Mathematical Modeling seminars

A central question in motor physiology has been whether motor cortex activity resembles muscle activity, and if not, why not? Over fifty years, extensive observations have failed to provide a concise answer, and the topic remains much debated. To provide a different perspective, we employed a novel behavioral paradigm that affords extensive comparison between time-evolving neural and muscle activity. Single motor-cortex neurons displayed many muscle-like properties, but the structure of population activity was not muscle-like. Unlike muscle activity, neural activity was structured to avoid ’trajectory tangling’: moments where similar activity patterns led to dissimilar future patterns. Avoidance of trajectory tangling was present across tasks and species. Network models revealed a potential reason for this consistent feature: low tangling confers noise robustness. Remarkably, we were able to predict motor cortex activity from muscle activity alone, by leveraging the hypothesis that muscle-like commands are embedded in additional structure that yields low tangling. Our results argue that motor cortex embeds descending commands in additional structure that ensure low tangling, and thus noise-robustness. The dominant structure in motor cortex may thus serve not a representational function (encoding specific variables) but a computational function: ensuring that outgoing commands can be generated reliably. Our results establish the utility of an emerging approach: understanding the structure of neural activity based on properties of population geometry that flow from normative principles such as noise robustness.

Spontaneous behavior in animals and humans shows a striking amount of variability both in the spatial domain (which actions to choose) and temporal domain (when to act). Concatenating actions into sequences and behavioral plans reveals the existence of a hierarchy of timescales ranging from hundreds of milliseconds to minutes. How do multiple timescales emerge from neural circuit dynamics? How do circuits modulate temporal responses to flexibly adapt to changing demands? In this talk, we will present recent results from experiments and theory suggesting a new computational mechanism generating the temporal variability underlying naturalistic behavior. We will show how neural activity from premotor areas unfolds through temporal sequences of attractors, which predict the intention to act. These sequences naturally emerge from recurrent cortical networks, where correlated neural variability plays a crucial role in explaining the observed variability in action timing. We will then discuss how reaction times in these recurrent circuits can be accelerated or slowed down via gain modulation, induced by neuromodulation or perturbations. Finally, we will present a general mechanism producing a reservoir of multiple timescales in recurrent networks.

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.

Cortical circuits receive multiple inputs from upstream populations with non-overlapping stimulus tuning preferences. Both the feedforward and recurrent architectures of the receiving cortical layer will reflect this diverse input tuning. We study how population-wide neuronal variability propagates through a hierarchical cortical network receiving multiple, independent, tuned inputs. We present new analysis of in vivo neural data from the primate visual system showing that the number of latent variables (dimension) needed to describe population shared variability is smaller in V4 populations compared to those of its downstream visual area PFC. We successfully reproduce this dimensionality expansion from our V4 to PFC neural data using a multi-layer spiking network with structured, feedforward projections and recurrent assemblies of multiple, tuned neuron populations. We show that tuning-structured connectivity generates attractor dynamics within the recurrent PFC current, where attractor competition is reflected in the high dimensional shared variability across the population. Indeed, restricting the dimensionality analysis to activity from one attractor state recovers the low-dimensional structure inherited from each of our tuned inputs. Our model thus introduces a framework where high-dimensional cortical variability is understood as ``time-sharing’’ between distinct low-dimensional, tuning-specific circuit dynamics.

Neurons in the primary visual cortex (V1) encode the orientation and contrast of visual stimuli through changes in firing rate (Hubel and Wiesel, 1962). Their activity typically peaks at a preferred orientation and decays to zero at the orientations that are orthogonal to the preferred. This activity pattern is re-scaled by contrast but its shape is preserved, a phenomenon known as contrast invariance. Contrast-invariant selectivity is also observed at the population level in V1 (Carandini and Sengpiel, 2004). The mechanisms supporting the emergence of contrast-invariance at the population level remain unclear. How does the activity of different neurons with diverse orientation selectivity and non-linear contrast sensitivity combine to give rise to contrast-invariant population selectivity? Theoretical studies have shown that in the balance limit, the properties of single-neurons do not determine the population activity (van Vreeswijk and Sompolinsky, 1996). Instead, the synaptic dynamics (Mongillo et al., 2012) as well as the intracortical connectivity (Rosenbaum and Doiron, 2014) shape the population activity in balanced networks. We report that short-term plasticity can change the synaptic strength between neurons as a function of the presynaptic activity, which in turns modifies the population response to a stimulus. Thus, the same circuit can process a stimulus in different ways –linearly, sublinearly, supralinearly – depending on the properties of the synapses. We found that balanced networks with excitatory to excitatory short-term synaptic plasticity cannot be contrast-invariant. Instead, short-term plasticity modifies the network selectivity such that the tuning curves are narrower (broader) for increasing contrast if synapses are facilitating (depressing). Based on these results, we wondered whether balanced networks with plastic synapses (other than short-term) can support the emergence of contrast-invariant selectivity. Mathematically, we found that the only synaptic transformation that supports perfect contrast invariance in balanced networks is a power-law release of neurotransmitter as a function of the presynaptic firing rate (in the excitatory to excitatory and in the excitatory to inhibitory neurons). We validate this finding using spiking network simulations, where we report contrast-invariant tuning curves when synapses release the neurotransmitter following a power- law function of the presynaptic firing rate. In summary, we show that synaptic plasticity controls the type of non-linear network response to stimulus contrast and that it can be a potential mechanism mediating the emergence of contrast invariance in balanced networks with orientation-dependent connectivity. Our results therefore connect the physiology of individual synapses to the network level and may help understand the establishment of contrast-invariant selectivity.

The dream of programmable matter is to create materials whose physical properties (shape, moduli, response to perturbations, etc.) can be changed on the fly. For many years, my group has been thinking about how to program flat sheets that fold up into three dimensional shapes, most recently by exploiting the principles of origami design. Unfortunately, a combinatorial explosion of folding pathways makes robust folding particularly challenging. In this talk, I will discuss how this pluripotency arises from the topology of the configuration space. This suggests a broader understanding of a larger class of materials spanning from folding forms to spring networks to mechanical structures that perform computational logic.

Neural integrators are circuits that are able to code analog information such as spatial location or amplitude. Storing amplitude requires the network to have a large number of attractors. In classic models with recurrent excitation, such networks require very careful tuning to behave as integrators and are not robust to small mistuning of the recurrent weights. In this talk, I introduce a circuit with recurrent connectivity that is subjected to a slow subthreshold oscillation (such as the theta rhythm in the hippocampus). I show that such a network can robustly maintain many discrete attracting states. Furthermore, the firing rates of the neurons in these attracting states are much closer to those seen in recordings of animals. I show the mechanism for this can be explained by the instability regions of the Mathieu equation. I then extend the model in various ways and, for example, show that in a spatially distributed network, it is possible to code location and amplitude simultaneously. I show that the resulting mean field equations are equivalent to a certain discontinuous differential equation.

In a cross wind, the direction a fly moves through the air may differ from its heading direction, the direction defined by its body axis. I will present a model based on experimental results that reveals how a heading direction “compass” signal is combined with optic flow to compute and represent the direction that a fly is traveling. This provides a general framework for understand how flies perform vector computations.

When was the last time you ran into a giant? Chances are never. Almost 100 years ago, JBS Haldane posed an outwardly simple yet complex question – what is the most optimal size (for a biological system)? The living world around us contains a huge diversity of organisms, each with its own characteristic size. Even the size of subcellular organelles is tightly controlled. In absence of physical rulers, how do cells and organisms truly “know” how large is large enough? What are the mechanisms in place to enforce size control? Many of these questions have motivated generations of scientists to look for physical principles underlying size control in biological systems. In the next edition of Emory's Theory and Modeling of Living Systems (TMLS) workshop series, our panel of speakers will take a close look at these questions, across the entire scale - from the molecular, all the way to the ecosystem.

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.

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.

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.