The development of neuromorphic processors has opened a new field of their application – solving classical algorithmic and optimization problems using spike neural networks. The low power consumption, high performance and natural parallelism of neuromorphic processors make their use for such tasks very effective in some cases. For example, discrete optimization problems arise when planning robot movements or controlling a group of UAVs. The report examines the traveling salesman problem and its solution using a spike neural network. We use a special architecture network based on winner takes all (WTA) blocks proposed earlier in the literature. Stable network states form a local solution to the traveling salesman problem by adjusting the connection weights so that they reflect the conditions of the problem. Adding noise to the membrane potential of a neuron leads to random switching of network states and, as a result, convergence to a more efficient solution. We investigate two cases: with constant noise intensity and with the noise decaying over time. The efficiency of both approaches is examined by simulations with an instance of TSP problem available in an open dataset.