Mathematical Modeling seminars

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.

Animals depend on fast and reliable detection of novel stimuli in their environment. Neurons in multiple sensory areas respond more strongly to novel in comparison to familiar stimuli. Yet, it remains unclear which circuit, cellular, and synaptic mechanisms underlie those responses. Here, we show that spike-timing-dependent plasticity of inhibitory-to-excitatory synapses generates novelty responses in a recurrent spiking network model. Inhibitory plasticity increases the inhibition onto excitatory neurons tuned to familiar stimuli, while inhibition for novel stimuli remains low, leading to a network novelty response. The generation of novelty responses does not depend on the periodicity but rather on the distribution of presented stimuli. By including tuning of inhibitory neurons, the network further captures stimulus-specific adaptation. Finally, we suggest that disinhibition can control the amplification of novelty responses. Therefore, inhibitory plasticity provides a flexible, biologically plausible mechanism to detect the novelty of bottom-up stimuli, enabling us to make experimentally testable predictions.

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

Like last year this online workshop brings together researchers in the field to present their work and discuss ways of translating these findings into a better understanding of neural circuits. Topics include artificial and biologically plausible learning algorithms and the dissection of trained spiking circuits toward understanding neural processing. We have a manageable number of talks with ample time for discussions. This year’s executive committee comprises Chiara Bartolozzi, Sander Bohté, Dan Goodman, and Friedemann Zenke.

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.

Human vision is full of puzzles. Observers can grasp the essence of a scene in an instant, yet when probed for details they are at a loss. People have trouble finding their keys, yet they may be quite visible once found. How does one explain this combination of marvelous successes with quirky failures? I will describe our attempts to develop a unifying theory that brings a satisfying order to multiple phenomena. One key is to understand peripheral vision. A visual system cannot process everything with full fidelity, and therefore must lose some information. Peripheral vision must condense a mass of information into a succinct representation that nonetheless carries the information needed for vision at a glance. We have proposed that the visual system deals with limited capacity in part by representing its input in terms of a rich set of local image statistics, where the local regions grow — and the representation becomes less precise — with distance from fixation. This scheme trades off computation of sophisticated image features at the expense of spatial localization of those features. What are the implications of such an encoding scheme? Critical to our understanding has been the use of methodologies for visualizing the equivalence classes of the model. These visualizations allow one to quickly see that many of the puzzles of human vision may arise from a single encoding mechanism. They have suggested new experiments and predicted unexpected phenomena. Furthermore, visualization of the equivalence classes has facilitated the generation of testable model predictions, allowing us to study the effects of this relatively low-level encoding on a wide range of higher-level tasks. Peripheral vision helps explain many of the puzzles of vision, but some remain. By examining the phenomena that cannot be explained by peripheral vision, we gain insight into the nature of additional capacity limits in vision. In particular, I will suggest that decision processes face general-purpose limits on the complexity of the tasks they can perform at a given time.

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.

Homeostatic plasticity plays a key role in stabilizing neural network activity. But what is its role in neural information processing? We showed analytically how homeostasis changes collective dynamics and consequently information flow - depending on the input to the network. We then studied how input and homeostasis on a recurrent network of LIF neurons impacts information flow and task performance. We showed how we can tune the working point of the network, and found that, contrary to previous assumptions, there is not one optimal working point for a family of tasks, but each task may require its own working point.

Choosing among spatially distributed options is a central challenge for animals, from deciding among alternative potential food sources or refuges, to choosing with whom to associate. Here, using an integrated theoretical and experimental approach (employing immersive Virtual Reality), with both invertebrate and vertebrate models—the fruit fly, desert locust and zebrafish—we consider the recursive interplay between movement and collective vectorial integration in the brain during decision-making regarding options (potential ‘targets’) in space. We reveal that the brain repeatedly breaks multi-choice decisions into a series of abrupt (critical) binary decisions in space-time where organisms switch, spontaneously, from averaging vectorial information among, to suddenly excluding one of, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains. Close to each bifurcation the ‘susceptibility’ of the system exhibits a sharp increase, inevitably causing small differences among the remaining options to become amplified; a property that both comes ‘for free’ and is highly desirable for decision-making. This mechanism facilitates highly effective decision-making, and is shown to be robust both to the number of options available, and to context, such as whether options are static (e.g. refuges) or mobile (e.g. other animals). In addition, we find evidence that the same geometric principles of decision-making occur across scales of biological organisation, from neural dynamics to animal collectives, suggesting they are fundamental features of spatiotemporal computation.

Spike-driven adaptation involves intracellular mechanisms that are initiated by spiking and lead to the subsequent reduction of spiking rate. One of its consequences is the temporal patterning of spike trains, as it imparts serial correlations between interspike intervals in baseline activity. Surprisingly the hidden adaptation states that lead to these correlations themselves exhibit quasi-independence. This talk will first discuss recent findings about the role of such adaptation in suppressing noise and extending sensory detection to weak stimuli that leave the firing rate unchanged. Further, a matching of the post-synaptic responses to the pre-synaptic adaptation time scale enables a recovery of the quasi-independence property, and can explain observations of correlations between post-synaptic EPSPs and behavioural detection thresholds. We then consider the involvement of spike-driven adaptation in the representation of intervals between sensory events. We discuss the possible link of this time-stamping mechanism to the conversion of egocentric to allocentric coordinates. The heterogeneity of the population parameters enables the representation and Bayesian decoding of time sequences of events which may be put to good use in path integration and hilus neuron function in hippocampus.

The success of deep learning sparked interest in whether the brain learns by using similar techniques for assigning credit to each synaptic weight for its contribution to the network output. However, the majority of current attempts at biologically-plausible learning methods are either non-local in time, require highly specific connectivity motives, or have no clear link to any known mathematical optimization method. Here, we introduce Deep Feedback Control (DFC), a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of feedback connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing. By combining dynamical system theory with mathematical optimization theory, we provide a strong theoretical foundation for DFC that we corroborate with detailed results on toy experiments and standard computer-vision benchmarks.

Biological systems can self-organize in complex structures, able to evolve and adapt to widely varying environmental conditions. Despite the importance of fluid flow for transporting and organizing populations, few laboratory systems exist to systematically investigate the impact of advection on their spatial evolutionary dynamics. In this talk, I will discuss how we can address this problem by studying the morphology and genetic spatial structure of microbial colonies growing on the surface of a viscous substrate. When grown on a liquid, I will show that S. cerevisiae (baker’s yeast) can behave like “active matter” and collectively generate a fluid flow many times larger than the unperturbed colony expansion speed, which in turn produces mechanical stresses and fragmentation of the initial colony. Combining laboratory experiments with numerical modeling, I will demonstrate that the coupling between metabolic activity and hydrodynamic flows can produce positive feedbacks and drive preferential growth phenomena leading to the formation of microbial jets. Our work provides rich opportunities to explore the interplay between hydrodynamics, growth and competition within a versatile system.

Networks are widely used as mathematical models of complex systems across many scientific disciplines, and in particular within neuroscience. In this talk, we introduce two aspects of our collaborative research: (1) machine learning and networks, and (2) graph dimensionality. Machine learning and networks. Decades of work have produced a vast corpus of research characterising the topological, combinatorial, statistical and spectral properties of graphs. Each graph property can be thought of as a feature that captures important (and sometimes overlapping) characteristics of a network. We have developed hcga, a framework for highly comparative analysis of graph data sets that computes several thousands of graph features from any given network. Taking inspiration from hctsa, hcga offers a suite of statistical learning and data analysis tools for automated identification and selection of important and interpretable features underpinning the characterisation of graph data sets. We show that hcga outperforms other methodologies (including deep learning) on supervised classification tasks on benchmark data sets whilst retaining the interpretability of network features, which we exemplify on a dataset of neuronal morphologies images. Graph dimensionality. Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. Deviating from approaches based on fractals, here, we present a new framework to define intrinsic notions of dimension on networks, the relative, local and global dimension. We showcase our method on various physical systems.