The excerpt note is from the Song et al., 2005 paper. The example code is implemented by Fangwen. The whole example code is here.

Pengcheng Song, and Xiao-Jing Wang. “Angular path integration by moving “hill of activity”: a spiking neuron model without recurrent excitation of the head-direction system.” Journal of Neuroscience 25, no. 4 (2005): 1002-1014.

In the model, the peak location of the bell-shaped activity profile of the excitatory population provides a natural choice of the internal representation of the animal’s instantaneous head direction. It was determined by a population vector scheme (Georgopoulos et al., 1986), as follows:

Where is the firing rate of neuron , of which the preferred direction is . When a “hill of activity” moves at a constant propagation speed, the latter was used as a natural internal representation of the animal’s angular head velocity (AHV).

Georgopoulos, Apostolos P., Andrew B. Schwartz, and Ronald E. Kettner. “Neuronal population coding of movement direction.” Science (1986): 1416-1419.

[value, z] = max(HDCELLS);
z_HDCELLS=zeros(1,HD_DIMENSION_THETA);
z_HDCELLS(HD_AVG_TH_WRAP(z:z+HD_CELLS_TO_AVG*2)) = HDCELLS(HD_AVG_TH_WRAP(z:z+HD_CELLS_TO_AVG*2));
out_theta = mod(atan2(sum(HD_TH_SUM_SIN_LOOKUP*(z_HDCELLS')), sum(HD_TH_SUM_COS_LOOKUP * (z_HDCELLS')))/HD_CELL_SIZE_THETA, HD_DIMENSION_THETA);