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U ovom diplomskom radu smo uveli osnovne pojmove vezane za mrežne automate te dokazali teoreme koji pokazuju snagu neuronskih mreža i staničnih automata. Posebno bitan je taj da postoji univerzalan stanični automat, te da za svaki stanični automat postoji neuronska mreža koja ga simulira. Pomoću Lyapunovljevih funkcionala smo dokazali teoreme u kojima su dane granice za duljinu prijelaza i duljinu ciklusa. Za simetrične neuronske mreže smo pokazali da poprimaju cikluse perioda 1 i najviše 2, za sekvencijalnu i sinkronu iteraciju redom, dok su za antisimetrične mreže duljine 2 i 4 za sekvencijalnu i sinkronu iteraciju redom. Na posljetku, dokazana su dva teorema koja osiguravaju da postoje neuronske mreže s eksponencijalnim duljinama prijelaza, i to konstruirajući takvu neuronsku mrežu., In this master’s thesis we have introduced basic terms related to automata networks and proved theorems that show the power of neural networks and cellular automata. Especially important are the ones which show that there exists universal cellular automata and that for every cellular automata exists neural network which simulates it. Using Lyapunov functionals we have proved theorems where we give bounds for transient length and cycle length. For symmetrical neural networks we have proved that their cycle lengths are 1 for sequential, and 1 or 2 for synchronous iteration, while in case of antisymmetrical neural networks cycle lengths are 2 and 4, for sequential and synchronous iteration respectively. At the end, we have proved two theorems which insure that there exist neural networks with exponential transient length, constructing network with that property.