A growing need in the field of cognitive science is to develop mathematical tools that relate the functioning of the brain to perception, memory, learning and decision making. In this study, an integrated dynamical model consisting of coupled nonlinear differential equations, neural activation functions, synaptic plasticity and external stimuli, representing cognitive states is proposed. The framework provides a set of biologically interpretable parameters for excitation, inhibition, connectivity, decay, adaptation and noise and stability and equilibrium analyses describe long term behavior. The numerical simulations provide an assessment of activity propagation, state transitions, convergence and sensitivity with different parameter sets. Results show that balanced excitation and inhibition lead to a stable cognitive representation while stronger coupling and/or noise lead to oscillatory and unstable responses. The computational power of the model provides a framework for understanding the neural mechanisms, comparing cognitive scenarios and informing the development of brain-inspired intelligent systems and clinically relevant investigations of cognition and behavior.