In this work we reveal and explore a new class of attractor neural networks, based on inborn connections provided by model molecular markers, the molecular marker based attractor neural networks (MMBANN). We have explored conditions for the existence of attractor states, critical relations between their parameters and the spectrum of single neuron models, which can implement the MMBANN. Besides, we describe functional models (perceptron and SOM) which obtain significant advantages, while using MMBANN. In particular, the perceptron based on MMBANN, gets specificity gain in orders of error probabilities values, MMBANN SOM obtains real neurophysiological meaning, the number of possible grandma cells increases 1000- fold with MMBANN. Each set of markers has a metric, which is used to make connections between neurons containing the markers. The resulting neural networks have sets of attractor states, which can serve as finite grids for representation of variables in computations. These grids may show dimensions of d = 0, 1, 2,... We work with static and dynamic attractor neural networks of dimensions d = 0 and d = 1. We also argue that the number of dimensions which can be represented by attractors of activities of neural networks with the number of elements N=104 does not exceed 8.
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