Experimental results suggest that there are two distinct mechanisms of inhibition in cortical networks: subtractive and divisive inhibition. Subtractive inhibition shifts the neuronal input-output function to the right without changing the slope, whereas divisive inhibition causes a reduction in slope thus providing a gain control mechanism. Notably, recent experiments done using optogenetics show that these mechanisms are delivered by different populations of interneurons with a well understood connectivity between them and the pyramidal population [1,2]. While most research has focussed on understanding the mechanism of gain control, the role of these inhibitory mechanisms in regulating the dynamics of the network is less well understood.
This work presents a novel mathematical model of this basic neocortical circuitry, which incorporates the two inhibitory mechanisms. We investigated the role of these inhibitory mechanisms in terms of network dynamics, and particularly focussing on the transition from ordered to chaotic behaviour. We show that the model incorporating divisive inhibition exhibits quite different behaviour compared to an equivalent model without divisive inhibition. The presence of divisive inhibition in the network prevents the abrupt transition from regular to chaotic dynamics across the parameter space. In contrast, in models which only have subtractive inhibitory elements, there are many cases where small changes in synaptic strength result in abrupt transitions to chaos either through a period-doubling cascade or as a result of hysteresis. In the case of period-doubling cascades, it was found that the presence of divisive inhibition ensures a typical Feigenbaum transition  whereas the absence of it can lead to arbitrarily abrupt cascades depending on the specific connectivity parameters of the network.
Synaptic plasticity has been postulated as a mechanism by which the brain learns and stores information. Our results have interesting implications, therefore, for how divisive inhibition can ensure stability of dynamic network activity during such plastic changes. It is also relevant to other network transitions such as from physiological to pathological (epileptic) brain dynamics.