Dynamical Systems seminars

The brain exhibits a rich repertoire of oscillatory patterns organized in space, time and frequency. However, despite ever more-detailed characterizations of spectrally-resolved network patterns, the principles governing oscillatory activity at the system-level remain unclear. Here, we propose that the transient emergence of spatially organized brain rhythms are signatures of weakly stable synchronization between subsets of brain areas, naturally occurring at reduced collective frequencies due to the presence of time delays. To test this mechanism, we build a reduced network model representing interactions between local neuronal populations (with damped oscillatory response at 40Hz) coupled in the human neuroanatomical network. Following theoretical predictions, weakly stable cluster synchronization drives a rich repertoire of short-lived (or metastable) oscillatory modes, whose frequency inversely depends on the number of units, the strength of coupling and the propagation times. Despite the significant degree of reduction, we find a range of model parameters where the frequencies of collective oscillations fall in the range of typical brain rhythms, leading to an optimal fit of the power spectra of magnetoencephalographic signals from 89 heathy individuals. These findings provide a mechanistic scenario for the spontaneous emergence of frequency-specific long-range phase-coupling observed in magneto- and electroencephalographic signals as signatures of resonant modes emerging in the space-time structure of the Connectome, reinforcing the importance of incorporating realistic time delays in network models of oscillatory brain activity.

A myriad of pathological changes associated with epilepsy, including the loss of specific cell types, improper expression of individual ion channels, and synaptic sprouting, can be recast as decreases in cell and circuit heterogeneity. In recent experimental work, we demonstrated that biophysical diversity is a key characteristic of human cortical pyramidal cells, and past theoretical work has shown that neuronal heterogeneity improves a neural circuit’s ability to encode information. Viewed alongside the fact that seizure is an information-poor brain state, these findings motivate the hypothesis that epileptogenesis can be recontextualized as a process where reduction in cellular heterogeneity renders neural circuits less resilient to seizure onset. By comparing whole-cell patch clamp recordings from layer 5 (L5) human cortical pyramidal neurons from epileptogenic and non-epileptogenic tissue, we present the first direct experimental evidence that a significant reduction in neural heterogeneity accompanies epilepsy. We directly implement experimentally-obtained heterogeneity levels in cortical excitatory-inhibitory (E-I) stochastic spiking network models. Low heterogeneity networks display unique dynamics typified by a sudden transition into a hyper-active and synchronous state paralleling ictogenesis. Mean-field analysis reveals a distinct mathematical structure in these networks distinguished by multi-stability. Furthermore, the mathematically characterized linearizing effect of heterogeneity on input-output response functions explains the counter-intuitive experimentally observed reduction in single-cell excitability in epileptogenic neurons. This joint experimental, computational, and mathematical study showcases that decreased neuronal heterogeneity exists in epileptogenic human cortical tissue, that this difference yields dynamical changes in neural networks paralleling ictogenesis, and that there is a fundamental explanation for these dynamics based in mathematically characterized effects of heterogeneity. These interdisciplinary results provide convincing evidence that biophysical diversity imbues neural circuits with resilience to seizure and a new lens through which to view epilepsy, the most common serious neurological disorder in the world, that could reveal new targets for clinical treatment.

Large-scale recordings of neural activity are providing new opportunities to study network-level dynamics with unprecedented detail. However, the sheer volume of data and its dynamical complexity are major barriers to uncovering and interpreting these dynamics. I will present machine learning frameworks that enable inference of dynamics from neuronal population spiking activity on single trials and millisecond timescales, from diverse brain areas, and without regard to behavior. I will then demonstrate extensions that allow recovery of dynamics from two-photon calcium imaging data with surprising precision. Finally, I will discuss our efforts to facilitate comparisons within our field by curating datasets and standardizing model evaluation, including a currently active modeling challenge, the 2021 Neural Latents Benchmark [neurallatents.github.io].

Cognitive effort has long been an important explanatory factor in the study of human behavior in health and disease. Yet, the biophysical nature of cognitive effort remains far from understood. In this talk, I will offer a network perspective on cognitive effort. I will begin by canvassing a recent perspective that casts cognitive effort in the framework of network control theory, developed and frequently used in systems engineering. The theory describes how much energy is required to move the brain from one activity state to another, when activity is constrained to pass along physical pathways in a connectome. I will then turn to empirical studies that link this theoretical notion of energy with cognitive effort in a behaviorally demanding task, and with a metabolic notion of energy as accessible to FDG-PET imaging. Finally, I will ask how this structurally-constrained activity flow can provide us with insights about the brain’s non-equilibrium nature. Using a general tool for quantifying entropy production in macroscopic systems, I will provide evidence to suggest that states of marked cognitive effort are also states of greater entropy production. Collectively, the work I discuss offers a complementary view of cognitive effort as a dynamical process occurring atop a complex network.

This talk will focus on the generative modelling of resting state time series or endogenous neuronal activity. I will survey developments in modelling distributed neuronal fluctuations – spectral dynamic causal modelling (DCM) for functional MRI – and how this modelling rests upon functional connectivity. The dynamics of brain connectivity has recently attracted a lot of attention among brain mappers. I will also show a novel method to identify dynamic effective connectivity using spectral DCM. Further, I will summarise the development of the next generation of DCMs towards large-scale, whole-brain schemes which are computationally inexpensive, to the other extreme of the development using more sophisticated and biophysically detailed generative models based on the canonical microcircuits.

Behavior relies on the ability of sensory systems to infer changing properties of the environment from incoming sensory stimuli. However, the demands that detecting and adjusting to changes in the environment place on a sensory system often differ from the demands associated with performing a specific behavioral task. This necessitates neural coding strategies that can dynamically balance these conflicting needs. I will discuss our ongoing theoretical work to understand how this balance can best be achieved. We connect ideas from efficient coding and Bayesian inference to ask how sensory systems should dynamically allocate limited resources when the goal is to optimally infer changing latent states of the environment, rather than reconstruct incoming stimuli. We use these ideas to explore dynamic tradeoffs between the efficiency and speed of sensory adaptation schemes, and the downstream computations that these schemes might support. Finally, we derive families of codes that balance these competing objectives, and we demonstrate their close match to experimentally-observed neural dynamics during sensory adaptation. These results provide a unifying perspective on adaptive neural dynamics across a range of sensory systems, environments, and sensory tasks.

Neuronal dendritic trees display a wide range of nonlinear input integrations due to their voltage-dependent active calcium channels. We reveal that in vivo-like fluctuating input enhances nonlinearity substantially in a single dendritic compartment and shifts the input-output relation to exhibiting nonmonotonous or bistable dynamics. In particular, with the slow activation of calcium dynamics, we analyze noise-induced bistability and its timescales. We show bistability induces long-timescale fluctuation that can account for observed dendritic plateau potentials in vivo conditions. In a multicompartmental model neuron with realistic synaptic input, we show that noise-induced bistability persists in a wide range of parameters. Using Fredholm's theory to calculate the spiking rate of multivariable neurons, we discuss how dendritic bistability shifts the spiking dynamics of single neurons and its implications for network phenomena in the processing of in vivo–like fluctuating input.

There is a recent surge of interest in the behavior of active particles that can at the same time align their direction of movement and synchronize their oscillations, known as swarmalators. While analytical and numerical models of such systems are now abundant, no real-life examples have been shown to date. I will present an experimental investigation of the collective motion of the nematode Turbatrix aceti, which self-propel by body undulation. I will show that under favorable conditions these nematodes can synchronize their body oscillations, forming striking traveling metachronal waves which, similar to the case of beating cilia, produce strong fluid flows. I will demonstrate that the location and strength of this collective state can be controlled through the shape of the confining structure; in our case the contact angle of a droplet. This opens a way for producing controlled work such as on-demand flows or displacement of objects. I will illustrate this by a practical example: showing that the force generated by the collectively moving nematodes is sufficient to change the mode of evaporation of fluid droplets, by counteracting the surface-tension force, which allow us to estimate its strength.

The ability to reach, grasp, and manipulate objects is a remarkable expression of motor skill, and the loss of this ability in injury, stroke, or disease can be devastating. These behaviors are controlled by the coordinated activity of tens of millions of neurons distributed across many CNS regions, including the primary motor cortex. While many studies have characterized the activity of single cortical neurons during reaching, the principles governing the dynamics of large, distributed neural populations remain largely unknown. Recent work in primates has suggested that during the execution of reaching, motor cortex may autonomously generate the neural pattern controlling the movement, much like the spinal central pattern generator for locomotion. In this seminar, I will describe recent work that tests this hypothesis using large-scale neural recording, high-resolution behavioral measurements, dynamical systems approaches to data analysis, and optogenetic perturbations in mice. We find, by contrast, that motor cortex requires strong, continuous, and time-varying thalamic input to generate the neural pattern driving reaching. In a second line of work, we demonstrate that the cortico-cerebellar loop is not critical for driving the arm towards the target, but instead fine-tunes movement parameters to enable precise and accurate behavior. Finally, I will describe my future plans to apply these experimental and analytical approaches to the adaptive control of locomotion in complex environments.

Modular structures in myriad forms — genetic, structural, functional — are ubiquitous in the brain. While modularization may be shaped by genetic instruction or extensive learning, the mechanisms of module emergence are poorly understood. Here, we explore complementary mechanisms in the form of bottom-up dynamics that push systems spontaneously toward modularization. As a paradigmatic example of modularity in the brain, we focus on the grid cell system. Grid cells of the mammalian medial entorhinal cortex (mEC) exhibit periodic lattice-like tuning curves in their encoding of space as animals navigate the world. Nearby grid cells have identical lattice periods, but at larger separations along the long axis of mEC the period jumps in discrete steps so that the full set of periods cluster into 5-7 discrete modules. These modules endow the grid code with many striking properties such as an exponential capacity to represent space and unprecedented robustness to noise. However, the formation of discrete modules is puzzling given that biophysical properties of mEC stellate cells (including inhibitory inputs from PV interneurons, time constants of EPSPs, intrinsic resonance frequency and differences in gene expression) vary smoothly in continuous topographic gradients along the mEC. How does discreteness in grid modules arise from continuous gradients? We propose a novel mechanism involving two simple types of lateral interaction that leads a continuous network to robustly decompose into discrete functional modules. We show analytically that this mechanism is a generic multi-scale linear instability that converts smooth gradients into discrete modules via a topological “peak selection” process. Further, this model generates detailed predictions about the sequence of adjacent period ratios, and explains existing grid cell data better than existing models. Thus, we contribute a robust new principle for bottom-up module formation in biology, and show that it might be leveraged by grid cells in the brain.

Entorhinal grid cells, so-called because of their hexagonally tiled spatial receptive fields, are organized in modules which, collectively, are believed to form a population code for the animal’s position. Here, we apply topological data analysis to simultaneous recordings of hundreds of grid cells and show that joint activity of grid cells within a module lies on a toroidal manifold. Each position of the animal in its physical environment corresponds to a single location on the torus, and each grid cell is preferentially active within a single “field” on the torus. Toroidal firing positions persist between environments, and between wakefulness and sleep, in agreement with continuous attractor models of grid cells.

Gradient-based optimization powered by the backpropagation algorithm proved to be the pivotal method in the training of non-spiking artificial neural networks. At the same time, spiking neural networks hold the promise for efficient processing of real-world sensory data by communicating using discrete events in continuous time. We derive the backpropagation algorithm for a recurrent network of spiking (leaky integrate-and-fire) neurons with hard thresholds and show that the backward dynamics amount to an event-based backpropagation of errors through time. Our derivation uses the jump conditions for partial derivatives at state discontinuities found by applying the implicit function theorem, allowing us to avoid approximations or substitutions. We find that the gradient exists and is finite almost everywhere in weight space, up to the null set where a membrane potential is precisely tangent to the threshold. Our presented algorithm, EventProp, computes the exact gradient with respect to a general loss function based on spike times and membrane potentials. Crucially, the algorithm allows for an event-based communication scheme in the backward phase, retaining the potential advantages of temporal sparsity afforded by spiking neural networks. We demonstrate the optimization of spiking networks using gradients computed via EventProp and the Yin-Yang and MNIST datasets with either a spike time-based or voltage-based loss function and report competitive performance. Our work supports the rigorous study of gradient-based optimization in spiking neural networks as well as the development of event-based neuromorphic architectures for the efficient training of spiking neural networks. While we consider the leaky integrate-and-fire model in this work, our methodology generalises to any neuron model defined as a hybrid dynamical system.

How can neural networks learn to efficiently represent complex and high-dimensional inputs via local plasticity mechanisms? Classical models of representation learning assume that input weights are learned via pairwise Hebbian-like plasticity. Here, we show that pairwise Hebbian-like plasticity only works under specific requirements on neural dynamics and input statistics. To overcome these limitations, we derive from first principles a learning scheme based on voltage-dependent synaptic plasticity rules. Here, inhibition learns to locally balance excitatory input in individual dendritic compartments, and thereby can modulate excitatory synaptic plasticity to learn efficient representations. We demonstrate in simulations that this learning scheme works robustly even for complex, high-dimensional and correlated inputs. It also works in the presence of inhibitory transmission delays, where Hebbian-like plasticity typically fails. Our results draw a direct connection between dendritic excitatory-inhibitory balance and voltage-dependent synaptic plasticity as observed in vivo, and suggest that both are crucial for representation learning.

Being fundamentally a non-equilibrium process, synchronization comes with unavoidable energy costs and has to be maintained under the constraint of limited resources. Such resource constraints are often reflected as a finite coupling budget available in a network to facilitate interaction and communication. In this talk, I will show that introducing temporal variation in the network structure can lead to efficient synchronization even when stable synchrony is impossible in any static network under the given budget. Our strategy is based on an open-loop control scheme and alludes to a fundamental advantage of temporal networks. Whether this advantage of temporality can be utilized in the brain is an interesting open question.

The basal ganglia are a collection of brain areas that are connected by a variety of synaptic pathways and are a site of significant reward-related dopamine release. These properties suggest a possible role for the basal ganglia in action selection, guided by reinforcement learning. In this talk, I will discuss a framework for how this function might be performed. I will also present some recent experimental results and theory that call for a re-evaluation of certain aspects of this framework. Next, I will turn to the changes in basal ganglia activity observed to occur with the dopamine depletion associated with Parkinson’s disease. I will discuss some of the potential functional implications of some of these changes and, if time permits, will conclude with some new results that focus on delta oscillations under dopamine depletion.

After more than 4 million deaths worldwide, the ongoing vaccination to conquer the COVID-19 disease is now competing with the emergence of increasingly contagious mutations, repeatedly supplanting earlier strains. Following the near-absence of historical examples of the long-time evolution of infectious diseases under similar circumstances, models are crucial to exemplify possible scenarios. Accordingly, in the present work we systematically generalize the popular susceptible-infected-recovered model to account for mutations leading to repeatedly occurring new strains, which we coarse grain based on tools from statistical mechanics to derive a model predicting the most likely outcomes. The model predicts that mutations can induce a super exponential growth of infection numbers at early times, which self-amplify to giant infection waves which are caused by a positive feedback loop between infection numbers and mutations and lead to a simultaneous infection of the majority of the population. At later stages -- if vaccination progresses too slowly -- mutations can interrupt an ongoing decrease of infection numbers and can cause infection revivals which occur as single waves or even as whole wave trains featuring alternative periods of decreasing and increasing infection numbers. Our results might be useful for discussions regarding the importance of a release of vaccine-patents to reduce the risk of mutation-induced infection revivals but also to coordinate the release of measures following a downwards trend of infection numbers.

Hydra is an extraordinary creature. Continuously replacing itself, it can live indefinitely, performing a stable repertoire of reasonably sophisticated behaviors. This remarkable stability under plasticity may be due to the uniform nature of its nervous system, which consists of two apparently noncommunicating nerve net layers. We use modeling to understand the role of active muscles and biomechanics interact with neural activity to shape Hydra behaviour. We will discuss our findings and thoughts on how this simple nervous system may self-organize to produce purposeful behavior.

Homeostatic plasticity plays a key role in stabilizing neural network activity. But what is its role in neural information processing? We showed analytically how homeostasis changes collective dynamics and consequently information flow - depending on the input to the network. We then studied how input and homeostasis on a recurrent network of LIF neurons impacts information flow and task performance. We showed how we can tune the working point of the network, and found that, contrary to previous assumptions, there is not one optimal working point for a family of tasks, but each task may require its own working point.

Choosing among spatially distributed options is a central challenge for animals, from deciding among alternative potential food sources or refuges, to choosing with whom to associate. Here, using an integrated theoretical and experimental approach (employing immersive Virtual Reality), with both invertebrate and vertebrate models—the fruit fly, desert locust and zebrafish—we consider the recursive interplay between movement and collective vectorial integration in the brain during decision-making regarding options (potential ‘targets’) in space. We reveal that the brain repeatedly breaks multi-choice decisions into a series of abrupt (critical) binary decisions in space-time where organisms switch, spontaneously, from averaging vectorial information among, to suddenly excluding one of, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains. Close to each bifurcation the ‘susceptibility’ of the system exhibits a sharp increase, inevitably causing small differences among the remaining options to become amplified; a property that both comes ‘for free’ and is highly desirable for decision-making. This mechanism facilitates highly effective decision-making, and is shown to be robust both to the number of options available, and to context, such as whether options are static (e.g. refuges) or mobile (e.g. other animals). In addition, we find evidence that the same geometric principles of decision-making occur across scales of biological organisation, from neural dynamics to animal collectives, suggesting they are fundamental features of spatiotemporal computation.

In nature, sensory inputs are often highly structured, and statistical regularities of these signals can be extracted to form expectation about future sensorimotor associations, thereby optimizing behavior. One of the fundamental questions in neuroscience concerns the neural computations that underlie these probabilistic sensorimotor processing. Through a recurrent neural network (RNN) model and human psychophysics and electroencephalography (EEG), the present study investigates circuit mechanisms for processing probabilistic structures of sensory signals to guide behavior. We first constructed and trained a biophysically constrained RNN model to perform a series of probabilistic decision-making tasks similar to paradigms designed for humans. Specifically, the training environment was probabilistic such that one stimulus was more probable than the others. We show that both humans and the RNN model successfully extract information about stimulus probability and integrate this knowledge into their decisions and task strategy in a new environment. Specifically, performance of both humans and the RNN model varied with the degree to which the stimulus probability of the new environment matched the formed expectation. In both cases, this expectation effect was more prominent when the strength of sensory evidence was low, suggesting that like humans, our RNNs placed more emphasis on prior expectation (top-down signals) when the available sensory information (bottom-up signals) was limited, thereby optimizing task performance. Finally, by dissecting the trained RNN model, we demonstrate how competitive inhibition and recurrent excitation form the basis for neural circuitry optimized to perform probabilistic information processing.