2009 — 2013 
Curto, Carina 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Stimulus Representation and Spontaneous Activity in Recurrent Networks @ University of NebraskaLincoln
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 1115).
This research consists of three projects that will develop new theoretical approaches and experimental predictions about recurrent networks in the mammalian brain. The first two projects are primarily concerned with stimulus representation via recurrent networks, while the third project focuses on spontaneous activity. Project 1 formalizes and develops an algebrageometric framework previously introduced in joint work with Vladimir Itskov. The main idea is to associate geometry and topology to population spike train data  primarily from electrophysiological recordings in cortex and hippocampus. Project 2 explores the relationship between stimulus space topology and recurrent network connectivity. The goal of this project is to find experimental predictions about the connectivity of recurrent networks in cases where either the stimulus space structure or the topographic structure of the underlying network is known. Project 3 investigates the interaction between the activity of single neurons and the population activity of the local network.
Building accurate representations of the world is one of the basic functions of the brain. When a stimulus is paired with pleasure or pain, an animal quickly learns the association. However, we also learn the (neutral) relationships between stimuli of the same type. For example, a bar held at a 45degree angle seems closer to one held at 50 degrees than to a perfectly vertical one. Upon hearing a pair of distinct pure tones, one seems higher than the other. We do not perceive the world as a stream of unrelated stimuli; rather, our brains organize different types of stimuli into structured stimulus spaces. Regardless of immediate relevance to survival, it appears to be beneficial for the brain to reflect as much structure as possible of the outside world. The main goal of this research is to improve our understanding of how stimulus spaces are represented via recurrent networks in the brain. To this end, we develop ideas and techniques coming from geometry and topology to extract novel insights from neuroscience experiments. How the brain represents the world is a fundamental and ageold question. Progress in understanding how the structure of neural circuits underlies representational functions may also provide clues in the study of neurological disease.

1 
2011 — 2018 
Hartke, Stephen (coPI) [⬀] Hunter, Amber Toundykov, Daniel (coPI) [⬀] Rebarber, Richard [⬀] Marley, Thomas (coPI) [⬀] Woodward, Gordon Curto, Carina 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Nebraska Math Scholars @ University of NebraskaLincoln
The Nebraska Math Scholars Program provides scholarships to financially disadvantaged undergraduate students who major in mathematics at the University of NebraskaLincoln (UNL). The program attracts and retains talented students by encouraging them to be part of a network of students who are excited about college and about mathematics. Nebraska Math Scholars participate in an active Math Club, an undergraduate seminar series, regular casual group meetings with faculty and graduate students, and enhanced advising. Math Scholars attend special workshops and recitation sections where they work in groups on problems designed to strengthen the skills and understanding of concepts necessary for success in their precalculus and calculus courses. During their undergraduate careers, the Math Scholars are encouraged to engage in supervised research projects with faculty mentors. In addition, the Nebraska Math Scholars Program includes a bridge program for underprepared graduate students. The SSTEM Bridge to Graduate School program permits meritorious students to take an extra senior year of mathematics before beginning graduate studies. The Nebraska Math Scholars Program is increasing the percentage of disadvantaged and/or underrepresented students who succeed academically in STEM fields while expanding the pipeline of students applying to graduate school.

1 
2012 — 2016 
Curto, Carina 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Memory Encoding in Spatially Structured Networks: Dynamics, Discrete Geometry & Topology @ University of NebraskaLincoln
Hippocampal networks are believed to be a major center of associative learning due to the central role of the hippocampus in learning and memory, as well as the relatively high levels of recurrent connectivity and synaptic plasticity. The lack of topographic structure in hippocampus has made it a natural inspiration for associative memory models, such as the Hopfield model, for encoding memories in unstructured recurrent networks. At the same time, studies in rodents have uncovered the critical role of hippocampus in spatial navigation and, more recently, timetracking. In contrast to associative memory encoding, these functions have been successfully modeled using spatially structured networks. How can these viewpoints be reconciled? The central goal of this research is to develop a mathematical theory of memory encoding in spatially structured networks, and to study the neural codes that arise from such networks. Specifically, the research will develop mathematical theory to answer the following questions: (1) How can overlapping memory patterns be encoded precisely as attractors of an unstructured neural network, without introducing unwanted "spurious states"? (2) How can memories be encoded in a spatially structured network, such as a bump attractor network, while maintaining functions that depend on the network's spatial organization? (3) Aside from error correction, what are the advantages of redundancy in a neural code, such as the hippocampal place field code, that is characterized by heavily overlapping receptive fields? This last question will also be explored via the analysis of cortical and hippocampal data sets provided by collaborating labs.
The hippocampus is often thought of as a "Swiss knife" in the brain. Decades of experimental work have uncovered its essential role in learning and memory, as well as in spatial navigation. From a theoretical standpoint, it is puzzling how the same neural network can achieve such disparate functions. In particular, mathematical models of memory encoding are fundamentally quite different from models of spatial navigation. This work will integrate these two major types of neural network models, with the goal of understanding how the hippocampus can support multiple important functions. At its core, the research will advance the mathematical theory behind our understanding of networklevel computation in the brain.

1 
2015 — 2018 
Curto, Carina 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Theory of ThresholdLinear Networks and Combinatorial Neural Codes. @ Pennsylvania State Univ University Park
How do connections between neurons store memories and shape the dynamics of neural activity in the brain? How do firing patterns of neurons represent our sensory experiences? The advent of technologies that facilitate simultaneous recordings of large populations of neurons present new opportunities to answer these classical questions of neuroscience. There are mathematical models that are frequently used in network simulations and data analyses that can be employed, but whose mathematical properties are still poorly understood. To guide these efforts, a better understanding of theoretical models of recurrent networks and population codes is essential. This research will focus on two such examples: thresholdlinear networks and combinatorial neural codes. The goal is to produce major advances in the mathematical theory of these models, with an eye towards neuroscience applications. Part of the research will involve the analyses of neural activity in the cortex and hippocampus, in collaboration with experimentalists. Despite the focus on neuroscience, the mathematical results have the potential to be sufficiently general so as to be useful in a variety of broader contexts in the biological and social sciences.
A thresholdlinear network is a common firing rate model for a recurrent network, with a threshold nonlinearity. These networks generically exhibit multiple stable fixed points, and multistability makes them attractive as models for memory storage and retrieval. Preliminary results have shown that the equilibria possess a rich combinatorial structure, and can be analyzed using ideas from classical distance geometry. The first project will build on this understanding in order to develop a more complete picture of the structure of fixed points and higherdimensional attractors of these networks. A combinatorial neural code is a collection of binary patterns for a population of neurons. The second project will develop an algebraic classification of combinatorial codes, using the recently developed framework of the neural ring. The neural ring encodes information about a neural code in a manner that makes properties such as receptive field organization most transparent. The resulting methods will be tested and refined using electrophysiological recordings of place cells in the hippocampus. This research will also generate new and interesting problems at the interface of neuroscience with applied algebra, combinatorics, and geometry.

0.943 
2016 — 2018 
Curto, Carina 
R01Activity Code Description: To support a discrete, specified, circumscribed project to be performed by the named investigator(s) in an area representing his or her specific interest and competencies. 
Emergent Dynamics From Network Connectivity: a Minimal Model @ Pennsylvania State UniversityUniv Park
Project Summary Even in the absence of changing sensory inputs, many networks in the brain exhibit emer gent dynamics: that is, they display patterns of neural activity that are shaped by the intrinsic structure of the network, rather than modulated by an external input. Such dynamics are be lieved to underlie central pattern generators (CPGs) for locomotion, oscillatory activity in cortex and hippocampus, and the complex interplay between sensorydriven responses and ongoing spontaneous activity. The goal of this research is to develop a theory of how emergent dynamics can arise solely from the structure of connectivity between neurons. We will do this in the con text of a simple but fundamentally nonlinear model: the Combinatorial ThresholdLinear Network (CTLN) model. This model has binary synapses and simple, perceptronlike neurons, ensuring that any emergent dynamics arise purely from the structure of connections, as described by a directed graph. Despite its simplicity, the CTLN model captures the full range of nonlinear dynamic behav iors observed in neural systems within a single model framework. Crucially, the model is also mathematically tractable, allowing us to prove very general results (theorems) that help guide the applications. We have already obtained theoretical results that enable us to reason about the underlying connectivity graph and obtain valid (and intuitive) predictions about the resulting dynamics. Our speci?c aims will extend these results and develop several applications. Specif ically, we will further develop the theory of emergent dynamics in the CTLN model, completing our classi?cation of ?xed point attractors and investigating the transition to chaos as a function of connection sparsity. We will use the same framework to study pattern generation in small net works, in order to understand how sequences and complex rhythms emerge from the structure of connectivity. Finally, we will apply our theoretical results to investigate a variety of dynamic phenomena in hippocampus in a single unifying model.

0.943 