A neural model of self-organizing grid cells
6.2 Single neuron model of a grid cell
6.2.3 Interpretation of the model and results
The coordinatexi directly corresponds to a location in two dimensional space due to considerations of simplicity of the simulations. However it is postulated that, on the one hand, xi could correspond to co-activity of spatially modulated neurons in a real neural network which are not necessarily place cells. Several boundary vector cells are likely candidates to provide input which yields unique spatial identification [13]. On the other hand, xi could correspond to a normalized physical location of a dendritic branch.
The weight update given in Equation (6.13) forms a Reaction-Diffusion System (RDS), known to be capable of producing Turing patterns [71]. Similar functions were used in CAN models of grid cells to form hexagonal firing fields [39, 70]. The major difference of these models to the model presented here is that the equation is not directly applied to the neural recurrent dynamics, i.e. the activity of neurons in a network, but to the excitability of a neuron based on its dendritic weights.
Furthermore, these models typically include constants to tune the network dynamics and are susceptible to erroneous settings, whereas constants required in the equations above are a direct consequence of the error function given in Equation (6.5).
The results show a numerical problem of the simulation, observable in Figure 6.5.
As soon as the receptive field size becomes too small, i.e.σ1 ≤0.07, the receptive field deteriorates and is not circular anymore due to the discretized square input bins. This can also be seen in the first tile of Figure 6.7. Thus, it is impossible for the dynamics to form a hexagonally dense arrangement of circular fields. A related issue is visible in the data shown for 0.15≤σ1 ≤0.18. Here, the weight distribution forms one or two main blobs which are not detectable as a grid-like arrangement.
The remarkable results forσ1 ≥0.19 are only because a singular response field is
5 10 15 20 Receptive field sizeσ0 (×10−2)
−1.5
−1.0
−0.5 0.0 0.5 1.0 1.5
Gridnessscore
Gridness score over receptive field size (λ= 0.50)
N = 48 N = 64
5 10 15 20
Receptive field sizeσ0(×10−2)
−1.5
−1.0
−0.5 0.0 0.5 1.0 1.5
Gridnessscore
Gridness score over receptive field size (λ= 0.65)
N = 48 N = 64
Figure 6.5 – Single grid cell model, gridness scores over receptive field sizes. Gridness scores were computed directly on the emerging weight maps after5000iterations. The relative error importance was set toλ= 0.50in all simulations presented in the top row. The bottom row contains results forλ= 0.65. Weights were initialized with a chance of10% totanh (1)and 0otherwise. Each depicted configuration was simulated for 40times.
6.2 Single neuron model of a grid cell 63
0.0 0.25 0.50 0.75 1.0
Relative error importanceλ -1.5
-1.0 -0.5 0.0 0.5 1.0 1.5
Gridnessscore
Gridness score over error importance (σ1= 0.10)
N = 48 N = 64
0.0 0.25 0.50 0.75 1.0
Relative error importanceλ -1.5
-1.0 -0.5 0.0 0.5 1.0 1.5
Gridnessscore
Gridness score over error importance (σ1= 0.13)
N = 48 N = 64
Figure 6.6–Single grid cell model, gridness scores over error importances. Gridness scores were computed directly on the emerging weight maps after5000iterations. The receptive field size was set to σ1 = 0.10 in all simulations of the top row, and toσ1 = 0.13in the bottom row. Weights were initialized with a chance of10% totanh (1)and0 otherwise. Each depicted configuration was simulated for 40times.
Figure 6.7–Single grid cell model, state of convergence after5000iterations for varying σ1. The simulated cell hadN = 48dendrites andλ= 0.50. Large values ofσ1 show problems of the simplified rectangular receptive field. Results forN = 64are qualitatively identical.
formed, which leads to perfectly hexagonal arrangements in the auto-correlogram used to compute the gridness score. Furthermore, the small number of dendritic weights, which was chosen due to reduce simulation times to an acceptable duration, introduce small but mostly negligible problems. Nevertheless, the results depicted in Figure 6.6 and Figure 6.5 demonstrate that the model is able to form stable hexagonal fields in most of the cases despite these numerical inaccuracies.
The square pre-synaptic input space is biologically unlikely. However, spatially modulated neurons with response fields localized only in single or few locations have been observed, e.g. in form of place cells [265]. Furthermore, the boundary vector space in combination with head direction information allows to represent arbitrary locations unambiguously in a square environment [13]. Thus, spatially modulated neurons which are confined to the square experimental environment are not unlikely, thereby providing a limited input-space which is also quadratic.
It is proposed that one of the primary elements of the input space to grid cells is boundary vector information. Such a space, anchored egocentrically, provides information about distances to geometrical boundaries and appears to be an ideal candidate to generate hexagonal fields. Boundary information was used in a model for place cell formation [13], whereby it was shown that boundary vector cells provide sufficient information to identify locations. The latter is crucial in the sampling process for the dendritic tree, presented here. Furthermore it was reported that grid fields arrange due to the geometry of an environment [197, 199], clearly in favor of the proposition.
In the model, a single grid cell associates with multiple input state representations.
Thus, it is proposed that local dendritic computation in grid cells is more involved than what is suggested in most other models for grid cells, with the notable exception of the model by Kerdels et al. [178]. It is likely that novel techniques, such as protein calcium imaging [57], will be able to determine if dendritic computations are indeed performed by grid cells, or if individual dendritic spines of grid cells perform distributed computations. Specificity towards individual inputs and distributed computation were already reported for other cells on the level of dendritic branches,