Sampling motion trajectories during hippocampal theta sequences

Figures

Figure 1

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Figure 2

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Figure 3 with 3 supplements

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Figure 3—figure supplement 1

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Figure 3—figure supplement 3

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Figure 4 with 3 supplements

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Figure 5 with 3 supplements

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Figure 6 with 2 supplements

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Tables

Table 1

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)

Table 2

Parameter	Strong +	Weak +	Independent	Weak -	Strong -
${\displaystyle {\gamma }_{\vartheta }}$(slope)	-8	-5	0	5	5
${\displaystyle {\vartheta }_0}$(threshold)	${\displaystyle \pi /8}$	${\displaystyle \pi /4}$	${\displaystyle \pi /2}$	${\displaystyle \pi /4}$	${\displaystyle \pi /2}$

Table 3

Panel: f
regular	jitter: 0	5	10	20	30	40
8.9e-10	1e-06	4.4e-07	1.5e-07	1e-05	5.7e-07	1.8e-07
panel: g
regular	jitter: 0	5	10	20	30	40
0.0001	4e-15	3.9e-15	1e-13	3.4e-11	1.5e-11	2.4e-06
panel: k
rat1 day1	rat1 day2	rat2 day1	rat2 day2	rat3 day1	rat3 day2	rat4 day1	rat4 day2
5e-05	2.5e-18	2.5e-05	0.0001	2.1e-10	2.8e-07	1.4e-08	2.4e-06
panel: l
rat1 day1	rat1 day2	rat2 day1	rat2 day2	rat3 day1	rat3 day2	rat4 day1	rat4 day2
<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001

Table 4

Panel: e
regular	jitter: 0	5	10	20	30	40
1.8e-05	1.4e-05	9e-05	4.8e-05	0.0016	0.0025	0.0006
panel: f
regular	jitter: 0	5	10	20	30	40
0.01	4e-05	7e-05	0.0007	0.0006	0.014	0.16
panel: j
rat1 day1	rat1 day2	rat2 day1	rat2 day2	rat3 day1	rat3 day2	rat4 day1	rat4 day2
0.0018	4.5e-09	6e-08	4.9e-07	3.7e-19	4e-11	3.3e-25	2.4e-07
panel: k
rat1 day1	rat1 day2	rat2 day1	rat2 day2	rat3 day1	rat3 day2	rat4 day1	rat4 day2
<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001	<0.001

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