(a,b,c) Columns from left to right: Spatial tuning and head direction tuning (polar plot) of excitatory and inhibitory input neurons (one example each); spatial firing rate map of the output neuron before learning and after spatial exploration of 10 hr with corresponding autocorrelogram; head direction tuning of the output neuron after learning. The numbers in the polar plots indicate the peak firing rate at the preferred head direction after averaging over space. (a) Wider spatial tuning of inhibitory input neurons than of excitatory input neurons combined with narrower head direction tuning of inhibitory input neurons leads to a grid cell-like firing pattern in space with invariance to head direction, that is, the output neuron fires like a pure grid cell. (b) The same spatial input characteristics combined with head direction-invariant inhibitory input neurons leads to grid cell-like activity in space and a preferred head direction, that is, the output neuron fires like a conjunctive cell. (c) If the spatial tuning of inhibitory input neurons is less smooth than that of excitatory neurons and the concurrent head direction tuning is wider for inhibitory than for excitatory neurons, the output neuron is not tuned to space but to a single head direction, that is, the output neuron fires like a pure head direction cell. (d) Head direction tuning and grid score of 10 simulations of the three cell types. Each symbol represents one realization with random input tuning. The markers correspond to the tuning properties of the input neurons as depicted in (a,b,c): grid cell (triangles), conjunctive cell (squares), head direction cell (circles). The values that correspond to the output cells in (a,b,c) are shown as filled symbols. (e) In our model, the head direction tuning of individual grid fields is sharper than the overall head direction tuning of the conjunctive cell. Depicted is a rate map of a conjunctive cell (left) and the corresponding head direction tuning (right, dashed). For three individual grid fields, indicated with colored squares, the head direction tuning is shown in the same polar plot. The overall tuning of the grid cell (dashed) is a superposition of the tuning of all grid fields. Numbers indicate the peak firing rate (in Hz) averaged individually within each of the four rectangles in the rate map.