Dynamical Systems seminars

Mikaela Iacobelli introduces kinetic theory and the mathematical description of charged particles in plasmas through Vlasov-type equations. The colloquium examines stability and instability, well-posedness, and how solutions behave in singular limits. It also presents a new family of Wasserstein-type distances that offers tools for studying the stability of kinetic equations. The exposition is intended for a broad mathematical audience.

A fundamental problem in neuroscience is understanding how the properties of individual neurons and synapses contribute to neuronal circuit dynamics and behavior. In recent years we have done both computational and experimental studies that demonstrate that the same physiological output can arise from multiple, degenerate solutions, and that individual animals with similar behavior can nonetheless have quite different sets of underlying circuit parameters. Most recently, we have been studying the resilience of individual animals to perturbations such as temperature and high potassium concentrations. This has revealed that extreme environmental experiences can produce long-term changes in circuit performance that can be hidden, or “cryptic” unless the animals are again challenged or perturbed. Our present experimental and computational work is designed to understand differential resilience in natural, wild-caught animals in response to climate change, and shows long-lasting influences of the animals’ temperature history. VVTNS Fourth Season Closing Lecture. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-06-26. Recording duration: 00:49:33.

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.

Persistent, stimulus-dependent neuronal activity has been observed in numerous brain areas during tasks that require the temporary maintenance of information. Several competing hypotheses for the neuronal mechanisms underlying persistent activity have been proposed. We have employed data-driven models in conjunction with optogenetic disruptions of neural circuits within memory-guided motor tasks. Our findings revealed a mechanism governed by dynamic attractors, pivotal in sustaining neuronal activity. This mechanism, shaped by time-varying inputs reflecting temporal predictions, is instrumental in regulating the impact of sensory information on the premotor cortex, thereby preserving memory traces from distracting stimuli. We then asked how persistent activity driven by attractor dynamics emerges during motor learning. It has been proposed that activity-dependent synaptic plasticity underpins motor learning, as it can reconfigure network architectures to produce the appropriate neural dynamics for specific behaviors. To verify this hypothesis, we investigated how the mouse premotor cortex acquires specific neural dynamics that govern the planning of movement at different stages of motor learning. We developed network models that replicated the effects of acute manipulations of synaptic plasticity. The models, which display attractor dynamics, also explain flexible behavior after learning has ended. By leveraging the model's predictions, we can formulate testable hypotheses regarding the distinct mechanisms governing movement planning at various stages of the learning process. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-05-01. Recording duration: 00:39:52.

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.

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.

Deep neural network learning dynamics are very complex with large numbers of learnable weights and many sources of disorder. In this talk, I will discuss mean field approaches to analyze the learning dynamics of neural networks in large system size limits when starting from random initial conditions. The result of this analysis is a dynamical mean field theory (DMFT) where all neurons obey independent stochastic single site dynamics. Correlation functions (kernels) and response functions for the features and gradients at each layer can be computed self-consistently from these stochastic processes. Depending on the choice of scaling of the network output, the network can operate in a kernel regime or a feature learning regime in the infinite width limit. I will discuss how this theory can be used to analyze various learning rules for deep architectures (backpropagation, feedback alignment based rules, Hebbian learning etc), where the weight updates do not necessarily correspond to gradient descent on an energy function. I will then present recent extensions of this theory to residual networks at infinite depth and discuss the utility of deriving scaling limits to obtain consistent optimal hyperparameters (such as learning rate) across widths and depths. Feature learning in other types of architectures will be discussed if time permits. Lastly, I will discuss open problems and challenges associated with this theoretical approach to neural network learning dynamics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2023-11-29. Recording duration: 00:41:18.

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.

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.

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.

The large-scale activity of the human brain exhibits rich and complex patterns, but the spatiotemporal dynamics of these patterns and their functional roles in cognition remain unclear. Here by characterizing moment-by-moment fluctuations of human cortical functional magnetic resonance imaging signals, we show that spiral-like, rotational wave patterns (brain spirals) are widespread during both resting and cognitive task states. These brain spirals propagate across the cortex while rotating around their phase singularity centres, giving rise to spatiotemporal activity dynamics with non-stationary features. The properties of these brain spirals, such as their rotational directions and locations, are task relevant and can be used to classify different cognitive tasks. We also demonstrate that multiple, interacting brain spirals are involved in coordinating the correlated activations and de-activations of distributed functional regions; this mechanism enables flexible reconfiguration of task-driven activity flow between bottom-up and top-down directions during cognitive processing. Our findings suggest that brain spirals organize complex spatiotemporal dynamics of the human brain and have functional correlates to cognitive processing.

The traditional view of a neuron as a feature detector or an efficient encoder has difficulties in explaining the function of motor neurons and experimentally observed variable and context-dependent response properties of neurons. We put forward an alternative perspective, modeling each neuron as a feedback controller within a closed loop that includes other neurons and the external environment. Based on the recently developed Direct Data-Driven Control (DD-DC) approach, we propose a biologically plausible controller which implicitly identifies the dynamics of the rest of the loop, infers its latent state and optimizes control. The DD-DC model of a neuron accounts for multiple neurophysiological observations, including the switch from potentiation to depression in Spike-Timing-Dependent Plasticity (STDP) and its asymmetry; temporally extended feedforward and feedback neuronal filters and their adaptation to input statistics; imprecision of the neuronal spike-generation mechanism under constant input; as well as the prevalence of variability and/or noise in brain operation. The DD-DC neuron offers an alternative to the feedforward, instantaneously responding McCulloch-Pitts-Rosenblatt unit as a primitive for constructing biologically-inspired neural networks. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2023-06-28. Recording duration: 00:52:20.

The nervous system is fundamentally a closed loop control device: the output of actions continually influences the internal state and subsequent actions. This is true at the single cell and even the molecular level, where “actions” take the form of signals that are fed back to achieve a variety of functions, including homeostasis, excitability and various kinds of multistability that allow switching and storage of memory. It is also true at the behavioural level, where an animal’s motor actions directly influence sensory input on short timescales, and higher level information about goals and intended actions are continually updated on the basis of current and past actions. Studying the brain in a closed loop setting requires a multidisciplinary approach, leveraging engineering and theory as well as advances in measuring and manipulating the nervous system. I will describe our recent attempts to achieve this fusion of approaches at multiple levels in the nervous system, from synaptic signalling to closed loop brain machine interfaces.

The critical brain hypothesis states that the brain can benefit from operating close to a second-order phase transition. While it has been shown that several computational aspects of sensory information processing (e.g., sensitivity to input) are optimal in this regime, it is still unclear whether these computational benefits of criticality can be leveraged by neural systems performing behaviorally relevant computations. To address this question, we investigate signatures of criticality in networks optimized to perform efficient encoding. We consider a network of leaky integrate-and-fire neurons with synaptic transmission delays and input noise. Previously, it was shown that the performance of such networks varies non-monotonically with the noise amplitude. Interestingly, we find that in the vicinity of the optimal noise level for efficient coding, the network dynamics exhibits signatures of criticality, namely, the distribution of avalanche sizes follows a power law. When the noise amplitude is too low or too high for efficient coding, the network appears either super-critical or sub-critical, respectively. This result suggests that two influential, and previously disparate theories of neural processing optimization—efficient coding, and criticality—may be intimately related

Modern electrophysiological recordings simultaneously capture single-unit spiking activities of hundreds of neurons. These neurons exhibit highly complex coordination patterns. Where does this complexity stem from? One candidate is the ubiquitous heterogeneity in connectivity of local neural circuits. Studying neural network dynamics in the linearized regime and using tools from statistical field theory of disordered systems, we derive relations between structure and dynamics that are readily applicable to subsampled recordings of neural circuits: Measuring the statistics of pairwise covariances allows us to infer statistical properties of the underlying connectivity. Applying our results to spontaneous activity of macaque motor cortex, we find that the underlying network operates in a strongly recurrent regime. In this regime, network connectivity is highly heterogeneous, as quantified by a large radius of bulk connectivity eigenvalues. Being close to the point of linear instability, this dynamical regime predicts a rich correlation structure, a large dynamical repertoire, long-range interaction patterns, relatively low dimensionality and a sensitive control of neuronal coordination. These predictions are verified in analyses of spontaneous activity of macaque motor cortex and mouse visual cortex. Finally, we show that even microscopic features of connectivity, such as connection motifs, systematically scale up to determine the global organization of activity in neural circuits.

Cortical variations in cytoarchitecture form a sensory-fugal axis that shapes regional profiles of extrinsic connectivity and is thought to guide signal propagation and integration across the cortical hierarchy. While neuroimaging work has shown that this axis constrains local properties of the human connectome, it remains unclear whether it also shapes the asymmetric signaling that arises from higher-order topology. Here, we used network control theory to examine the amount of energy required to propagate dynamics across the sensory-fugal axis. Our results revealed an asymmetry in this energy, indicating that bottom-up transitions were easier to complete compared to top-down. Supporting analyses demonstrated that asymmetries were underpinned by a connectome topology that is wired to support efficient bottom-up signaling. Lastly, we found that asymmetries correlated with differences in communicability and intrinsic neuronal time scales and lessened throughout youth. Our results show that cortical variation in cytoarchitecture may guide the formation of macroscopic connectome topology.