Mathematical Modeling seminars

Lenka Zdeborová introduces the Rules-and-Facts model, in which some observations follow a shared rule while others are isolated exceptions requiring memorization. The framework studies when a learner can acquire the rule and retain those exceptions simultaneously. Its results emphasize how capacity is organized and deployed, rather than capacity alone. The talk examines how regularization and the geometry of kernels or learned feature maps can reserve resources for memorization without undermining rule learning. It connects the balance between abstraction and memory to architectural and algorithmic choices, providing a theoretical account of neural networks that both generalize and recall specific facts.

This seminar discusses a proof-of-concept gravitational-waveform model described in arXiv:2511.10522. It augments post-Newtonian equations using physics-informed neural-network methods and universal differential equations. The talk also covers ongoing development of a precessing surrogate model designed to evaluate in less than one millisecond while covering a broader portion of the LVK frequency band.

Understanding how brains learn requires bridging evidence across scales—from behaviour and neural circuits to cells, synapses, and molecules. In our work, we use computational modelling and data analysis to explore how the physical properties of neurons and neural circuits constrain learning. These include limits imposed by brain wiring, energy availability, molecular noise, and the 3D structure of dendritic spines. In this talk I will describe one such project testing if wiring motifs from fly brain connectomes can improve performance of reservoir computers, a type of recurrent neural network. The hope is that these insights into brain learning will lead to improved learning algorithms for artificial systems.

Brain Emulation Challenge workshop will tackle cutting-edge topics such as ground-truthing for validation, leveraging artificial datasets generated from virtual brain tissue, and the transformative potential of virtual brain platforms, such as applied to the forthcoming Brain Emulation Challenge.

Brain Emulation Challenge workshop will tackle cutting-edge topics such as ground-truthing for validation, leveraging artificial datasets generated from virtual brain tissue, and the transformative potential of virtual brain platforms, such as applied to the forthcoming Brain Emulation Challenge.

Brain Emulation Challenge workshop will tackle cutting-edge topics such as ground-truthing for validation, leveraging artificial datasets generated from virtual brain tissue, and the transformative potential of virtual brain platforms, such as applied to the forthcoming Brain Emulation Challenge.

This webinar addressed computational perspectives on how animals and humans make decisions, spanning normative, descriptive, and mechanistic models. Sam Gershman (Harvard) presented a capacity-limited reinforcement learning framework in which policies are compressed under an information bottleneck constraint. This approach predicts pervasive perseveration, stimulus‐independent “default” actions, and trade-offs between complexity and reward. Such policy compression reconciles observed action stochasticity and response time patterns with an optimal balance between learning capacity and performance. Jonathan Pillow (Princeton) discussed flexible descriptive models for tracking time-varying policies in animals. He introduced dynamic Generalized Linear Models (Sidetrack) and hidden Markov models (GLM-HMMs) that capture day-to-day and trial-to-trial fluctuations in choice behavior, including abrupt switches between “engaged” and “disengaged” states. These models provide new insights into how animals’ strategies evolve under learning. Finally, Kenji Doya (OIST) highlighted the importance of unifying reinforcement learning with Bayesian inference, exploring how cortical-basal ganglia networks might implement model-based and model-free strategies. He also described Japan’s Brain/MINDS 2.0 and Digital Brain initiatives, aiming to integrate multimodal data and computational principles into cohesive “digital brains.”

In this talk, I will explore how social affective biases arise even in the absence of motivational factors as an emergent outcome of the basic structure of social learning. In several studies, we found that initial negative interactions with some members of a group can cause subsequent avoidance of the entire group, and that this avoidance perpetuates stereotypes. Additional cognitive modeling discovered that approach and avoidance behavior based on biased beliefs not only influences the evaluative (positive or negative) impressions of group members, but also shapes the depth of the cognitive representations available to learn about individuals. In other words, people have richer cognitive representations of members of groups that are not avoided, akin to individualized vs group level categories. I will end presenting a series of multi-agent reinforcement learning simulations that demonstrate the emergence of these social-structural feedback loops in the development and maintenance of affective biases.

Developing scientifically-grounded theories for representation learning and generalization in artificial deep neural networks remains a grand challenge of fundamental interest to theoretical neuroscience and machine learning. I will discuss our work on one facet of this challenge — namely understanding generalization or “scaling laws” in learned neural networks as a function of basic control variables. I’ll discuss a taxonomy we develop that classifies different regimes of scaling behavior. We identify regimes where generalization exhibits universal scaling behavior and others where it can be traced back to properties of the data and neural architecture. The theoretical analysis is enabled by leveraging exactly solvable models of deep neural networks that arise naturally in the limit of large hidden layers. Along the way, I’ll also discuss our work on these theoretical models, which have been a useful starting point for theoretical descriptions of neural network dynamics. Finally, I’ll discuss our findings connecting generalization in neural networks to properties of the learned data manifold. I’ll close by discussing future directions and new hypotheses that emerge from our findings Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-06-19. Recording duration: 00:46:53.

Learning in biological systems takes place in contexts and with dynamics not often accounted for by simple models. I will describe the learning dynamics of two model systems that incorporate either architectural or dynamic constraints from biological observations. In the first case, inspired by the observed mesoscopic structure of the mouse brain as revealed by the Allen Mouse Brain Connectivity Atlas, as well as multiple examples of parallel pathways in mammalian brains, I present a mathematical analysis of learning dynamics in networks that have parallel computational pathways driven by the same cost function. We use the approximation of deep linear networks with large hidden layer sizes to show that, as the depth of the parallel pathways increases, different features of the training set (defined by the singular values of the input-output correlation) will typically concentrate in one of the pathways. This result is derived analytically and demonstrated with numerical simulation with both linear and non-linear networks. Thus, rather than sharing stimulus and task features across multiple pathways, parallel network architectures learn to produce sharply diversified representations with specialized and specific pathways, a mechanism which may hold important consequences for codes in both biological and artificial systems. In the second case, I discuss learning dynamics in a generalization of Hebbian rules and show that these rules allow a neuron to learn tensor decompositions of higher-order input correlations. Unlike the case of the Oja rule and PCA, the resulting learned representation is not unique but selects amongst the tensor eigenvectors according to initial conditions. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-04-10. Recording duration: 00:30:39.

Cognitive maps confer animals with flexible intelligence by representing spatial, temporal, and abstract relationships that can be used to shape thought, planning, and behavior. Cognitive maps have been observed in the hippocampus, but their algorithmic form and the processes by which they are learned remain obscure. Here, we employed large-scale, longitudinal two-photon calcium imaging to record activity from thousands of neurons in the CA1 region of the hippocampus while mice learned to efficiently collect rewards from two subtly different versions of linear tracks in virtual reality. The results provide a detailed view of the formation of a cognitive map in the hippocampus. Throughout learning, both the animal behavior and hippocampal neural activity progressed through multiple intermediate stages, gradually revealing improved task representation that mirrored improved behavioral efficiency. The learning process led to progressive decorrelations in initially similar hippocampal neural activity within and across tracks, ultimately resulting in orthogonalized representations resembling a state machine capturing the inherent struture of the task. We show that a Hidden Markov Model (HMM) and a biologically plausible recurrent neural network trained using Hebbian learning can both capture core aspects of the learning dynamics and the orthogonalized representational structure in neural activity. In contrast, we show that gradient-based learning of sequence models such as Long Short-Term Memory networks (LSTMs) and Transformers do not naturally produce such orthogonalized representations. We further demonstrate that mice exhibited adaptive behavior in novel task settings, with neural activity reflecting flexible deployment of the state machine. These findings shed light on the mathematical form of cognitive maps, the learning rules that sculpt them, and the algorithms that promote adaptive behavior in animals. The work thus charts a course toward a deeper understanding of biological intelligence and offers insights toward developing more robust learning algorithms in artificial intelligence.

We build upon and expand the efficient coding and predictive information models of neurons, presenting a novel perspective that neurons not only predict but also actively influence their future inputs through their outputs. We introduce the concept of neurons as feedback controllers of their environments, a role traditionally considered computationally demanding, particularly when the dynamical system characterizing the environment is unknown. By harnessing a novel data-driven control framework, we illustrate the feasibility of biological neurons functioning as effective feedback controllers. This innovative approach enables us to coherently explain various experimental findings that previously seemed unrelated. Our research has profound implications, potentially revolutionizing the modeling of neuronal circuits and paving the way for the creation of alternative, biologically inspired artificial neural networks.

Changes in ocular hemodynamics may be indicative of pathological conditions in the eye (e.g. glaucoma, age-related macular degeneration), but also elsewhere in the body (e.g. systemic hypertension, diabetes, neurodegenerative disorders). Thanks to its transparent fluids and structures that allow the light to go through, the eye offers a unique window on the circulation from large to small vessels, and from arteries to veins. Deciphering the causes that lead to changes in ocular hemodynamics in a specific individual could help prevent vision loss as well as aid in the diagnosis and management of diseases beyond the eye. In this talk, we will discuss how mathematical and computational modelling can help in this regard. We will focus on two main factors, namely blood pressure (BP), which drives the blood flow through the vessels, and intraocular pressure (IOP), which compresses the vessels and may impede the flow. Mechanism-driven models translates fundamental principles of physics and physiology into computable equations that allow for identification of cause-to-effect relationships among interplaying factors (e.g. BP, IOP, blood flow). While invaluable for causality, mechanism-driven models are often based on simplifying assumptions to make them tractable for analysis and simulation; however, this often brings into question their relevance beyond theoretical explorations. Data-driven models offer a natural remedy to address these short-comings. Data-driven methods may be supervised (based on labelled training data) or unsupervised (clustering and other data analytics) and they include models based on statistics, machine learning, deep learning and neural networks. Data-driven models naturally thrive on large datasets, making them scalable to a plethora of applications. While invaluable for scalability, data-driven models are often perceived as black- boxes, as their outcomes are difficult to explain in terms of fundamental principles of physics and physiology and this limits the delivery of actionable insights. The combination of mechanism-driven and data-driven models allows us to harness the advantages of both, as mechanism-driven models excel at interpretability but suffer from a lack of scalability, while data-driven models are excellent at scale but suffer in terms of generalizability and insights for hypothesis generation. This combined, integrative approach represents the pillar of the interdisciplinary approach to data science that will be discussed in this talk, with application to ocular hemodynamics and specific examples in glaucoma research.

Humans and animals are not always rational. They not only rationally exploit rewards but also explore an environment owing to their curiosity. However, the mechanism of such curiosity-driven irrational behavior is largely unknown. Here, we developed a decision-making model for a two-choice task based on the free energy principle, which is a theory integrating recognition and action selection. The model describes irrational behaviors depending on the curiosity level. We also proposed a machine learning method to decode temporal curiosity from behavioral data. By applying it to rat behavioral data, we found that the rat had negative curiosity, reflecting conservative selection sticking to more certain options and that the level of curiosity was upregulated by the expected future information obtained from an uncertain environment. Our decoding approach can be a fundamental tool for identifying the neural basis for reward–curiosity conflicts. Furthermore, it could be effective in diagnosing mental disorders.

When interacting with complex environments, humans can rapidly adapt their behavior to changes in task or context. To facilitate this adaptation, we often spend substantial periods of time contemplating possible futures before acting. For such planning to be rational, the benefits of planning to future behavior must at least compensate for the time spent thinking. Here we capture these features of human behavior by developing a neural network model where not only actions, but also planning, are controlled by prefrontal cortex. This model consists of a meta-reinforcement learning agent augmented with the ability to plan by sampling imagined action sequences drawn from its own policy, which we refer to as 'rollouts'. Our results demonstrate that this agent learns to plan when planning is beneficial, explaining the empirical variability in human thinking times. Additionally, the patterns of policy rollouts employed by the artificial agent closely resemble patterns of rodent hippocampal replays recently recorded in a spatial navigation task, in terms of both their spatial statistics and their relationship to subsequent behavior. Our work provides a new theory of how the brain could implement planning through prefrontal-hippocampal interactions, where hippocampal replays are triggered by - and in turn adaptively affect - prefrontal dynamics.

I will discuss work with Jack Lindsey modeling reinforcement learning for action selection in the basal ganglia. I will argue that the presence of multiple brain regions, in addition to the basal ganglia, that contribute to motor control motivates the need for an off-policy basal ganglia learning algorithm. I will then describe a biological implementation of such an algorithm that predicts tuning of dopamine neurons to a quantity we call "action surprise," in addition to reward prediction error. In the same model, an implementation of learning from a motor efference copy also predicts a novel solution to the problem of multiplexing feedforward and efference-related striatal activity. The solution exploits the difference between D1 and D2-expressing medium spiny neurons and leads to predictions about striatal dynamics.

What is the relationship between task, network architecture, and population activity in nonlinear deep networks? I will describe the Gated Deep Linear Network framework, which schematizes how pathways of information flow impact learning dynamics within an architecture. Because of the gating, these networks can compute nonlinear functions of their input. We derive an exact reduction and, for certain cases, exact solutions to the dynamics of learning. The reduction takes the form of a neural race with an implicit bias towards shared representations, which then govern the model’s ability to systematically generalize, multi-task, and transfer. We show how appropriate network architectures can help factorize and abstract knowledge. Together, these results begin to shed light on the links between architecture, learning dynamics and network performance.

Humans and animals learn and plan with flexibility and efficiency well beyond that of modern Machine Learning methods. This is hypothesized to owe in part to the ability of animals to build structured representations of their environments, and modulate these representations to rapidly adapt to new settings. In the first part of this talk, I will discuss theoretical work describing how learned representations in hippocampus enable rapid adaptation to new goals by learning predictive representations, while entorhinal cortex compresses these predictive representations with spectral methods that support smooth generalization among related states. I will also cover recent work extending this account, in which we show how the predictive model can be adapted to the probabilistic setting to describe a broader array of generalization results in humans and animals, and how entorhinal representations can be modulated to support sample generation optimized for different behavioral states. In the second part of the talk, I will overview some of the ways in which we have combined many of the same mathematical concepts with state-of-the-art deep learning methods to improve efficiency and performance in machine learning applications like physical simulation, relational reasoning, and design.

Cognitive science and cognitive neuroscience researchers have agreed that the replication of findings is important for establishing which ideas (or theories) are integral to the study of cognition across the lifespan. Recently, high-profile papers have called into question findings that were once thought to be unassailable. Much attention has been paid to how p-hacking, publication bias, and sample size are responsible for failed replications. However, much less attention has been paid to the fidelity by which researchers enact study protocols. Researchers conducting education or clinical trials are aware of the importance in fidelity – or the extent to which the protocols are delivered in the same way across participants. Nevertheless, this idea has not been applied to cognitive contexts. This seminar discusses factors that impact the replicability of findings alongside recent models suggesting that even small fidelity deviations have real impacts on the data collected.