A Recurrent Neural Circuit Mechanism of Temporal-scaling Equivariant Representation

Time perception is critical in our daily life. An important feature of time perception is temporal scaling (TS): the ability to generate temporal sequences (e.g., motor actions) at different speeds. However, it is largely unknown about the math principle underlying temporal scaling in recurrent circuits in the brain. To shed insight, the present study investigates the 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. Furthermore, we found the TS group operators can be explicitly represented by a control input fed into the recurrent circuit, where the input gain determines the temporal scaling factor (group parameter), and the spatial offset between the control input and network state emerges the generator. The neuronal responses in the recurrent circuit are also consistent with experimental findings. We illustrated that the recurrent circuit can drive a feedforward circuit to generate complex temporal sequences with different time scales, even in the case of negative time scaling (''time reversal''). Our work for the first time analytically links the abstract temporal scaling group and concrete neural circuit dynamics.