Figure 9. Encoding of future head direction in thalamus helps to better encode present head speed in retrosplenial LR cells.

(A) The schematic depicts tuning curves for an HD cell with a preferred direction of 180 degrees. Top: the tuning curves of this cell if it displayed no anticipatory firing (ATI = 0 ms). Note that clockwise and counterclockwise turns produce identical tuning curves in this case. Bottom: the tuning curves of this cell if it had an ATI = 50 ms. Note that now, during head turns in either direction, the cell will fire 50 ms prior to when the animal faces 180 degrees. Our convention takes positive angular head velocity to denote counterclockwise turning. (B) Anticipatory firing of presynaptic HD cells improves speed coding in the postsynaptic LR cell. To quantify strength of speed coding independent of latency between head speed and postsynaptic firing rate, we used the maximum correlation between head speed and postsynaptic firing rate across all time lags. Inset shows a similar relationship, now for the maximum of cross-mutual information. (C) Anticipatory firing of presynaptic HD cells improves the latency between current head speed and postsynaptic firing rate, enabling more temporally precise speed coding.

Figure 9—figure supplement 1. Analytical calculations independently confirm the improvement of head speed coding with anticipatory firing of head direction inputs.

(A) Simulations utilized the continuously varying Gaussian tuning curves, but our mean-field analysis employed the more analytically tractable, but discontinuous, step-function tuning curves. (B) The mean-field model replicates the behavior observed in the spiking simulations, including improvement of postsynaptic speed coding with the introduction of anticipatory firing. Moreover, analytical calculations enabled us to compute the optimal value of anticipatory time interval (ATI) for this parameter set, which came out to 86 ms; the dotted gray line shows that mean-field simulation results indeed agree with the calculation.