Mathematical Modeling seminars

The ability to record large neural populations—hundreds to thousands of cells simultaneously—is a defining feature of modern systems neuroscience. Aside from improved experimental efficiency, what do these technologies fundamentally buy us? I'll argue that they provide an exciting opportunity to move beyond studying the "average" neural response. That is, by providing dense neural circuit measurements in individual subjects and moments in time, these recordings enable us to track changes across repeated behavioral trials and across experimental subjects. These two forms of variability are still poorly understood, despite their obvious importance to understanding the fidelity and flexibility of neural computations. Scientific progress on these points has been impeded by the fact that individual neurons are very noisy and unreliable. My group is investigating a number of customized statistical models to overcome this challenge. I will mention several of these models but focus particularly on a new framework for quantifying across-subject similarity in stochastic trial-by-trial neural responses. By applying this method to noisy representations in deep artificial networks and in mouse visual cortex, we reveal that the geometry of neural noise correlations is a meaningful feature of variation, which is neglected by current methods (e.g. representational similarity analysis).

A complete neuroscience requires multi-level theories that address phenomena ranging from higher-level cognitive behaviors to activities within a cell. Unfortunately, we don't have cognitive models of behavior whose components can be decomposed into the neural dynamics that give rise to behavior, leaving an explanatory gap. Here, we decompose SUSTAIN, a clustering model of concept learning, into neuron-like units (SUSTAIN-d; decomposed). Instead of abstract constructs (clusters), SUSTAIN-d has a pool of neuron-like units. With millions of units, a key challenge is how to bridge from abstract constructs such as clusters to neurons, whilst retaining high-level behavior. How does the brain coordinate neural activity during learning? Inspired by algorithms that capture flocking behavior in birds, we introduce a neural flocking learning rule to coordinate units that collectively form higher-level mental constructs ("virtual clusters"), neural representations (concept, place and grid cell-like assemblies), and parallels recurrent hippocampal activity. The decomposed model shows how brain-scale neural populations coordinate to form assemblies encoding concept and spatial representations, and why many neurons are required for robust performance. Our account provides a multi-level explanation for how cognition and symbol-like representations are supported by coordinated neural assemblies formed through learning.

Neural computations are currently investigated using two separate approaches: sorting neurons into functional subpopulations or examining the low-dimensional dynamics of collective activity. Whether and how these two aspects interact to shape computations is currently unclear. Using a novel approach to extract computational mechanisms from networks trained on neuroscience tasks, here we show that the dimensionality of the dynamics and subpopulation structure play fundamentally com- plementary roles. Although various tasks can be implemented by increasing the dimensionality in networks with fully random population structure, flexible input–output mappings instead require a non-random population structure that can be described in terms of multiple subpopulations. Our analyses revealed that such a subpopulation structure enables flexible computations through a mechanism based on gain-controlled modulations that flexibly shape the collective dynamics. Our results lead to task-specific predictions for the structure of neural selectivity, for inactivation experiments and for the implication of different neurons in multi-tasking.

A long standing challenge in biological and artificial intelligence is to understand how new knowledge can be constructed from known building blocks in a way that is amenable for computation by neuronal circuits. Here we focus on the task of storage and recall of structured knowledge in long-term memory. Specifically, we ask how recurrent neuronal networks can store and retrieve multiple knowledge structures. We model each structure as a set of binary relations between events and attributes (attributes may represent e.g., temporal order, spatial location, role in semantic structure), and map each structure to a distributed neuronal activity pattern using a vector symbolic architecture (VSA) scheme. We then use associative memory plasticity rules to store the binarized patterns as fixed points in a recurrent network. By a combination of signal-to-noise analysis and numerical simulations, we demonstrate that our model allows for efficient storage of these knowledge structures, such that the memorized structures as well as their individual building blocks (e.g., events and attributes) can be subsequently retrieved from partial retrieving cues. We show that long-term memory of structured knowledge relies on a new principle of computation beyond the memory basins. Finally, we show that our model can be extended to store sequences of memories as single attractors.

Our perception of time isn’t like a clock; it varies depending on other aspects of experience, such as what we see and hear in that moment. However, in everyday life, the properties of these simple features can change frequently, presenting a challenge to understanding real-world time perception based on simple lab experiments. We developed a computational model of human time perception based on tracking changes in neural activity across brain regions involved in sensory processing, using fMRI. By measuring changes in brain activity patterns across these regions, our approach accommodates the different and changing feature combinations present in natural scenarios, such as walking on a busy street. Our model reproduces people’s duration reports for natural videos (up to almost half a minute long) and, most importantly, predicts whether a person reports a scene as relatively shorter or longer–the biases in time perception that reflect how natural experience of time deviates from clock time

Artificial neural networks can be useful for studying brain functions. In cognitive neuroscience, recurrent neural networks are often used to model cognitive functions. I will first offer my opinion on what is missing in the classical use of recurrent neural networks. Then I will discuss two lines of ongoing efforts in our group to move beyond the classical recurrent neural networks by studying multi-system neural networks (the talk will focus on two-system networks). These are networks that combine modules for several neural systems, such as vision, audition, prefrontal, hippocampal systems. I will showcase how multi-system networks can potentially be constrained by experimental data in fundamental ways and at scale.

The predictive map hypothesis provides a promising framework to model representations in the hippocampal formation. I will introduce a tractable implementation of a predictive map called the successor representation (SR), before presenting data showing that rats and humans display SR-like navigational choices on a novel open-field maze. Next, I will show how such a predictive map could be implemented using spatial representations found in the hippocampal formation, before finally presenting how such learning might be well approximated by phenomena that exist in the spatial memory system - namely spike-timing dependent plasticity and theta phase precession.

To understand neural computation and the dynamics in the brain, we usually focus on the connectivity among neurons. In contrast, the properties of single neurons are often thought to be negligible, at least as far as the activity of networks is concerned. In this talk, I will contradict this notion and demonstrate how the biophysics of action-potential generation can have a decisive impact on network behaviour. Our recent theoretical work shows that, among regularly firing neurons, the somewhat unattended homoclinic type (characterized by a spike onset via a saddle homoclinic orbit bifurcation) particularly stands out: First, spikes of this type foster specific network states - synchronization in inhibitory and splayed-out/frustrated states in excitatory networks. Second, homoclinic spikes can easily be induced by changes in a variety of physiological parameters (like temperature, extracellular potassium, or dendritic morphology). As a consequence, such parameter changes can even induce switches in network states, solely based on a modification of cellular voltage dynamics. I will provide first experimental evidence and discuss functional consequences of homoclinic spikes for the design of efficient pattern-generating motor circuits in insects as well as for mammalian pathologies like febrile seizures. Our analysis predicts an interesting role for homoclinic action potentials as an integral part of brain dynamics in both health and disease.

New tasks are often similar in structure to old ones. Animals that take advantage of such conserved or “abstract” task structures can master new tasks with minimal training. To understand the neural basis of this abstraction, we developed a novel behavioural paradigm for mice: the “ABCD” task, and recorded from their medial frontal neurons as they learned. Animals learned multiple tasks where they had to visit 4 rewarded locations on a spatial maze in sequence, which defined a sequence of four “task states” (ABCD). Tasks shared the same circular transition structure (… ABCDABCD …) but differed in the spatial arrangement of rewards. As well as improving across tasks, mice inferred that A followed D (i.e. completed the loop) on the very first trial of a new task. This “zero-shot inference” is only possible if animals had learned the abstract structure of the task. Across tasks, individual medial Frontal Cortex (mFC) neurons maintained their tuning to the phase of an animal’s trajectory between rewards but not their tuning to task states, even in the absence of spatial tuning. Intriguingly, groups of mFC neurons formed modules of coherently remapping neurons that maintained their tuning relationships across tasks. Such tuning relationships were expressed as replay/preplay during sleep, consistent with an internal organisation of activity into multiple, task-matched ring attractors. Remarkably, these modules were anchored to spatial locations: neurons were tuned to specific task space “distances” from a particular spatial location. These newly discovered “Spatially Anchored Task clocks” (SATs), suggest a novel algorithm for solving abstraction tasks. Using computational modelling, we show that SATs can perform zero-shot inference on new tasks in the absence of plasticity and guide optimal policy in the absence of continual planning. These findings provide novel insights into the Frontal mechanisms mediating abstraction and flexible behaviour.

How do children learn to read in different countries? How do deficits in various components of the reading network affect learning outcomes? What are the consequences of such deficits in different languages? In this talk, I will present a full-blown developmentally plausible computational model of reading acquisition that has been implemented in English, French, Italian and German. The model can simulate individual learning trajectories and intervention outcomes on the basis of three component skills: orthography, phonology, and vocabulary. I will use the model to show how cross-language differences affect the learning-to-read process in different languages and to investigate to what extent similar deficits will produce similar or different manifestations of dyslexia in different languages.

Electrophysiological recordings during perceptual decision tasks in monkeys suggest that the degree of confidence in a decision is based on a simple neural signal produced by the neural decision process. Attractor neural networks provide an appropriate biophysical modeling framework, and account for the experimental results very well. However, it remains unclear whether attractor neural networks can account for confidence reports in humans. We present the results from an experiment in which participants are asked to perform an orientation discrimination task, followed by a confidence judgment. Here we show that an attractor neural network model quantitatively reproduces, for each participant, the relations between accuracy, response times and confidence. We show that the attractor neural network also accounts for confidence-specific sequential effects observed in the experiment (participants are faster on trials following high confidence trials), as well as non confidence-specific sequential effects. Remarkably, this is obtained as an inevitable outcome of the network dynamics, without any feedback specific to the previous decision (that would result in, e.g., a change in the model parameters before the onset of the next trial). Our results thus suggest that a metacognitive process such as confidence in one’s decision is linked to the intrinsically nonlinear dynamics of the decision-making neural network.

A central goal in neuroscience is to understand how orchestrated computations in the brain arise from the properties of single neurons and networks of such neurons. Answering this question requires theoretical advances that shine light into the ‘black box’ of representations in neural circuits. In this talk, we will demonstrate theoretical approaches that help describe how cognitive and behavioral task implementations emerge from the structure in neural populations and from biologically plausible neural networks. First, we will introduce an analytic theory that connects geometric structures that arise from neural responses (i.e., neural manifolds) to the neural population’s efficiency in implementing a task. In particular, this theory describes a perceptron’s capacity for linearly classifying object categories based on the underlying neural manifolds’ structural properties. Next, we will describe how such methods can, in fact, open the ‘black box’ of distributed neuronal circuits in a range of experimental neural datasets. In particular, our method overcomes the limitations of traditional dimensionality reduction techniques, as it operates directly on the high-dimensional representations, rather than relying on low-dimensionality assumptions for visualization. Furthermore, this method allows for simultaneous multi-level analysis, by measuring geometric properties in neural population data, and estimating the amount of task information embedded in the same population. These geometric frameworks are general and can be used across different brain areas and task modalities, as demonstrated in the work of ours and others, ranging from the visual cortex to parietal cortex to hippocampus, and from calcium imaging to electrophysiology to fMRI datasets. Finally, we will discuss our recent efforts to fully extend this multi-level description of neural populations, by (1) investigating how single neuron properties shape the representation geometry in early sensory areas, and by (2) understanding how task-efficient neural manifolds emerge in biologically-constrained neural networks. By extending our mathematical toolkit for analyzing representations underlying complex neuronal networks, we hope to contribute to the long-term challenge of understanding the neuronal basis of tasks and behaviors.

Efficient navigation requires animals to track their position, velocity and heading direction (HD). Some animals’ behavior suggests that they also track uncertainties about these navigational variables, and make strategic use of these uncertainties, in line with a Bayesian computation. Ring-attractor networks have been proposed to estimate and track these navigational variables, for instance in the HD system of the fruit fly Drosophila. However, such networks are not designed to incorporate a notion of uncertainty, and therefore seem unsuited to implement dynamic Bayesian inference. Here, we close this gap by showing that specifically tuned ring-attractor networks can track both a HD estimate and its associated uncertainty, thereby approximating a circular Kalman filter. We identified the network motifs required to integrate angular velocity observations, e.g., through self-initiated turns, and absolute HD observations, e.g., visual landmark inputs, according to their respective reliabilities, and show that these network motifs are present in the connectome of the Drosophila HD system. Specifically, our network encodes uncertainty in the amplitude of a localized bump of neural activity, thereby generalizing standard ring attractor models. In contrast to such standard attractors, however, proper Bayesian inference requires the network dynamics to operate in a regime away from the attractor state. More generally, we show that near-Bayesian integration is inherent in generic ring attractor networks, and that their amplitude dynamics can account for close-to-optimal reliability weighting of external evidence for a wide range of network parameters. This only holds, however, if their connection strengths allow the network to sufficiently deviate from the attractor state. Overall, our work offers a novel interpretation of ring attractor networks as implementing dynamic Bayesian integrators. We further provide a principled theoretical foundation for the suggestion that the Drosophila HD system may implement Bayesian HD tracking via ring attractor dynamics.

Animals exhibit remarkable learning capabilities with little direct supervision. Likewise, self-supervised learning is an emergent paradigm in artificial intelligence, closing the performance gap to supervised learning. In the context of biology, self-supervised learning corresponds to a setting where one sense or specific stimulus may serve as a supervisory signal for another. After learning, the latter can be used to predict the former. On the implementation level, it has been demonstrated that such predictive learning can occur at the single neuron level, in compartmentalized neurons that separate and associate information from different streams. We demonstrate the power such self-supervised learning over unsupervised (Hebb-like) learning rules, which depend heavily on stimulus statistics, in two examples: First, in the context of animal navigation where predictive learning can associate internal self-motion information always available to the animal with external visual landmark information, leading to accurate path-integration in the dark. We focus on the well-characterized fly head direction system and show that our setting learns a connectivity strikingly similar to the one reported in experiments. The mature network is a quasi-continuous attractor and reproduces key experiments in which optogenetic stimulation controls the internal representation of heading, and where the network remaps to integrate with different gains. Second, we show that incorporating global gating by reward prediction errors allows the same setting to learn conditioning at the neuronal level with mixed selectivity. At its core, conditioning entails associating a neural activity pattern induced by an unconditioned stimulus (US) with the pattern arising in response to a conditioned stimulus (CS). Solving the generic problem of pattern-to-pattern associations naturally leads to emergent cognitive phenomena like blocking, overshadowing, saliency effects, extinction, interstimulus interval effects etc. Surprisingly, we find that the same network offers a reductionist mechanism for causal inference by resolving the post hoc, ergo propter hoc fallacy.

Current SNNs studies frequently ignore dendrites, the thin membranous extensions of biological neurons that receive and preprocess nearly all synaptic inputs in the brain. However, decades of experimental and theoretical research suggest that dendrites possess compelling computational capabilities that greatly influence neuronal and circuit functions. Notably, standard point-neuron networks cannot adequately capture most hallmark dendritic properties. Meanwhile, biophysically detailed neuron models are impractical for large-network simulations due to their complexity, and high computational cost. For this reason, we introduce Dendrify, a new theoretical framework combined with an open-source Python package (compatible with Brian2) that facilitates the development of bioinspired SNNs. Dendrify, through simple commands, can generate reduced compartmental neuron models with simplified yet biologically relevant dendritic and synaptic integrative properties. Such models strike a good balance between flexibility, performance, and biological accuracy, allowing us to explore dendritic contributions to network-level functions while paving the way for developing more realistic neuromorphic systems.

Neuronal network computation and computation by avalanche supporting networks are of interest to the fields of physics, computer science (computation theory as well as statistical or machine learning) and neuroscience. Here we show that computation of complex Boolean functions arises spontaneously in threshold networks as a function of connectivity and antagonism (inhibition), computed by logic automata (motifs) in the form of computational cascades. We explain the emergent inverse relationship between the computational complexity of the motifs and their rank-ordering by function probabilities due to motifs, and its relationship to symmetry in function space. We also show that the optimal fraction of inhibition observed here supports results in computational neuroscience, relating to optimal information processing.

An object’s colour, brightness and pattern are all influenced by its surroundings, and a number of visual phenomena and “illusions” have been discovered that highlight these often dramatic effects. Explanations for these phenomena range from low-level neural mechanisms to high-level processes that incorporate contextual information or prior knowledge. Importantly, few of these phenomena can currently be accounted for when measuring an object’s perceived colour. Here we ask to what extent colour appearance is predicted by a model based on the principle of coding efficiency. The model assumes that the image is encoded by noisy spatio-chromatic filters at one octave separations, which are either circularly symmetrical or oriented. Each spatial band’s lower threshold is set by the contrast sensitivity function, and the dynamic range of the band is a fixed multiple of this threshold, above which the response saturates. Filter outputs are then reweighted to give equal power in each channel for natural images. We demonstrate that the model fits human behavioural performance in psychophysics experiments, and also primate retinal ganglion responses. Next we systematically test the model’s ability to qualitatively predict over 35 brightness and colour phenomena, with almost complete success. This implies that contrary to high-level processing explanations, much of colour appearance is potentially attributable to simple mechanisms evolved for efficient coding of natural images, and is a basis for modelling the vision of humans and other animals.

The cytoskeleton has the remarkable ability to self-organize into active materials which underlie diverse cellular processes ranging from motility to cell division. Actomyosin is a canonical example of an active material, which generates cellularscale contractility in part through the forces exerted by myosin motors on actin filaments. While the molecular players underlying actomyosin contractility have been well characterized, how cellular-scale deformation in disordered actomyosin networks emerges from filament-scale interactions is not well understood. In this talk, I’ll present work done in collaboration with Sebastian Fürthauer and Nikta Fakhri addressing this question in vivo using the meiotic surface contraction wave seen in oocytes of the bat star Patiria miniata as a model system. By perturbing actin polymerization, we find that the cellular deformation rate is a nonmonotonic function of cortical actin density peaked near the wild type density. To understand this, we develop an active fluid model coarse-grained from filament-scale interactions and find quantitative agreement with the measured data. The model makes further predictions, including the surprising prediction that deformation rate decreases with increasing motor concentration. We test these predictions through protein overexpression and find quantitative agreement. Taken together, this work is an important step for bridging the molecular and cellular length scales for cytoskeletal networks in vivo.

The rising and setting of the sun is accompanied by changes in both the irradiance and the spectral distribution of the sky. Since the discovery of intrinsically photosensitive retinal ganglion cells (ipRGCs) 20 years ago, considerable progress has been made in understanding melanopsin's contributions to encoding irradiance. Much less is known about the cone inputs to ipRGCs and how they could encode changes in the color of the sky. I will summarize our recent connectomic investigation into the cone-opponent inputs to primate ipRGCs and the implications of this work on our understanding of circadian photoentrainment and the evolution of color vision.