### Our Research

I am a computational scientist and applied mathematician
with an interest in * networked dynamical systems*,
in particular

*. These systems are often adaptive, meaning they can learn from their environment, and exhibit complex behavior.*

**neural systems**Methodologies that I apply include:

- Statistical inference & machine learning
- Dimensionality reduction
- Dynamical systems
- Graph & network theory.

- Computer modeling
- Linear algebra & matrix computations
- Differential equations
- Probability & analysis.

*of the systems we study. See my Publications page for papers.*

**know the biology**Slides from a number of my talks are available here.

### Current Projects

**Theoretical neural networks:**
Artificial neural networks have enjoyed a huge renaissance in the last decade.
Theory is starting to catch up to the enormous empirical success of these networks.
A promising connection is the similarity of wide neural networks to kernel methods,
which has shown success in explaining partially untrained and fully-trained networks
in certain regimes ("random features" and "neural tangent kernel" are keywords).

These theories have mostly been developed for very simple neural architectures, like when the weights are drawn as independent Gaussians with equal variance. We have found it illuminating to ask, What changes when you add structures that are found in successful biological neuronal systems?

For example sparsity, an important conserved property of cerebellum and mushroom body brain areas, leads to the network approximating additive functions. This constraint leads to faster learning from fewer examples in such networks. Similarly, allowing the weights to reproduce known neuronal tuning properties from V1 leads to improved performance on simple image recognition tasks (work with Biraj Pandey and Bing Brunton).

Our view of networks as function approximators is uncommon in computational neuroscience,
but offers many advantages.
The kernel associated with a network architecture tells you important *geometrical*
properties about the functions that it can approximate.

**Brain network inference:**
Together with Stefan Mihalas and others at the
Allen Institute for Brain Science,
I am improving the mathematical techniques used to
analyze tracing experiments.
In these experiments,
mice are injected with a modified virus which travels along
axons and causes them to express green fluourescent protein,
highlighting which areas are targets of the injection site.
We use the spatial regularities and symmetries
of projections to "fill in" undersampled areas,
leading to unprecedentedly detailed whole-brain connectivity matrices.

- Download the voxel network or the Python package mcmodels
- Here is the poster I presented at NIPS, 2016.
- I was a TA for the Summer Workshop on the Dynamic Brain in 2016: video

**Respiratory rhythms:**
The pre-Botzinger complex (preBot) is an area of the brainstem
which is essential for breathing...
which is obviously an essential system for us animals!
This area contains a group of neurons which produce coordinated
rhythmic bursts of activity which drive motoneurons
to produce inspiration (the inward breath).
Nearby areas produce the other phases of breathing,
such as expiration, but none of them are as essential as preBot.
Thus, preBot is often called the *kernel* of the respiratory central
pattern generator.

Significant research has gone into identifying the properties of neurons in this area, but there continues to be disagreement about the mechanism of rhythmic bursting in preBot. We are using models to test whether the preBot rhythm is an example of emergent synchronization of neural activity. We specifically look at the role of inhibition and network connectivity in shaping the resulting rhythm.

- This work is a collaboration with Nino Ramirez's group at Seattle Children's Research Institute.

**Random graphs and percolation/contagion:**
We examine the topology of random networks with
arbitrary correlations between nodes of different types.
From this, we can determine whether spreading is possible
for different generalized contagion/diffusion processes.
Inhomogeneous response functions among nodes as well as the
ability for nodes to "turn off" can lead to more complex
behavior.

Working with Ioana Dumitriu and Gerandy Brito, we've found semi-analytical techniques for computing eigenvalue spectra of random graphs. We can numerically solve recursive equations to find the spectral distribution. This technique works even for random graph models with complicated block structure, cases where the equations appear too difficult to solve by hand. We have also studied the spectral gap in certain bipartite, biregular random graph families, with applications to matrix completion, community detection, and expander codes.

### Older Research

These projects were started before my PhD.

**Happiness of online interactions:**
Here we look at a basic measure of the happiness
in large-scale texts. A psychological study called
"Associative Norms for English Words" (ANEW) asked a
number of adults to assign a happiness value or
"valence" between 1 and 9 to 1035 common words. When considering
a large enough sample of text, the average valence
of the ANEW words contained therein can give a rough
measure of the text's emotional content. This was recently
published in a
paper
by Danforth and Dodds in the Journal of Happiness Studies.
Together with the group at the
Computational Story Lab
we expanded this study to the social networking site Twitter.

- Hedonometer is a website I helped design which allows users to interactively explore these data

**Chaotic convection, a platform for data assimilation:**
I worked with
Chris Danforth,
Darren Hitt,
Nick Allgaier,
and El Hassan Ridouane
to build and analyze a physical analog of
the 1963 Lorenz system, the first example of deterministic chaos.
It's called a thermosyphon and is a type of non-mechanical
heat pump or convection loop.
We use a combination of data assimilation and
ensemble forecasting methods to predict the occurance of flow
reversals we call regime chnages. I've uploaded a
cool video
that shows one measure of the system's stability on the attractor.
Here is a presentation
outlining our research that was
given at the UVM Student Research Conference.

- My undergraduate honors thesis

**Leaf vascular networks:**
Also known as venation patterns, a
number of mathematical models have been proposed to explain the
branching structures of leaves.
Qinglan Xia
proposed a model in 2007 which grows each leaf while
simultaneously minimizing a cost function.
This approach produces both the vascular network and
boundary of the leaf. Varying the parameters of the
model can produce many of the shapes observed in nature,
including that of the maple leaf (left, my reporduction).
I began this as a project in the networks classes of
Peter Dodds.

- Term paper on the project

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