Mathematical Modeling

Leonid Berlyand discusses mathematical models for the onset of cell motion driven by myosin contraction. A two-dimensional free-boundary PDE model links cell-shape evolution to diffusion and Keller–Segel-type transport. The talk examines linear stability, a stability-determining eigenvalue and the way nonlinear diffusion changes the bifurcation from supercritical to subcritical. It also considers the curvature of the bifurcation curve, connections to bistability, and the role of non-self-adjoint operators. An example illustrates why a spectral gap alone need not guarantee stability. This is an in-person Analysis of Fluids and Related Topics seminar at Princeton.

This seminar derives how dark-matter halo density profiles emerge from hierarchical assembly using the Boltzmann–Poisson equations for a coarse-grained distribution of subhalos in an expanding universe. An exact self-similar solution relates density slope γ to mass growth M proportional to tˢ through γ = 6/(s + 2). Initial collapse gives a prompt γ = 3/2 cusp; hierarchical accretion recovers the Syer–White relation and an attractive n = −2 fixed point with γ = 1; stalled accretion approaches γ = 3. These stages provide a dynamical account of the inner and outer Navarro–Frenk–White profile.

ISEC brings together ecologists, statisticians and quantitative researchers working on ecological inference and modelling. The main conference runs 10–15 January 2027 in Mérida, Mexico; separate pre-conference workshops run 8–9 January. Registration is open, with early rates through 30 September 2026 and reduced rates for eligible Global South participants.

SIAM’s computational science and engineering meeting covers mathematical models, scalable algorithms, scientific software, high-performance computing and scientific machine learning. It takes place in Pittsburgh on 22–26 February 2027. Travel-support applications are open until 30 November 2026. Abstract deadlines have passed; registration is scheduled to open in December 2026. The meeting is co-located with ACDA27 and the International Meshing Roundtable.

Large recurrent networks are important models in several fields, including neuroscience, machine learning, physics, and applied mathematics. Yet their dynamics are difficult to study directly, because high-dimensional nonlinear systems can exhibit rich behavior that is hard to summarize in terms of individual trajectories. In this talk, I will discuss an approach that seeks to understand such dynamics through the structure of the network’s equilibria. I will focus on a random balanced network of threshold-linear units that undergoes a transition from a single stable equilibrium to extensive chaos as the disorder strength crosses a critical value. Using a combination of Kac–Rice theory, replica calculations, numerical root-finding, and dynamical mean-field theory, we show that the chaotic regime contains an exponentially large number of equilibria. These equilibria are all saddles, but with only a fractionally small number of unstable directions. Surprisingly, despite the completely random connectivity, the equilibria are not scattered randomly through phase space. Instead, they are strongly correlated and confined to a comparatively small region. The chaotic attractor lies within this same region, suggesting a direct geometric link between the organization of unstable equilibria and the collective structure of the dynamics. This picture helps explain why networks with extensive chaos can nevertheless display dynamics dominated by a relatively small number of collective modes. More broadly, the results suggest that the geometry of equilibria provides a useful complementary perspective to dynamical mean-field theory for understanding high-dimensional neural dynamics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-05-27. Recording duration: 00:46:40.

Networks of interconnected neurons display diverse patterns of activity. Relating these patterns to the structure of the network is a central goal of theoretical neuroscience. Classic neural field and rate models have been powerful tools for this purpose due to their analytical tractability. Here, we show that the recently-developed combinatorial threshold-linear network (CTLN) model is a mean-field theory for excitatory-inhibitory Hawkes networks, with clustered connectivity, in an inhibition-stabilized regime. This mapping allows us to leverage powerful analytical results for CTLN networks to predict diverse macroscopic dynamics of clustered Hawkes networks, including metastability between various macroscopic fixed points, limit cycles, and chaotic attractors. We will then examine an extension of this approach to models with nonlinear dendritic dynamics, focusing on dendritic calcium spikes.We uncover a marked point process mean-field theory for these n etworks and use this to examine how somatic vs dendritic-targeting connectivity shapes the mean-field equilibrium phase diagram. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-05-13. Recording duration: 00:54:14.

How does a cat gracefully walk and suddenly freeze when spotting a mouse? In this talk, we look at how networks in the spinal cord generate movement. In particular, we address the fundamental yet poorly understood question of motor control: How can rhythmic movements, such as walking, be generated and stopped at any point in the cycle while posture is preserved? Since conventional models of spinal motor function rely on alternation between flexor and extensor modules, which are limited to just two phases, this question exposes the essential shortcoming of the conventional understanding: How can a system with only two phases generate and stop walking in any phase? To address this question and better understand the generation and stopping of motor activity, we use Neuropixels probes in the rat spinal cord during voluntary, freely moving locomotion. We utilize optogenetic activation of a brainstem nucleus to induce stopping. During locomotion, neuronal manifold activity exhibits robust rotational patterns that are topologically invariant with respect to speed (Linden 2022). Furthermore, this trajectory converges on a stable point-attractor precisely at the moment of arrest, and it persists until the movement is resumed. Through computational modeling, we propose that the walk-to-stop represents a bifurcation from a limit cycle to a fixed point attractor. We also propose a structural network mechanism for its physical implementation (Komi 2026). The structural mechanism entails a longitudinal projectome with a skewed Mexican hat topology, i.e., primarily local recurrent excitation and longer-range inhibition. Such a network can generate motor patterns via traveling waves, with frequency and amplitude controlled independently, and rhythm induced without requiring cellular pacemaker mechanisms. Together, our experimental observations support a new theory for the mechanism behind the generation of movement by networks in the spinal cord. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-03-25. Recording duration: 00:37:44.

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.

Sabbatical fellowship providing salary replacement (up to 50% of academic-year salary, capped at $125,000) plus up to $10,000 in leave-related expenses so mathematicians can extend a sabbatical term to a full academic year; maximum total budget $162,000 including 20% overhead.

This NIH BRAIN Initiative funding opportunity supports new or substantially advanced theories, mechanistic or predictive models, and computational or statistical methods that improve quantitative understanding of brain function across scales. Tools must address complex neural and behavioral data and be made broadly available to the neuroscience research community.

Develop an analytical and computational research project with Arne Traulsen’s Department of Theoretical Biology. Possible directions include population structure, evolutionary game theory, microbial populations, ecological stochasticity, complex life cycles and infectious-disease models. The project will be developed with the incoming researcher to reflect their interests and skills. The institute’s working language is English. This vacancy belongs to the September 2026 Max Planck Postdoc Program call, which accepts applications until 13 October 2026 at 12:00 CEST through the official application platform.

Leonid Berlyand discusses mathematical models for the onset of cell motion driven by myosin contraction. A two-dimensional free-boundary PDE model links cell-shape evolution to diffusion and Keller–Segel-type transport. The talk examines linear stability, a stability-determining eigenvalue and the way nonlinear diffusion changes the bifurcation from supercritical to subcritical. It also considers the curvature of the bifurcation curve, connections to bistability, and the role of non-self-adjoint operators. An example illustrates why a spectral gap alone need not guarantee stability. This is an in-person Analysis of Fluids and Related Topics seminar at Princeton.

ISEC brings together ecologists, statisticians and quantitative researchers working on ecological inference and modelling. The main conference runs 10–15 January 2027 in Mérida, Mexico; separate pre-conference workshops run 8–9 January. Registration is open, with early rates through 30 September 2026 and reduced rates for eligible Global South participants.

SIAM’s computational science and engineering meeting covers mathematical models, scalable algorithms, scientific software, high-performance computing and scientific machine learning. It takes place in Pittsburgh on 22–26 February 2027. Travel-support applications are open until 30 November 2026. Abstract deadlines have passed; registration is scheduled to open in December 2026. The meeting is co-located with ACDA27 and the International Meshing Roundtable.

The Society for Industrial and Applied Mathematics convenes researchers developing and applying dynamical-systems methods across biology, chemistry, physics, climate science, social science, industry and data science. Themes include computational, experimental, theoretical and data-driven methods; AI-informed systems; fluid and climate dynamics; materials; networks; pattern formation; population dynamics; stochastic systems and tipping points. Minisymposium proposals are due 26 October 2026, followed by contributed lecture, poster and minisymposium-presentation abstracts on 23 November 2026.