Mathematical Modeling seminars

Galaxy images and spectra map large-scale cosmic structure and constrain fundamental physics. While galaxy-clustering bias models are well developed, forthcoming datasets can also trace the intergalactic medium and recover complementary cosmic-web information. Jamie Sullivan discusses models for both neutral and ionized intergalactic material. Neutral-hydrogen work includes the environments of Lyman-alpha emitters at redshifts 2.5–5, relevant to future DESI extensions and Spec-S5, together with the measured Lyman-alpha forest and an observationally practical compression of its bispectrum. Ionized-gas research studies momentum through an effective-field-theory kinetic Sunyaev–Zel’dovich bispectrum, and density through fast-radio-burst clustering and cross-correlations. Together, these approaches aim to map intergalactic modes alongside galaxy tracers and improve precision cosmological constraints.

How do humans stabilize an inverted pendulum, and why does a balanced pole eventually fall? Drawing on 25 years of fingertip pole-balancing research, this talk examines neural correction delays: longer poles move more slowly relative to the nervous system’s response time. Delay-differential models can stabilize the upright position, yet skilled people still experience falls. The proposed explanation is microchaos arising from interactions among delay, sensory dead zones and frequency-dependent force encoding. A region of transient falling solutions lies next to stable microchaotic dynamics. Such microchaos is absent in virtual frontal-plane balancing tasks, while models of standing postural sway lack the corresponding transient regime. The comparison suggests that human falls, unlike pole falls, are more plausibly associated with medical events or slips and trips.

To orchestrate complex behaviors in mammals, the cerebral cortex must continuously transmit its processing results to subcortical regions. While internal cortical processing relies on highly selective sparse codes, these descending cortical output streams employ an enigmatic dense code, characterized by high firing rates and low feature selectivity. This dense population code presents a profound paradox: how can downstream regions extract precise, cognitively relevant signals when most cortical output neurons can be active at any moment? We address this paradox by testing the hypothesis that cortex does not lose information when switching to a dense code. Instead, it utilizes a mechanism for sparse-to-dense coding transformations discovered by my laboratory: thalamocortical synapses target specifically the dendritic initiation zone for calcium action potentials (APs), which enables cortical output neurons to transmit multiple information streams simultaneously via a multiplexed 1-2-3 AP syntax. I will provide first evidence that this remarkable synaptic specificity and coding syntax generalize across long-range pathways. Information coupling via an anatomically and biophysically distinct dendritic nexus may hence be a ubiquitous mechanism for sparse-to-dense coding transformations in cortex. Dissecting these mechanistic origins of cortical output streams is only now possible due to a unique in vivo – in silico approach, perfected over two decades, that my laboratory developed for bridging the gaps between dendritic and population-level computations

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.

Much is known about the neural representations of current environmental location and direction within the hippocampal formation, but use of such a “cognitive map” requires the online representation of desired locations and how to get there, and the neural basis for this function has been more elusive. I will discuss how “theta sweeps” of place and grid cell firing encode the current location (at early phases of each theta cycle) while, at later phases, sampling around the forward direction during exploration and indicating the direction to desired locations during goal-directed navigation. I will show how a relatively simple attractor model captures these results, but requires inputs signalling movement-direction and goal-direction.I will discuss why it is useful to consider the hippocampus as a generative model (in which head-direction, rather than movement-direction, is required, to translate egocentric sensory inputs to allocentric latent representations and back again) in explaining its roles in both spatial cognition and memory consolidation. “Replay sequences” are thought to support offline consolidation, and likely resemble theta sweeps more than behavioural experience. I will finish (given time) by considering how human memory consolidation can be seen as extraction of latent variables from replay via self-supervised learning, and how this perspective explains aspects of human memory such as gist-based distortions, imagination and planning. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-02-25. Recording duration: 00:46:52.

n the neurosciences the need for some 'overarching' theory is sometimes expressed, but it is not always obvious what is meant by this. One can perhaps agree that in modern science observation and experimentation is normally complemented by 'theory', i.e. the development of theoretical concepts that help guiding and evaluating experiments and measurements. A deeper discussion of 'brain theory' will require the clarification of some further distictions, in particular: theory vs. model and brain research (and its theory) vs. neuroscience. Other questions are: Does a theory require mathematics? Or even differential equations? Today it is often taken for granted that the whole universe including everything in it, for example humans, animals, and plants, can be adequately treated by physics and therefore theoretical physics is the overarching theory. Even if this is the case, it has turned out that in some particular parts of physics (the historical example is thermodynamics) it may be useful to simplify the theory by introducing additional theoretical concepts that can in principle be 'reduced' to more complex descriptions on the 'microscopic' level of basic physical particals and forces. In this sense, brain theory may be regarded as part of theoretical neuroscience, which is inside biophysics and therefore inside physics, or theoretical physics. Still, in neuroscience and brain research, additional concepts are typically used to describe results and help guiding experimentation that are 'outside' physics, beginning with neurons and synapses, names of brain parts and areas, up to concepts like 'learning', 'motivation', 'attention'. Certainly, we do not yet have one theory that includes all these concepts. So 'brain theory' is still in a 'pre-newtonian' state. However, it may still be useful to understand in general the relations between a larger theory and its 'parts', or between microscopic and macroscopic theories, or between theories at different 'levels' of description. This is what I plan to do.

How can the hippocampal system transition from episodic one-shot learning to a multi-shot learning regime and what is the utility of the resultant neural representations? This talk will explore the role of the dentate gyrus (DG) anatomy in this context. The canonical DG model suggests it performs pattern separation. More recent experimental results challenge this standard model, suggesting DG function is more complex and also supports the precise binding of objects and events to space and the integration of information across episodes. Very recent studies attribute pattern separation and pattern integration to anatomically distinct parts of the DG (the suprapyramidal blade vs the infrapyramidal blade). We propose a computational model that investigates this distinction. In the model the two processing streams (potentially localized in separate blades) contribute to the storage of distinct episodic memories, and the integration of information across episodes, respectively. The latter forms generalized expectations across episodes, eventually forming a cognitive map. We train the model with two data sets, MNIST and plausible entorhinal cortex inputs. The comparison between the two streams allows for the calculation of a prediction error, which can drive the storage of poorly predicted memories and the forgetting of well-predicted memories. We suggest that differential processing across the DG aids in the iterative construction of spatial cognitive maps to serve the generation of location-dependent expectations, while at the same time preserving episodic memory traces of idiosyncratic events.

The centre of memory is the medial temporal lobe (MTL) and especially the hippocampus. In our research, a more flexible brain-inspired computational microcircuit of the CA1 region of the mammalian hippocampus was upgraded and used to examine how information retrieval could be affected under different conditions. Six models (1-6) were created by modulating different excitatory and inhibitory pathways. The results showed that the increase in the strength of the feedforward excitation was the most effective way to recall memories. In other words, that allows the system to access stored memories more accurately.

This webinar brings together three leaders in theoretical and computational neuroscience—Larry Abbott, Haim Sompolinsky, and Terry Sejnowski—to discuss how neural circuits generate fundamental aspects of the mind. Abbott illustrates mechanisms in electric fish that differentiate self-generated electric signals from external sensory cues, showing how predictive plasticity and two-stage signal cancellation mediate a sense of self. Sompolinsky explores attractor networks, revealing how discrete and continuous attractors can stabilize activity patterns, enable working memory, and incorporate chaotic dynamics underlying spontaneous behaviors. He further highlights the concept of object manifolds in high-level sensory representations and raises open questions on integrating connectomics with theoretical frameworks. Sejnowski bridges these motifs with modern artificial intelligence, demonstrating how large-scale neural networks capture language structures through distributed representations that parallel biological coding. Together, their presentations emphasize the synergy between empirical data, computational modeling, and connectomics in explaining the neural basis of cognition—offering insights into perception, memory, language, and the emergence of mind-like processes.

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.”

This webinar on learning and memory features three experts—Nicolas Brunel, Ashok Litwin-Kumar, and Julijana Gjorgieva—who present theoretical and computational approaches to understanding how neural circuits acquire and store information across different scales. Brunel discusses calcium-based plasticity and how standard “Hebbian-like” plasticity rules inferred from in vitro or in vivo datasets constrain synaptic dynamics, aligning with classical observations (e.g., STDP) and explaining how synaptic connectivity shapes memory. Litwin-Kumar explores insights from the fruit fly connectome, emphasizing how the mushroom body—a key site for associative learning—implements a high-dimensional, random representation of sensory features. Convergent dopaminergic inputs gate plasticity, reflecting a high-dimensional “critic” that refines behavior. Feedback loops within the mushroom body further reveal sophisticated interactions between learning signals and action selection. Gjorgieva examines how activity-dependent plasticity rules shape circuitry from the subcellular (e.g., synaptic clustering on dendrites) to the cortical network level. She demonstrates how spontaneous activity during development, Hebbian competition, and inhibitory-excitatory balance collectively establish connectivity motifs responsible for key computations such as response normalization.

In recent years, my lab and others have demonstrated the value of vector symbolic algebras (VSAs) for capturing a wide variety of neural and behavioural results. In this talk I discuss the surprising and compelling variety of tasks and styles of reasoning that are well-suited to descriptions using a specific VSA. These tasks include path integration, navigation, Bayesian reasoning, sampling, memorization, and logical inference. The resulting spiking neural network models capture various hippocampal cell types (grid, place, border, etc.), behavioural errors, and a variety of observed neural dynamics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-11-20. Recording duration: 00:47:23.

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.

There is a fundamental tension between storing discrete traces of individual experiences, which allows recall of particular moments in our past without interference, and extracting regularities across these experiences, which supports generalization and prediction in similar situations in the future. One influential proposal for how the brain resolves this tension is that it separates the processes anatomically into Complementary Learning Systems, with the hippocampus rapidly encoding individual episodes and the neocortex slowly extracting regularities over days, months, and years. But this does not explain our ability to learn and generalize from new regularities in our environment quickly, often within minutes. We have put forward a neural network model of the hippocampus that suggests that the hippocampus itself may contain complementary learning systems, with one pathway specializing in the rapid learning of regularities and a separate pathway handling the region’s classic episodic memory functions. This proposal has broad implications for how we learn and represent novel information of specific and generalized types, which we test across statistical learning, inference, and category learning paradigms. We also explore how this system interacts with slower-learning neocortical memory systems, with empirical and modeling investigations into how the hippocampus shapes neocortical representations during sleep. Together, the work helps us understand how structured information in our environment is initially encoded and how it then transforms over time.

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.

Remembering events in the past is crucial to intelligent behaviour. Flexible memory retrieval, beyond simple recall, requires a model of how events relate to one another. Two key brain systems are implicated in this process: the hippocampal episodic memory (EM) system and the prefrontal working memory (WM) system. While an understanding of the hippocampal system, from computation to algorithm and representation, is emerging, less is understood about how the prefrontal WM system can give rise to flexible computations beyond simple memory retrieval, and even less is understood about how the two systems relate to each other. Here we develop a mathematical theory relating the algorithms and representations of EM and WM by showing a duality between storing memories in synapses versus neural activity. In doing so, we develop a formal theory of the algorithm and representation of prefrontal WM as structured, and controllable, neural subspaces (termed activity slots). By building models using this formalism, we elucidate the differences, similarities, and trade-offs between the hippocampal and prefrontal algorithms. Lastly, we show that several prefrontal representations in tasks ranging from list learning to cue dependent recall are unified as controllable activity slots. Our results unify frontal and temporal representations of memory, and offer a new basis for understanding the prefrontal representation of WM

First, a new measure of MI for reaction times is proposed that takes the entire RT distribution into account. Second, we present some recent developments in TWIN modeling, including a new proposal for the sound-induced flash illusion (SIFI).

In the current virtual journal club Dr Di Luca will present findings from a series of psychophysical investigations where he measured sensitivity and bias in the perception of the timing of stimuli. He will present how improved detection with longer sequences and biases in reporting isochrony can be accounted for by optimal statistical predictions. Among his findings was also that the timing of stimuli that occasionally deviate from a regularly paced sequence is perceptually distorted to appear more regular. Such change depends on whether the context these sequences are presented is also regular. Dr Di Luca will present a Bayesian model for the combination of dynamically updated expectations, in the form of a priori probability, with incoming sensory information. These findings contribute to the understanding of how the brain processes temporal information to shape perceptual experiences.