A topological deep learning framework for neural spike decoding

Abstract

The brain's spatial orientation system uses different neuron ensembles to aid in environment-based navigation. Two of the ways brains encode spatial information are through head direction cells and grid cells. Brains use head direction cells to determine orientation, whereas grid cells consist of layers of decked neurons that overlay to provide environment-based navigation. These neurons fire in ensembles where several neurons fire at once to activate a single head direction or grid. We want to capture this firing structure and use it to decode head direction and animal location from head direction and grid cell activity. Understanding, representing, and decoding these neural structures require models that encompass higher-order connectivity, more than the one-dimensional connectivity that traditional graph-based models provide. To that end, in this work, we develop a topological deep learning framework for neural spike train decoding. Our framework combines unsupervised simplicial complex discovery with the power of deep learning via a new architecture we develop herein called a simplicial convolutional recurrent neural network. Simplicial complexes, topological spaces that use not only vertices and edges but also higher-dimensional objects, naturally generalize graphs and capture more than just pairwise relationships. Additionally, this approach does not require prior knowledge of the neural activity beyond spike counts, which removes the need for similarity measurements. The effectiveness and versatility of the simplicial convolutional neural network is demonstrated on head direction and trajectory prediction via head direction and grid cell datasets.

Figure 1

The framework for the SCRNN. First, captured neural spiking data are recorded in a spike matrix. Next, the data are converted into a simplicial complex for a series of time windows. Then, each simplicial complex is fed into the simplicial convolutional layers. After flattening the simplicial information into a single vector, the vector is fed through an RNN, which predicts the location (or head direction) of the mouse based on the neural firing data. To see this figure in color, go online.

Figure 2

An example of the preprocessing procedure. First, neural spiking data are represented as a raster plot. Next, the data are binned and converted to a spike count matrix. A row-wise thresholding procedure, given in Eq. 1, binarizes the matrix. In this figure, each colored box denotes a 1, and each white box denotes a 0. Then, each neuron is represented by a node of the simplicial complex, color coded to match the corresponding matrix row. To construct the simplicial complex, the colored nodes are connected by the appropriate dimensional simplex to capture the neurons that fire together. For example, we see that the second column of the binarized matrix has three active neurons (green, orange, and blue). This generates a 2 simplex on the corresponding nodes. To see this figure in color, go online.

Figure 3

A diagram of $L=2$ simplicial convolutional layers each equipped with two filters, $H_k^1(\_)$ and $H_k^2(\_)$, for each simplicial dimension $k=0,1,2$ and $F=2$ filters for each dimension of the input simplicial complex. In the first layer, we see three orange and three blue filters indicating the three dimensions of the input simplicial complex. The features extracted using these filters result in new orange and blue simplicial complexes, respectively. In the second simplicial convolutional layer, the process is repeated with two new filters, depicted by yellow and dark blue, giving us 4 simplicial complexes. In order to prevent exponential growth, features extracted from the same input from the previous layer are summed, resulting in a new orange and blue simplicial complex. Finally, all extracted features are summed and flattened to create one feature vector for the RNN. To see this figure in color, go online.

Figure 4

Plots depicting the true and the predicted head angles for the second minute of the testing portion of four different networks using 100 ms time bins, (a) FFNN, (b) RNN, (c) GNN, and (d) SCRNN, and the corresponding test AAEs for the first 5 min. Each network was generated with the RayTune-optimized hyperparameters listed in Table 1. To see this figure in color, go online.

Figure 5

Grid cell decoding task. (a–d) Plots showing results from 2 min of the grid cell decoding task. (a and b) Comparison of decoded versus ground-truth x coordinate and y coordinate. (c) Error for each time bin measured by Eq. 13. (d) In gray, the ground-truth position of the rat’s trajectory in the environment. For visual purposes, we include colorized paths showing a 5 s comparison of decoded versus ground-truth position. To see this figure in color, go online.