Figures and data
Theta sequence and theta phase shift of place cell firing. a, An illustration of an animal running on a linear track. A group of place cells each represented by a different color are aligned according to their firing fields on the linear track. b, An illustration of the forward theta sequences of the neuron population (upper panel), and the theta phase precession of the 4th place cell (represented by the green color, lower panel). c, An illustration of both forward and reverse theta sequences (upper panel), and the corresponding theta phase precession and procession of the 4th place cell (lower panel). The sinusoidal trace illustrates the theta rhythm of local field potential (LFP), with individual theta cycles separated by vertical dashed lines.
The network architecture and tracking dynamics. a, A 1D continuous attractor neural network formed by place cells. Neurons are aligned according to the locations of their firing fields on the linear track. The recurrent connection strength J (x, x′) (blue arrows) between two neurons decays with their distance on the linear track. Each neuron receives an daptation current −V (x, t) (red dashed arrows). The external input Iext(x, t), represented by a Gaussian-shaped bump, conveys location-dependent sensory inputs to the network. b, An illustration of the state space of the CANN. The CANN holds a family of bump attractors which form a continuous valley in the energy space. c, The smooth tracking state. The network bump (hot colors) smoothly tracks the external moving input (the white line). The red (blue) color represents high (low) firing rate. d, The travelling wave state when the CANN has strong firing rate adaptation. The network bump moves spontaneously with a speed much faster than the external moving input. e, The intrinsic speed of the travelling wave versus the adaptation strength. f, The oscillatory tracking state. The bump position sweeps around the external input (black line) with an offset d0. g, The phase diagram of the tracking dynamics with respect to the adaptation strength m and the external input strength α. The colored area shows the parameter regime for the oscillatory tracking state. Yellow (blue) color represents fast (slow) oscillation frequency. h-i, Simulated (red points) and theoretical (blue line) oscillation frequency as a function of the adaptation strength (h) or the external input strength (i).
Oscillatory tracking accounts for theta sweeps and theta phase shift. a, Snapshots of the bump oscillation along the linear track in one theta cycle (0 ms - 140 ms). Red triangles indicate the location of the external moving input. b, Decoded relative positions based on place cell population activities. Upper panel: experimental data, adapted from Wang et al. (2020). Lower panel: the relative locations of the bump center (shown by the 10 most active neurons at each timestamp) with respect to the location of the external input (horizontal line) in five theta cycles. c, Upper panel: the process of the animal running through the firing field of the probe neuron (large black dot) is divided into three stages: the entry stage (green), the phase shift stage (red) and the departure stage (blue). Lower panel: the displacement between the bump center and the probe neuron as the animal runs through the firing field. The horizontal line represents the location of the probe neuron, which is x = 0. d, The firing rates of the probe neuron as the animal runs through the firing field. Colored points indicate firing peaks. The trace of the firing rate in the phase shift stage (the dashed box) is enlarged in the sub-figure on the right hand-side, which exhibits both phase precession (red points) and procession (blue points) in successive theta cycles. e, The firing phase shift of the probe neuron in successive theta cycles. Red points progress to earlier phases from π/2 to −π/2 and blues points progress to later phases from π/2 to 3π/2. The color of the dots represent the peak firing rates, which is also shown in d.
Different adaptation strengths account for the emergence of bimodal and unimodal cells. a, The firing rate trace of a typical bimodal cell in our model (upper panel) and the experiment data (lower panel, adapted from (Skaggs et al., 1996)). Blue boxes mark the phase shift stage. Note that there are two peaks in each theta cycle. b, The firing rate trace of a typical unimodal cell. Note that there is only one firing peak in each theta cycle. c, The averaged bump heights in the forward (blue curve) and backward windows (red curve) as a function of the adaptation strength m. d, Variation of the bump height when the adaptation strength is relatively small (blue line) or large (red line). e-f, Relative location of the bump center in a theta cycle when adaptation strength is relatively small (e) or large (f). Dashed line separate the forward and backward windows. g-h, Theta phase as a function of the normalized position of the animal in place field, averaged over all bimodal cells (g) or over all unimodal cells (h). −1 indicates that the animal just enters the place field, and 1 represents that the animal is about to leave the place field. Dashed lines separate the forward and backward windows. The lower panels in both g and h present the rescaled colormaps only in the backward window.
Constant cycling of future positions in a T-maze environment. a, An illustration of an animal navigating a T-maze environment with two possible upcoming choices (the left and right arms). b, Upper panel: Snapshots of constant cycling of theta sweeps on two arms when the animal is approaching the choice point. Red triangle marks the location of the external input. Note that the red triangle moves slightly towards the choice point in the 200 ms duration. Lower panel: Constant cycling of two possible future locations. The black, red and blue traces represent the bump location on the center, left and right arms, respectively. The green line marks the location of the external moving input. c, Left panel: the firing rate trace of a neuron A on the left arm when the animal approaches the choice point. Right panel: the firing rate traces of a pair of neurons when the animal approaches the choice point, with neuron A (red) on the left arm and neuron B (blue) on the right arm. Dashed lines separate theta cycles. d, Upper panel: the auto-correlogram of the firing rate trace of probe neuron A. Lower panel: the cross-correlogram between the firing rate trace of neuron A and the firing rate trace of neuron B.
Robust phase coding of position. a, speed modulation of place cell firing frequency, adapted from Geisler et al. (2007). Oscillation frequency of the place cell is higher in the faster run trial (blue) then that in the slower run trail (green). The frequency difference is linearly increased with the running speed. b, Phase precession is preserved after stimulation-induced perturbation (grey shaded area of the yellow part), adapted from Zugaro et al. (2005). c, left: normalized spectrum of bump oscillation (black curve) and the oscillation of a unimodal cell (blue curve). Right: linear relationship between the frequency difference and the running speed. d, same as c but for a bimodal cell. e, silencing the network activity for 100 ms (grey shaded area) when the external moving input passes through the center part of the place field of a unimodal cell. Theta phase shifts of the unimodal cell are shown with (black points) or without (blue curve) silencing the network.
Au have two real solutions (indicated by the ± sign in Eq. 18), i.e., the dynamic system (Eqs. 16&17) has two fixed points. It can be checked that only 
Commonly used parameter values in the simulation of the linear track environment.
Figure specific parameter values for input strength α and adaptation strength m.
