• Unsupervised Learning Techniques for Large-Scale, Multi-Trial Neural Data
• An increasingly common paradigm in neuroscience is to simultaneously record the activity of many neurons over repeated experimental trials (e.g., multiple presentations of a sensory stimulus, or a repeated motor action). The resulting datasets can be very large (in some cases, containing thousands of neurons and trials). I'm interested in finding general statistical approaches for understanding datasets of this form.
• Tensor Components Analysis (TCA) — Commonly used methods for dimensionality reduction (such as PCA) identify low-dimensional features of within-trial neural dynamics, but do not model changes in neural activity across trials. To better understand processes like learning and trial-to-trial variability, I'm exploring tensor decomposition methods to find reduced representations of multi-trial datasets.
• TensorTools — My Python toolbox for fitting tensor decompositions to neural data.
• Time Warped Dimensionality Reduction — Analysis of neural data often relies on a strict alignment of neural activity to a stimulus or behavioral event on each trial. However, alignment to external events is not always possible (e.g., in cases where neural activity is locked to internal cognitive states or decisions or other unobserved latent factors). I've developed a method to align trials by time warping, while jointly fitting a dimensionality reduction model. This was done in close collaboration with Ben Poole and Niru Maheswaranathan.



• 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.



• Miscellaneous Codes
• I've written a few packages in Julia for statistics and optimization. These are not in active development at the moment, but I'd like to return to them someday.
• 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.
• 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 (2016). eLife. 5:e20556
• 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.