Mathematical Modeling seminars

Higher cortical areas carry a wide range of sensory, cognitive, and motor signals supporting complex goal-directed behavior. These signals mix in heterogeneous responses of single neurons, making it difficult to untangle underlying mechanisms. I will present two approaches for revealing interpretable circuit mechanisms from heterogeneous neural responses during cognitive tasks. First, I will show a flexible nonparametric framework for simultaneously inferring population dynamics on single trials and tuning functions of individual neurons to the latent population state. When applied to recordings from the premotor cortex during decision-making, our approach revealed that populations of neurons encoded the same dynamic variable predicting choices, and heterogeneous firing rates resulted from the diverse tuning of single neurons to this decision variable. The inferred dynamics indicated an attractor mechanism for decision computation. Second, I will show an approach for inferring an interpretable network model of a cognitive task—the latent circuit—from neural response data. We developed a theory to causally validate latent circuit mechanisms via patterned perturbations of activity and connectivity in the high-dimensional network. This work opens new possibilities for deriving testable mechanistic hypotheses from complex neural response data.

Activity dependent synaptic plasticity is considered to be a primary mechanism underlying learning and memory. Yet it is unclear whether plasticity rules such as STDP measured in vitro apply in vivo. Network models with STDP predict that activity patterns (e.g., place-cell spatial selectivity) should change much faster than observed experimentally. We address this gap by investigating a nonlinear calcium-based plasticity rule fit to experiments done in physiological conditions. In this model, LTP and LTD result from intracellular calcium transients arising almost exclusively from synchronous coactivation of pre- and postsynaptic neurons. We analytically approximate the full distribution of nonlinear calcium transients as a function of pre- and postsynaptic firing rates, and temporal correlations. This analysis directly relates activity statistics that can be measured in vivo to the changes in synaptic efficacy they cause. Our results highlight that both high-firing rates and temporal correlations can lead to significant changes to synaptic efficacy. Using a mean-field theory, we show that the nonlinear plasticity rule, without any fine-tuning, gives a stable, unimodal synaptic weight distribution characterized by many strong synapses which remain stable over long periods of time, consistent with electrophysiological and behavioral studies. Moreover, our theory explains how memories encoded by strong synapses can be preferentially stabilized by the plasticity rule. We confirmed our analytical results in a spiking recurrent network. Interestingly, although most synapses are weak and undergo rapid turnover, the fraction of strong synapses are sufficient for supporting realistic spiking dynamics and serve to maintain the network’s cluster structure. Our results provide a mechanistic understanding of how stable memories may emerge on the behavioral level from an STDP rule measured in physiological conditions. Furthermore, the plasticity rule we investigate is mathematically equivalent to other learning rules which rely on the statistics of coincidences, so we expect that our formalism will be useful to study other learning processes beyond the calcium-based plasticity rule.

One of Tali's cherished goals was to transform biology into physics. In his view, biologists were far too enamored by the details of the specific models they studied, losing sight of the big principles that may govern the behavior of these models. One such big principle that he suggested was the 'information bottleneck (IB) principle'. The iIB principle is an information-theoretical approach for extracting the relevant information that one random variable carries about another. Tali applied the IB principle to numerous problems in biology, gaining important insights in the process. Here I will describe two applications of the IB principle to neurobiological data. The first is the formalization of the notion of surprise that allowed us to rigorously estimate the memory duration and content of neuronal responses in auditory cortex, and the second is an application to behavior, allowing us to estimate 'optimal policies under information constraints' that shed interesting light on rat behavior.

In the past several years, I have been involved in building a biologically realistic model of the monkey visual cortex. Work on one of the input layers (4Ca) of the primary visual cortex (V1) is now nearly complete, and I would like to share some of what I have learned with the community. After a brief overview of the model and its capabilities, I would like to focus on three sets of results that represent three different aspects of the modeling. They are: (i) emergent E-I dynamics in local circuits; (ii) how visual cortical neurons acquire their ability to detect edges and directions of motion, and (iii) a view across the cortical surface: nonequilibrium steady states (in analogy with statistical mechanics) and beyond.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no cure. Yet, a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages associated with various cognitive deficits and pathologies. How can we use mathematical modelling to gain insight into this process and, doing so, gain understanding about how the brain works? In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions.

Classic studies show that in many species – from leech and cricket to primate – responses of neural populations can be quite successfully read out using a measure neural population activity termed the population vector. However, despite its successes, detailed analyses have shown that the standard population vector discards substantial amounts of information contained in the responses of a neural population, and so is unlikely to accurately describe how signal communication between parts of the nervous system. I will describe recent theoretical results showing how to modify the population vector expression in order to read out neural responses without information loss, ideally. These results make it possible to quantify the contribution of weakly tuned neurons to perception. I will also discuss numerical methods that can be used to minimize information loss when reading out responses of large neural populations.

Birdsong emerges when a set of highly interconnected brain areas manage to generate a complex output. The similarities between birdsong production and human speech have positioned songbirds as unique animal models for studying learning and production of this complex motor skill. In this work, we developed a low dimensional model for a neural network in which the variables were the average activities of different neural populations within the nuclei of the song system. This neural network is active during production, perception and learning of birdsong. We performed electrophysiological experiments to record neural activity from one of these nuclei and found that the low dimensional model could reproduce the neural dynamics observed during the experiments. Also, this model could reproduce the respiratory motor patterns used to generate song. We showed that sparse activity in one of the neural nuclei could drive a more complex activity downstream in the neural network. This interdisciplinary work shows how low dimensional neural models can be a valuable tool for studying the emergence of complex motor tasks

This conference is a free event that includes a range of activities to stimulate the cross-fertilization of ideas, including invited speaker talks, workshops, micro talks, an undergraduate research competition, a contest to discover mathematical questions in biology, and plenty of networking opportunities. Today's speakers: Cassandra Extavour, William Bialek, Amy Shyer, Ankur Saxena, Jie Liang

This conference is a free event that includes a range of activities to stimulate the cross-fertilization of ideas, including invited speaker talks, workshops, micro talks, an undergraduate research competition, a contest to discover mathematical questions in biology, and plenty of networking opportunities. Today's speakers: Nathalie Dostatni, Christopher Obara, Hernan Garcia, Aaron Dinner, David Lubensky, Jianping Fu

Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are constrained by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics, such as emergent sequences and dynamic attractors, from properties of the graph. These rules provide a direct link between the structure and function of these networks, and may provide new insights into how connectivity shapes dynamics in real neural circuits.

Many of us came to biology from physics. There we have been trained on such classic examples as muon g-2, where experimental data and theoretical predictions agree to many significant digits. Now, working in biology, we routinely hear that it is messy, most details matter, and that the best hope for theory in biology is to be semi-qualitative, predict general trends, and to forgo the hope of ever making quantitative predictions with the precision that we are used to in physics. Colloquially, we should be satisfied even if data and models differ so much that plotting them on the same plot makes little sense. However, some of us won’t be satisfied by this. So can we take theory in biology seriously and predict experimental outcomes within (small) error bars? Certainly, we won’t be able to predict everything, but this is never required, even in traditional physics. But we should be able to choose some features of data that are nontrivial and interesting, and focus on them. We also should be able to find different classes of models --- maybe even null models --- that match biology better, and thus allow for a better agreement. It is even possible that large-dimensional datasets of modern high-throughput experiments, and the ensuing “more is different” statistical physics style models will make quantitative, precise theory easier. To explore the role of quantitative theory in biology, in this workshop, eight speakers will address some of the following general questions based on their specific work in different corners of biology: Which features of biological data are predictable? Which types of models are best suited to making quantitative predictions in different fields? Should theorists interested in quantitative predictions focus on different questions, not typically asked by biologists? Do large, multidimensional datasets make theories (and which theories?) more or less likely to succeed? This will be an unapologetically theoretical physics workshop — we won’t focus on a specific subfield of biology, but will explore these questions across the fields, hoping that the underlying theoretical frameworks will help us find the missing connections.

The Ca2+ modulated pulsatile secretions of glucagon and insulin by pancreatic α and β cells play a key role in glucose metabolism and homeostasis. However, how different types of cells in the islet couple and coordinate to give rise to various Ca2+ oscillation patterns and how these patterns are being tuned by paracrine regulation are still elusive. Here we developed a microfluidic device to facilitate long-term recording of islet Ca2+ activity at single cell level and found that islets show heterogeneous but intrinsic oscillation patterns. The α and β cells in an islet oscillate in antiphase and are globally phase locked to display a variety of oscillation modes. A mathematical model of islet oscillation maps out the dependence of the oscillation modes on the paracrine interactions between α and β cells. Our study reveals the origin of the islet oscillation patterns and highlights the role of paracrine regulation in tuning them.