Grid cell modeling with mapping representation of self-motion for path integration

Abstract

The representation of grid cells in the medial entorhinal cortex region is crucial for path integration. In this paper, we proposed a grid cell modeling mechanism by mapping the agent’s self-motion in Euclidean space to the neuronal activity of grid cells. Our representational model can achieve multi-scale hexagonal patterns of grid cells from recurrent neural network (RNN) and enables path integration for 1D, 2D and 3D spaces. Different from the existing works which need to learn weights of RNN to get the vector representation of grid cells, our method can obtain weights by direct matrix operations. Moreover, compared with the classical models based on continuous attractor network, our model avoids the connection matrix’s symmetry limitation and spatial representation redundancy problems. In this paper, we also discuss the connection pattern between grid cells and place cells to demonstrate grid cells’ functioning as a metric for coding space.

Appendices

Appendices

1.1 Appendix 1: Grid cell modeling in 1D space

The position in 1D space is denoted as r, which is a scalar value. There are N grid cells in the neural network and ${\varvec{s}}(r)= [s_1(r),s_2(r),\ldots ,s_N(r)]^T$(or ${\varvec{s}}=[s_1,s_2,\ldots ,s_N]^T$) represents the grid cell population activity. The formulaic description of grid cell firing patterns we used as below:

$$\begin{aligned} s_i(r)=\frac{1}{2}\cos (k_0(r-\varDelta r))+\frac{1}{2} \end{aligned}$$

where $\varDelta r$ and $k_0$, respectively, determine the grid pattern phase and spatial scale. The weight can be calculated as follows:

$$\begin{aligned} {\varvec{W}}={\varvec{M}}{\varvec{K}}_0{\varvec{M}}^\dagger \end{aligned}$$

where

$$\begin{aligned} {\varvec{K}}_0&=\begin{bmatrix} 0&{} -k_0 \\ k_0&{}0 \end{bmatrix} \\ {\varvec{M}}&=\begin{bmatrix} \mathrm{cos}k_0\varDelta r_1 &{}\mathrm{sin}k_0\varDelta r_1\\ \vdots &{} \vdots \\ \mathrm{cos}k_0\varDelta r_N &{}\mathrm{sin}k_0\varDelta r_N \end{bmatrix} \end{aligned}$$

${\varvec{M}}^\dagger$ denotes the pseudo-inverse matrix of ${\varvec{M}}$ and can be obtained through the singular value decomposition of matrix ${\varvec{M}}$. Finally, the dynamics of grid cells in 1D space:

$$\begin{aligned} \tau {\mathrm {d} {\varvec{s}}}/{\mathrm {d} t}=f(v_t{\varvec{W}}{\varvec{s}}) \end{aligned}$$

(8)

where $f(x) = 0$ if $x \le 0$ otherwise $f(x)=1$ and $v_t$ is the moving velocity in 1D space.

1.2 Appendix 2: Grid cell modeling in 2D space

The position in 2D space is denoted as ${\varvec{r}}$=(x, y). There are N grid cells in the neural network and ${\varvec{s}}({\varvec{r}})= [s_1( {\varvec{r}}),s_2({\varvec{r}}),\ldots ,s_N({\varvec{r}})]^T$(or ${\varvec{s}}=[s_1,s_2,\ldots ,s_N]^T$) represents the grid cell population activity. The formulaic description of grid cell firing patterns we used as below:

$$\begin{aligned} s_i({\varvec{r}})=\frac{1}{3}\sum _{j=1}^{3}\cos ({\varvec{k}}_j({\varvec{r}}-\varDelta {\varvec{r}}))+\frac{1}{3} \end{aligned}$$

where $s_i({\varvec{r}})$ is the ith grid cell’s activity. $\varDelta {\varvec{W}}$=$(\varDelta x,\varDelta y)$ determines the grid pattern phase. ${\varvec{r}}_j$ can be given as the row vector of the matrix ${\varvec{K}}$:

$$\begin{aligned} {\varvec{K}}=\begin{bmatrix} {\varvec{k}}_1 \\ {\varvec{k}}_2 \\ {\varvec{k}}_3 \end{bmatrix}=k_0 \begin{bmatrix} \cos {( \frac{\pi }{6}-\theta )} &{} \sin {( \frac{\pi }{6}-\theta )} \\ \cos {(-\frac{\pi }{6}-\theta )} &{} \sin {(-\frac{\pi }{6}-\theta )} \\ \cos {(-\frac{\pi }{2}-\theta )} &{} \sin {(-\frac{\pi }{2}-\theta )} \end{bmatrix} \end{aligned}$$

where $k_0=2\pi /T$. T and $\theta$, respectively, represent the grid scale and direction, which are same for grid cells in a neural network. The weight can be calculated as follows:

$$\begin{aligned} {\varvec{W}}^m={\varvec{M}}{\varvec{B}}_m{\varvec{M}}^\dagger ~~(m=1,2) \end{aligned}$$

where

$$\begin{aligned} {\varvec{B}}_m&=\begin{bmatrix} \text {0 } &{} {\varvec{B}}_{m1}\\ {\varvec{B}}_{m2} &{} \text { 0} \end{bmatrix} \\ {\varvec{B}}_{m1}&=diag(\begin{bmatrix}-K_{1m},-K_{2m},-K_{3m}\end{bmatrix}) \\ {\varvec{B}}_{m2}&=diag(\begin{bmatrix}K_{1m}, K_{2m},K_{3m}\end{bmatrix})\\ g(\varDelta {\varvec{r}})&=\begin{bmatrix} \cos {\varvec{k}}_1 \varDelta {\varvec{r}}&\cdots&\cos {\varvec{k}}_3 \varDelta {\varvec{r}}&\sin {\varvec{k}}_1 \varDelta {\varvec{r}}&\cdots&\sin {\varvec{k}}_3 \varDelta {\varvec{r}} \end{bmatrix} \\ {\varvec{M}}&=\frac{1}{3}\begin{bmatrix} g(\varDelta {\varvec{r}}_1)\\ \vdots \\ g(\varDelta {\varvec{r}}_N) \end{bmatrix} \end{aligned}$$

It should be noted that ${\varvec{M}}^\dagger$ denotes the pseudo-inverse matrix of ${\varvec{M}}$ and can be obtained through the singular value decomposition of matrix ${\varvec{M}}$. Finally, the dynamics of grid cells in 2D space is:

$$\begin{aligned} \tau {\mathrm {d} {\varvec{s}}}/{\mathrm {d} t}=f\left( \sum _{i=1}^{2} v_m^t{\varvec{W}}^m{\varvec{r}}\right) \end{aligned}$$

(9)

where $f(x) = 0$ if $x \le 0$, otherwise $f(x) = 0$ and ${\varvec{v}}_t=[v_1^t, v_2^t]^T$ is the moving velocity vector in 2D space.

1.3 Appendix 3: Grid coding performance analysis in 2D space

Grid coding performance analysis based on different grid scales in 2D space is illustrated in Fig. 10.

Fig. 10

Grid coding performance analysis based on different grid scales in 2D space. a The phase distribution of sub-RNNs with different scales. b The agent’s exploring trajectory in a $25\,\mathrm{m}^2$ area. c Histogram showing accumulated coding error of each sub-RNN after path integration following the simulation trajectory in b. d The comparison between the coding result derived from our model and the ground truth activity of the selected grid cells in a

Fig. 11

Grid coding performance analysis based on different phase distributions in 2D space. a The phase distributions, respectively, generated by uniform random distribution, square tiling and hexagonal tiling. b The comparison between the ground truth and grid coding derived form our model for three grid cells randomly selected in the cases of different phase distribution. c Histogram showing, with different network sizes, accumulated grid coding error of the whole model in the case of different phase distributions

Grid coding performance analysis based on different phase distributions in 2D space is illustrated in Fig. 11.