The statistical relationship between connectivity and neural activity in fractionally connected feed-forward networks

It is desirable to have a statistical description of neuronal connectivity in developing tractable theories on the development of biological neural networks and in designing artificial neural networks. In this paper, we bring out a relationship between the statistics of the input environment, the degree of network connectivity, and the average postsynaptic activity. These relationships are derived using simple neurons whose inputs are only feed-forward, excitatory and whose activity is a linear function of its inputs. In particular, we show that only the empirical mean of the pairwise input correlations, rather than the full matrix of all such correlations, is needed to produce an accurate estimate of the number of inputs necessary to attain a prespecified average postsynaptic activity level. Predictions from this work also include distributional aspects of connectivity and activity as shown by a combination of analysis and simulations.