Saccade Velocity Driven Oscillatory Network Model of Grid Cells

Abstract

Grid cells and place cells are believed to be cellular substrates for the spatial navigation functions of hippocampus as experimental animals physically navigated in 2D and 3D spaces. However, a recent saccade study on head fixated monkey has also reported grid-like representations on saccadic trajectory while the animal scanned the images on a computer screen. We present two computational models that explain the formation of grid patterns on saccadic trajectory formed on the novel Images. The first model named Saccade Velocity Driven Oscillatory Network -Direct PCA (SVDON-DPCA) explains how grid patterns can be generated on saccadic space using Principal Component Analysis (PCA) like learning rule. The model adopts a hierarchical architecture. We extend this to a network model viz. Saccade Velocity Driven Oscillatory Network-Network PCA (SVDON-NPCA) where the direct PCA stage is replaced by a neural network that can implement PCA using a neurally plausible algorithm. This gives the leverage to study the formation of grid cells at a network level. Saccade trajectory for both models is generated based on an attention model which attends to the salient location by computing the saliency maps of the images. Both models capture the spatial characteristics of grid cells such as grid scale variation on the dorso-ventral axis of Medial Entorhinal cortex. Adding one more layer of LAHN over the SVDON-NPCA model predicts the Place cells in saccadic space, which are yet to be discovered experimentally. To the best of our knowledge, this is the first attempt to model grid cells and place cells from saccade trajectory.

Keywords: grid cells; hippocampus; oscillator; principal component analysis-PCA; saccades; salience map.

Figure 1

Model Architecture of SVDON: The model consists of a saccade generation stage, Saccade Direction Encoding stage, Path Integration stage and the output SC layer. (No copyright permission is required).

Figure 2

Two sample images used (A1,A2) and overlapped trajectory (Yellow) generated on them (B1–B2) by bottom-up model of attention. Sample images are taken from: Caltech-256 Object Category Dataset. (No copyright permission is required).

Figure 3

Spatial representations from SC layer: (A–F) Firing fields (Left): Blue is the trajectory of the Saccade and red dots are the spike locations; Firing Rate map (Middle): red is peak rate and blue is no firing; Autocorrelation map (Right) of SC layer neuron in SVDON-PCA model.

Figure 4

Spatial representations from sixth SC layer neuron: Firing field (A): Blue is the trajectory of the Saccade and red dots are the firing locations; Firing map (B): red is peak rate and blue is no firing; Autocorrelation map (C) of the sixth SC layer neuron in SVDON- PCA model.

Figure 5

Oscillation power modulation (A): loss of grid field formation on reducing oscillation power (A1); grid field reemerged as the oscillation power was restored (A2). β modulation (B): loss of grid field when β = 0 (B1), grid field reemerged when β = 0.2.

Figure 6

Spatial representations of three different neurons (A–C) from LAHN (SC) layer. Left Column is firing field of the neurons (A1–C1); Middle Column is the firing rate of the neurons (A2–C2); and the Right column is autocorrelation map of the neurons (A3–C3).

Figure 7

Effect of modulation factor β. (A) Grid Scale gradient is captured by varying spatial scale parameter β. (B) Comparison between the model and the experiment. (B1) Grid scale variation in the model by varying β (Grid scale is averaged for 10 trajectories). (B2) Empirically observed grid scale variation at different locations of medial to dorsal rhinal sulcus axis (Killian et al., 2012).

Figure 8

LAHN (PC) layer activity of six different neurons. (A1–F1) firing field (blue is the trajectory of the Saccade and red dots are the spike locations) and (A2–F2) firing maps (red is peak rate and blue is no firing) of 6 different neurons in LAHN(PC). Characterization of LAHN (PC) layer showing the % of cells type vs. the number of neurons in layer. Here PC is Place cells, GC is Grid Cells and NC is non-spatial cells.

Figure 9

Spatial cell response to a single image. (A1,B1) Images given as the input. (A2,B2) are the outputs of LAHN(SC) and (A3,B3) are the outputs of LAHN(PC). Red dots are the firing locations on the image (Images source is ImageNet: A Large-Scale Hierarchical Image Database).