Symbol	Description
${\varphi }_v,{\varphi }_h,{\psi }_v,{\psi }_h$	Angles in two elongated diamonds depicted in Figure 2C and D
$k_i,{\mu }_i$	Kappa sets the tuning width and Mu the mean of the von Mises function for the i-th neuron
${\gamma }_h,{\gamma }_v$	Angles around the horizontal axis and the vertical axis
${\theta }_a,{\theta }_b$	Orientation of two lines that form an angle of $\gamma$
$n_i$	Number of cells for i-th neuron
$r_{ai},r_{bi}$	Feedforward response of i-th cell given orientations of ${\theta }_a$ and ${\theta }_b$ respectively
$g_i$	Divisive gain control of i-th cell
$R_{ai},R_{bi}$	Normalized response of i-th cell
$\widehat{{\theta }_a},\widehat{{\theta }_b}$	Decoded orientations
$\widehat{{\gamma }_v},\widehat{{\gamma }_h}$	Decoded angles given ${\gamma }_h$ and ${\gamma }_v$ respectively
$G_{i,\mathit{odd}},G_{i,\mathit{even}}$	Odd and even 3D Gabor filters of i-th cell
$Q_i$	Normalization constant for the Gabor filter
$c_{x,i},c_{y,i},c_{t,i}$	Spatial and temporal means of the Gabor filter
${\sigma }_{s,i},{\sigma }_{t,i}$	Spatial and temporal envelopes of the Gabor filter
${\omega }_{x0},{\omega }_{y0},{\omega }_{t0}$	Spatial and temporal frequencies of the sine component of the Gabor filter
${\eta }_s,{\eta }_t$	Spatial and temporal orientations of the Gabor filter
$F_0$	Frequency of the Gabor filter
$\mathrm{\Delta }{\omega }_{\frac{1}{2}}$	Half-height orientation bandwidths
$\hat{u},\hat{v}$	The best estimate of local velocity components
${\mathfrak{N}}_i(u,v)$	Predicted response of filter $i$ to velocity $(u,v)$
$m_i$	i-th motion energy output
${\bar{m}}_l,\overline{{\mathfrak{N}}_l}(u,v)$	The sum of ${\mathfrak{N}}_i$ and the sum of $m_i$ for filters sharing the same spatial orientation
${\mathbb{R}}_y,{\mathbb{R}}_z$	Rotational matrices around Y and Z axis respectively
$\overrightarrow{P}(\theta )$	The position of a point on the ring lying on the X – Z plane
$\tau$	Tilt of the ring
$\omega$	Angular velocity of the rotation of the ring
$k$	The wobbling weight ($k=0\sim rotation,k=1\sim wobbling$)
$Proj(\cdot )$	The projection function
$\overrightarrow{P_k}(\theta ,t)$	The position of a point at a space and time for a moving ring with the wobbling weight of $k$
$u(\theta ,t),v(\theta ,t)$	The $x$ and $y$ components for $\overrightarrow{P_k}(\theta ,t)$
$F_k$	The velocity field for the moving ring with the wobbling weight of $k$