| In this dissertation I develop both general results on the dynamics of neural oscillators and integrators and specific applications of these results to brain areas involved in simple cognitive tasks. The scientific motivation is broad: neural networks inside our brains are able to adapt to changing information processing demands by exercising cognitive control, for example focussing on salient components of noisy sensory inputs when making specific decisions based on these inputs, but relaxing this focus when other needs become prominent. But what free variables or parameters can account for the observed adaptability? And does this adaptation occur optimally, with respect to simple economic metrics and physiological limitations? Here I address these questions via reduced models of neurons and populations near bifurcations, which characterize the dynamics of a brainstem nucleus involved in adaptive cognitive control, and via variational problems arising from neural signal processing, which clarify the role of this nucleus, and other dynamical mechanisms in decision tasks.; First, I study and apply nonlinear oscillator dynamics. I develop and extend phase reductions for single compartment ordinary differential equation neuron models that show how both type of, and distance from, the four codimension-one bifurcations to periodic firing affect responses of neural populations to stimuli. I also extend results from equivariant dynamics which describe how coupling functions determine the existence and stability of phase-locked states in which subgroups of oscillators are synchronized. These results are then applied to the firing rate dynamics of the locus coeruleus (LC) brainstem nucleus, thereby characterizing the inputs that drive the LC and suggesting a new biophysical mechanism for transitions among LC-mediated states of cognitive performance.; LC-driven neuromodulation transiently adjusts the sensitivity ("gain") of integrator units believed to determine simple cognitive decisions in response to sensory stimuli, and these gain effects are the focus of the second part of this dissertation. I study how transient parameter adjustments can optimize decision tasks for speed and accuracy in the presence of noise. The results indicate a surprising match between empirical data on the time course of LC firing rates and optimal gain trajectories found via variational methods, providing an explicit hypothesis for the role of the LC in decision tasks. |
