Mathematical Modeling seminars

Large-scale recordings of neural activity are providing new opportunities to study network-level dynamics with unprecedented detail. However, the sheer volume of data and its dynamical complexity are major barriers to uncovering and interpreting these dynamics. I will present machine learning frameworks that enable inference of dynamics from neuronal population spiking activity on single trials and millisecond timescales, from diverse brain areas, and without regard to behavior. I will then demonstrate extensions that allow recovery of dynamics from two-photon calcium imaging data with surprising precision. Finally, I will discuss our efforts to facilitate comparisons within our field by curating datasets and standardizing model evaluation, including a currently active modeling challenge, the 2021 Neural Latents Benchmark [neurallatents.github.io].

This talk will focus on the generative modelling of resting state time series or endogenous neuronal activity. I will survey developments in modelling distributed neuronal fluctuations – spectral dynamic causal modelling (DCM) for functional MRI – and how this modelling rests upon functional connectivity. The dynamics of brain connectivity has recently attracted a lot of attention among brain mappers. I will also show a novel method to identify dynamic effective connectivity using spectral DCM. Further, I will summarise the development of the next generation of DCMs towards large-scale, whole-brain schemes which are computationally inexpensive, to the other extreme of the development using more sophisticated and biophysically detailed generative models based on the canonical microcircuits.

Brain microstructure plays a key role in driving the transport of drug molecules directly administered to the brain tissue as in Convection-Enhanced Delivery procedures. This study reports the first systematic attempt to characterize the cytoarchitecture of commissural, long association and projection fiber, namely: the corpus callosum, the fornix and the corona radiata. Ovine samples from three different subjects have been imaged using scanning electron microscope combined with focused ion beam milling. Particular focus has been given to the axons. For each tract, a 3D reconstruction of relatively large volumes (including a significant number of axons) has been performed. Namely, outer axonal ellipticity, outer axonal cross-sectional area and its relative perimeter have been measured. This study [1] provides useful insight into the fibrous organization of the tissue that can be described as composite material presenting elliptical tortuous tubular fibers, leading to a workflow to enable accurate simulations of drug delivery which include well-resolved microstructural features. As a demonstration of the use of these imaging and reconstruction techniques, our research analyses the hydraulic permeability of two white matter (WM) areas (corpus callosum and fornix) whose three-dimensional microstructure was reconstructed starting from the acquisition of the electron microscopy images. Considering that the white matter structure is mainly composed of elongated and parallel axons we computed the permeability along the parallel and perpendicular directions using computational fluid dynamics [2]. The results show a statistically significant difference between parallel and perpendicular permeability, with a ratio about 2 in both the white matter structures analysed, thus demonstrating their anisotropic behaviour. This is in line with the experimental results obtained using perfusion of brain matter [3]. Moreover, we find a significant difference between permeability in corpus callosum and fornix, which suggests that also the white matter heterogeneity should be considered when modelling drug transport in the brain. Our findings, that demonstrate and quantify the anisotropic and heterogeneous character of the white matter, represent a fundamental contribution not only for drug delivery modelling but also for shedding light on the interstitial transport mechanisms in the extracellular space. These and many other discoveries will be discussed during the talk." "1. https://www.researchsquare.com/article/rs-686577/v1, 2. https://www.pnas.org/content/118/36/e2105328118, 3. https://ieeexplore.ieee.org/abstract/document/9198110

Macromolecular protein complexes perform essential biological functions across life forms. A fundamental, though yet unsolved question in biology is how the function of such complexes is regulated by intracellular or extracellular signals. For instance, we have little understanding of how forces affect multi-protein machines whose function is often mechanical in nature. We address this question by studying the bacterial flagellar motor, a large complex that powers swimming motility in many bacteria. This rotary motor autonomously adapts to changes in mechanical load by adding or removing force-generating ‘stator’ units that power rotation. In the bacterium Escherichia coli, up to 11 units drive the motor at high load while all the units are released at low load. We manipulate motor load using electrorotation, a technique in which a rapidly rotating electric field applies an external torque on the motor. This allows us to change motor load at will and measure the resulting stator dynamics at single-unit resolution. We found that the force generated by the stator units controls their unbinding, forming a feedback loop that leads to autoregulation of the assembly. We complemented our experiments with theoretical models that provide insight into the underlying molecular interactions. Torque-dependent remodeling takes place within seconds, making it a highly responsive control mechanism, one that is mediated by the mechano-chemical tuning of protein interactions.

Behavior relies on the ability of sensory systems to infer changing properties of the environment from incoming sensory stimuli. However, the demands that detecting and adjusting to changes in the environment place on a sensory system often differ from the demands associated with performing a specific behavioral task. This necessitates neural coding strategies that can dynamically balance these conflicting needs. I will discuss our ongoing theoretical work to understand how this balance can best be achieved. We connect ideas from efficient coding and Bayesian inference to ask how sensory systems should dynamically allocate limited resources when the goal is to optimally infer changing latent states of the environment, rather than reconstruct incoming stimuli. We use these ideas to explore dynamic tradeoffs between the efficiency and speed of sensory adaptation schemes, and the downstream computations that these schemes might support. Finally, we derive families of codes that balance these competing objectives, and we demonstrate their close match to experimentally-observed neural dynamics during sensory adaptation. These results provide a unifying perspective on adaptive neural dynamics across a range of sensory systems, environments, and sensory tasks.

Neuronal dendritic trees display a wide range of nonlinear input integrations due to their voltage-dependent active calcium channels. We reveal that in vivo-like fluctuating input enhances nonlinearity substantially in a single dendritic compartment and shifts the input-output relation to exhibiting nonmonotonous or bistable dynamics. In particular, with the slow activation of calcium dynamics, we analyze noise-induced bistability and its timescales. We show bistability induces long-timescale fluctuation that can account for observed dendritic plateau potentials in vivo conditions. In a multicompartmental model neuron with realistic synaptic input, we show that noise-induced bistability persists in a wide range of parameters. Using Fredholm's theory to calculate the spiking rate of multivariable neurons, we discuss how dendritic bistability shifts the spiking dynamics of single neurons and its implications for network phenomena in the processing of in vivo–like fluctuating input.

Macroscopic brain organisation emerges early in life, even prenatally, and continues to develop through adolescence and into early adulthood. The emergence and continual refinement of large-scale brain networks, connecting neuronal populations across anatomical distance, allows for increasing functional integration and specialisation. This process is thought crucial for the emergence of complex cognitive processes. But how and why is this process so diverse? We used structural neuroimaging collected from a large diverse cohort, to explore how different features of macroscopic brain organisation are associated with diverse cognitive trajectories. We used diffusion-weighted imaging (DWI) to construct whole-brain white-matter connectomes. A simulated attack on each child's connectome revealed that some brain networks were strongly organized around highly connected 'hubs'. The more children's brains were critically dependent on hubs, the better their cognitive skills. Conversely, having poorly integrated hubs was a very strong risk factor for cognitive and learning difficulties across the sample. We subsequently developed a computational framework, using generative network modelling (GNM), to model the emergence of this kind of connectome organisation. Relatively subtle changes within the wiring rules of this computational framework give rise to differential developmental trajectories, because of small biases in the preferential wiring properties of different nodes within the network. Finally, we were able to use this GNM to implicate the molecular and cellular processes that govern these different growth patterns.

It is generally agreed upon that qualities of conscious experience instantiate structural properties, usually called relations. They furnish a representation of qualities (or qualia, in fact) in terms of a mathematical space Q (rather than a set), which is crucial to both modelling and measuring of conscious experience." "What is usually disregarded is that “only such structural properties generalize across individuals” (Austen Clark), but that qualities themselves as differentiated by stimulus specifications, behavior or reports do not. We show that this implies that only the part of Q which is invariant with respect to the automorphism group has a well-defined referent, while individual elements do not. This poses a prima facie limitation of any theory or experiment that aims to address individual qualities. We show how mathematical theories of consciousness can overcome this limitation via symmetry groups and group actions, making accessible to science what is properly called private language.

Modular structures in myriad forms — genetic, structural, functional — are ubiquitous in the brain. While modularization may be shaped by genetic instruction or extensive learning, the mechanisms of module emergence are poorly understood. Here, we explore complementary mechanisms in the form of bottom-up dynamics that push systems spontaneously toward modularization. As a paradigmatic example of modularity in the brain, we focus on the grid cell system. Grid cells of the mammalian medial entorhinal cortex (mEC) exhibit periodic lattice-like tuning curves in their encoding of space as animals navigate the world. Nearby grid cells have identical lattice periods, but at larger separations along the long axis of mEC the period jumps in discrete steps so that the full set of periods cluster into 5-7 discrete modules. These modules endow the grid code with many striking properties such as an exponential capacity to represent space and unprecedented robustness to noise. However, the formation of discrete modules is puzzling given that biophysical properties of mEC stellate cells (including inhibitory inputs from PV interneurons, time constants of EPSPs, intrinsic resonance frequency and differences in gene expression) vary smoothly in continuous topographic gradients along the mEC. How does discreteness in grid modules arise from continuous gradients? We propose a novel mechanism involving two simple types of lateral interaction that leads a continuous network to robustly decompose into discrete functional modules. We show analytically that this mechanism is a generic multi-scale linear instability that converts smooth gradients into discrete modules via a topological “peak selection” process. Further, this model generates detailed predictions about the sequence of adjacent period ratios, and explains existing grid cell data better than existing models. Thus, we contribute a robust new principle for bottom-up module formation in biology, and show that it might be leveraged by grid cells in the brain.

Entorhinal grid cells, so-called because of their hexagonally tiled spatial receptive fields, are organized in modules which, collectively, are believed to form a population code for the animal’s position. Here, we apply topological data analysis to simultaneous recordings of hundreds of grid cells and show that joint activity of grid cells within a module lies on a toroidal manifold. Each position of the animal in its physical environment corresponds to a single location on the torus, and each grid cell is preferentially active within a single “field” on the torus. Toroidal firing positions persist between environments, and between wakefulness and sleep, in agreement with continuous attractor models of grid cells.

Gradient-based optimization powered by the backpropagation algorithm proved to be the pivotal method in the training of non-spiking artificial neural networks. At the same time, spiking neural networks hold the promise for efficient processing of real-world sensory data by communicating using discrete events in continuous time. We derive the backpropagation algorithm for a recurrent network of spiking (leaky integrate-and-fire) neurons with hard thresholds and show that the backward dynamics amount to an event-based backpropagation of errors through time. Our derivation uses the jump conditions for partial derivatives at state discontinuities found by applying the implicit function theorem, allowing us to avoid approximations or substitutions. We find that the gradient exists and is finite almost everywhere in weight space, up to the null set where a membrane potential is precisely tangent to the threshold. Our presented algorithm, EventProp, computes the exact gradient with respect to a general loss function based on spike times and membrane potentials. Crucially, the algorithm allows for an event-based communication scheme in the backward phase, retaining the potential advantages of temporal sparsity afforded by spiking neural networks. We demonstrate the optimization of spiking networks using gradients computed via EventProp and the Yin-Yang and MNIST datasets with either a spike time-based or voltage-based loss function and report competitive performance. Our work supports the rigorous study of gradient-based optimization in spiking neural networks as well as the development of event-based neuromorphic architectures for the efficient training of spiking neural networks. While we consider the leaky integrate-and-fire model in this work, our methodology generalises to any neuron model defined as a hybrid dynamical system.

I will argue that the circuit architecture of the early olfactory system provides an adaptive, efficient mechanism for compressing the vast space of odor mixtures into the responses of a small number of sensors. In this view, the olfactory sensory repertoire employs a disordered code to compress a high dimensional olfactory space into a low dimensional receptor response space while preserving distance relations between odors. The resulting representation is dynamically adapted to efficiently encode the changing environment of volatile molecules. I will show that this adaptive combinatorial code can be efficiently decoded by systematically eliminating candidate odorants that bind to silent receptors. The resulting algorithm for 'estimation by elimination' can be implemented by a neural network that is remarkably similar to the early olfactory pathway in the brain. Finally, I will discuss how diffuse feedback from the central brain to the bulb, followed by unstructured projections back to the cortex, can produce the convergence and divergence of the cortical representation of odors presented in shared or different contexts. Our theory predicts a relation between the diversity of olfactory receptors and the sparsity of their responses that matches animals from flies to humans. It also predicts specific deficits in olfactory behavior that should result from optogenetic manipulation of the olfactory bulb and cortex, and in some disease states.

How can neural networks learn to efficiently represent complex and high-dimensional inputs via local plasticity mechanisms? Classical models of representation learning assume that input weights are learned via pairwise Hebbian-like plasticity. Here, we show that pairwise Hebbian-like plasticity only works under specific requirements on neural dynamics and input statistics. To overcome these limitations, we derive from first principles a learning scheme based on voltage-dependent synaptic plasticity rules. Here, inhibition learns to locally balance excitatory input in individual dendritic compartments, and thereby can modulate excitatory synaptic plasticity to learn efficient representations. We demonstrate in simulations that this learning scheme works robustly even for complex, high-dimensional and correlated inputs. It also works in the presence of inhibitory transmission delays, where Hebbian-like plasticity typically fails. Our results draw a direct connection between dendritic excitatory-inhibitory balance and voltage-dependent synaptic plasticity as observed in vivo, and suggest that both are crucial for representation learning.

Cortical circuits process information by rich recurrent interactions between excitatory neurons and inhibitory interneurons. One of the prime functions of interneurons is to stabilize the circuit by feedback inhibition, but the level of specificity on which inhibitory feedback operates is not fully resolved. We hypothesized that inhibitory circuits could enable separate feedback control loops for different synaptic input streams, by means of specific feedback inhibition to different neuronal compartments. To investigate this hypothesis, we adopted an optimization approach. Leveraging recent advances in training spiking network models, we optimized the connectivity and short-term plasticity of interneuron circuits for compartment-specific feedback inhibition onto pyramidal neurons. Over the course of the optimization, the interneurons diversified into two classes that resembled parvalbumin (PV) and somatostatin (SST) expressing interneurons. The resulting circuit can be understood as a neural decoder that inverts the nonlinear biophysical computations performed within the pyramidal cells. Our model provides a proof of concept for studying structure-function relations in cortical circuits by a combination of gradient-based optimization and biologically plausible phenomenological models

The talk will give a tour of Guided Search 6.0 (GS6), the latest evolution of the Guided Search model of visual search. Part 1 describes The Mechanics of Search. Because we cannot recognize more than a few items at a time, selective attention is used to prioritize items for processing. Selective attention to an item allows its features to be bound together into a representation that can be matched to a target template in memory or rejected as a distractor. The binding and recognition of an attended object is modeled as a diffusion process taking > 150 msec/item. Since selection occurs more frequently than that, it follows that multiple items are undergoing recognition at the same time, though asynchronously, making GS6 a hybrid serial and parallel model. If a target is not found, search terminates when an accumulating quitting signal reaches a threshold. Part 2 elaborates on the five sources of Guidance that are combined into a spatial “priority map” to guide the deployment of attention (hence “guided search”). These are (1) top-down and (2) bottom-up feature guidance, (3) prior history (e.g. priming), (4) reward, and (5) scene syntax and semantics. Finally, in Part 3, we will consider the internal representation of what we are searching for; what is often called “the search template”. That search template is really two templates: a guiding template (probably in working memory) and a target template (in long term memory). Put these pieces together and you have GS6.

Tissue folding is a ubiquitous shape change event during development whereby a cell sheet bends into a curved 3D structure. This mechanical process is remarkably robust, and the correct final form is almost always achieved despite internal fluctuations and external perturbations inherent in living systems. While many genetic and molecular strategies that lead to robust development have been established, much less is known about how mechanical patterns and movements are ensured at the population level. I will describe how quantitative imaging, physical modeling and concepts from network science can uncover collective interactions that govern tissue patterning and shape change. Actin and myosin are two important cytoskeletal proteins involved in the force generation and movement of cells. Both parts of this talk will be about the spontaneous organization of actomyosin networks and their role in collective tissue dynamics. First, I will present how out-of-plane curvature can trigger the global alignment of actin fibers and a novel transition from collective to individual cell migration in culture. I will then describe how tissue-scale cytoskeletal patterns can guide tissue folding in the early fruit fly embryo. I will show that actin and myosin organize into a network that spans a domain of the embryo that will fold. Redundancy in this supracellular network encodes the tissue’s intrinsic robustness to mechanical and molecular perturbations during folding.

After more than 4 million deaths worldwide, the ongoing vaccination to conquer the COVID-19 disease is now competing with the emergence of increasingly contagious mutations, repeatedly supplanting earlier strains. Following the near-absence of historical examples of the long-time evolution of infectious diseases under similar circumstances, models are crucial to exemplify possible scenarios. Accordingly, in the present work we systematically generalize the popular susceptible-infected-recovered model to account for mutations leading to repeatedly occurring new strains, which we coarse grain based on tools from statistical mechanics to derive a model predicting the most likely outcomes. The model predicts that mutations can induce a super exponential growth of infection numbers at early times, which self-amplify to giant infection waves which are caused by a positive feedback loop between infection numbers and mutations and lead to a simultaneous infection of the majority of the population. At later stages -- if vaccination progresses too slowly -- mutations can interrupt an ongoing decrease of infection numbers and can cause infection revivals which occur as single waves or even as whole wave trains featuring alternative periods of decreasing and increasing infection numbers. Our results might be useful for discussions regarding the importance of a release of vaccine-patents to reduce the risk of mutation-induced infection revivals but also to coordinate the release of measures following a downwards trend of infection numbers.

Hydra is an extraordinary creature. Continuously replacing itself, it can live indefinitely, performing a stable repertoire of reasonably sophisticated behaviors. This remarkable stability under plasticity may be due to the uniform nature of its nervous system, which consists of two apparently noncommunicating nerve net layers. We use modeling to understand the role of active muscles and biomechanics interact with neural activity to shape Hydra behaviour. We will discuss our findings and thoughts on how this simple nervous system may self-organize to produce purposeful behavior.