Methods and simulation results

Virtual input trajectories with movement statistics close to real data of rodents were used. A novel trajectory was generated for each simulation. An example of a trajectory as well as statistics on angular velocity and running speed are depicted in Figure 6.8. The location of the animated animal was presented to the network every 10 ms. The duration of one training session was 3 h to observe long term effects.

Furthermore, N_g = 3 and N_d = 48. The weights of each neuron were initialized such that they had a 10% chance to be set to 1, or were set to 0 otherwise. The receptive field size was set with parametersσ1 = 0.10 and σ2 = 2σ1. The network was simulated for a total of 400 times. Note thatλwas dropped from the equations due to the results of the previous section. The receptive field sizes were determined by σ1 = 0.10 andσ2= 2σ1.

As discussed in Subsection 6.2.2, the gridness scores were computed directly on the dendritic weights and not on intermediate spike response plots. Certainly, the ongoing plasticity of the weight distribution may introduce changes in the location of formed grid fields. However, it was observed that once the fields formed, they remained stable throughout the rest of the simulations except for subtle re-arrangement of the fields. The increase of the gridness score over time time as depicted in Figure 6.9 and the stability of the fields can be observed in the examples presented in Figure 6.11.

This means that although the cells were subject to persistent plasticity, the dendritic weights stayed at their peak locations once the network formed pronounced grid fields. Furthermore and although the cells were subject to continued competition within the network, the fields only moved towards an improved packing of fields.

This is also expressed in the steady increase of the gridness score over time.

In addition to the gridness score, relative orientation errors between the cells of each simulation were computed. Thereby it is possible to assess if the grid cells are aligned or form random alignments to each other. Likewise the characterization by Hafting et al. [135], the alignment of the dendritic weights of each grid cell was assumed to be in the range from 0 to 60 degrees. Subsequently, the relative orientation between cells was computed which gives results in the range from 0 to 30 degrees.

Finally, the average error of each simulation time-step was computed. The calculation

6.3 Competitive network model of grid cells 69

0 20 40 60 80 100 120 140 160 180

Time [min]

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

Gridnessscore

Gridness scores over time (400 simulations)

mean median

standard deviation

Figure 6.9– Competitive grid cell network, gridness score over time (400 simulations).

The data was computed using400 simulations of a network ofNg = 3cells with competitive dynamics,Nd= 48dendritic weights each, andσ1= 0.10. The median and mean of the gridness scores stay above zero after about2.5 or4minutes, respectively. The standard deviation of the gridness scores is comparably large, but may be due to numerical issues introduced by setting values for smallNg andσ1.

0 20 40 60 80 100 120 140 160 180

Time [min]

0 5 10 15 20

Orientationerror[degrees]

Relative grid orientation error (400 simulations)

mean median

standard deviation

Figure 6.10 – Competitive grid cell network, relative orientation error over time (400 simulations). The first four minutes of data are cut off because an orientation could not be computed with certainty. Over time, the median and mean of the orientation error approach zero but stay at an approximately 2.5degrees offset. So far it is unclear if the offset is introduced by the network dynamics or numerical resolution of the auto-correlogram which was used to compute the orientation.

6.3 Competitive network model of grid cells 71

of the orientation error is described in detail in Appendix B. Orientation errors over time for all 400 simulations are depicted in Figure 6.10.

Wall-offset orientations were automatically computed for all neurons using their primary orientation extracted from their auto-correlograms. So far, a clear preference for a specific value was not observable in the data. Some cells showed an alignment of their weight fields in perfect alignment of the walls. Yet, most of the cells settled for an orientation offset in the range from 5 to 12 degrees. However, results from visual inspection and manual analysis for many simulations indicate that the weight fields are subject to skewing and shearing effects near walls. Consequently, the orientation of the fields along walls is slightly different to the orientation of fields in more central areas of the arena. Examples for the skewing and wall-offset are observable in Figure 6.11. In all simulations the response fields at the end of the simulations (time-step t= 180 min) appear to be slightly curved. The effect is especially prominent in simulation 311 (top row), neuron 0 and in simulation 60 (middle row), neuron 2. However, further studies are required to characterize the effects and investigate if the model indeed generates wall-offsets comparable to the findings presented by Stensola et al. [337].

The average gridness score was above zero at the end of the simulations in all simulations, and only in 11% of the simulations one single cells had gridness score below zero. In addition, the mean gridness score reached a gridness of 0.0 after only about 4.0 min of simulated time, and the median was permanently above zero already after approximately 2.5 min. The median and mean gridness scores as well as the standard deviation for all 400 simulations are depicted in Figure 6.9. The standard deviation appears to be quite large but may be due to numerical issues discussed previously and introduced by the small number of dendritic weights as well as the receptive field size σ₁.

The evolution of dendritic weights of exemplary simulations with high, average, and low gridness scores are depicted in Figure 6.11.

6.3.3 Discussion of the model and its results, predictions, and future