This paper is concerned with the sliding mode control problem for a class of continuous-time switched neural networks with mode-dependent average dwell time (MDADT). The considered continuous-time switched neural networks are motivated by biological neural networks which contain a nonlinear term and a changeable switched signal. The concept of MDADT is introduced, in which every subsystem has its own dwell time before switching to another subsystem. Moreover, a novel sliding mode controller is designed by an event-triggered mechanism which is based on the observer error and the system mode, where its triggered condition can be more conservative and practical than the existing triggered conditions. Sufficient conditions are derived to ensure that the closed-loop system is stochastically exponentially stable in terms of linear matrix inequalities. The designed sliding mode controller can promote the sliding mode motion of the system state. Finally, an illustrative example is provided to demonstrate the effectiveness and merits of the proposed method.