On Wang $k$ WTA With Input Noise, Output Node Stochastic, and Recurrent State Noise

In this paper, the effect of input noise, output node stochastic, and recurrent state noise on the Wang kWTA is analyzed. Here, we assume that noise exists at the recurrent state y(t) and it can either be additive or multiplicative. Besides, its dynamical change (i.e., dy/dt) is corrupted by noise as well. In sequel, we model the dynamics of y(t) as a stochastic differential equation and show that the stochastic behavior of y(t) is equivalent to an Ito diffusion. Its stationary distribution is a Gibbs distribution, whose modality depends on the noise condition. With moderate input noise and very small recurrent state noise, the distribution is single modal and hence y(∞) has high probability varying within the input values of the k and k + 1 winners (i.e., correct output). With small input noise and large recurrent state noise, the distribution could be multimodal and hence y(∞) could have probability varying outside the input values of the k and k + 1 winners (i.e., incorrect output). In this regard, we further derive the conditions that the kWTA has high probability giving correct output. Our results reveal that recurrent state noise could have severe effect on Wang kWTA. But, input noise and output node stochastic could alleviate such an effect.