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Organizing Committee
Allen Institute
The Pennsylvania State University
University of Chicago
University of Colorado Boulder
Rutgers University
University of Northern Colorado
Abstract
One of the fundamental questions in neuroscience is to understand how network connectivity shapes neural activity. Over the last 10 years, tremendous progress has been made in collecting neural activity and connectivity data, but theoretical advances have lagged behind. This workshop will focus on identifying mathematical challenges that arise in studying the dynamics of learning, memory, plasticity, decision-making, sequence generation, and central pattern generator circuits. Mathematical ideas and approaches from dynamical systems, statistical mechanics, linear algebra, graph theory, topology, and traditional areas of applied mathematics are all expected to play an important role.
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Workshop Schedule
Monday, September 18, 2023
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Session Chair
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Neural dynamics on sparse networks—pruning, error correction, and signal reconstruction
11th Floor Lecture Hall
Speaker
Rishidev Chaudhuri, University of California, Davis
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Many networks in the brain are sparsely connected, and the brain eliminates connections during development and learning. This talk will focus on questions related to computation and dynamics on these sparse networks.
We will first focus on pruning redundant network connections while preserving dynamics and function. In a recurrent network, determining the importance of a connection between two neurons is a difficult computational problem, depending on the role that both neurons play and on all possible pathways of information flow between them. Noise is ubiquitous in neural systems, and often considered an irritant to be overcome. We suggest that noise could instead play a functional role in pruning, allowing the brain to probe network structure and determine which connections are redundant. We construct a simple, local, unsupervised rule that either strengthens or prunes synapses using only connection weight and the noise-driven covariance of the neighboring neurons. For a subset of linear and rectified-linear networks, we adapt matrix concentration of measure arguments from the field of graph sparsification to prove that this rule preserves the spectrum of the original matrix and hence preserves network dynamics even when the fraction of pruned connections asymptotically approaches 1. The plasticity rule is biologically-plausible and may suggest a new role for noise in neural computation.
Time permitting, we will then discuss the application of sparse expander graphs to modeling dynamics on neural networks. Expander graphs combine the seemingly contradictory properties of being sparse and well-connected. Among other remarkable properties, they allow efficient communication, credit assignment and error correction with simple greedy dynamical rules. We suggest that these applications might provide new ways of thinking about neural dynamics, and provide several proofs of principle.
10:00 - 10:15 AM EDT
Coffee Break
11th Floor Collaborative Space
10:15 - 11:00 AM EDT
Local breakdown of the balance of excitation and inhibition accounts for divisive normalization
11th Floor Lecture Hall
Speaker
Yashar Ahmadian, Cambridge University
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Excitatory and inhibitory (E & I) inputs to cortical neurons remain balanced across different conditions. This is captured in the balanced network model in which neural populations dynamically adjust their rates to yield tightly balanced E and I inputs and a state in which all neurons are active at levels observed in cortex. But global tight E-I balance predicts linear stimulus dependence for population responses, and does not account for systematic cortical response nonlinearities such as divisive normalization, a canonical brain computation. However, when necessary connectivity conditions for global balance fail, states arise in which a subset of neurons are inhibition dominated and inactive. Here, we show analytically that the emergence of such localized balance states robustly leads to normalization, including sublinear integration and winner-take-all behavior. An alternative model that exhibits normalization is the Stabilized Supralinear Network (SSN), in which the E-I balance is generically loose, but becomes tight asymptotically for strong inputs. However, an understanding of the causal relationship between E-I balance and normalization in SSN are lacking. Here we show that when tight E-I balance in the asymptotic, strongly driven regime of SSN is not global, the network does not exhibit normalization at any input strength; thus, in SSN too, significant normalization requires the breakdown of global balance. In summary, we causally and quantitatively connect a fundamental feature of cortical dynamics with a canonical brain computation.
11:15 - 11:45 AM EDT
Open Problems Discussion
Problem Session - 11th Floor Lecture Hall
Session Chair
Carina Curto, The Pennsylvania State University
Session Chair
Katie Morrison, University of Northern Colorado
11:45 AM - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:15 PM EDT
Discovering dynamical patterns of activity from single-trial neural data
11th Floor Lecture Hall
Speaker
Rodica Curtu, The University of Iowa
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
In this talk I will discuss a data-driven method that leverages time-delayed coordinates, diffusion maps, and dynamic mode decomposition, to identify neural features in large scale brain recordings that correlate with subject-reported perception. The method captures the dynamics of perception at multiple timescales and distinguishes attributes of neural encoding of the stimulus from those encoding the perceptual states. Our analysis reveals a set of latent variables that exhibit alternating dynamics along a low-dimensional manifold, like trajectories of attractor-based models. I will conclude by proposing a phase-amplitude-coupling-based model that illustrates the dynamics of data.
2:30 - 2:35 PM EDT
Synaptic mechanisms for resisting distractors in neural fields
Lightning Talks - 11th Floor Lecture Hall
Speaker
Heather Cihak, University of Colorado Boulder
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Persistent neural activity has been observed in the non-human primate cortex when making delayed estimates. Organized activity patterns according to cell feature preference reveals "bumps" that represent analog variables during the delay. Continuum neural field models support bump attractors whose stochastic dynamics can be linked to response statistics (estimate bias and error). Models often ignore the distinct dynamics of bumps in both excitatory/inhibitory population activity, but recent neural and behavioral recordings suggest both play a role in delayed estimate codes and responses. In past work, we developed new methods in asymptotic and multiscale analyses for stochastic and spatiotemporal systems to understand how network architecture determines bump dynamics in networks with distinct E/I populations and short term plasticity. The inhibitory bump dynamics as well as facilitation and diffusion impact the stability and wandering motion of the excitatory bump. Our current work moves beyond studying ensemble statistics like variance to examine potential mechanisms underlying the robustness of working memory to distractors (irrelevant information) presented during the maintenance period wherein the relative timescales of the E/I populations, synaptic vs activity dynamics, as well as short term plasticity may play an important role.
2:35 - 2:40 PM EDT
Convex optimization of recurrent neural networks for rapid inference of neural dynamics
Lightning Talks - 11th Floor Lecture Hall
Speaker
Fatih Dinc, Stanford University
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Advances in optical and electrophysiological recording technologies have made it possible to record the dynamics of thousands of neurons, opening up new possibilities for interpreting and controlling large neural populations. A promising way to extract computational principles from these large datasets is to train data-constrained recurrent neural networks (dRNNs). However, existing training algorithms for dRNNs are inefficient and have limited scalability, making it a challenge to analyze large neural recordings even in offline scenarios. To address these issues, we introduce a training method termed Convex Optimization of Recurrent Neural Networks (CORNN). In studies of simulated recordings of hundreds of cells, CORNN attained training speeds ~ 100-fold faster than traditional optimization approaches while maintaining or enhancing modeling accuracy. We further validated CORNN on simulations with thousands of cells that performed simple computations such as those of a 3-bit flip-flop or the execution of a timed response. Finally, we showed that CORNN can robustly reproduce network dynamics and underlying attractor structures despite mismatches between generator and inference models, severe subsampling of observed neurons, or mismatches in neural time-scales. Overall, by training dRNNs with millions of parameters in subminute processing times on a standard computer, CORNN constitutes a first step towards real-time network reproduction constrained on large-scale neural recordings and a powerful computational tool for advancing the understanding of neural computation. My talk focuses on how dRNNs enabled by CORNN can help us reverse engineer the neural code in the mammalian brain.
2:40 - 2:45 PM EDT
Recall tempo of Hebbian sequences depends on the interplay of Hebbian kernel with tutor signal timing
Lightning Talks - 11th Floor Lecture Hall
Speaker
Matthew Farrell, Harvard University
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Understanding how neural circuits generate sequential activity is a longstanding challenge. While foundational theoretical models have shown how sequences can be stored as memories with Hebbian plasticity rules, these models considered only a narrow range of Hebbian rules. In this talk I introduce a model for arbitrary Hebbian plasticity rules, capturing the diversity of spike-timing-dependent synaptic plasticity seen in experiments, and show how the choice of these rules and of neural activity patterns influences sequence memory formation and retrieval. In particular, I will present a general theory that predicts the speed of sequence replay. This theory lays a foundation for explaining how cortical tutor signals might give rise to motor actions that eventually become ``automatic''. This theory also captures the impact of changing the speed of the tutor signal. Beyond shedding light on biological circuits, this theory has relevance in artificial intelligence by laying a foundation for frameworks whereby slow and computationally expensive deliberation can be stored as memories and eventually replaced by inexpensive recall.
2:45 - 2:50 PM EDT
Modeling human temporal EEG responses subject to VR visual stimuli
Lightning Talks - 11th Floor Lecture Hall
Speaker
Richard Foster, Virginia Commonwealth University
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
When subject to visual stimuli flashing at a constant temporal frequency, it is well-known that the EEG response has a sharp peak in the power spectrum at the driving frequency. But the EEG response with random frequency stimuli and corresponding biophysical mechanisms are largely unknown. We present a phenomenological model framework in hopes of eventually capturing these EEG responses and unveiling the biophysical mechanisms. Based on observed heterogeneous temporal frequency selectivity curves in V1 cells (Hawken et al. ‘96, Camillo et al ‘20, Priebe et al. ‘06), we endow individual units with these response properties. Preliminary simulation results show that particular temporal frequency selectivity curves can be more indicative of the EEG response. Future directions include the construction of network architecture with interacting units to faithfully model the EEG response.
2:50 - 2:55 PM EDT
RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Leo Kozachkov, Massachusetts Institute of Technology
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.
3:15 - 3:45 PM EDT
Coffee Break
11th Floor Collaborative Space
3:45 - 4:30 PM EDT
Universal Properties of Strongly Coupled Recurrent Networks
11th Floor Lecture Hall
Speaker
Robert Rosenbaum, University of Notre Dame
Session Chair
Carina Curto, The Pennsylvania State University
Abstract
Balanced excitation and inhibition is widely observed in cortex. How does this balance shape neural computations and stimulus representations? This question is often studied using computational models of neuronal networks in a dynamically balanced state. But balanced network models predict a linear relationship between stimuli and population responses. So how do cortical circuits implement nonlinear representations and computations? We show that every balanced network architecture admits stimuli that break the balanced state and these breaks in balance push the network into a “semi-balanced state” characterized by excess inhibition to some neurons, but an absence of excess excitation. The semi-balanced state produces nonlinear stimulus representations and nonlinear computations, is unavoidable in networks driven by multiple stimuli, is consistent with cortical recordings, and has a direct mathematical relationship to artificial neural networks.
4:30 - 6:00 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, September 19, 2023
9:00 - 9:45 AM EDT
Multilayer Networks in Neuroscience
11th Floor Lecture Hall
Speaker
Mason Porter, UCLA
Session Chair
Brent Doiron, University of Chicago
Abstract
I will discuss multilayer networks in neuroscience. I will introduce the idea of multilayer networks and discuss some uses of multilayer networks in dneuroscience. I will present some interesting challenges.
10:00 - 10:15 AM EDT
Coffee Break
11th Floor Collaborative Space
10:15 - 11:00 AM EDT
State modulation in spatial networks of multiple interneuron subtypes
11th Floor Lecture Hall
Speaker
Chengcheng Huang, University of Pittsburgh
Session Chair
Brent Doiron, University of Chicago
Abstract
Neuronal responses to sensory stimuli can be strongly modulated by animal's brain state. Three distinct subtypes of inhibitory interneurons, parvalbumin (PV), somatostatin (SOM), and vasoactive intestinal peptide (VIP) expressing cells, have been identified as key players of flexibly modulating network activity. The three interneuron populations have specialized local microcircuit motifs and are targeted differentially by neuromodulators and top-down inputs from higher-order cortical areas. In this work, we systematically study the function of each interneuron cell type at modulating network dynamics in a spatially ordered spiking neuron network. We analyze the changes in firing rates and network synchrony as we apply static current to each cell population. We find that the modulation pattern by activating E or PV cells is distinct from that by activating SOM or VIP cells. In particular, we identify SOM cells as the main driver of network synchrony.
11:15 - 11:45 AM EDT
Open Problems Discussion
Problem Session - 11th Floor Lecture Hall
Session Chair
Brent Doiron, University of Chicago
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
11:50 AM - 12:00 PM EDT
Group Photo (Immediately After Talk)
11th Floor Lecture Hall
12:00 - 1:30 PM EDT
Working Lunch
11th Floor Collaborative Space
1:30 - 2:15 PM EDT
Plasticity in balanced neuronal networks
11th Floor Lecture Hall
Speaker
Kresimir Josic, University of Houston
Session Chair
Brent Doiron, University of Chicago
Abstract
I will first describe how to extend the theory of balanced networks to account for synaptic plasticity. This theory can be used to show when a plastic network will maintain balance, and when it will be driven into an unbalanced state. I will next discuss how this approach provides evidence for a novel form of rapid compensatory inhibitory plasticity using experimental evidence obtained using optogenetic activation of excitatory neurons in primate visual cortex (area V1). The theory explains how such activation induces a population-wide dynamic reduction in the strength of neuronal interactions over the timescale of minutes during the awake state, but not during rest. I will shift gears in the final part of the talk, and discuss how community detection algorithms can help uncover the large scale organization of neuronal networks from connectome data, using the Drosophila hemibrain dataset as an example.
2:35 - 2:40 PM EDT
Q-Phase reduction of multi-dimensional stochastic Ornstein-Uhlenbeck process networks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Maxwell Kreider, Case Western Reserve University
Session Chair
Brent Doiron, University of Chicago
Abstract
Phase reduction is an effective tool to study the network dynamics of deterministic limit-cycle oscillators. The recent introduction of stochastic phase concepts allows us to extend these tools to stochastic oscillators; of particular utility is the asymptotic stochastic phase, derived from the eigenfunction decomposition of the system's probability density. Here, we study networks of coupled oscillatory two-dimensional Ornstein-Uhlenbeck processes (OUPs) with complex eigenvalues. We characterize system dynamics by providing an exact expression for the asymptotic stochastic phase for OUP networks of any dimension and arbitrary coupling structure. Furthermore, we introduce an order parameter quantifying the synchrony of networks of stochastic oscillators, and apply it to our OUP model. We argue that the OUP network provides a new, analytically tractable approach to analysis of large scale electrophysiological recordings.
2:40 - 2:45 PM EDT
Feedback Controllability as a Normative Theory of Neural Dynamics
Lightning Talks - 11th Floor Lecture Hall
Speaker
Ankit Kumar, UC Berkeley
Session Chair
Brent Doiron, University of Chicago
Abstract
Brain computations emerge from the collective dynamics of distributed neural populations. Behaviors including reaching and speech are explained by principles of optimal feedback control, yet if and how this normative description shapes neural population dynamics is unknown. We created dimensionality reduction methods that identify subspaces of dynamics that are most feedforward controllable (FFC) vs. feedback controllable (FBC). We show that FBC and FFC subspaces diverge for dynamics generated by non-normal connectivity. In neural recordings from monkey M1 and S1 during reaching, FBC subspaces are better decoders of reach velocity, particularly during reach acceleration, and that FBC provides a first principles account of the observation of rotational dynamics. Overall, our results demonstrate feedback controllability is a novel, normative theory of neural population dynamics, and reveal how the structure of high dynamical systems shape their ability to be controlled.
2:45 - 2:50 PM EDT
Adaptive whitening with fast gain modulation and slow synaptic plasticity
Lightning Talks - 11th Floor Lecture Hall
Speaker
David Lipshutz, Flatiron Institute
Session Chair
Brent Doiron, University of Chicago
Abstract
Neurons in early sensory areas rapidly adapt to changing sensory statistics, both by normalizing the variance of their individual responses and by reducing correlations between their responses. Together, these transformations may be viewed as an adaptive form of statistical whitening. In this talk, I will present a normative multi-timescale mechanistic model of adaptive whitening with complementary computational roles for gain modulation and synaptic plasticity. Gains are modified on a fast timescale to adapt to the current statistical context, whereas synapses are modified on a slow timescale to learn structural properties of the input statistics that are invariant across contexts.
2:50 - 2:55 PM EDT
The combinatorial code and the graph rules of Dale networks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Nikola Milicevic, Pennsylvania State University
Session Chair
Brent Doiron, University of Chicago
Abstract
We describe the combinatorics of equilibria and steady states of neurons in threshold-linear networks that satisfy the Dale’s law. The combinatorial code of a Dale network is characterized in terms of two conditions: (i) a condition on the network connectivity graph, and (ii) a spectral condition on the synaptic matrix. We find that in the weak coupling regime the combinatorial code depends only on the connectivity graph, and not on the particulars of the synaptic strengths. Moreover, we prove that the combinatorial code of a weakly coupled network is a sublattice, and we provide a learning rule for encoding a sublattice in a weakly coupled excitatory network. In the strong coupling regime we prove that the combinatorial code of a generic Dale network is intersection-complete and is therefore a convex code, as is common in some sensory systems in the brain.
2:55 - 3:00 PM EDT
Decomposed Linear Dynamical Systems for Studying Inter and Intra-Region Neural Dynamics
Lightning Talks - 11th Floor Lecture Hall
Speaker
Noga Mudrik, The Johns Hopkins University
Session Chair
Brent Doiron, University of Chicago
Abstract
Understanding the intricate relationship between recorded neural activity and behavior is a pivotal pursuit in neuroscience. However, existing models frequently overlook the non-linear and non-stationary behavior evident in neural data, opting instead to center their focus on simplified projections or overt dynamical systems. We introduce a Decomposed Linear Dynamical Systems (dLDS) approach to capture these complex dynamics by representing them as a sparse time-varying linear combination of interpretable linear dynamical components. dLDS is trained using an expectation maximization procedure where the obscured dynamical components are iteratively inferred using dictionary learning. This approach enables the identification of overlapping circuits, while the sparsity applied during the training maintains the model interpretability. We demonstrate that dLDS successfully recovers the underlying linear components and their time-varying coefficients in both synthetic and neural data examples, and show that it can learn efficient representations of complex data. By leveraging the rich data from the International Brain Laboratory’s Brain Wide Map dataset, we extend dLDS to model communication among ensembles within and between brain regions, drawing insights from multiple non-simultaneous recording sessions.
3:00 - 3:05 PM EDT
Characterizing Neural Spike Train Data for Chemosensory Coding Analysis
Lightning Talks - 11th Floor Lecture Hall
Speaker
Audrey Nash, Florida State University
Session Chair
Brent Doiron, University of Chicago
Abstract
In this presentation, we explore neural spike train data to discern a neuron's ability to distinguish between various stimuli. By examining both the spiking rate and the temporal distribution of spikes (phase of spiking), we aim to unravel the intricacies of chemosensory coding in neurons. We will provide a concise overview of our methodology for identifying chemosensory coding neurons and delve into the application of metric-based analysis techniques in conjunction with optimal transport methods. This combined approach allows us to uncover emerging patterns in tastant coding across multiple neurons and quantify the respective impacts of spiking rate and temporal phase in taste decoding.
3:05 - 3:10 PM EDT
Infinite-dimensional Dynamics in a Model of EEG Activity in the Neocortex
Lightning Talks - 11th Floor Lecture Hall
Speaker
Farshad Shirani, Georgia Institute of Technology
Session Chair
Brent Doiron, University of Chicago
Abstract
I present key analytical and computational results on a mean field model of electroencephalographic activity in the neocortex, which is composed of a system of coupled ODEs and PDEs. I show that for some sets of biophysical parameter values the equilibrium set of the model is not compact, which further implies that the global attracting set of the model is infinite-dimensional. I also present computational results on generation and spatial propagation of transient gamma oscillations in the solutions of the model. The results identify important challenges in interpreting and modelling the temporal pattern of EEG recordings, caused by low spatial resolution of EEG electrodes.
3:10 - 3:15 PM EDT
What is the optimal topology of setwise connections for a memory network?
Lightning Talks - 11th Floor Lecture Hall
Speaker
Thomas Burns, ICERM
Session Chair
Brent Doiron, University of Chicago
Abstract
Simplicial Hopfield networks (Burns & Fukai, 2023) explicitly model setwise connections between neurons based on a simplicial complex to store memory patterns. Randomly diluted networks -- where only a randomly chosen fraction of the simplices, i.e., setwise connections, have non-zero weights -- show performance above traditional associative memory networks with only pairwise connections between neurons but the same total number of non-zero weighted connections. However, could there be a cleverer choice of connections to weight given known memory patterns we want to store? I suspect so, and in this talk I will to formally pose the problem for others to consider.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Reliability and robustness of oscillations in some slow-fast chaotic systems
11th Floor Lecture Hall
Speaker
Jonathan Jaquette, New Jersey Institute of Technology
Session Chair
Brent Doiron, University of Chicago
Abstract
A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator, we propose a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales.
We show that systems displaying relaxation cycles going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales, thus consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may generate erratic activity also at slow timescales. We analyze in detail these phenomena in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics, and we validate this numerically in other models.
5:30 - 7:00 PM EDT
Networking event with Carney Institute for Brain Science
External Event - Carney Institute for Brain Science - 164 Angell St, Providence RI, 02906
Wednesday, September 20, 2023
9:00 - 9:45 AM EDT
Modeling in neuroscience: the challenges of biological realism and computability
11th Floor Lecture Hall
Speaker
Lai-Sang Young, Courant Institute
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Biologically realistic models of the brain have the potential to offer insight into neural mechanisms; they have predictive power, the ultimate goal of biological modeling. These benefits, however, come at considerable costs: network models that involve hundreds of thousands of neurons and many (unknown) parameters are unwieldy to build and to test, let alone to simulate and to analyze. Reduced models have obvious advantages, but the farther removed from biology a model is, the harder it is to draw meaningful inferences. In this talk, I propose a modeling strategy that aspires to be both realistic and computable. Two crucial ingredients are (i) we track neuronal dynamics on two spatial scales: coarse-grained dynamics informed by local activity, and (ii) we compute a family of potential local responses in advance, eliminating the need to perform similar computations at each spatial location in each update. I will illustrate this computational strategy using a model of the monkey visual cortex, which is very similar to that of humans.
10:00 - 10:15 AM EDT
Coffee Break
11th Floor Collaborative Space
10:15 - 11:00 AM EDT
Uncertainty Quantification for Neurobiological Networks.
11th Floor Lecture Hall
Speaker
Daniele Avitabile, Vrije Universiteit Amsterdam
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
This talk presents a framework for forward uncertainty quantification problems in spatially-extended neurobiological networks. We will consider networks in which the cortex is represented as a continuum domain, and local neuronal activity evolves according to an integro-differential equation, collecting inputs nonlocally, from the whole cortex. These models are sometimes referred to as neural field equations.
Large-scale brain simulations of such models are currently performed heuristically, and the numerical analysis of these problems is largely unexplored. In the first part of the talk I will summarise recent developments for the rigorous numerical analysis of projection schemes for deterministic neural fields, which sets the foundation for developing Finite-Element and Spectral schemes for large-scale problems.
The second part of the talk will discuss the case of networks in the presence of uncertainties modelled with random data, in particular: random synaptic connections, external stimuli, neuronal firing rates, and initial conditions. Such problems give rise to random solutions, whose mean, variance, or other quantities of interest have to be estimated using numerical simulations. This so-called forward uncertainty quantification problem is challenging because it couples spatially nonlocal, nonlinear problems to large-dimensional random data.
I will present a family of schemes that couple a spatial projector for the spatial discretisation, to stochastic collocation for the random data. We will analyse the time- dependent problem with random data and the schemes from a functional analytic viewpoint, and show that the proposed methods can achieve spectral accuracy, provided the random data is sufficiently regular. We will showcase the schemes using several examples.
Acknowledgements This talk presents joint work with Francesca Cavallini (VU Amsterdam), Svetlana Dubinkina (VU Amsterdam), and Gabriel Lord (Radboud University).
11:15 AM - 12:00 PM EDT
Open Problems Discussion
Problem Session - 11th Floor Lecture Hall
Session Chair
Konstantin Mischaikow, Rutgers University
Session Chair
Katie Morrison, University of Northern Colorado
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Dynamics of stochastic integrate-and-fire networks
11th Floor Lecture Hall
Speaker
Gabe Ocker, Boston University
Session Chair
Katie Morrison, University of Northern Colorado
3:00 - 3:05 PM EDT
A Step Towards Uncovering The Structure of Multistable Neural Networks
Lightning Talks - 11th Floor Lecture Hall
Speaker
Magnus Tournoy, Flatiron Institute
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
With the experimental advances in the recording of large populations of neurons, theorists are in the humbling position of making sense of a staggering amount of data. One question that will become more into reach is how network structure relates to function. But going beyond explanatory models and becoming more predictive will require a fundamental approach.
In this talk we’ll take the view of a physicist and formulate exact results within a simple, yet general, toy model called Glass networks. Named after its originator Leon Glass, they are the infinite gain limit of well-known circuit models like continuous-time Hopfield networks. We’ll show that, within this limit, stability conditions reduce to semipositivity constraints on the synaptic weight matrix.
Having a clear link between structure and function in possession, the consequences of multistability on the network architecture can be explored. One finding is the factorization of the weight matrix in terms of nonnegative matrices. Interestingly this factorization completely identifies the existence of stable states. Another result is the reduction of allowed sign patterns for the connections. A consequence hereof are lower bounds on the number of excitatory and inhibitory connections. At last we will discuss the special case of “sign stability”, where stability is guaranteed by the topology of the network. Derivations of these results will be supplemented by a number of examples.
3:05 - 3:10 PM EDT
Clustering and Distribution of the Adaptation Variable
Lightning Talks - 11th Floor Lecture Hall
Speaker
Ka Nap Tse, University of Pittsburgh
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Brain wave is an important phenomenon in neuroscience. Besides synchronous spiking, excitatory cells with adaptation can spike in clusters to cause a rhythmic activity of the network. In previous works, the adaptation variable is usually eliminated for further analysis. In this talk, a way to study this clustering behaviour through the evolution of the distribution of the adaptation variable will be discussed. We then transform the distribution to the time-to-spike coordinate for further explorations.
3:10 - 3:15 PM EDT
Low-dimensional manifold of neural oscillations revealed by data-driven model reduction
Lightning Talks - 11th Floor Lecture Hall
Speaker
Zhuo-Cheng Xiao, New York University
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Neural oscillations across various frequency bands are believed to underlie essential brain functions, such as information processing and cognitive activities. However, the emergence of oscillatory dynamics from spiking neuronal networks—and the interplay among different cortical rhythms—has seldom been theoretically explored, largely due to the strong nonlinearity and high dimensionality involved. To address this challenge, we have developed a series of data-driven model reduction methods tailored for spiking network dynamics.
In this talk I will present nearly two-dimensional manifolds in the reduced coordinates that successfully capture the emergence of gamma oscillations. Specifically, we find that the initiation phases of each oscillation cycle are the most critical. Subsequent cycles are more deterministic and lie on the aforementioned two-dimensional manifold. The Poincaré mappings between these initiation phases reveal the structure of the dynamical system and successfully explain the bifurcation from gamma oscillations to multi-band oscillations.
3:15 - 3:20 PM EDT
Sensitivity to control signals in triphasic rhythmic neural systems: a comparative mechanistic analysis via infinitesimal local timing response curves
Lightning Talks - 11th Floor Lecture Hall
Speaker
Zhuojun Yu, Case Western Reserve University
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
Similar activity patterns may arise from model neural networks with distinct coupling properties and individual unit dynamics. These similar patterns may, however, respond differently to parameter variations and, specifically, to tuning of inputs that represent control signals. In this work, we analyze the responses resulting from modulation of a localized input in each of three classes of model neural networks that have been recognized in the literature for their capacity to produce robust three-phase rhythms: coupled fast-slow oscillators, near-heteroclinic oscillators, and threshold-linear networks. Triphasic rhythms, in which each phase consists of a prolonged activation of a corresponding subgroup of neurons followed by a fast transition to another phase, represent a fundamental activity pattern observed across a range of central pattern generators underlying behaviors critical to survival, including respiration, locomotion, and feeding. To perform our analysis, we extend the recently developed local timing response curve (lTRC), which allows us to characterize the timing effects due to perturbations, and we complement our lTRC approach with model-specific dynamical systems analysis. Interestingly, we observe disparate effects of similar perturbations across distinct model classes. Thus, this work provides an analytical framework for studying control of oscillations in nonlinear dynamical systems, and may help guide model selection in future efforts to study systems exhibiting triphasic rhythmic activity.
3:20 - 3:25 PM EDT
Modeling the effects of cell-type specific lateral inhibition
Lightning Talks - 11th Floor Lecture Hall
Speaker
Soon Ho Kim, Georgia Institute of Technology
Session Chair
Katie Morrison, University of Northern Colorado
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Computing the Global Dynamics of Parameterized Families of ODEs
11th Floor Lecture Hall
Speaker
Marcio Gameiro, Rutgers University
Session Chair
Katie Morrison, University of Northern Colorado
Abstract
We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics is extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to systems of ODEs arising from gene regulatory networks.
Thursday, September 21, 2023
9:00 - 9:45 AM EDT
Multiple timescale respiratory dynamics and effect of neuromodulation
11th Floor Lecture Hall
Speaker
Yangyang Wang, Brandeis University
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
Abstract
Respiration is an involuntary process in all living beings required for our survival. The preBötzinger complex (preBötC) in the mammalian brainstem is a neuronal network that drives inspiratory rhythmogenesis, whose activity is constantly modulated by neuromodulators in response to changes in the environment. In this talk, we will discuss chanllenges involved in the analysis of bursting dynamics in preBötC neurons and how these dynamics change during prenatal development. We will also combine insights from in vitro recordings and dynamical systems modeling to investigate the effect of norepinephrine (NE), an excitatory neuromodulator, on respiratory dynamics. Our investigation employs bifurcation analysis to reveal the mechanisms by which NE differentially modulates different types of preBötC bursting neurons.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Enhancing Neuronal Classification Capacity via Nonlinear Parallel Synapses
11th Floor Lecture Hall
Speaker
Marcus Benna, UC San Diego
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
Abstract
We discuss models of a neuron that has multiple synaptic contacts with the same presynaptic axon. We show that a diverse set of learned nonlinearities in these parallel synapses leads to a substantial increase in the neuronal classification capacity.
11:30 AM - 1:30 PM EDT
Working Lunch: Open Problems Session
Working Lunch - 11th Floor Collaborative Space
1:30 - 2:15 PM EDT
Combinatorial structure of continuous dynamics in gene regulatory networks
11th Floor Lecture Hall
Speaker
Tomas Gedeon, Montana State University
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
Abstract
Gene network dynamics and neural network dynamics face similar challenges of high dimensionality of both phase space and parameter space, and a lack of reliable experimental data to infer parameters.
We first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a combinatorial description of global dynamics of a network over its parameter space. Finite description allows comparison of parameterized dynamics between hundreds of networks to discard networks that are not compatible with experimental data. We also describe a close connection of DSGRN to Boolean network models that allows us to view DSGRN as a connection between parameterized continuous time dynamics and discrete dynamics of Boolean modets.
If time allows, we discuss several applications of this methodology to systems biology.
2:30 - 3:15 PM EDT
A model of the mammalian neural motor architecture elucidates the mechanisms underlying efficient and flexible control of network dynamics
11th Floor Lecture Hall
Speaker
Laureline Logiaco, Massachusetts Institute of Technology
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
Abstract
One of the fundamental functions of the brain is to flexibly plan and control movement production at different timescales in order to efficiently shape structured behaviors. I will present research investigating how these complex computations are performed in the mammalian brain, with an emphasis on autonomous motor control. Specifically, I will focus on the mechanisms supporting efficient interfacing between 'higher-level' planning commands and 'lower-level' motor cortical dynamics that ultimately drive muscles. I will take advantage of the fact that the anatomy of the circuits underlying motor control is well known. It notably involves the primary motor cortex, a recurrent network that generates learned commands to drive muscles while interacting through loops with thalamic neurons that lack recurrent excitation. Using an analytically tractable model that incorporates these architectural constraints, I will explain how this motor circuit can implement a form of efficient modularity by combining (i) plastic thalamocortical loops that are movement-specific and (ii) shared hardwired circuits. I will show that this modular architecture can balance two different objectives: first, supporting the flexible recombination of an extensible library of re-usable motor primitives; and second, promoting the efficient use of neural resources by taking advantage of shared connections between modules. I will end by mentioning some open avenues for further mathematical analyses related to this framework.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Low-rank neural connectivity for the discrimination of temporal patterns.
11th Floor Lecture Hall
Speaker
Sean Escola, Columbia University
Session Chair
Zachary Kilpatrick, University of Colorado Boulder
Friday, September 22, 2023
9:00 - 9:45 AM EDT
Mean-field theory of learning dynamics in deep neural networks
11th Floor Lecture Hall
Speaker
Cengiz Pehlevan, Harvard University
Session Chair
Konstantin Mischaikow, Rutgers University
Abstract
Learning dynamics of deep neural networks is complex. While previous approaches made advances in mathematical analysis of the dynamics of two-layer neural networks, addressing deeper networks have been challenging. In this talk, I will present a mean field theory of the learning dynamics of deep networks and discuss its implications.
10:00 - 10:45 AM EDT
Multi-level measures for understanding and comparing biological and artificial neural networks
11th Floor Lecture Hall
Speaker
SueYeon Chung, New York University
Session Chair
Konstantin Mischaikow, Rutgers University
Abstract
I will share recent theoretical advances on how representation's population level properties such as high-dimensional geometries and spectral properties can be used to capture (1) the classification capacity of neural manifolds, and (2) prediction error of neural data from network model representations.
11:00 - 11:30 AM EDT
Coffee Break
11th Floor Collaborative Space
11:30 AM - 12:15 PM EDT
A Sparse-coding Model of Category-specific Functional Organization in IT Cortex
11th Floor Lecture Hall
Speaker
Demba Ba, Harvard University
Session Chair
Konstantin Mischaikow, Rutgers University
Abstract
Primary sensory areas in the brain of mammals may have evolved to compute efficient representations of natural scenes. In the late 90s, Olhausen and Field proposed a model that expresses the components of a natural scene, e.g. natural-image patches, as sparse combinations of a common set of patterns. Applied to a dataset of natural images, this so-called sparse coding model learns patterns that resemble the receptive fields of V1 neurons. Recordings from the monkey infero-temporal (IT) cortex suggest the presence, in this region, of a sparse code for natural-image categories. The recordings also suggest that, physically, IT neurons form spatial clusters, each of which preferentially responds to images from certain categories. Taken together, this evidence suggests that neurons in IT cortex form functional groups that reflect the grouping of natural images into categories. My talk will introduce a new sparse-coding model that exhibits this categorical form of functional grouping.
12:30 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Final Open Problems Discussion
Problem Session - 11th Floor Lecture Hall
Session Chair
Carina Curto, The Pennsylvania State University
Session Chair
Konstantin Mischaikow, Rutgers University
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
