Latent dynamical systems have been widely used to characterize the dynamics of neural population activity in the brain. However, these models typically ignore the fact that the brain contains multiple cell types. This limits their ability to capture the functional roles of distinct cell classes, or to accurately predict the effects of cell-specific optogenetic perturbations on neural activity or behavior. To overcome these limitations, we introduce the ``cell-type dynamical systems'' (CTDS) model. This model extends latent linear dynamical systems to contain distinct latent variables for each cell class, with biologically inspired constraints on both dynamics and emissions. To illustrate our approach, we consider neural recordings with distinct excitatory (E) and inhibitory (I) populations. The CTDS model defines separate latents for E and I cells, and constrains the dynamics so that E (I) latents have a strictly positive (negative) effects on other latents. We applied CTDS to recordings from rat frontal orienting fields (FOF) and anterior dorsal striatum (ADS) during an auditory decision-making task. The model achieved higher accuracy than a standard linear dynamical system (LDS), and revealed that both E and I latents could be used to decode the animal's choice, showing that choice-related information is not restricted to a single cell class. We also performed in-silico optogenetic perturbation experiments in the FOF and ADS, and found that CTDS was able to replicate the causal effects of different perturbations on behavior, whereas a standard LDS model---which lacks the ability to capture cell-specific perturbations---did not. Crucially, our model allowed us to understand the effects of these perturbations by revealing the dynamics of different cell-specific latents. Finally, CTDS can also be used to identify cell types for neurons whose class labels are unknown in electrophysiological recordings. These results illustrate the power of the CTDS model to provide more accurate and more biologically interpretable descriptions of neural population dynamics and their relationship to behavior.