Mathematical Modeling seminars

Starting with the work of Hodgkin and Huxley in the 1950s, we now have a fairly good understanding of how the spiking activity of neurons can be modelled mathematically. For cortical circuits the understanding is much more limited. Most network studies have considered stylized models with a single or a handful of neuronal populations consisting of identical neurons with statistically identical connection properties. However, real cortical networks have heterogeneous neural populations and much more structured synaptic connections. Unlike typical simplified cortical network models, real networks are also “multipurpose” in that they perform multiple functions. Historically the lack of computational resources has hampered the mathematical exploration of cortical networks. With the advent of modern supercomputers, however, simulations of networks comprising hundreds of thousands biologically detailed neurons are becoming feasible (Einevoll et al, Neuron, 2019). Further, a large-scale biologically network model of the mouse primary visual cortex comprising 230.000 neurons has recently been developed at the Allen Institute for Brain Science (Billeh et al, Neuron, 2020). Using this model as a starting point, I will discuss how we can move towards multipurpose models that incorporate the true biological complexity of cortical circuits and faithfully reproduce multiple experimental observables such as spiking activity, local field potentials or two-photon calcium imaging signals. Further, I will discuss how such validated comprehensive network models can be used to gain insights into the functioning of cortical circuits.

Hydra is a fascinating model organism for neuroscience. It is transparent; new genetic lines allow one to image activity in both neurons (Dupre and Yuste, 2017) and muscle cells (Szymanski and Yuste, 2019) ; it exhibits rich behavior, and it continually rebuilds itself. Hydra’s fairly simply physical structure as a two-layered fluid-filled hydrostat and the accessibility of information about neural and muscle activity opens the possibility of a complete model of neural control of behavior. This requires understanding the transformations that occur in the muscle cell layers and a biomechanical model of the body column. We show that we can use this modeling to reverse engineer how neural activity drives behavior.

Evidence regarding protein structure and function manifest the imperative role that dynamics play in proteins, underlining reconsideration of the unanimated sequence-to-structure-to-function paradigm. Structural dynamics portray a heterogeneous energy landscape described by conformational ensembles where each structural representation can be responsible for unique functions or enable macromolecular assemblies. Using the human p27/Cdk2/Cyclin A ternary complex as an example, we highlight the vital role of intra- and intermolecular dynamics for target recognition, binding, and inhibition as a critical modulator of cell division. Rapidly sampling configurations is critical for the population of different conformational ensembles encoding functional roles. To garner this knowledge, we present how the integration of (sub)ensemble and single-molecule fluorescence spectroscopy with molecular dynamic simulations can characterize structural dynamics linking the heterogeneous ensembles to function. The incorporation of dynamics into the sequence-to-structure-to-function paradigm promises to assist in tackling various challenges, including understanding the formation and regulation of mesoscale assemblies inside cells.

Both computational and experimental results in single neurons and small networks demonstrate that very similar network function can result from quite disparate sets of neuronal and network parameters. Using the crustacean stomatogastric nervous system, we study the influence of these differences in underlying structure on differential resilience of individuals to a variety of environmental perturbations, including changes in temperature, pH, potassium concentration and neuromodulation. We show that neurons with many different kinds of ion channels can smoothly move through different mechanisms in generating their activity patterns, thus extending their dynamic range. The talk will be simultaneously translated to spanish by the interpreter Liliana Viera, MSc. Los resultados tanto computacionales como experimentales en neuronas individuales y redes pequeñas demuestran que funcionamientos de redes muy similares pueden pueden resultar de conjuntos bastante dispares de parámetros neuronales y de las redes. Utilizando el sistema nervioso estomatogástrico de los crustáceos, estudiamos la influencia de estas diferencias en la estructura subyacente en la resistencia diferencial de los individuos a una variedad de perturbaciones ambientales, incluidos los cambios de temperatura, pH, concentración de potasio y neuromodulación. Mostramos que neuronas con muchos tipos diferentes de canales iónicos pueden moverse suavemente a través de diferentes mecanismos para generar sus patrones de actividad, extendiendo así su rango dinámico. La conferencia será traducida simultáneamente al español por la intérprete Liliana Viera MSc.

Inhibitory interneurons come in different classes and form intricate circuits. While our knowledge of these circuits has advanced substantially over the last decades, it is not fully understood how the structure of these circuits relates to their function. I will present some of our recent attempts to “understand” the structure of interneuron circuits by means of computational modeling. Surprisingly (at least for us), we found that prominent features of inhibitory circuitry can be accounted for by an optimisation for excitation-inhibition (E/I) balance. In particular, we find that such an optimisation generates networks that resemble mouse V1 in terms of the structure of synaptic efficacies between principal cells and parvalbumin-positive interneurons. Moreover, an optimisation for E/I balance across neuronal compartments promotes a functional diversification of interneurons into two classes that resemble parvalbumin and somatostatin-positive interneurons. Time permitting, I may briefly touch on recent work in which we link E/I balance to prediction error coding in V1.

Many networks in the brain are sparsely connected, and the brain eliminates synapses during development and learning. How could the brain decide which synapses to prune? In a recurrent network, determining the importance of a synapse 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. In the first part of this talk, I will suggest that noise could play a functional role in synaptic pruning, allowing the brain to probe network structure and determine which synapses are redundant. I will introduce a simple, local, unsupervised plasticity rule that either strengthens or prunes synapses using only synaptic weight and the noise-driven covariance of the neighboring neurons. For a subset of linear and rectified-linear networks, this rule provably preserves the spectrum of the original matrix and hence preserves network dynamics even when the fraction of pruned synapses asymptotically approaches 1. The plasticity rule is biologically-plausible and may suggest a new role for noise in neural computation. Time permitting, I will then turn to the problem of extracting structure from neural population data sets using dimensionality reduction methods. I will argue that nonlinear structures naturally arise in neural data and show how these nonlinearities cause linear methods of dimensionality reduction, such as Principal Components Analysis, to fail dramatically in identifying low-dimensional structure.

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.

Paramecium is a unicellular organism that swims in fresh water using cilia. When it is stimulated (mechanically, chemically, optically, thermally, etc), it often swims backward then turns and swims forward again: this is called the avoiding reaction. This reaction is triggered by a calcium-based action potential. For this reason, it enjoyed a period of glory in the 1970s as a model organism for neuroscience. I will describe the behavior and electrophysiology of this “swimming neuron”, then I will present our ongoing attempts at developing an integrative quantitative model of Paramecium.

Synaptic inputs on cortical dendrites are organized with remarkable subcellular precision at the micron level. This organization emerges during early postnatal development through patterned spontaneous activity and manifests both locally where synapses with similar functional properties are clustered, and globally along the axis from dendrite to soma. Recent experiments reveal species-specific differences in the local and global synaptic organization in mouse, ferret and macaque visual cortex. I will present a computational framework that implements functional and structural plasticity from spontaneous activity patterns to generate these different types of organization across species and scales. Within this framework, a single anatomical factor - the size of the visual cortex and the resulting magnification of visual space - can explain the observed differences. This allows us to make predictions about the organization of synapses also in other species and indicates that the proximal-distal axis of a dendrite might be central in endowing a neuron with powerful computational capabilities.

The Smoluchowski equation has often been used as the starting point of many continuum models of active suspensions. However, its six-dimensional nature depending on time, space and orientation requires a huge computational cost, fundamentally limiting its use for large-scale problems, such as mixing and transport of active fluids in turbulent flows. Despite the singular nature in strain-dominant flows, the generalised Taylor dispersion (GTD) theory (Frankel & Brenner 1991, J. Fluid Mech. 230:147-181) has been understood to be one of the most promising ways to reduce the Smoluchowski equation into an advection-diffusion equation, the mean drift and diffusion tensor of which rely on ‘local’ flow information only. In this talk, we will introduce an exact transformation of the Smoluchowski equation into such an advection-diffusion equation requiring only local flow information. Based on this transformation, a new advection-diffusion equation will subsequently be proposed by taking an asymptotic analysis in the limit of small particle velocity. With several examples, it will be demonstrated that the new advection-diffusion model, non-singular in strain-dominant flows, outperforms the GTD theory.

Epithelial cell sheets form a fundamental role in the developing embryo, and also in adult tissues including the gut and the cornea of the eye. Soft and active matter provides a theoretical and computational framework to understand the mechanics and dynamics of these tissues.I will start by introducing the simplest useful class of models, active brownian particles (ABPs), which incorporate uncoordinated active crawling over a substrate and mechanical interactions. Using this model, I will show how the extended ’swirly’ velocity fluctuations seen in sheets on a substrate can be understood using a simple model that couples linear elasticity with disordered activity. We are able to quantitatively match experiments using in-vitro corneal epithelial cells.Adding a different source of activity, cell division and apoptosis, to such a model leads to a novel 'self-melting' dense fluid state. Finally, I will discuss a direct application of this simple particle-based model to the steady-state spiral flow pattern on the mouse cornea.

Recurrent neural networks (RNNs) are powerful dynamical models, widely used in machine learning (ML) for processing sequential data, and also in neuroscience, to understand the emergent properties of networks of real neurons. Prior theoretical work in understanding the properties of RNNs has focused on models with additive interactions. However, real neurons can have gating i.e. multiplicative interactions, and gating is also a central feature of the best performing RNNs in machine learning. Here, we develop a dynamical mean-field theory (DMFT) to study the consequences of gating in RNNs. We use random matrix theory to show how gating robustly produces marginal stability and line attractors – important mechanisms for biologically-relevant computations requiring long memory. The long-time behavior of the gated network is studied using its Lyapunov spectrum, and the DMFT is used to provide a novel analytical expression for the maximum Lyapunov exponent demonstrating its close relation to relaxation-time of the dynamics. Gating is also shown to give rise to a novel, discontinuous transition to chaos, where the proliferation of critical points (topological complexity) is decoupled from the appearance of chaotic dynamics (dynamical complexity), contrary to a seminal result for additive RNNs. Critical surfaces and regions of marginal stability in the parameter space are indicated in phase diagrams, thus providing a map for principled parameter choices for ML practitioners. Finally, we develop a field-theory for gradients that arise in training, by incorporating the adjoint sensitivity framework from control theory in the DMFT. This paves the way for the use of powerful field-theoretic techniques to study training/gradients in large RNNs.

Recently a powerful example of a replicating nano-machine entered our society. In principle, it’s just a normal disease, that one attempts to model with 3 or 4 simple coupled equations with 2 important parameters: a timescale, and a replication factor (the famous R0). Then one tries to guess how changes in society change R0 and perhaps adopt some more or less strong lock-down measures. However, this virus has more “personality” than that. It behaves differently in different persons, and persons behave differently. Presumably, only a few of us infect a lot, while most do not infect so much. This assumption is supported by the observation that couples living together only infect each other with about 15 percent probability, indicating that most infected people are not really infectious. I will discuss this and other aspects of Covid-19 in the perspective of models that describe heterogeneous individuals in a society. In particular, we suggest that limiting superspreading opportunities is a cost-effective strategy to mitigate Covid-19.

Spontaneous firing, observed in many neurons, is often attributed to ion channel or network level noise. Cortical cells during slow wave sleep exhibit transitions between so called Up and Down states. In this sleep state, with limited sensory stimuli, neurons fire in the Up state. Spontaneous firing is also observed in slices of cholinergic interneurons, cerebellar Purkinje cells and even brainstem inspiratory neurons. In such in vitro preparations, where the functional relevance is long lost, neurons continue to display a rich repertoire of firing properties. It is perplexing that these neurons, instead of saving their energy during information downtime and functional irrelevance, are eager to fire. We propose that spontaneous firing is not a chance event but instead, a vital activity for the well-being of a neuron. We postulate that neurons, in anticipation of synaptic inputs, keep their ATP levels at maximum. As recovery from inputs requires most of the energy resources, neurons are ATP surplus and ADP scarce during synaptic quiescence. With ADP as the rate-limiting step, ATP production stalls in the mitochondria when ADP is low. This leads to toxic Reactive Oxygen Species (ROS) formation, which are known to disrupt many cellular processes. We hypothesize that spontaneous firing occurs at these conditions - as a release valve to spend energy and to restore ATP production, shielding the neuron against ROS. By linking a mitochondrial metabolism model to a conductance-based neuron model, we show that spontaneous firing depends on baseline ATP usage and on ATP-cost-per-spike. From our model, emerges a mitochondrial mediated homeostatic mechanism that provides a recipe for different firing patterns. Our findings, though mostly affecting intracellular dynamics, may have large knock-on effects on the nature of neural coding. Hitherto it has been thought that the neural code is optimised for energy minimisation, but this may be true only when neurons do not experience synaptic quiescence.

Synaptic plasticity is believed to be a key physiological mechanism for learning. It is well-established that it depends on pre and postsynaptic activity. However, models that rely solely on pre and postsynaptic activity for synaptic changes have, to date, not been able to account for learning complex tasks that demand hierarchical networks. Here, we show that if synaptic plasticity is regulated by high-frequency bursts of spikes, then neurons higher in the hierarchy can coordinate the plasticity of lower-level connections. Using simulations and mathematical analyses, we demonstrate that, when paired with short-term synaptic dynamics, regenerative activity in the apical dendrites, and synaptic plasticity in feedback pathways, a burst-dependent learning rule can solve challenging tasks that require deep network architectures. Our results demonstrate that well-known properties of dendrites, synapses, and synaptic plasticity are sufficient to enable sophisticated learning in hierarchical circuits.

Temporal lobe epilepsy is associated with memory deficits but the circuit mechanisms underlying these cognitive disabilities are not understood. We used electrophysiological recordings, open-source wire-free miniaturized microscopy and computational modeling to probe these deficits in a model of temporal lobe epilepsy. We find desynchronization of dentate gyrus interneurons with CA1 interneurons during theta oscillations and a loss of precision and stability of place fields. We also find that emergence of place cell dysfunction is delayed, providing a potential temporal window for treatments. Computation modeling shows that desynchronization rather than interneuron cell loss can drive place cell dysfunction. Future studies will uncover cell types driving these changes and transcriptional changes that may be driving dysfunction.

Understanding what drives foraging decisions in animals requires careful manipulation of the value of available options while monitoring animal choices. Value-based decision-making tasks, in combination with formal learning models, have provided both an experimental and theoretical framework to study foraging decisions in lab settings. While these approaches were successfully used in the past to understand what drives choices in mammals, very little work has been done on fruit flies. This is even though fruit flies have served as a model organism for many complex behavioural paradigms. To fill this gap we developed a single-animal, trial-based decision-making task, where freely walking flies experienced optogenetic sugar-receptor neuron stimulation. We controlled the value of available options by manipulating the probabilities of optogenetic stimulation. We show that flies integrate a reward history of chosen options and forget value of unchosen options. We further discover that flies assign higher values to rewards experienced early in the behavioural session, consistent with formal reinforcement learning models. Finally, we show that the probabilistic rewards affect walking trajectories of flies, suggesting that accumulated value is controlling the navigation vector of flies in a graded fashion. These findings establish the fruit fly as a model organism to explore the genetic and circuit basis of value-based decisions.

Single-cell measurements of mRNA copy numbers inform our understanding of stochastic gene expression, but these measurements coarse-grain over the individual copies of the gene, where transcription and its regulation take place stochastically. We recently combined single-molecule quantification of mRNA and gene loci to measure the transcriptional activity of an endogenous gene in individual Escherichia coli bacteria. When interpreted using a theoretical model for mRNA dynamics, the single-cell data allowed us to obtain the probabilistic rates of promoter switching, transcription initiation and elongation, mRNA release and degradation. Unexpectedly, we found that gene activity can be strongly coupled to the transcriptional state of another copy of the same gene present in the cell, and to the event of gene replication during the bacterial cell cycle. These gene-copy and cell-cycle correlations demonstrate the limits of mapping whole-cell mRNA numbers to the underlying stochastic gene activity and highlight the contribution of previously hidden variables to the observed population heterogeneity.