Your measurements are actually discrete. You are not examining the mice every millisecond. If the number of studied moments of time is not large, you can try Repeated Measures ANOVA, with Group being a between-subject effect.

If the number of studied moments of time is large, you can estimate a time series model for each group and compare the relevant coefficients (e.g. long-run mean, trend etc) between the groups using a likelihood ratio test (or a similar test).