Figure 1.
Weights of two example units on a neural manifold of position and velocity (parameters: ).
The weight profile has periods along the spatial dimension. (A) The outgoing weight (left panel) and the incoming weight (right panel) of the unit at
(marked by white dots). The outgoing weight profile of the unit is not centered at its own location in the position dimension, but rotated 0.15 radians to the right (white triangle). The amount of the shift is determined by the velocity label of the unit, as indicated by the black arrow. In the velocity dimension, the connections show broad modulation (the peak of the weight profile marked by the white triangle). The incoming weights (right panel) to the same unit (white circle) is tilted, since the unit receives strong connections from units in the left/right with a positive/negative shift determined by the projection units, among which the maximal activation comes from the unit 0.15 to the left (marked by white triangle) due to the modulation in the velocity dimension; (B) The outgoing weight (left panel) and the incoming weight (right panel) of the unit with negative velocity label,
(marked by white dots). The outgoing weight profile is centered (white triangle) to the left of the unit in the spatial dimension, due to the negative velocity label of the unit. The incoming weight of the unit is tilted, with the maximal connection coming from the right.
Figure 2.
Depending on the parameters, the network operates in different regimes.
(A) The amplitude instability (A) is separated from the homogeneous regime (H) and localized activity regimes (S and T) by Eq. 7 and 12. Localized regimes are separated from the homogeneous regime by Eq. 8. The regime of traveling bumps (T) is separated from the regime of static bumps (S) by Eq. 35; (B) An example of the network state in localized activity regime (). (C) An example of traveling bumps (
); (D–E) Fixed point solutions of order parameters
and
for various
. The square markers correspond to the order parameters of the examples shown in (A). With larger
, the bumps in the network are less tilted (larger
) and smaller (smaller
).
Figure 3.
The instantaneous velocity of the traveling bumps is well described by the approximation (solid curve).
The bumps are put at 11 different positions on the velocity axis. Each circle shows the instantaneous velocity of the bumps during a 1 ms step in the one-second simulation. Overlapping circles demonstrate stable intrinsic velocity of the bumps. For the parameters used, the bumps cannot be put to positions on the velocity axis since the bumps touch the border
.
Figure 4.
The units develop stable position-by-velocity maps on a two-meter linear track in a simulation of 20 minutes (parameters: ).
(A) Part of the trajectory of the virtual animal. (B) One snapshot of the network activity during the simulation. (C) The velocity of the bumps is linearly related to the velocity of the virtual rat. Every 100 ms, the instantaneous velocities of the bumps and the animal during 1 ms interval is shown by a dot in the plot. The line shows the slope , ref. Eq. 15; (D) The tracking error (the difference between the estimated position and the actual position of the animal) is small compared to the spacing (
cm). (E) Position-by-velocity maps of two conjunctive units (top two rows) and a grid unit (bottom). The coordinate
in the neural space is indicated at the top of each panel. Non-sampled bins are represented by white color.
Figure 5.
The network performs robust path-integration against perturbations in weights (parameters: ).
(A–F) Perturbation by Gaussian random noise with zero mean and standard deviation 2% or 10% relative to the weight range. A,C: Scatter plots of the velocity of the bumps with respect to the velocity of the virtual animal for 2% perturbation (A) or 10% perturbation (C). Every 100 ms in the simulation, the instantaneous velocities of the bumps and the animal during 1 ms interval is marked by a dot. The line indicates the slope derived from Eq. 15; B,D: Spatial fields of two example units in the network with 2% (B) or 10% Gaussian perturbation (D); E: Tracking error, i.e. the difference between the estimated position from the network activity and the actual position of the animal; F: Drift, defined as the absolute value of tracking error, averaged across eight independent simulations. (G–L) Dilution of connectivity by
or
. The weights are rescaled by
after the dilution to keep the strength of the connections comparable to the original connections. G,I: The relation between the velocity of bumps and the velocity of the animal. The same legends are used as in A; H,J: Spatial fields of two example units from the network with 20% (H) or 40% dilution; K: Tracking error; L: Drift.
Figure 6.
The network is able to perform accurate path-integration even when the firing response is nonlinear in the input and the velocity input is of finite resolution (parameters: ).
(A) Snapshot of the network activity at one example step in the simulation. The firing of the units in the network saturates due to nonlinearity of the transfer function; (B) Firing maps of the units, as a function of the actual position and velocity of the simulated rat, show that the top two units are conjunctive grid units while the unit at the bottom is a pure positional grid unit. The coordinate in the neural space is indicated at the top of each panel. The spacing is 30 cm, determined by the parameter
put in the simulation. Non-sampled bins are represented by white color.
Figure 7.
Weight matrix in four dimensional neural space .
Only the slices at of the outgoing weights (A) from and incoming weights (B) to the example unit
are shown. (A) The asymmetry in the outgoing weights is determined by the projecting unit (white dot). The triangle marks the unit that is maximally activated among the units in the slice by the projecting unit. (B) The asymmetry in the incoming weights depends on the velocity labels of presynaptic units. Among the unit in the slice, the unit marked by the triangle has the strongest connection to the example unit (white dot).
Figure 8.
The network activity changes from homogeneous to localized profile in the velocity dimensions with increasing (parameters:
).
(A) The maximal activity of the units with the same labels for different
; (B) The maximal activity of the units with the same
labels for different
. Due to the symmetry in velocity labels, the plots in (A) and (B) are the same.
Figure 9.
A sample trajectory of the simulated animal in a two-dimensional square environment.
The animal is not allowed to move beyond the boundary of the environment. The speed of the animal varies between [0, 100] cm/s.
Figure 10.
A snapshot of the network activity of the units that prefer zero velocity in y direction () when the animal runs with velocity.
cm/s and
cm/s. Each panel shows the activity of the units on the slice with the fixed velocity labels. The velocity labels of the slice are shown at the top of each panel.
Figure 11.
Mean activity of three example units in the network during 20-minute exploration depicted as a function of position (left), head direction (middle) and velocity (right) of the simulated animal (parameters: ).
(A,B) Conjunctive units; (C) Grid unit.
Figure 12.
Elliptical grids form if the mapping is different for each velocity component (parameters:
).
The scaling factor is 30 cm for the mapping in x direction, and is 24 cm for y direction, reduced by 20%. A–C: three different units.
Figure 13.
Robustness of path-integration in two dimensional environments when the weights are perturbed by Gaussian random numbers (A) or are deleted randomly (B).
Parameters: cm. (A) One simulation with the weights perturbed by 10% Gaussian random numbers. Left: drift. middle: fields of a unit in the network after three minutes of exploration. Right: fields of an example unit in the network after six minutes of exploration; (B) One simulation with the weights diluted by 20%. Left: drift. middle: fields of a unit in the network after three minutes of exploration. Right: fields of an example unit in the network after six minutes of exploration; (C) Averaged drift across 8 independent simulations; the network is able to path integrate for 2 minutes (the mean drift within 15 cm, i.e. half of the grid spacing, light gray line) with 10% Gaussian perturbation in the weights, relative to the range of the weights. The black and dark gray lines show the drifts with no and 2% perturbation respectively. Error bars show
standard deviations; (D) When 20% of the weights are set to zero, the network is able to path-integration for 2 minutes on average (the mean drift across 8 independent simulations kept within half of the grid spacing, dark gray line). The black and light gray lines show the drifts with zero and 40% dilution respectively.
Figure 14.
Estimated order parameters of the traveling multiple bumps.
(A) Estimated velocity of the bumps (filled circles) for different matches the theoretical values (solid line, Eq. 55). The linear approximation of the velocity of the bumps is plotted as the dashed line; (B) When the absolute value of
goes to the limit, the network has homogeneous activity, with finite mean activity
(square markers) and vanishing amplitude of the bumps
(triangular markers).