Mathematical Modeling seminars

At small length-scales, capillary effects are significant, and thus the mechanics of soft material interfaces may be dominated by solid surface stresses or liquid surface tensions. The balance between surface and bulk properties is described by an elasto-capillary length-scale, in which equilibrium interfacial energies are constant. However, at small length-scales in biological materials, including living cells and tissues, interfacial energies are not constant but are actively regulated and driven far from equilibrium. Thus, the balance between surface and bulk properties depends upon the distance from equilibrium. Here, we model the spreading (wetting) of living cell aggregates as ‘active droplets’, with a non-equilibrium surface tension that depends upon internal stress generated by the actomyosin cytoskeleton. Depending upon the extent of activity, droplet surface properties adapt to the mechanics of their surroundings. The impact of this adaptation challenges contemporary models of interfacial mechanics, including extensively used models of contact mechanics and wetting.

Our group is interested in discovering design principles that govern the structure and function of neurons and neural circuits. We record from well-defined neurons, mainly in flies’ visual systems, to measure the molecular and cellular factors that determine relevant measures of performance, such as representational capacity, dynamic range and accuracy. We combine this empirical approach with modelling to see how the basic elements of neural systems (ion channels, second messengers systems, membranes, synapses, neurons, circuits and codes) combine to determine performance. We are investigating four general problems. How are circuits designed to integrate information efficiently? How do sensory adaptation and synaptic plasticity contribute to efficiency? How do the sizes of neurons and networks relate to energy consumption and representational capacity? To what extent have energy costs shaped neurons, sense organs and brain regions during evolution?

Spontaneous behavior in animals and humans shows a striking amount of variability both in the spatial domain (which actions to choose) and temporal domain (when to act). Concatenating actions into sequences and behavioral plans reveals the existence of a hierarchy of timescales ranging from hundreds of milliseconds to minutes. How do multiple timescales emerge from neural circuit dynamics? How do circuits modulate temporal responses to flexibly adapt to changing demands? In this talk, we will present recent results from experiments and theory suggesting a new computational mechanism generating the temporal variability underlying naturalistic behavior and cortical activity. We will show how neural activity from premotor areas unfolds through temporal sequences of attractors, which predict the intention to act. These sequences naturally emerge from recurrent cortical networks, where correlated neural variability plays a crucial role in explaining the observed variability in action timing. We will then discuss how reaction times can be accelerated or slowed down via gain modulation, flexibly induced by neuromodulation or perturbations; and how gain modulation may control response timing in the visual cortex. Finally, we will present a new biologically plausible way to generate a reservoir of multiple timescales in cortical circuits.

The SARS-CoV-2 pandemic is awash in data, including daily, spatially-resolved COVID case data, virus sequence data, patients `omics data, and mobility data. Journals are now also awash in studies that make use of quantitative modeling approaches to gain insight into the geographic spread of SARS-CoV-2 and its temporal dynamics, as well as studies that predict the impact of control strategies on SARS-CoV-2 circulation. Some, but by no means all, of these studies are informed by the massive amounts of available data. Some, but by no means all, of these studies have been useful — in that their predictions revealed something beyond simple back of the envelope calculations. To summarize some of these findings, in this symposium, we will address questions such as: What do we want from models of disease spread? What can and should be predicted? Which data are the most useful for predictions? When do we need mechanistic models? What have we learned about how to model disease spread from unmet and/or conflicting predictions? The workshop speakers will explore these questions from different perspectives on what data need to be considered and how models can be evaluated. As at other TMLS workshops, each speaker will deliver a 10-minute talk with ample time set aside for moderated questions/discussion. We expect the talks to be provocative and bold, while respecting different perspectives.

My talk will discuss the use of some of these four, simple Matlab functions to simulate models of timing, and to fit models to empirical data. Feel free to examine the code and the relatively brief book chapter that explains the code before the talk if you would like to learn more about computational/mathematical modeling.

Spike trains recorded from the cortex of behaving animals can be complex, highly variable from trial to trial, and therefore challenging to interpret. A fraction of cells exhibit trial-averaged responses with obvious task-related features such as pure tone frequency tuning in auditory cortex. However, a substantial number of cells (including cells in primary sensory cortex) do not appear to fire in a task-related manner and are often neglected from analysis. We recently used a novel single-trial, spike-timing-based analysis to show that both classically responsive and non-classically responsive cortical neurons contain significant information about sensory stimuli and behavioral decisions suggesting that non-classically responsive cells may play an underappreciated role in perception and behavior. We now expand this investigation to explore the synaptic origins and potential contribution of these cells to network function. To do so, we trained a novel spiking recurrent neural network model that incorporates spike-timing-dependent plasticity (STDP) mechanisms to perform the same task as behaving animals. By leveraging excitatory and inhibitory plasticity rules this model reproduces neurons with response profiles that are consistent with previously published experimental data, including classically responsive and non-classically responsive neurons. We found that both classically responsive and non-classically responsive neurons encode behavioral variables in their spike times as seen in vivo. Interestingly, plasticity in excitatory-to-excitatory synapses increased the proportion of non-classically responsive neurons and may play a significant role in determining response profiles. Finally, our model also makes predictions about the synaptic origins of classically and non-classically responsive neurons which we can compare to in vivo whole-cell recordings taken from the auditory cortex of behaving animals. This approach successfully recapitulates heterogeneous response profiles measured from behaving animals and provides a powerful lens for exploring large-scale neuronal dynamics and the plasticity rules that shape them.

Many psychological theories assume that qualitatively different cognitive processes can result in identical responses. Multinomial processing tree (MPT) models allow researchers to disentangle latent cognitive processes based on observed response frequencies. Recently, MPT models have been extended to explicitly account for participant and item heterogeneity. These hierarchical Bayesian MPT models provide the opportunity to connect two traditionally isolated disciplines. Whereas cognitive psychology has often focused on the experimental validation of MPT model parameters on the group level, psychometrics provides the necessary concepts and tools for measuring differences in MPT parameters on the item or person level. Moreover, MPT parameters can be regressed on covariates to model latent processes as a function of personality traits or other person characteristics.

The link between behavior, learning and the underlying connectome is a fundamental open problem in neuroscience. In my talk I will show how it is possible to develop a theory that bridges across these three levels (animal behavior, learning and network connectivity) based on the geometrical properties of neural activity. The central tool in my approach is the dimensionality of neural activity. I will link animal complex behavior to the geometry of neural representations, specifically their dimensionality; I will then show how learning shapes changes in such geometrical properties and how local connectivity properties can further regulate them. As a result, I will explain how the complexity of neural representations emerges from both behavioral demands (top-down approach) and learning or connectivity features (bottom-up approach). I will build these results regarding neural dynamics and representations starting from the analysis of neural recordings, by means of theoretical and computational tools that blend dynamical systems, artificial intelligence and statistical physics approaches.

Interval timing is a fundamental component action, and is susceptible to motor-related temporal distortions. Previous studies have shown that movement biases temporal estimates, but have primarily considered self-modulated movement only. However, real-world encounters often include situations in which movement is restricted or perturbed by environmental factors. In the following experiments, we introduced viscous movement environments to externally modulate movement and investigated the resulting effects on temporal perception. In two separate tasks, participants timed auditory intervals while moving a robotic arm that randomly applied four levels of viscosity. Results demonstrated that higher viscosity led to shorter perceived durations. Using a drift-diffusion model and a Bayesian observer model, we confirmed these biasing effects arose from perceptual mechanisms, instead of biases in decision making. These findings suggest that environmental perturbations are an important factor in movement-related temporal distortions, and enhance the current understanding of the interactions of motor activity and cognitive processes. https://www.biorxiv.org/content/10.1101/2020.10.26.355396v1

In daily life, we consistently make decisions in pursuit of some goal. Many decisions are informed by multiple sources of information. Unfortunately, these sources often provide ambiguous information about what course of action to take. Therefore, determining how the brain integrates information to resolve this ambiguity is key to understanding the neural mechanisms of decision-making. In the domain of time, this topic can be studied by training subjects to predict when a future event will occur based on distinct cues (e.g., tone, light, etc.). If multiple cues are presented simultaneously and their cue-to-event intervals differ (e.g., tone-10s + light-30s), subjects will often expect the event to occur at the average of their intervals. This ‘temporal averaging’ effect is presumably how the timing system resolves ambiguous time-information. The neural mechanisms of temporal averaging are currently unclear. Here, we will propose how temporal averaging could emerge in cortical circuits using a simple modification of a ‘drift-diffusion’ model of timing.

How do people learn in real-world environments where the space of possible actions can be vast or even infinite? The study of human learning has made rapid progress in past decades, from discovering the neural substrate of reward prediction errors, to building AI capable of mastering the game of Go. Yet this line of research has primarily focused on learning through repeated interactions with the same stimuli. How are humans able to rapidly adapt to novel situations and learn from such sparse examples? I propose a theory of how generalization guides human learning, by making predictions about which unobserved options are most promising to explore. Inspired by Roger Shepard’s law of generalization, I show how a Bayesian function learning model provides a mechanism for generalizing limited experiences to a wide set of novel possibilities, based on the simple principle that similar actions produce similar outcomes. This model of generalization generates predictions about the expected reward and underlying uncertainty of unexplored options, where both are vital components in how people actively explore the world. This model allows us to explain developmental differences in the explorative behavior of children, and suggests a general principle of learning across spatial, conceptual, and structured domains.

Inspired by ongoing experiments on three dimensional active gels composed of sliding microtubule bundles, we study a few idealized problems in a minimal hydrodynamic model for active liquid crystals. Our aim is to use flow to determine the value of the coefficient of activity in a continuum theory. We consider the case of apolar active particles that form a disordered phase in the absence of flow, and study how activity affects the swimming speed of a prescribed swimmer, as well as the stability of a fluid interface. We also consider flows of active matter in channels or past immersed objects.

The remarkable properties of information-processing of biological and of artificial neuronal networks alike arise from the interaction of large numbers of neurons. A central quest is thus to characterize their collective states. The directed coupling between pairs of neurons and their continuous dissipation of energy, moreover, cause dynamics of neuronal networks outside thermodynamic equilibrium. Tools from non-equilibrium statistical mechanics and field theory are thus instrumental to obtain a quantitative understanding. We here present progress with this recent approach [1]. On the experimental side, we show how correlations between pairs of neurons are informative on the dynamics of cortical networks: they are poised near a transition to chaos [2]. Close to this transition, we find prolongued sequential memory for past signals [3]. In the chaotic regime, networks offer representations of information whose dimensionality expands with time. We show how this mechanism aids classification performance [4]. Together these works illustrate the fruitful interplay between theoretical physics, neuronal networks, and neural information processing.

Physics offers countless examples for which theoretical predictions are astonishingly powerful. But it’s hard to imagine a similar precision in complex systems where the number and interdependencies between components simply prohibits a first-principles approach, look no further than the challenge of the billions of neurons and trillions of connections within our own brains. In such settings how do we even identify the important theoretical questions? We describe a systems-scale perspective in which we integrate information theory, dynamical systems and statistical physics to extract understanding directly from measurements. We demonstrate our approach with a reconstructed state space of the behavior of the nematode C. elegans, revealing a chaotic attractor with symmetric Lyapunov spectrum and a novel perspective of motor control. We then outline a maximally predictive coarse-graining in which nonlinear dynamics are subsumed into a linear, ensemble evolution to obtain a simple yet accurate model on multiple scales. With this coarse-graining we identify long timescales and collective states in the Langevin dynamics of a double-well potential, the Lorenz system and in worm behavior. We suggest that such an ``inverse’’ approach offers an emergent, quantitative framework in which to seek rather than impose effective organizing principles of complex systems.

In the past decade evidence has accumulated in favor of the hypothesis that multiple sub-networks in the medial entorhinal cortex (MEC) are characterized by low-dimensional, continuous attractor dynamics. Much has been learned about the joint activity of grid cells within a module (a module consists of grid cells that share a common grid spacing), but little is known about the interactions between them. Under typical conditions of spatial exploration in which sensory cues are abundant, all grid-cells in the MEC represent the animal’s position in space and their joint activity lies on a two-dimensional manifold. However, if the grid cells in a single module mechanistically constitute independent attractor networks, then under conditions in which salient sensory cues are absent, errors could accumulate in the different modules in an uncoordinated manner. Such uncoordinated errors would give rise to catastrophic readout errors when attempting to decode position from the joint grid-cell activity. I will discuss recent theoretical works from our group, in which we explored different mechanisms that could impose coordination in the different modules. One of these mechanisms involves coordination with the hippocampus and must be set up such that it operates across multiple spatial maps that represent different environments. The other mechanism is internal to the entorhinal cortex and independent of the hippocampus.

Reliable and validated measures of emotion in animals are of great import; they are crucial to better understanding and developing treatments for human mood disorders, and they are necessary for ensuring good animal welfare. We have developed a novel measure of emotion in animals that is grounded in theory and psychological research – decision-making under ambiguity. Specifically, we consider that more ‘optimistic’ decisions about ambiguous stimuli reflect more positive emotional states, while the opposite is true for more ‘pessimistic’ decisions. In this talk, we will outline the background behind and implementation of this measure, meta-analyses that have been conducted to validate the measure, and discuss how computational modelling has been used to further understand the cognitive processes underlying ‘optimistic’ and ‘pessimistic’ decision-making as an indicator of animal emotion and welfare.

Cross-sectional and longitudinal multimodal brain imaging studies using positron emission tomography (PET) and magnetic resonance imaging (MRI) have provided detailed insight into the pathophysiological progression of Alzheimer’s disease. It starts at an asymptomatic stage with widespread gradual accumulation of beta-amyloid and spread of pathological tau deposits. Subsequently changes of functional connectivity and glucose metabolism associated with mild cognitive impairment and brain atrophy may develop. However, the rate of progression to a symptomatic stage and ultimately dementia varies considerably between individuals. Mathematical models have been developed to describe disease progression, which may be used to identify markers that determine the current stage and likely rate of progression. Both are very important to improve the efficacy of clinical trials. In this lecture, I will provide an overview on current research and future perspectives in this area.