GABAergic interneurons can be subdivided into three subclasses: parvalbumin positive (PV), somatostatin positive (SOM) and serotonin positive neurons. With principal cells (PCs) they form complex networks. We examine PCs and PV responses in mouse anterior lateral motor cortex (ALM) and barrel cortex (S1) upon PV photostimulation in vivo . In layer 5, the PV response is paradoxical: photoexcitation reduces their activity. This is not the case in ALM layer 2/3. We combine analytical calculations and numerical simulations to investigate how these results constrain the architecture. Twopopulation models cannot account for the results. Networks with three inhibitory populations and V1like architecture account for the data in ALM layer 2/3. Our data in layer 5 can be accounted for if SOM neurons receive inputs only from PCs and PV neurons. In both fourpopulation models, the paradoxical effect implies not too strong recurrent excitation. It is not evidence for stabilization by inhibition.sufficiently strong, namely ifThe denominator in is the strength of the connection from the SOM population to J EE the VIP population. The numerator is the gain of the pathway which connects these two populations via the PCs. When the negative contribution of the disinhibitory J EE > J EE loop PCVIPSOMPC dominates in the expression of . It is the opposite when χ II . The stability of the balanced state provides other necessary conditions that J EE < J EE the interactions must satisfy (Materials and Methods). In particular, the determinant of the interaction matrix, , must be positive. J of the positive feedback loop, SOMXPCSOM, is sufficiently strong (Supplementary Materials, SMD). Obviously, this condition simplifies and reads J J J EX XS > J XX ES (4) Remarkably, this inequality does not depend on . This is in contrast to what J EE happens in Model 1 where the paradoxical effect occurs only if is small enough J EE (see Eq. (2)). https://docs.google.com/document/d/1UaHERm5J7ZX8fMIoyG42_mFfOzWPZq32QDjE_gxdSgk/edit?ts=5c9dfa46# 24/46 was already substantial for small light intensities, where the PCs were still significantly active. The twopopulation model cannot account for this feature. In Model 1, whether the network exhibits a paradoxical effect depends on the value of the ratio where . Here, , is theThe requirement that at baseline the network state is fully balanced and stable implies that J I0 Therefore, the PC population activity increases upon PC stimulation if J J J J IX XS > IS XX (SM61)