Mathematical Modeling seminars

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.

Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals. Unfortunately, continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law that defines them. This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations. We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms. Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar. We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors. Fast-slow decomposition analysis uncovers the existence of a persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error of these approximations of continuous attractors. Finally, we train recurrent neural networks on analog memory tasks to support the appearance of these systems as solutions and their generalization capabilities. Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-11-27. Recording duration: 00:59:01.

Theoretical neuroscience aims to understand the relationship between neuron dynamics and connectivity in recurrent circuits. This has been intensively studied at the local level, where dynamics is described by pairwise correlations. Recent advances in simultaneous recordings of many neurons have allowed researchers to address the question at the global level, such as for the dimensionality of population dynamics. Our work contributes to this effort by analyzing the impact of connectivity statistics, including certain motifs, on the bulk and outlier covariance eigenvalues. By considering linearized dynamics around a steady state, we obtained analytically the covariance spectrum which exhibits a signature long tail robust to model variants and matches zebrafish calcium imaging data. This provides a local circuit mechanism for shaping the geometry of population dynamics and a quantitative benchmark for interpreting data. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-05-22. Recording duration: 00:53:15.

Neural computation is inextricably bound to decisions that must be made under time pressure and uncertainty. At the level of neural circuits, single neurons need to decide whether to spike. On longer timescales, the component circuitry needs to decide whether to reconfigure to store memories and adapt to novel situations. In this talk I will focus on two fun ideas in each of these contexts by showing how nonlinearities in neural components naturally form excitable switches that enable reliable decisions to be made in fluctuating environments. I will also issue propaganda that the kind of high level, cognitive faculties that we normally associate with decision making apply equally well and are understudied at the level of neural and synaptic populations. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-04-17. Recording duration: 00:41:08.

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.

Inhibition has long been thought to stabilize the activity of cortical networks at low rates, and to shape significantly their response to sensory inputs. In this talk, I will describe three recent collaborative projects that shed light on these issues. (1) I will show how optogenetic excitation of inhibition neurons is consistent with cortex being inhibition stabilized even in the absence of sensory inputs, and how this data can constrain the coupling strengths of E-I cortical network models. (2) Recent analysis of the effects of optogenetic excitation of pyramidal cells in V1 of mice and monkeys shows that in some cases this optogenetic input reshuffles the firing rates of neurons of the network, leaving the distribution of rates unaffected. I will show how this surprising effect can be reproduced in sufficiently strongly coupled E-I networks. (3) Another puzzle has been to understand the respective roles of different inhibitory subtypes in network stabilization. Recent data reveal a novel, state dependent, paradoxical effect of weakening AMPAR mediated synaptic currents onto SST cells. Mathematical analysis of a network model with multiple inhibitory cell types shows that this effect tells us in which conditions SST cells are required for network stabilization.

In this talk, I will explain a theory of connectome-constrained neural networks in which a “student” networks is trained to reproduce the activity of a ground-truth “teacher”, representing a neural system for which a connectome is available. Unlike standard paradigms with unconstrained connectivity, here both networks have the same connectivity but they have different biophysical parameters, reflecting uncertainty in neuronal and synaptic properties. We find that the connectome is often insufficient to constrain the dynamics of networks that perform a specific task, illustrating the difficulty of inferring function from connectivity alone. However, recordings from a small subset of neurons can remove this degeneracy, producing dynamics in the student that agree with the teacher. Our theory can prioritize which neurons to record from to most efficiently unmeasured network activity. The analysis shows that the solution spaces of connectome-constrained and unconstrained models are qualitatively different, and provides a framework to determine when such models yield consistent dynamics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-04-03. Recording duration: 00:42:28.

The connectivity structure of many biological systems, including neural circuits, is highly non-uniform. Recent technologies allow detailed mapping of these irregularities, but our understanding of their effect on the overall circuit dynamics is still lacking. By developing complex system analytical tools that perform reduction of the network, I determine the impact of connectivity features on network dynamics. I will demonstrate the use of these tools on neural assemblies (clusters), a ubiquitous non-uniform structure in our brains. I will show how neural assemblies of different sizes naturally generate multiple timescales of activity spanning several orders of magnitude. I will demonstrate how the analytical theory we develop for rate networks, supported by spiking network simulations, reveals the dependency between neural timescales and assembly sizes and how new recordings of spontaneous activity from a million neurons support this analysis. I will also show how our model can naturally explain the particular long-tailed timescale distribution observed in the awake primate cortex. In olfactory cortex, neural assemblies represent odor stimuli. Previously, I showed how the diffuse recurrent excitation among these assemblies allows the conversion of time-encoded inputs from the bulb to spacial neural assembly representations in olfactory cortex. Here, I will show how changes in the dynamical properties of these assemblies alter both their timing response and properties of the time-encoded inputs via feedback. This demonstrates the role of neural assemblies in time-sensitive modulation needed for cognitive tasks, such as attention. Our results offer a biologically plausible mechanism of assemblies in network connectivity for explaining multiple puzzling dynamical phenomena: The ability of neural circuits to transform external simultaneous temporal fluctuations into spatial representations and alter them; and the ability of neuronal circuits to generate simultaneous temporal fluctuations across a large range of timescales; Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-03-20. Recording duration: 00:41:28.

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.

Over the past decade we have demonstrated that the fusion of subject-specific structural information of the human brain with mathematical dynamic models allows building biologically realistic brain network models, which have a predictive value, beyond the explanatory power of each approach independently. The network nodes hold neural population models, which are derived using mean field techniques from statistical physics expressing ensemble activity via collective variables. Our hybrid approach fuses data-driven with forward-modeling-based techniques and has been successfully applied to explain healthy brain function and clinical translation including aging, stroke and epilepsy. Here we illustrate the workflow along the example of epilepsy: we reconstruct personalized connectivity matrices of human epileptic patients using Diffusion Tensor weighted Imaging (DTI). Subsets of brain regions generating seizures in patients with refractory partial epilepsy are referred to as the epileptogenic zone (EZ). During a seizure, paroxysmal activity is not restricted to the EZ, but may recruit other healthy brain regions and propagate activity through large brain networks. The identification of the EZ is crucial for the success of neurosurgery and presents one of the historically difficult questions in clinical neuroscience. The application of latest techniques in Bayesian inference and model inversion, in particular Hamiltonian Monte Carlo, allows the estimation of the EZ, including estimates of confidence and diagnostics of performance of the inference. The example of epilepsy nicely underwrites the predictive value of personalized large-scale brain network models. The workflow of end-to-end modeling is an integral part of the European neuroinformatics platform EBRAINS and enables neuroscientists worldwide to build and estimate personalized virtual brains. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2023-12-06. Recording duration: 00:50:07.

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.

A central question in neuroscience is how the structure of brain circuits determines their activity and function. To explore this systematically, we developed a 230,000-neuron model of mouse primary visual cortex (area V1). The model integrates a broad array of experimental data:Distribution and morpho-electric properties of different neuron types in V1.

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.

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.

Over the past decade we have demonstrated that the fusion of subject-specific structural information of the human brain with mathematical dynamic models allows building biologically realistic brain network models, which have a predictive value, beyond the explanatory power of each approach independently. The network nodes hold neural population models, which are derived using mean field techniques from statistical physics expressing ensemble activity via collective variables. Our hybrid approach fuses data-driven with forward-modeling-based techniques and has been successfully applied to explain healthy brain function and clinical translation including aging, stroke and epilepsy. Here we illustrate the workflow along the example of epilepsy: we reconstruct personalized connectivity matrices of human epileptic patients using Diffusion Tensor weighted Imaging (DTI). Subsets of brain regions generating seizures in patients with refractory partial epilepsy are referred to as the epileptogenic zone (EZ). During a seizure, paroxysmal activity is not restricted to the EZ, but may recruit other healthy brain regions and propagate activity through large brain networks. The identification of the EZ is crucial for the success of neurosurgery and presents one of the historically difficult questions in clinical neuroscience. The application of latest techniques in Bayesian inference and model inversion, in particular Hamiltonian Monte Carlo, allows the estimation of the EZ, including estimates of confidence and diagnostics of performance of the inference. The example of epilepsy nicely underwrites the predictive value of personalized large-scale brain network models. The workflow of end-to-end modeling is an integral part of the European neuroinformatics platform EBRAINS and enables neuroscientists worldwide to build and estimate personalized virtual brains.

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.

The neurobiological mechanisms of arousal and anesthesia remain poorly understood. Recent evidence highlights the key role of interactions between the cerebral cortex and the diffusely projecting matrix thalamic nuclei. Here, we interrogate these processes in a whole-brain corticothalamic neural mass model endowed with targeted and diffusely projecting thalamocortical nuclei inferred from empirical data. This model captures key features seen in propofol anesthesia, including diminished network integration, lowered state diversity, impaired susceptibility to perturbation, and decreased corticocortical coherence. Collectively, these signatures reflect a suppression of information transfer across the cerebral cortex. We recover these signatures of conscious arousal by selectively stimulating the matrix thalamus, recapitulating empirical results in macaque, as well as wake-like information processing states that reflect the thalamic modulation of largescale cortical attractor dynamics. Our results highlight the role of matrix thalamocortical projections in shaping many features of complex cortical dynamics to facilitate the unique communication states supporting conscious awareness.

Myopia is predicted to affect 50% of all people worldwide by 2050, and is a risk factor for significant, potentially blinding ocular pathologies, such as retinal detachment and glaucoma. Thus, there is significant motivation to better understand the process of myopigenesis and to develop effective anti-myopigenic treatments. In nearly all cases of human myopia, scleral remodeling is an obligate step in the axial elongation that characterizes the condition. Here I will describe the development of a biomechanical assay based on transient unconfined compression of scleral samples. By treating the scleral as a poroelastic material, one can determine scleral biomechanical properties from extremely small samples, such as obtained from the mouse eye. These properties provide proxy measures of scleral remodeling, and have allowed us to identify all-trans retinoic acid (atRA) as a myopigenic stimulus in mice. I will also describe nascent collaborative work on modeling the transport of atRA in the eye.

To effectively move in complex and changing environments, animals must control locomotor speed and gait, while precisely coordinating and adapting limb movements to the terrain. The underlying neuronal control is facilitated by circuits in the spinal cord, which integrate supraspinal commands and afferent feedback signals to produce coordinated rhythmic muscle activations necessary for stable locomotion. I will present a series of computational models investigating dynamics of central neuronal interactions as well as a neuromechanical model that integrates neuronal circuits with a model of the musculoskeletal system. These models closely reproduce speed-dependent gait expression and experimentally observed changes following manipulation of multiple classes of genetically-identified neuronal populations. I will discuss the utility of these models in providing experimentally testable predictions for future studies.

The dominant theoretical framework to account for reinforcement learning in the brain is temporal difference (TD) reinforcement learning. The TD framework predicts that some neuronal elements should represent the reward prediction error (RPE), which means they signal the difference between the expected future rewards and the actual rewards. The prominence of the TD theory arises from the observation that firing properties of dopaminergic neurons in the ventral tegmental area appear similar to those of RPE model-neurons in TD learning. Previous implementations of TD learning assume a fixed temporal basis for each stimulus that might eventually predict a reward. Here we show that such a fixed temporal basis is implausible and that certain predictions of TD learning are inconsistent with experiments. We propose instead an alternative theoretical framework, coined FLEX (Flexibly Learned Errors in Expected Reward). In FLEX, feature specific representations of time are learned, allowing for neural representations of stimuli to adjust their timing and relation to rewards in an online manner. In FLEX dopamine acts as an instructive signal which helps build temporal models of the environment. FLEX is a general theoretical framework that has many possible biophysical implementations. In order to show that FLEX is a feasible approach, we present a specific biophysically plausible model which implements the principles of FLEX. We show that this implementation can account for various reinforcement learning paradigms, and that its results and predictions are consistent with a preponderance of both existing and reanalyzed experimental data.