Mathematical Modeling seminars

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.

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.

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

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.

Subsecond timing underlies nearly all sensory and motor activities across species and is critical to survival. While subsecond temporal information has been found across cortical and subcortical regions, it is unclear if it is generated locally and intrinsically or if it is a read out of a centralized clock-like mechanism. Indeed, mechanisms of subsecond timing at the circuit level are largely obscure. Primary sensory areas are well-suited to address these question as they have early access to sensory information and provide minimal processing to it: if temporal information is found in these regions, it is likely to be generated intrinsically and locally. We test this hypothesis by training mice to perform an audio-visual temporal pattern sensory discrimination task as we use 2-photon calcium imaging, a technique capable of recording population level activity at single cell resolution, to record activity in primary visual cortex (V1). We have found significant changes in network dynamics through mice’s learning of the task from naive to middle to expert levels. Changes in network dynamics and behavioral performance are well accounted for by an intrinsic model of timing in which the trajectory of q network through high dimensional state space represents temporal sensory information. Conversely, while we found evidence of other temporal encoding models, such as oscillatory activity, we did not find that they accounted for increased performance but were in fact correlated with the intrinsic model itself. These results provide insight into how subsecond temporal information is encoded mechanistically at the circuit level.

Working memory (WM) is a cognitive function that allows the short-term maintenance and manipulation of information when no longer accessible to the senses. It relies on temporarily storing stimulus features in the activity of neuronal populations. To preserve these dynamics from distraction it has been proposed that pre and post-distraction population activity decomposes into orthogonal subspaces. If orthogonalization is necessary to avoid WM distraction, it should emerge as performance in the task improves. We sought evidence of WM orthogonalization learning and the underlying mechanisms by analyzing calcium imaging data from the prelimbic (PrL) and anterior cingulate (ACC) cortices of mice as they learned to perform an olfactory dual task. The dual task combines an outer Delayed Paired-Association task (DPA) with an inner Go-NoGo task. We examined how neuronal activity reflected the process of protecting the DPA sample information against Go/NoGo distractors. As mice learned the task, we measured the overlap between the neural activity onto the low-dimensional subspaces that encode sample or distractor odors. Early in the training, pre-distraction activity overlapped with both sample and distractor subspaces. Later in the training, pre-distraction activity was strictly confined to the sample subspace, resulting in a more robust sample code. To gain mechanistic insight into how these low-dimensional WM representations evolve with learning we built a recurrent spiking network model of excitatory and inhibitory neurons with low-rank connections. The model links learning to (1) the orthogonalization of sample and distractor WM subspaces and (2) the orthogonalization of each subspace with irrelevant inputs. We validated (1) by measuring the angular distance between the sample and distractor subspaces through learning in the data. Prediction (2) was validated in PrL through the photoinhibition of ACC to PrL inputs, which induced early-training neural dynamics in well-trained animals. In the model, learning drives the network from a double-well attractor toward a more continuous ring attractor regime. We tested signatures for this dynamical evolution in the experimental data by estimating the energy landscape of the dynamics on a one-dimensional ring. In sum, our study defines network dynamics underlying the process of learning to shield WM representations from distracting tasks.

Higher cortical areas carry a wide range of sensory, cognitive, and motor signals supporting complex goal-directed behavior. These signals mix in heterogeneous responses of single neurons, making it difficult to untangle underlying mechanisms. I will present two approaches for revealing interpretable circuit mechanisms from heterogeneous neural responses during cognitive tasks. First, I will show a flexible nonparametric framework for simultaneously inferring population dynamics on single trials and tuning functions of individual neurons to the latent population state. When applied to recordings from the premotor cortex during decision-making, our approach revealed that populations of neurons encoded the same dynamic variable predicting choices, and heterogeneous firing rates resulted from the diverse tuning of single neurons to this decision variable. The inferred dynamics indicated an attractor mechanism for decision computation. Second, I will show an approach for inferring an interpretable network model of a cognitive task—the latent circuit—from neural response data. We developed a theory to causally validate latent circuit mechanisms via patterned perturbations of activity and connectivity in the high-dimensional network. This work opens new possibilities for deriving testable mechanistic hypotheses from complex neural response data.

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.

One of the fundamental functions of the brain is to flexibly plan and control movement production at different timescales in order to efficiently shape structured behaviors. I will present research investigating how these complex computations are performed in the mammalian brain, with an emphasis on autonomous motor control. Specifically, I will focus on the mechanisms supporting efficient interfacing between 'higher-level' planning commands and 'lower-level' motor cortical dynamics that ultimately drive muscles. I will take advantage of the fact that the anatomy of the circuits underlying motor control is well known. It notably involves the primary motor cortex, a recurrent network that generates learned commands to drive muscles while interacting through loops with thalamic neurons that lack recurrent excitation. Using an analytically tractable model that incorporates these architectural constraints, I will explain how this motor circuit can implement a form of efficient modularity by combining (i) plastic thalamocortical loops that are movement-specific and (ii) shared hardwired circuits. I will show that this modular architecture can balance two different objectives: first, supporting the flexible recombination of an extensible library of re-usable motor primitives; and second, promoting the efficient use of neural resources by taking advantage of shared connections between modules. I will finally show that these insights are relevant for designing artificial neural networks able to flexibly and robustly compose hierarchical analog behaviors from a library of motor primitives. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2023-01-11. Recording duration: 00:49:52.

Several species of insects have been observed to perform accurate path integration, constantly updating a vector memory of their location relative to a starting position, which they can use to take a direct return path. Foraging insects such as bees and ants are also able to store and recall the vectors to return to food locations, and to take novel shortcuts between these locations. Other insects, such as dung beetles, are observed to integrate multimodal directional cues in a manner well described by vector addition. All these processes appear to be functions of the Central Complex, a highly conserved and strongly structured circuit in the insect brain. Modelling this circuit, at the single neuron level, suggests it has general capabilities for vector encoding, vector memory, vector addition and vector rotation that can support a wide range of directed and navigational behaviours.

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.

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.