Student Feature - Alexandria Volkening

By Invited Student Contributor

My research is in applied dynamical systems, and I am particularly interested in the study of self-organization and emergent behavior. Many natural and social phenomena can be viewed as involving small agents coming together to produce larger dynamics: whether at the cellular level in the case of animal skin patterns or at the level of individuals in the setting of political elections, I think the fact that patterns can emerge from the interactions of agents is fascinating, and I love to look for collective behavior in everyday life (e.g. traffic flow).

I completed my PhD in applied mathematics with Bjorn Sandstede at Brown University in May 2017. My thesis was on zebrafish skin patterns, and my interest in agent-based dynamics started when Bjorn suggested I read Andrew Bernoff and Chad Topaz’s work [1]. I think self-organization gives rise to both important applied questions and rich mathematical problems, and I strive to work closely with each application to develop and analyze predictive agent-based models and continuum limits.  Because patterns emerge from the actions of individuals, one natural question is how changing these rules of behavior alters the group dynamics.

Individual pigment cells direct skin patterning on zebrafish during early development, and if these interactions are altered, a mutated pattern appears. This provides an opportunity to link genome to phenome, and the goal of my research is to better understand wild-type cell interactions and then predict altered cell behaviors in mutations. My thesis with Bjorn began with developing an agent-based model that included two types of cells [2] and our more recent work [3] makes predictions about what cell behaviors may distinguish the patterns on zebrafish from those on closely related fish. 

From a mathematical perspective, I am very interested in linking agent-based models with more analytically tractable approaches (e.g. PDEs). For example, zebrafish are often described using the framework of Turing pattern formation [4], yet obtaining reaction-diffusion equations as a continuum limit requires scaling intrinsic length scales by 1/N as the number of cells N goes to infinity [5]. When the underlying dynamics are discrete and involve long-range cellular communication on growing domains, what PDE models are most appropriate? What length scales are important to preserve and how can we more closely relate the parameters in agent-based models with their reduced PDE equivalents? In the future, I plan to explore zebrafish patterning using Cahn-Hilliard equations [6,7] or non-local conservation law models [1,8].

As a postdoc at the Mathematical Biosciences Institute at Ohio State, I have also branched out into two new areas within self-organization and emergent behavior. First, with Daniel Linder, Mason Porter, and Grzegorz Rempala, I am developing models of political opinion dynamics and US election prediction. Our work uses polling data from HuffPost [9] to parametrize multi-compartmental SIS epidemic models to previous senate, gubernatorial, and presidential elections. I am looking forward to using our models to produce predictions on the upcoming midterm elections. While our initial models are simplified, in the future I am interested in taking into account more details and working at the level of individual voters, rather than state populations, using agent-based or network models.   

   In a second direction, together with Chuan Xue, I am developing agent-based models of self-organization within nerve cells: healthy neurons typically feature smooth axons, but the presence and persistence of swollen beads (varicosities) along axons is a signature of traumatic brain injury. New work by the Gu lab [10] has shown that temporary varicosity formation can be induced by applying mechanical stress to axons in vitro. It has been suggested that varicosities can be viewed as intracellular traffic jams: stress may destabilize and rapidly shorten intracellular roadways (microtubules), leading to localized build-up of cargoes. Our goal is to test this hypothesis and explore what determines where varicosities appear and whether they persist or disappear. 

In addition to my main research, I was fortunate to participate in an AMS Math Research Community (MRC) in 2015, and it led to a collaborative project with Yuxin Chen, John Gemmer, and Mary Silber. This work (on noise-induced tipping in periodically forced systems [11]) provided the opportunity for me to learn numerical continuation, and I recommend MRCs as a great way to extend your research into new areas. Over the last year, as part of a team of people headed by Andrew Bernoff, I co-organized an MRC on agent-based modeling in biology and social science that took place in June [12]. It was a terrific learning experience to help organize a workshop and propose/orchestrate projects, and I am really excited about the new research directions in social science (related to pedestrian dynamics and the anatomy of social movements on Twitter) stemming from this program.

Outside research, I am passionate about sharing math with the broader local community. At Brown I was part of the Math CoOp [13], a group headed by Kavita Ramanan that presents math to K-12 students in engaging ways. My presentation was on stability and instability: when do small changes make a difference? More recently, I have been giving interactive talks at Columbus elementary schools to help broaden how students view math: I use black and yellow perler beads (“pigment cells”) to guide students through making their own zebrafish pattern by following simple rules. My talk ends with the students changing the rules in my Matlab code to produce their own mutated patterns in silico. 

Future directions: much of my work will continue to focus on agent-based dynamics, and I am also very interested in the mathematical questions that come out of application-driven research. Whether the application is biological, social, or ecological, I like the challenge of finding connections across disciplines, and I see my future research program as combining agent-based, data-driven, and continuum modeling with dynamical systems and PDE techniques to explore problems collaboratively in self-organization and emergent behavior. I am fortunate to be part of the applied math community, and I hope to meet you at a conference in the future!


[1] A. J. Bernoff and C. M. Topaz. A primer of swarm equilibria, SIAM Journal of Applied Dynamical Systems, 10 (2011), pp. 212-250.

[2] A. Volkening and B. Sandstede. Modelling stripe formation in zebrafish: an agent-based approach, Journal of the Royal Society Interface, 12 (2015), DOI: 10.1098/rsif.2015.0812.

[3] A. Volkening and B. Sandstede. Iridophores as a source of robustness in zebrafish stripes and variability in Danio patterns. To appear in Nature Communications.

[4] A. M. Turing. The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B: Biological Sciences, 237 (1952), pp. 37-72.

[5] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probability Theory and Related Fields, 82 (1989), pp. 565-586.

[6] J.W. Cahn and J.E. Hilliard. Free energy of a non-uniform system I. Interfacial free energy, Journal of Chemical Physics, 28 (1958), pp. 258-267.

[7] Q.-X. Liu, A. Doelman, V. Rottschäfer, M. de Jager, P.M.J. Herman, M. Rietkerk, and J. van de Koppel. Phase separation explains a new class of self-organized spatial patterns in ecological systems, Proceedings of the National Academy of Sciences, 110 (2013), pp. 11905-11910.

[8] J. A. Carrillo, F. Filbet, and M. Schmidtchen. Convergence of a finite volume scheme for a system of interacting species with cross-diffusion. (2018), arXiv:1804.04385.

[9] HuffPost Pollster:

[10] Y. Gu, P. Jukkola, Q. Wang, T. Esparza, Y. Zhao, D. Brody, and C. Gu. Polarity of varicosity initiation in central neuron mechanosensation, Journal of Cellular Biology (2017), jcb.201606065.

[11] Y. Chen, J. Gemmer, M. Silber, and A. Volkening. Noise-induced tipping in a periodically-forced system: the noise-drift balanced regime. In review (2018). arXiv:1801.05395.

[12] AMS MRC on Agent-Based Modeling:

[13] Brown Math CoOp

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