Towards blueprints for network architecture, biophysical dynamics and signal transduction.

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on examples that combine insights from these three areas to expand our understanding of systems neuroscience. These range from single neuron coding to models of decision making and electrosensory discrimination by networks and populations and also coincidence detection in pairs of dendrites and dynamics of large networks of excitable dendritic spines. We conclude by describing some of the challenges that lie ahead as the applied mathematics community seeks to provide the tools which will ultimately underpin systems neuroscience.