly the natural number found in memory-related areas of the brain. Thus, in comparison to entire neurons, dendrites increase storage capacity by providing a larger number of better-sized learning units. Our model provides the first normative theory that explains how dendrites increase the brain's capacity for online learning; predicts which combinations of parameter settings we should expect to find in the brain under normal operating conditions; leads to novel interpretations of an array of existing experimental results; and provides a tool for understanding which changes associated with neurological disorders, aging, or stress are most likely to produce memory deficits-knowledge that could eventually help in the design of improved clinical treatments for memory loss.

The Information Bottleneck and Geometric Clustering.

The information bottleneck (IB) approach to clustering takes a joint distribution P(X,Y) and maps the data X to cluster labels T , which retain maximal information about Y (Tishby, Pereira, & Bialek, 1999 ). This objective results in an algorithm that clusters data points based on the similarity of their conditional distributions P(Y∣X) . This is in contrast to classic geometric clustering algorithms such as k -means and gaussian mixture models (GMMs), which take a set of observed data points {xi}i=1:N and cluster them based on their geometric (typically Euclidean) distance from one another. Here, we show how to use the deterministic information bottleneck (DIB) (Strouse & Schwab, 2017 ), a variant of IB, to perform geometric clustering by choosing cluster labels that preserve information about data point location on a smoothed data set. We also introduce a novel intuitive method to choose the number of clusters via kinks in the information curve. We apply this approach to a variety of simple clustering problems, showing that DIB with our model selection procedure recovers the generative cluster labels. We also show that, in particular limits of our model parameters, clustering with DIB and IB is equivalent to k -means and EM fitting of a GMM with hard and soft assignments, respectively. Thus, clustering with (D)IB generalizes and provides an information-theoretic perspective on these classic algorithms.

The Deterministic Information Bottleneck.

Lossy compression and clustering fundamentally involve a decision about which features are relevant and which are not. The information bottleneck method (IB) by Tishby, Pereira, and Bialek ( 1999 ) formalized this notion as an information-theoretic optimization problem and proposed an optimal trade-off between throwing away as many bits as possible and selectively keeping those that are most important. In the IB, compression is measured by mutual information. Here, we introduce an alternative formulation that replaces mutual information with entropy, which we call the deterministic information bottleneck (DIB) and argue better captures this notion of compression. As suggested by its name, the solution to the DIB problem turns out to be a deterministic encoder, or hard clustering, as opposed to the stochastic encoder, or soft clustering, that is optimal under the IB. We compare the IB and DIB on synthetic data, showing that the IB and DIB perform similarly in terms of the IB cost function, but that the DIB significantly outperforms the IB in terms of the DIB cost function. We also empirically find that the DIB offers a considerable gain in computational efficiency over the IB, over a range of convergence parameters. Our derivation of the DIB also suggests a method for continuously interpolating between the soft clustering of the IB and the hard clustering of the DIB.