Some Joys and Trials of Mathematical Neuroscience
I describe the basic components of the nervous system—neurons and their connections via chemical synapses and electrical gap junctions—and review the model for the action potential produced by a single neuron, proposed by Hodgkin and Huxley (HH) over 60 years ago. I then review simplifications of the HH model and extensions that address bursting behavior typical of motoneurons, and describe some models of neural circuits found in pattern generators for locomotion. Such circuits can be studied and modeled in relative isolation from the central nervous system and brain, but the brain itself (and especially the human cortex) presents a much greater challenge due to the huge numbers of neurons and synapses involved. Nonetheless, simple stochastic accumulator models can reproduce both behavioral and electrophysiological data and offer explanations for human behavior in perceptual decisions. In the second part of the paper I introduce these models and describe their relation to an optimal strategy for identifying a signal obscured by noise, thus providing a norm against which behavior can be assessed and suggesting reasons for suboptimal performance. Accumulators describe average activities in brain areas associated with the stimuli and response modes used in the experiments, and they can be derived, albeit non-rigorously, from simplified HH models of excitatory and inhibitory neural populations. Finally, I note topics excluded due to space constraints and identify some open problems.