Mathematical Modeling seminars

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

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.

In sensory systems, stimuli are represented through the diverse firing responses and receptive fields of neurons. These features emerge from the interaction between excitatory (E) and inhibitory (I) neuron populations within the network. Changes in sensory inputs alter this balance, leading to shifts in firing patterns and the input-output properties of individual neurons and the network. While these phenomena have been studied extensively with experiments and theory, the underlying principles for combining E and I inputs are still unclear. Here, the rules for probabilistically combining E and I inputs are derived that describe how neurons in a feedforward inhibitory circuit respond to stimuli. This simple model is broadly applicable, capturing a wide range of response features that would otherwise require multiple separate models and offers insights into the cellular and network mechanisms influencing the input-output properties of neurons, gain modulation, and the emergence of diverse temporal firing patterns. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-02-11. Recording duration: 00:48:34.

Many animals rely on persistent internal representations of continuous angular variables for working memory, motor control, and navigation. Theories have proposed that such representations are maintained by a class of recurrently connected networks called ring attractor networks. These networks rely on large numbers of neurons to maintain continuous and stable representations and to accurately integrate incoming signals. The head direction system of the fruit fly, however, seems to achieve these properties with a remarkably small network. These findings challenge our understanding of ring attractors and their putative implementation in neural circuits. In this talk, I will show analytically how small networks can overcome the constraints of their size to generate a ring attractor and are hence capable of stably maintaining an internal representation of a continuous, periodic variable. Further, I will show how ring attractors emerge in small threshold linear networks through the coordination of a discrete set of line attractors. More broadly, this work informs our understanding of the functional capabilities of small, discrete systems. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-02-04. Recording duration: 00:44:59.

Computations in neural circuits are often construed as being implemented through the coordinated dynamics of neurons. In this picture, the role of synaptic connectivity is to sculpt neuronal dynamics to implement computations of interest. Of course, synapses are not static but change on a variety of timescales, including fast timescales comparable to those of neurons. Thus, a more accurate view of computation in neural circuits may involve the coupled dynamics of neurons and synapses. This form of computation is closer to what is implemented by Transformers via an equivalence between ongoing synaptic plasticity and self-attention. I will first describe a nonlinear recurrent neural-network model with ongoing Hebbian dynamics of “fast” synapses atop unstructured “slow” synapses. I will then describe two computations implemented through neuronal-synaptic dynamics, which can be studied in this model using techniques including dynamical mean-field theory and random-matrix theory. First, there exists a novel phase termed “freezable chaos” in which a stable fixed point of neuronal dynamics is continuously destabilized by synaptic dynamics. This allows for the creation of a stable fixed point at any neuronal state visited by the network by halting synaptic plasticity. Second, I will describe an effect termed “persistent oscillations” in which, following stimulation by a periodic signal, a plastic network continues to autonomously reproduce a similar signal for a duration exceeding any intrinsic timescale in the system. Thus, ongoing Hebbian plasticity can provide a dynamic form of working memory, complementing the static form provided by freezable chaos. Ongoing experimental work suggests that this effect is realized in cortical organoids. Overall, this line of work suggests that synapses should be promoted to first-class dynamical degrees of freedom in our conceptual understanding of neural-circuit function. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-01-14. Recording duration: 00:43:14.

Recurrent networks with balanced excitation-inhibition explain a wide range of neurophysiological observations, but can only implement a limited set of transformations on their input. On the other hand networks of firing-rate units with low-rank connectivity have universal computational capabilities, but do not work with spikes or generate noise self-consistently. Although empirical approaches to merge these two computational frameworks have been constructed, there is no established theory describing their unification. Here we develop such a theory. We study analytically and numerically networks with connectivity comprising random “strong”, and low-rank “weak” components. When the low-rank connectivity is slow, a well-defined notion of instantaneous firing rate emerges which implies universal computation as previously shown. However, the fact that such time-varying rates are the result of E-I balance has important implications. We show that internally or externally generated fluctuations along particular latent modes tend to break the E-I balance. Its maintenance is obtained through the emergence of a spontaneous coupling between the mean and the variance of the membrane potential and the norm of the latent state driving these modes. This leads to several predictions, the most counterintuitive of which is that coherent global fluctuations in subthreshold membrane potential (Vm) should coexist with desynchronized activity at constant firing rates when the dynamics of these modes is excited. To test our theory, we show that the coupling between the average Vm and the latent state adds new non-linear dimensions to the low-dimensional manifold of the network, which lead to a frequency doubling when the input to the network is periodic, a prediction that is borne out in population recordings from mouse V1. Our results unify two prevalent frameworks for cortical computation and clarify the relationship between computation, dynamics and geometry in circuits of spiking neurons. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-11-26. Recording duration: 00:39:26.

Minimising cortical prediction errors is thought to be a key computation underlying perception, action, and learning. Yet, how the cortex represents and uses uncertainty in this process remains unclear. In the first part of this talk, I will present a normative framework showing how uncertainty can modulate prediction error activity to yield uncertainty-modulated prediction errors (UPEs), hypothesised to be represented by layer 2/3 pyramidal neurons. We propose that these UPEs are computed through inhibitory mechanisms involving SST and PV interneurons. A circuit model demonstrates how cortical cell types can locally compute means, variances, and UPEs, leading to adaptive learning rates. In the second part, I will discuss how uncertainty modulation could be controlled by higher-level representations. We formally derived neural dynamics that minimise prediction errors under the assumption that cortical areas must not only predict the activity in other areas and sensory streams but also jointly project their inverse expected uncertainty about their predictions, which we call “confidence”. This yields a confidence-weighted integration of bottom-up and top-down signals, consistent with Bayesian principles, and predicts the existence of second-order errors that compare confidence with performance. We predict that these second-order errors propagate alongside classical prediction errors through the cortical hierarchy, and simulations demonstrate that this mechanism enables nonlinear classification within a single cortical area. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-11-12. Recording duration: 00:29:21.

Applications are invited for our third edition of Systems Vision Science (SVS) summer school since 2023, designed for everyone interested in gaining a systems level understanding of biological vision. We plan a coherent, graduate-level, syllabus on the integration of experimental data with theory and models, featuring lectures, guided exercises and discussion sessions. The summer school will end with a Systems Vision Science symposium on frontier topics on August 20-22, with additional invited and contributed presentations and posters. Call for contributions and participations to the symposium will be sent out spring of 2025. All summer school participants are invited to attend, and welcome to submit contributions to the symposium.

Applications are invited for our third edition of Systems Vision Science (SVS) summer school since 2023, designed for everyone interested in gaining a systems level understanding of biological vision. We plan a coherent, graduate-level, syllabus on the integration of experimental data with theory and models, featuring lectures, guided exercises and discussion sessions. The summer school will end with a Systems Vision Science symposium on frontier topics on August 20-22, with additional invited and contributed presentations and posters. Call for contributions and participations to the symposium will be sent out spring of 2025. All summer school participants are invited to attend, and welcome to submit contributions to the symposium.

Applications are invited for our third edition of Systems Vision Science (SVS) summer school since 2023, designed for everyone interested in gaining a systems level understanding of biological vision. We plan a coherent, graduate-level, syllabus on the integration of experimental data with theory and models, featuring lectures, guided exercises and discussion sessions. The summer school will end with a Systems Vision Science symposium on frontier topics on August 20-22, with additional invited and contributed presentations and posters. Call for contributions and participations to the symposium will be sent out spring of 2025. All summer school participants are invited to attend, and welcome to submit contributions to the symposium.

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.

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.

Measurements from hippocampal place cells in small enclosures have led to a classical view of highly stereotyped neural tuning functions with smooth, unimodal tuning fields that uniformly tile the environment. However, recent experiments conducted in large spaces across multiple species have revealed a much more irregular neural code than suggested by the classical view. Indeed, place cells in large environments typically fire in multiple locations, and the multiple firing fields of individual cells, as well as those of the entire population, vary considerably in size and shape. We recently showed that a simple mathematical model, wherein firing fields are generated by thresholding realizations of a random Gaussian process, accounts for the statistical properties of place fields in precise quantitative detail. This model captures the observed statistics of field sizes and positions and generates new quantitative predictions on field shapes and topologies. Moreover, these statistics are universal across species, but sensitive to the size and dimensionality of the environment. We quantitatively verified these predictions using multiple recent datasets from bats and rodents in one, two, and three dimensions, across both small and large environments. Collectively, these findings imply that common mechanism underlie the diverse statistical features observed in different experiments and suggest that synaptic projections to CA1 are predominantly random. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-05-28. Recording duration: 00:42:45.

When we attend a demanding task, our performance is poor at low arousal (when drowsy) or high arousal (when anxious), but we achieve optimal performance at intermediate arousal. This celebrated Yerkes-Dodson inverted-U law relating performance and arousal is colloquially referred to as being "in the zone." In this talk, I will elucidate the behavioral and neural mechanisms linking arousal and performance under the Yerkes-Dodson law in a mouse model. During decision-making tasks, mice express an array of discrete strategies, whereby the optimal strategy occurs at intermediate arousal, measured by pupil, consistent with the inverted-U law. Population recordings from the auditory cortex (A1) further revealed that sound encoding is optimal at intermediate arousal. To explain the computational principle underlying this inverted-U law, we modeled the A1 circuit as a spiking network with excitatory/inhibitory clusters, based on the observed functional clusters in A1. Arousal induced a transition from a multi-attractor (low arousal) to a single attractor phase (high arousal), and performance is optimized at the transition point. The model also predicts stimulus- and arousal-induced modulations of neural variability, which we confirmed in the data. Our theory suggests that a single unifying dynamical principle, phase transitions in metastable dynamics, underlies both the inverted-U law of optimal performance and state-dependent modulations of neural variability.

All animal locomotion is rhythmic,whether it is achieved through undulatory movement of the whole body or the coordination of articulated limbs. Neurobiologists have long studied locomotor circuits that produce rhythmic activity with non-rhythmic input, also called central pattern generators (CPGs). However, the cellular and microcircuit implementation of a walking CPG has not been described for any limbed animal. New comprehensive connectomes of the fruit fly ventral nerve cord (VNC) provide an opportunity to study rhythmogenic walking circuits at a synaptic scale.We use a data-driven network modeling approach to identify and characterize a putative walking CPG in the Drosophila leg motor system.

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.