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.

Neural integrators are circuits that are able to code analog information such as spatial location or amplitude. Storing amplitude requires the network to have a large number of attractors. In classic models with recurrent excitation, such networks require very careful tuning to behave as integrators and are not robust to small mistuning of the recurrent weights. In this talk, I introduce a circuit with recurrent connectivity that is subjected to a slow subthreshold oscillation (such as the theta rhythm in the hippocampus). I show that such a network can robustly maintain many discrete attracting states. Furthermore, the firing rates of the neurons in these attracting states are much closer to those seen in recordings of animals. I show the mechanism for this can be explained by the instability regions of the Mathieu equation. I then extend the model in various ways and, for example, show that in a spatially distributed network, it is possible to code location and amplitude simultaneously. I show that the resulting mean field equations are equivalent to a certain discontinuous differential equation.

In mammals, the selective transformation of transient experience into stored memory occurs in the hippocampus, which develops representations of specific events in the context in which they occur. In this talk, I will focus on the development of hippocampal circuits and the self-organized dynamics embedded in them since the latter critically support the role of the hippocampus in memory. I will discuss evidence that adult hippocampal cells and circuits are remarkably sculpted by development, as early as embryonic neurogenesis. We argue that these primary developmental programs provide a scaffold onto which later experience of the external world can be grafted. Next, I will present data on the emergence of recurrent connectivity and self-organized dynamics in hippocampal circuits and outline the critical turn points and discontinuities in that developmental journey.