Figure 1.
Distance and allocentric direction of walls are encoded in the firing of boundary vector cells.
Here we use a model to determine if these cells could be influenced by optic flow. a) Shows a rat in a box estimating the distance and allocentric direction of a wall from the sensed patterns of light on its retina. b) If the rat moves its eye, e.g. by a forward body motion, these sensed patterns of light shift on the surface of the eyeball. This shift can be described as an angular displacement. c) Schematic drawing of the proposed template model. This model sets up flow templates for parameters of self-motion in combination with parameters of planes either describing ground or wall. In a cascade of estimation steps (max-operations) the self-motion and parameters that describe ground and walls are estimated by looking for the best match between flow templates and sensed input flow. d) We use a box with an arbitrary outline (gray shading) to display the variables used in the boundary vector cell model. These variables are the cell's preferred allocentric direction Φi and preferred distance Di and the estimated allocentric direction α and estimated distance d. All distances are measured with respect to the wall's surface normal.
Table 1.
Matching statistics of recorded and simulated rat trajectories.
Figure 2.
Shows the pseudo-code for the generation of simulated rat trajectories.
Figure 3.
Replication of the velocity statistics of recorded rat trajectories using simulated rat trajectories.
a) The linear velocities of the rat's body motion are fitted by a Rayleigh distribution. b) The yaw-rotational velocities are fitted by a normal distribution. c) Shows the first minute of the simulated rat trajectory and d) the first five minutes of the same trajectory in a circular box (79 cm diameter). The panels e) and f) show the fits for linear and rotational velocity for a simulation in a squared box (62 cm×62 cm). The first minute and the first five minutes of the simulated trajectory are shown in g) and h), respectively.
Figure 4.
Examples of wall-ground segmentation and drop-off detection by our model.
a) Depicts a circular box of diameter 79 cm and a 50 cm high wall with the camera at ,
,
, with orientation
, and self-motion
and
. b) Shows the same circular box with walls removed to simulate a platform. c) Wall-ground segmentation estimated by our model based on the analytical flow shown as black arrows in a). d) Drop-off detection based on the flow discontinuities. Note that distant boundary locations are not detected, but these usually do not play a role for behavior. e) Shows a square box 62 cm×62 cm with a 50 cm high wall. f) Shows the same box as in e) with walls removed. g) Estimated wall-ground segmentation for the square box. h) Detected drop-off at close distance. In all examples the camera had the same position, orientation, and self-motion as mentioned in a). All boxes are described by a triangular mesh.
Figure 5.
Examples and error statistics for the allocentric direction and distance estimation of walls for different boxes.
a) Top-view of a circular box with diameter 79 cm and 50 cm high walls. The rat's position is ,
,
and the camera coordinate system has the orientation
in the xz-plane depicted by the red and green arrows. b) Top-view of the same configuration as in a) for a square box with 62 cm×62 cm with 50 cm high walls. c) An additional wall has been added inside the square box of b) and the rat's position changed to be
,
,
and orientation
. In all cases the camera moved by the linear velocity
and the rotational yaw-velocity
. d) Match values for distance and allocentric direction of walls in the circular environment (shown in a) provided by our template model. Low match values are encoded by black and high match values by white. e) Shows the match values for the square box (shown in b) with same encoding as used in d). Multiple separate regions of high intensity with their peak encode multiple walls as shown in this example. f) Match values for the box with interior wall (shown in c). In the last row the mean distance errors over all estimates from 20 min long simulated rat trajectories are shown depending on the position of the rat. g) Distance errors for the circular and square box h) both range within two centimeters. i) For the box with interior wall the mean distance error ranges within six centimeters.
Figure 6.
Boundary vector cell (BVC) responses (rate maps) for a square (top) and circular (bottom) box for the model and from data.
a) Shows the square box and the occupancy that is high at edges for the simulated rat trajectory. b) Rate maps for the BVC model using ground-truth distance and direction of walls. For the model we used eight allocentric directions ranging from east, east-north, north, … to south-east combined with the three distance tunings 2 cm, 10 cm, and 25 cm. High firing is encoded as red color and low as blue color. This color encoding is the same for all plots showing firing rate maps. White numbers are the individual scaling parameters for each plot similar to the firing rate scaling used for plotting the experimental data. c) Example of recorded BVCs. These firing maps have been redrawn from Lever et al, J. of Neurosci. 29, 2009 from their Figure 3 on page 9774 [3]. The numbers in black denote the firing rate of the cells. d) The BVC model receives estimates about allocentric direction and distance from our template model. e) Shows the circular box and occupancy of the simulated trajectory. f) Shows the rate maps of the BVC model that uses distance and direction estimates of our template model. g) Data from recorded BVCs [3].
Figure 7.
Boundary vector cell (BVC) responses in a box with an additional interior wall and for a platform.
a) Square box with an additional interior wall and the occupancy of the simulated rat trajectory. b) Simulation of the BVC model using estimates from our template model. The tuning of model cells is the same as in Figure 6. c) Data of recorded BVC. Note that model and recorded BVCs respond to any wall of a certain distance and allocentric direction and not only, e.g. to the exterior walls of a box. d) Circular platform together with the occupancy of the simulated rat trajectory. For reasons of comparison we use the same trajectory as in the simulation with a circular box. e) Shows the rate map of model BVC supplied with estimates about distance and direction of walls. f) Data from recorded cells. Firing maps of BVC have been redrawn from Lever et al, J. of Neurosci. 29, 2009 from their Figure 3 on page 9774 [3].
Table 2.
Parameters of the models and their values used in the simulations.
Figure 8.
Drawing of the spherical camera model and an analytical flow vector that arises if the entire model is moving by the linear velocity
and rotating around the y-axis by
.
For this paper we use the left-handed-coordinate system with the x-axis pointing to the right, the y-axis
pointing upward, and the z-axis
pointing forward. The location
is described by the angles
and
together with its distance
from the origin. Our spherical model describes the flow vector
by its angular, temporal differentials
and
- not depicted in the drawing for clarity.
Figure 9.
Shows the pseudo-code for wall-ground segmentation, and estimation of self-motion, distance, and direction of walls.
The constructor “MatrixValueLinear(min,max,num)” provides a linear equidistant sampling between “min” and “max” of “num” samples. In contrast, the constructor “MatrixValuesLog(min,max,num)” implements a logarithmic sampling between “min” and “max” with “num” samples. The function “matchGround” implements Equation (6) and “matchWall” implements Equation (7). Both functions are used to compute a wall-ground segmentation. The linear velocity is estimated using Equation (8). Further, “matchRotation” implements Equation (9) and “matchSpeedDirection” implements Equation (10). The readout functions “readout1D” and “readout2D” are defined in Code-box 3.
Figure 10.
Shows pseudo-code for the interpretation of match values.
Figure 11.
Shows pseudo-code for the method that detects drop-offs.