This paper presents an example of how the Ptolemy environment can be used constructively to simulate prototypes of Artificial Neural Network algorithms, implemented by means of Systolic Array architectures. Initially, a number of well-known ANN algorithms, which all fall under the general concept of Associative Memory Artificial Neural Networks, is presented. Then, follows the discussion for the transformation and mapping of those algorithms onto a Linear Array Systolic architecture, capable of implementing all the discussed Associative Memory algorithms. Further, the Systolic Array architecture is designed and simulated using the Ptolemy Environment. Through graphical means, the user can easily obtain systolic circuit prototypes, at a level high enough to be comprehensive and, at the same time, low enough to present the design complexity of a potential hardware implementation. Finally, the ability to simulate the functioning of the Systolic Artificial Neural Network circuit ensures the correctness of the initial algorithms, as well as of the systematic transformation and mapping technique.