Electroencephalography (EEG) is a complex multivariate signal measuring brain electrophysiology in real time. While very informative in neurological studies, the EEG presents serious challenges - it is often corrupted by noise, artifacts, and is inherently a high-dimensional, non-stationary process due to the intrinsic dynamics of underlying brain activity. Understanding how to interpret and analyze the EEG is therefore an ongoing effort employing a range of techniques. This dissertation aims to investigate novel approaches to quantitatively measure the inherent complexity of the EEG, and use these measures effectively to better understand and track the progress of recovery and treatment of complex neurological conditions in longitudinal studies through the use of three case studies. Specifically, novel EEG analyses utilizing graphical theoretical and spectral analytic measures are developed and applied to: 1) Studying the recovery of consciousness in patients with severe brain injuries; 2) Characterizing the effect of subcallosal deep brain stimulation in treatment resistant depression; 3) Validating the measures through the characterization of the test-retest stability of each measure in a set of healthy volunteers, tested at multiple time points. The novel measures are shown to provide new insight in each patient study. In the context of disorders of consciousness, measures of spectral coherence show that the functional connectivity of the brain is closely linked to behavioral changes in consciousness, and specifically, that coherence network approaches may be used as markers of underlying functional and structural recovery of communication. In the case of treatment resistant depression and subcallosal DBS we find that spectral measures of the alpha band are predictors of the efficacy of treatment. Collectively, the studies carried out in this thesis show that our novel measures allow us to track global brain state changes that reflect inherent underlying neurological processes related to disorders of consciousness and neuropsychiatric disorders. Further development and use of these tools may to provide neurologists and psychiatrists with methods to track underlying functional brain changes in their patients, allowing for advances in treatment, prognosis, diagnosis and understanding of underlying mechanisms of disease.