• Principled Unsupervised Learning Techniques in Neuroscience
• Neuroscientists use "off-the-shelf" unsupervised techniques to make sense of their data and gain intuition about their experimental system. Doing this is extremely necessary given the explosion and growth of datasets, but also involves a lot of choices and qualitative interpretation that reasonable experts may disagree on. What statistical methods, techniques, and philosophies should we adopt to encourage transparency and consistency across research groups?
• Clustering — Neuroscientists use clustering algorithms to categorize neurons and glia into functional and genetic cell types. While the idea of cannonical cell types is empirically useful for building models and experimental tools, it is unclear how the choice of clustering algorithm impacts our understanding of the brain. I'm interested in situations where clusterings can be found with provable guarantees.
• Dimensionality Reduction — Dimensionality reduction is another common (and related) class of techniques commonly used by neuroscientists and biologists. I'm interested in developing and applying extensions of principal components analysis (PCA) that are tailored to the particulars of biological data.



• Theoretical Molecular Neurobiology
• Biology computes with both electrical and biochemical signals. I'm interested in modeling the interface of these two substrates of computation.
• Homeostatic Plasticity — Neurons alter ion channel and synaptic receptor expression/activity to maintain activity levels in physiologically stable regimes. This can be modeled from a control theoretic perspective, which provides perspectives on how noisy molecular processes can nevertheless support reliable physiological behaviors.
• Microtubular Transport in Complex Dendritic Trees — Neurons are remarkably complex cells. Given this, it seems an almost insurmountable challenge to transport molecular cargo reliably. I've studied a few simple models of how reliable transport can be accomplished.
• PyNeuronToolbox — A package I wrote to enable better NEURON simulations in Jupyter notebooks.



• Julia Statistics and Optimization Packages
• I've written a few packages in Julia for statistics and optimization applications. Some of these are still works in progress or under active development.
• Einsum.jl — Einstein summation notation for flexible and efficient multi-dimensional arrray computations.
• NonNegLeastSquares.jl — Active-set methods to efficiently solve nonnegative least-squares problems.
• HiddenMarkovModel.jl — A package for fitting hidden markov models with arbitrary emission probability distributions.
• ToyHMM.jl — A simple package for fitting hidden markov models with discrete emissions. Good for teaching!

Papers

• Dendritic trafficking faces physiologically critical speed-precision tradeoffs
• Williams AH, O’Donnell C, Sejnowski T, O’Leary T (submitted). BioRxiv Preprint.
• Distinct or shared actions of peptide family isoforms: II. Multiple pyrokinins exert similar effects in the lobster stomatogastric nervous system.
• Dickinson PS, Kurland SC, Qu X, Parker BO, Sreekrishnan A, Kwiatkowski MA, Williams AH, Ysasi AB, Christie AE (2015). J Exp Biol. 218:2905-17
• Summary of the DREAM8 parameter estimation challenge: Toward parameter identification for whole-cell models.
• Karr JR, Williams AH, Zucker JD, Raue A, Steiert B, Timmer J, Kreutz C, DREAM8 Parameter Estimation Challenge Consortium, Wilkinson S, Allgood BA, Bot BM, Hoff BR, Kellen MR, Covert MW, Stolovitzky GA, Meyer P (2015). PLoS Comput Biol. 11(5):e1004096
• Cell types, network homeostasis and pathological compensation from a biologically plausible ion channel expression model.
• O’Leary T, Williams AH, Franci A, Marder E (2014). Neuron. 82(4):809-21
• Many parameter sets in a multicompartment model oscillator are robust to temperature perturbations.
• Caplan JS, Williams AH, Marder E (2014). J Neurosci. 34(14):4963-75
• The neuromuscular transform of the lobster cardiac system explains the opposing effects of a neuromodulator on muscle output.
• Williams AH, Calkins A, O’Leary T, Symonds R, Marder E, Dickinson PS (2013). J Neurosci. 33(42):16565-75
• Correlations in ion channel expression emerge from homeostatic regulation mechanisms.
• O’Leary T, Williams AH, Caplan JS, Marder E (2013). Proc Natl Acad Sci USA. 110(28):E2645-54
• Animal-to-animal variability in the phasing of the crustacean cardiac motor pattern: an experimental and computational analysis.
• Williams AH, Kwiatkoswki MA, Mortimer AL, Marder E, Zeeman ML, Dickinson PS (2013). (2013). J Neurophysiol. 109:2451-65.

Reviews

• Neuromodulation in Small Networks.
• Williams AH, Hamood AW, Marder E (2015). Springer Encyclopedia of Computational Neuroscience.
• Homeostatic Regulation of Neuronal Excitability.
• Williams AH, O’Leary T, Marder E (2015). Scholarpedia. 8(1):1656

Notes (not peer-reviewed)

• Demixed PCA.
• Williams AH (2016). Stanford Comp Neuro Journal Club.