Mathematical Modeling seminars

Risk occupies a central role in both the theory and practice of decision-making. Although it is deeply implicated in many conditions involving dysfunctional behavior and thought, modern theoretical approaches to understanding and mitigating risk in either one-shot or sequential settings, which are derived largely from finance and economics, have yet to permeate fully the fields of neural reinforcement learning and computational psychiatry. I will discuss the use of dynamic and static versions of one prominent approach, namely conditional value-at-risk, to examine both the nature of risk avoidant choices, encompassing such things as justified gambler's fallacies, and the optimal planning that can lead to consideration of such choices, with implications for offline, ruminative, thinking.

The songbird zebra finch is an exemplary model system in which to study trial-and-error learning, as the bird learns its single song gradually through the production of many noisy renditions. It is also a good system in which to study the maintenance of motor skills, as the adult bird actively maintains its song and retains some residual plasticity. Motor learning occurs through the association of timing within the song, represented by sparse firing in nucleus HVC, with motor output, driven by nucleus RA. Here we show through modeling that the small level of observed variability in HVC can result in a network which is more easily able to adapt to change, and is most robust to cell damage or death, than an unperturbed network. In collaboration with Carlos Lois’ lab, we also consider the effect of directly perturbing HVC through viral injection of toxins that affect the firing of projection neurons. Following these perturbations, the song is profoundly affected but is able to almost perfectly recover. We characterize the changes in song acoustics and syntax, and propose models for HVC architecture and plasticity that can account for some of the observed effects. Finally, we suggest a potential role for inputs from nucleus Uva in helping to control timing precision in HVC.

Embryo morphogenesis is impacted by dynamic changes in tissue material properties, which have been proposed to occur via processes akin phase transitions (PTs). Here, we show that rigidity percolation provides a simple and robust theoretical framework to predict material/structural PTs of embryonic tissues from local cell connectivity. By using percolation theory, combined with directly monitoring dynamic changes in tissue rheology and cell contact mechanics, we demonstrate that the zebrafish blastoderm undergoes a genuine rigidity PT, brought about by a small reduction in adhesion-dependent cell connectivity below a critical value. We quantitatively predict and experimentally verify hallmarks of PTs, including power-law exponents and associated discontinuities of macroscopic observables at criticality. Finally, we show that this uniform PT depends on blastoderm cells undergoing meta-synchronous divisions causing random and, consequently, uniform changes in cell connectivity. Collectively, our theoretical and experimental findings reveal the structural basis of material PTs in an organismal context.

The Group Iterative Multiple Model Estimation (GIMME) framework has emerged as a promising method for characterizing connections between brain regions in functional neuroimaging data. Two of the most appealing features of this framework are its ability to estimate the directionality of connections between network nodes and its ability to determine whether those connections apply to everyone in a sample (group-level) or just to one person (individual-level). However, there are outstanding questions about the validity and stability of these estimates, including: 1) how recovery of connection directionality is affected by features of data sets such as scan length and autoregressive effects, which may be strong in some imaging modalities (resting state fMRI, fNIRS) but weaker in others (task fMRI); and 2) whether inferences about directionality at the group and individual levels are stable across time. This talk will provide an overview of the GIMME framework and describe relevant results from a large-scale simulation study that assesses directionality recovery under various conditions and a separate project that investigates the temporal stability of GIMME’s inferences in the Human Connectome Project data set. Analyses from these projects demonstrate that estimates of directionality are most precise when autoregressive and cross-lagged relations in the data are relatively strong, and that inferences about the directionality of group-level connections, specifically, appear to be stable across time. Implications of these findings for the interpretation of directional connectivity estimates in different types of neuroimaging data will be discussed.

Sylvia Serfaty studies the many-particle limit of systems driven by Coulomb or Riesz interactions through gradient flows or analogous conservative dynamics, possibly with added random diffusion. The Riesz interaction has inverse-distance exponent s between d−2 and d in dimension d. The talk presents convergence, as the number of particles tends to infinity, to the expected mean-field evolution. It also discusses deriving the Vlasov–Poisson equation from Newtonian dynamics in the monokinetic case and related results for Ginzburg–Landau vortex dynamics.

Constraints on control-dependent processing have become a fundamental concept in general theories of cognition that explain human behavior in terms of rational adaptations to these constraints. However, theories miss a rationale for why such constraints would exist in the first place. Recent work suggests that constraints on the allocation of control facilitate flexible task switching at the expense of the stability needed to support goal-directed behavior in face of distraction. We formulate this problem in a dynamical system, in which control signals are represented as attractors and in which constraints on control allocation limit the depth of these attractors. We derive formal expressions of the stability-flexibility tradeoff, showing that constraints on control allocation improve cognitive flexibility but impair cognitive stability. We provide evidence that human participants adapt higher constraints on the allocation of control as the demand for flexibility increases but that participants deviate from optimal constraints. In continuing work, we are investigating how collaborative performance of a group of individuals can benefit from individual differences defined in terms of balance between cognitive stability and flexibility.

Since the theories of Marr, Ito, and Albus, the cerebellum has provided an attractive well-characterized model system to investigate biological mechanisms of learning. In recent years, theories have been developed that provide a normative account for many features of the anatomy and function of cerebellar cortex and cerebellum-like systems, including the distribution of parallel fiber-Purkinje cell synaptic weights, the expansion in neuron number of the granule cell layer and their synaptic in-degree, and sparse coding by granule cells. Typically, these theories focus on the learning of random mappings between uncorrelated inputs and binary outputs, an assumption that may be reasonable for certain forms of associative conditioning but is also quite far from accounting for the important role the cerebellum plays in the control of smooth movements. I will discuss in-progress work with Marjorie Xie, Samuel Muscinelli, and Kameron Decker Harris generalizing these learning theories to correlated inputs and general classes of smooth input-output mappings. Our studies build on earlier work in theoretical neuroscience as well as recent advances in the kernel theory of wide neural networks. They illuminate the role of pre-expansion structures in processing input stimuli and the significance of sparse granule cell activity. If there is time, I will also discuss preliminary work with Jack Lindsey extending these theories beyond cerebellum-like structures to recurrent networks.

The cellular representations and computations that allow rodents to navigate in space have been described with beautiful precision. In this talk, I will show that some of these same computations can be found in humans doing tasks that appear very different from spatial navigation. I will describe some theory that allows us to think about spatial and non-spatial problems in the same framework, and I will try to use this theory to give a new perspective on the beautiful spatial computations that inspired it. The overall goal of this work is to find a framework where we can talk about complicated non-spatial inference problems with the same precision that is only currently available in space.

'Shape' of microorganisms are diverse. However, we sometimes approximate them as a sphere or a spheroid when we mathematically model the hydrodynamics of motile and non-motile cells. Such a geometrical simplification can be theoretically validated for motions in a linear background flow, since the dynamics, known as the Jeffery orbit, only contain a single geometric parameter, called the Bretherton constant. In this talk, we generalise the Jeffery equations for a chiral axisymmetric object using the low-Reynolds-number hydrokinetic symmetry and then demonstrate that the dynamics of a certain type of chiral object in a fluid flow are characterised by a new chiral parameter in addition to the Bretherton constant. We also discuss how the generalised Jeffery orbits are applied to biased locomotion of bacteria in a bulk shear flow and we will share the idea of hydrodynamic `shape' of microorganisms to simplify the description of their dynamics.

Foraging research aims at describing, understanding, and predicting resource-gathering behaviour. Optimal Foraging Theory (OFT) is a sub-discipline that emphasises that these aims can be aided by segmenting foraging behaviour into discrete problems that can be formally described and examined with mathematical maximization techniques. Examples of such segmentation are found in the isolated treatment of issues such as patch residence time, prey selection, information gathering, risky choice, intertemporal decision making, resource allocation, competition, memory updating, group structure, and so on. Since foragers face these problems simultaneously rather than in isolation, it is unsurprising that OFT models are ‘always wrong but sometimes useful’. I will argue that a progressive optimal foraging research program should have a defined strategy for dealing with predictive failure of models. Further, I will caution against searching for brain structures responsible for solving isolated foraging problems.

Neurons and neural circuits can produce stereotyped and reliable output activity on the basis of highly variable cellular, synaptic, and circuit properties. This is crucial for proper nervous system function throughout an animal’s life in the face of growth, perturbations, and molecular turnover. But how can reliable output arise from neurons and synapses whose parameter vary between individuals in a population, and within an individual over time? I will review how a combination of experimental and computational methods can be used to examine how neuron and network function depends on the underlying parameters, such as neuronal membrane conductances and synaptic strengths. Within the high-dimensional parameter space of a neural system, the subset of parameter combinations that produce biologically functional neuron or circuit activity is captured by the notion of a ‘solution space’. I will describe solution space structures determined from electrophysiology data, ion channel expression levels across populations of neurons and animals, and computational parameter space explorations. A key finding centers on experimental and computational evidence for parameter correlations that give structure to solution spaces. Computational modeling suggests that such parameter correlations can be beneficial for constraining neuron and circuit properties to functional regimes, while experimental results indicate that neural circuits may have evolved to implement some of these beneficial parameter correlations at the cellular level. Finally, I will review modeling work and experiments that seek to illuminate how neural systems can homeostatically navigate their parameter spaces to stably remain within their solution space and reliably produce functional output, or to return to their solution space after perturbations that temporarily disrupt proper neuron or network function.

The COINTOB (conditional intention and integration to bound) model provides a heuristic framework of processes in Libet-style experiments. The model is based on three assumptions. First, brain activation preceding conscious intentions in Libet-style experiments does not reflect an unconscious decision but rather the unfolding of a decision process. Second, the time of conscious decision (W) reflects the moment in time when the decision boundary is crossed. This interpretation of W is consistent with our apparent intuition that we decide in the moment we experience the conscious intention to act. Third, the decision process is configured by conscious intentions that participants form at the beginning of the experiment based on the experimental instruction. Brass and Mele discuss the model, conceptual background for it, and the model’s bearing on free will.

Distinct synapses influence one another when they undergo changes, with unclear consequences for neuronal dynamics and function. Here we show that synapses can interact such that excitatory currents are naturally normalised and balanced by inhibitory inputs. This happens when classical spike-timing dependent synaptic plasticity rules are extended by additional mechanisms that incorporate the influence of neighbouring synaptic currents and regulate the amplitude of efficacy changes accordingly. The resulting control of excitatory plasticity by inhibitory activation, and vice versa, gives rise to quick and long-lasting memories as seen experimentally in receptive field plasticity paradigms. In models with additional dendritic structure, we observe experimentally reported clustering of co-active synapses that depends on initial connectivity and morphology. Finally, in recurrent neural networks, rich and stable dynamics with high input sensitivity emerge, providing transient activity that resembles recordings from the motor cortex. Our model provides a general framework for codependent plasticity that frames individual synaptic modifications in the context of population-wide changes, allowing us to connect micro-level physiology with behavioural phenomena.

Regulation of information flow in the brain is critical for many forms of behavior. In the process of sensory based decision-making, decisions about future actions are held in memory until enacted, making them potentially vulnerable to distracting sensory input. Therefore, gating of information flow from sensory to motor areas could protect memory from interference during decision-making, but the underlying network mechanisms are not understood. I will present our recent experimental and modeling work describing how information flow from the sensory cortex can be gated by state-dependent frontal cortex dynamics during decision-making in mice. Our results show that communication between brain regions can be regulated via attractor dynamics, which control the degree of commitment to an action, and reveal a novel mechanism of gating of neural information.

Classic studies show that in many species – from leech and cricket to primate – responses of neural populations can be quite successfully read out using a measure neural population activity termed the population vector. However, despite its successes, detailed analyses have shown that the standard population vector discards substantial amounts of information contained in the responses of a neural population, and so is unlikely to accurately describe how signal communication between parts of the nervous system. I will describe recent theoretical results showing how to modify the population vector expression in order to read out neural responses without information loss, ideally. These results make it possible to quantify the contribution of weakly tuned neurons to perception. I will also discuss numerical methods that can be used to minimize information loss when reading out responses of large neural populations. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2021-03-03. Recording duration: 00:45:35.

Microorganisms can swim in a variety of environments, interacting with chemicals and other proteins in the fluid. In this talk, we will highlight recent computational methods and results for swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. Sperm are modeled via a centerline representation where forces are solved for using elastic rod theory. The method of regularized Stokeslets is used to solve the fluid-structure interaction where emergent swimming speeds can be compared to asymptotic analysis. In the case of fluids with extra proteins or cells that may act as friction, swimming speeds may be enhanced, and attraction may not occur. We will also highlight how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties.

Biologists have recently come to appreciate that eukaryotic cells are home to a multiplicity of non-membrane bound compartments, many of which form and dissolve as needed for the cell to function. These dynamical “condensates” enable many central cellular functions – from ribosome assembly, to RNA regulation and storage, to signaling and metabolism. While it is clear that these compartments represent a type of separated phase, what controls their formation, how specific biological components are included or excluded, and how these structures influence physiological and biochemical processes remain largely mysterious. I will discuss recent experiments on phase separated condensates both in vitro and in vivo, and will present theoretical results that highlight a novel “magic number” effect relevant to the formation and control of two-component phase separated condensates.

The emergence of large-scale brain networks, and their continual refinement, represent crucial developmental processes that can drive individual differences in cognition and which are associated with multiple neurodevelopmental conditions. But how does this organization arise, and what mechanisms govern the diversity of these developmental processes? There are many existing descriptive theories, but to date none are computationally formalized. We provide a mathematical framework that specifies the growth of a brain network over developmental time. Within this framework macroscopic brain organization, complete with spatial embedding of its organization, is an emergent property of a generative wiring equation that optimizes its connectivity by renegotiating its biological costs and topological values continuously over development. The rules that govern these iterative wiring properties are controlled by a set of tightly framed parameters, with subtle differences in these parameters steering network growth towards different neurodiverse outcomes. Regional expression of genes associated with the developmental simulations converge on biological processes and cellular components predominantly involved in synaptic signaling, neuronal projection, catabolic intracellular processes and protein transport. Together, this provides a unifying computational framework for conceptualizing the mechanisms and diversity of childhood brain development, capable of integrating different levels of analysis – from genes to cognition. (Pre-print: https://www.biorxiv.org/content/10.1101/2020.08.13.249391v1)