Table 1. Summary of the symbols used in the model.

Symbol	Meaning
$n$	index of time step in the generative model measured as the number of theta cycles
${\displaystyle x_n}$	position at theta cycle n (two-dimensional)
${\displaystyle y_n}$	sensory input (two-dimensional)
${\displaystyle u_n}$	motor command (two-dimensional)
${\displaystyle {\overline{\mathbf{x}}}_n}$	planned position ($2\times T$-dimensional)
${\overline{x}}_n$	planned position (two-dimensional)
$y_{1:n}$	past sensory input until theta cycle n
${\mu }_n$	mean of the filtering posterior
${\mathrm{\Sigma }}_n$	covariance of the filtering posterior
$\phi$	theta phase
${\displaystyle {\mathbf{x}}_n\equiv x_{(n-n_p):(n+n_f)}}$	trajectory of the animal around theta cycle n
${\displaystyle {\mathit{\mu }}_n(\phi )}$	posterior mean trajectory at theta cycle n
${\displaystyle {\mathbf{\Sigma }}_n(\phi )}$	posterior variance of trajectory at theta cycle n
${\displaystyle {\tilde{\mathbf{x}}}_n}$	trajectory sampled from ${\displaystyle P({\mathbf{x}}_n|y_{1:n},u_{1:n})}$
${\varphi }_i(x)$	encoding basis function of cell i - firing rate as a function of the encoded position
${\psi }_i(x)$	empirical tuning curve of cell i - firing rate as a function of the real position
${\lambda }_i$	firing rate of cell i
${\displaystyle {\mathbf{s}}_n(\phi )}$	spikes recorded in theta cycle n encoding trajectory xn
${\displaystyle {\hat{\mathbf{x}}}_n(\phi )}$	trajectory decoded from the observed spikes assuming direct encoding (Equation 18)
${\displaystyle {\hat{\mathit{\mu }}}_n(\phi )}$	estimated trajectory mean assuming DDC encoding (Equation 19)
${\displaystyle {\hat{\mathbf{\Sigma }}}_n(\phi )}$	estimated trajectory variance assuming DDC encoding (Equation 19)