Cortical inhibitory tuning reflects the Fourier components of locally encoded features

Inhibition plays an important role in shaping neural representations. Still, it is unclear whether inhibitory neurons merely refine the tuning of excitatory (EX) neurons or encode a parallel neural representation. We aimed to uncover the principles of inhibitory tuning in the cortex, using a simple system encoding a 1-dimensional sensory signal - the head-direction (HD) system. To this end, we recorded populations of neurons in the postsubiculum (PoSub), the primary cortical hub of the HD system. While excitatory HD cells had narrow receptive fields, fast-spiking (FS) interneurons had broad and multi-modal tuning curves. FS cell receptive fields rotated in concert with HD cells during environmental manipulation and their temporal coordination with HD cells during sleep was similar to wakefulness, indicating that they are functionally integrated into the HD circuit. On average, their tuning showed the same Fourier spectrum as HD cells, with the power centered in the first three components. However, while the average FS cell spectrum reflected the HD cell spectrum, individual spectra were heterogeneous. To determine the origin of FS tuning, we performed selective disinhibition of thalamic input to PoSub. We observed exclusively multiplicative gain in HD cell tuning and exclusively additive gain in FS cell tuning, indicating that FS cells receive homogenous HD input from the thalamus. These findings suggest that tuning of FS cells is a Fourier transformation of the signal encoded by EX cells and provides new constraints on biologically plausible neural network models involving inhibition.