The Dark Matter of the Entorhinal Cortex

The canonical neurons in the medial entorhinal cortex respond selectively to an animal's position in space. Some have spatial receptive fields that resemble stereotyped grids over the environment, others fire near borders of the environment. These cells can be identified using the Tuning Curve Score (TCS) method. They are identified by finding cells whose tuning curves align with pre-defined shapes in a statistically significant way. However, this approach misses a lot of cells which code for multiple variables. The paper "A Multiplexed, Heterogeneous, and Adaptive Code for Navigation in Medial Entorhinal Cortex" by Hardcastle, Maheswaranathan, Ganguli and Giacomo from Stanford in Neuron, 2017 uses a linear non-linear model (LN) to study the cells that have mixed selectivity with respect to position-encoding (P), head-direction (H), speed cells (S) and theta-rhythm (T). They found cells whose firing rate was modulated by some combination of P, H, S and T cells. The LN model didn't require for the tuning curves of neurons to match pre-defined shapes. 

Before explaining the modeling approach in the paper, the task from which the experimental data flowed has to be described. The mice were foraging for randomly scattered food in an open arena. There were 14 mice with 794 neurons in total. The spike trains from these neurons are the neural modality of interest in this paper. 

Let's take a look at the computational methods. The features going into the LN model (P, H, S, T) were binned into binary vectors (400 [20x20 grid], 18, 10, 18 bins) and concatenated. There were multiple models consisting of different features-- models considered included 4 single variable models (P, H, S, T), 6 double variable models (PH, PS, PT, HS, HT, ST), 4 triple variable models (PHS, PHT, PST, HST) and one full model (PHST). The 1-value in a particular vector represents which feature was active for a particular time point. These vectors are then linearly combined with learned parameters (L-stage, multiplication and summation) and passed through an exponential non-linearity (N-stage). This gives a number for a mean firing rate and the spike train is then generated in the forward mode from this rate by a Poisson process. The model can be inverted and the parameters learned from observed spike trains using maximum likelihood. 

To select the simplest model that best described the neural data, the authors first determined which single variable model had the highest performance. They then compared this model to all double variable models that included this single variable. If the highest performing double-variable model, on held out data, was significantly better than the single variable model, they then compared this model to the best triple variable model, and so forth. In all cases, if the more complex model was not significantly more predictive of neural spikes, the simpler model was preferred. Cells for which the selected model did not perform significantly better than a fixed mean firing rate model were marked unclassified. They called this procedure nested LN model selection. There's a histogram in the paper that shows the proportion of the different mixed selectivity cells among all studied cells (Figure 5, A). 

Figure 5, G shows the classified cell parameters projected using PCA. For this, all parameters of a given variable across cells (Figures 5D–5G), they built a nxm matrix X for each set of parameters, where n = number of cells in the analysis and m = number of parameters. So for position parameters, the matrix was 400x421 (since there are 421 cells that encode position). You can then use dimensionality reduction on this matrix (PCA). The figure shows salt and pepper profiles for various combinations of coding cells in the principal component space.

As a conclusion, authors say that the computational methods that they employed could be used to shed light on principles of neural coding that involve mixed selectivity:-)