To a mathematical biologist, the answer to both of these questions is (no surprise) 'of course'. The brain is after all an extremely complicated network with hundreds of billions of neurons interacting in highly nonlinear ways, generating complex firing patterns that depend nontrivially on parameters. How can one possibly understand mechanisms underlying those patterns, test hypotheses and interpret data without a computational model and mathematical analysis to understand the model? Many (most?) neuroscientists remain skeptical. How can such simple-minded equations possibly help explain something so complicated? Even if one accepts that the brain is simply a complicated network, then how can one construct a useful model when so little is known about the properties of individual neurons, how and which neurons communicate with each other, what the collective behavior of any neuronal system is, what the firing patterns mean, what any part of the brain does, etc... ?
In this talk, I will give examples in which issues raised in the study of specific neuronal systems, computational modeling and mathematical analysis have all benefited from each other. In particular, I will describe work on Parkinsonian rhythms generated in the basal ganglia, sensory processing in the insect's antennal lobe and models for working memory.