A Recurrent Neural Circuit Mechanism of Temporal-scaling Equivariant Representation

Time perception is fundamental in our daily life. An important feature of time perception is temporal scaling (TS): the ability to generate temporal sequences (e.g., movements) with different speeds. However, it is largely unknown about the mathematical principle underlying TS in the brain. The present theoretical study investigates temporal scaling from the Lie group point of view. We propose a canonical nonlinear recurrent circuit dynamics, modeled as a continuous attractor network, whose neuronal population responses embed a temporal sequence that is TS equivariant. We find the TS group operators can be explicitly represented by a time-invariant control input to the network, whereby the input gain determines the TS factor (group parameter), and the spatial offset between the control input and the network state on the continuous attractor manifold gives rise to the generator of the Lie group. The recurrent circuits neuronal responses are consistent with experimental data. The recurrent circuit can drive a feedforward circuit to generate complex sequences with different temporal scales, even in the case of negative temporal scaling ("time reversal"). Our work for the first time analytically links the abstract temporal scaling group and concrete neural circuit dynamics.