A model of path integration and representation of spatial context in the retrosplenial cortex

Abstract

Inspired by recent biological experiments, we simulate animals moving in different environments (open space, spiral mazes and on a treadmill) to test the performances of a simple model of the retrosplenial cortex (RSC) acting as a path integration (PI) and as a categorization mechanism. The connection between the hippocampus, RSC and the entorhinal cortex is revealed through a novel perspective. We suppose that the path integration is performed by the information coming from RSC. Grid cells in the entorhinal cortex then can be built as the result of a modulo projection of RSC activity. In our model, PI is performed by a 1D field of neurons acting as a simple low-pass filter of head direction (HD) cells modulated by the linear velocity of the animal. Our paper focuses on the constraints on the HD cells shape for a good approximation of PI. Recording of neurons on our 1D PI field shows these neurons would not be intuitively interpreted as performing PI. Using inputs coming from a narrow neighbouring projection of our PI field creates place cell-like activities in the RSC when the mouse runs on the treadmill. This can be the result of local self-organizing maps representing blobs of neurons in the RSC (e.g. cortical columns). Other simulations show that accessing the whole PI field would induce place cells whatever the environment is. Since this property is not observed, we conclude that the categorization neurons in the RSC should have access to only a small fraction of the PI field.

Appendix

1.1 Classical conditioning mechanism used as a STM

In the classical conditioning mechanism, $dW_{ij} = W_{ij}(t+dt)-W_{ij}(t)$, we substitute $dW_{ij}$ in Eq. (1). Thereby, the conditional output (vector field) $O_i$ can be represented as:

$$\begin{aligned} O_i(t+dt) = C_j \cdot W_{ij}(t)+\lambda \cdot C_j^2 \cdot (U_i-O_i) \end{aligned}$$

(4)

$O_i(t+dt)$ is inhibited to 0 when $V_i(t-dt)=0$, otherwise,

$$\begin{aligned} O_i(t+dt) = W_{ij}(t)+\lambda \cdot (U_i-O_i) \end{aligned}$$

(5)

when $C_j=1$, $O_i = C_j \cdot W_{ij} = W_{ij}$. Finally, the output can be rewritten as:

$$\begin{aligned} O_i(t+dt) = (1-\lambda ) \cdot O_i(t)+\lambda \cdot U_i \end{aligned}$$

(6)

where $\lambda$ is the learning rate altered between no stimulated rate 0.001 and strong stimulated rate 1 (when the animal meets the reward or the reset mechanism is triggered. When $\lambda$ is small enough as the one we use for the simulation ($\lambda$ = 0.001), the animal reserves the most of the state of the past PI and updates PI field step by step. While the reset is activated ($\lambda$ = 1), the animal erases all the past PI and update to the current state immediately.

1.2 Kohonen map algorithm using scalar product

The activity of neurons on the Kohonen map $S_k(t)$ is discretized from PI field. To cluster the activity on the map, we use the dot product to determine which pattern of weights $W_{ik}(t)$ is the most similar to the vector of input activities $P_i(t)$ (Fig.2). The number of the winner neuron $k^w$ is defined by:

$$\begin{aligned} k^w(t) = \arg \max (\frac{P_i(t) \cdot W_{ik}(t)}{\root \of {P_i(t)^2 \cdot W_{ik}(t)^2}}), k\in \left\{ 1,\ldots ,M \right\} \end{aligned}$$

(7)

M is the number of neurons on a 1D Kohonen map. The weight between the neurons on PI field and on the Kohonen map is updated by:

$$\begin{aligned} W_{ik}(t+dt) = \epsilon \cdot (P_i(t)-W_{ik}(t)) \cdot S_k(t) \end{aligned}$$

(8)

$\epsilon$ is the learning rate of self-organizing. With a neighbourhood function $h_{kk^w}(t)$, we have the activity on the Kohonen map:

$$\begin{aligned} S_k(t+1) = h_{kk^w}(t,d) \end{aligned}$$

(9)

d is the coordinate distance between the winner neuron $k^w$ and other neurons on a 1D Kohonen map. Figure 16 shows the shape of the neighbourhood function.

Fig. 16

Shape of the neighbourhood function