The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function. However, quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of rectified-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. We then use this analytical characterization to rigorously analyze the solution space geometry and derive certainty conditions guaranteeing a non-zero synapse between neurons.

In a few months, human infants develop complex capacities in numerous cognitive domains. They learn their native language, recognize their parents, refine their numerical capacities and their perception of the world around them but are they conscious and how can we study consciousness when no verbal report is possible? One way to approach this question is to rely on the neural responses correlated with conscious perception in adults (i.e. a global increase of activity in notably frontal regions with top-down amplification of the sensory levels). We can thus study at what age the developing anatomical architecture might be mature enough to allow this type of responses, but moreover we can use similar experimental paradigms than in adults in which we expect to observe a similar pattern of functional responses.