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).