Mathematical Modeling seminars

The cerebellum contains more than half of the brain’s neurons and it is essential for motor control. Its neural circuits have a distinctive architecture comprised of a large, sparse expansion from the input mossy fibres to the granule cell layer. For years, theories of how cerebellar architectural features relate to cerebellar function have been formulated. It has been shown that some of these features can facilitate pattern separation. However, these theories don’t consider the need for it to learn fast in order to control smooth and accurate movements. Here, we confront this gap. This talk will show that the expansion to the granule cell layer in the cerebellar cortex improves learning speed and performance in the context of motor control by considering a cerebellar-like network learning an internal model of a motor apparatus online. By expressing the general form of the learning rate for such a system, this talk will provide a calculation of how increasing the number of granule cells diminishes the effect of noise and increases the learning speed. The researchers propose that the particular architecture of cerebellar circuits modifies the geometry of the error function in a favourable way for learning faster. Their results illuminate a new link between cerebellar structure and function.

Dr. Rajan and her lab design neural network models based on experimental data, and reverse-engineer them to figure out how brain circuits function in health and disease. They recently developed a powerful framework for tracing neural paths across multiple brain regions— called Current-Based Decomposition (CURBD). This new approach enables the computation of excitatory and inhibitory input currents that drive a given neuron, aiding in the discovery of how entire populations of neurons behave across multiple interacting brain regions. Dr. Rajan’s team has applied this method to studying the neural underpinnings of behavior. As an example, when CURBD was applied to data gathered from an animal model often used to study depression- and anxiety-like behaviors (i.e., learned helplessness) the underlying biology driving adaptive and maladaptive behaviors in the face of stress was revealed. With this framework Dr. Rajan's team probes for mechanisms at work across brain regions that support both healthy and disease states-- as well as identify key divergences from multiple different nervous systems, including zebrafish, mice, non-human primates, and humans.

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.

Organisms modify their behavior in response to its consequences, a phenomenon referred to as operant learning. Contemporary modeling of this learning behavior is based on reinforcement learning algorithms. I will discuss some of the challenges that these models face, and proposed a new approach to model-selection that is based on testing their ability to engineer behavior. Finally, I will present the results of The Choice Engineering Competition – an academic competition that compared the efficacies of qualitative and quantitative models of operant learning in shaping behavior.

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.

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.

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.

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.

Bacteria exhibit a bewildering diversity of morphologies, but despite their impact on nearly all aspects of life, they are frequently classified into a few general categories, usually just “spheres” and “rods.” Curved-rod bacteria are one simple variation observed in many environments, particularly the ocean. However, why so many species have evolved this shape is unknown. We used a regularized Stokeslet Boundary Element Method to model the motility of flagellated, curved bacteria. We show that curvature can increase swimming efficiency, revealing a widely applicable selective advantage. Furthermore, we show that the distribution of cell lengths and curvatures observed across bacteria in nature is predicted by evolutionary trade-offs between three tasks influenced by shape: efficient swimming, the ability to detect chemical gradients, and reduced cost of cell construction. We therefore reveal shape as an important component of microbial fitness.

We study the dynamics of (inhibitory) balanced networks at varying (i) the level of symmetry in the synaptic connectivity; and (ii) the ariance of the synaptic efficacies (synaptic gain). We find three regimes of activity. For suitably low synaptic gain, regardless of the level of symmetry, there exists a unique stable fixed point. Using a cavity-like approach, we develop a quantitative theory that describes the statistics of the activity in this unique fixed point, and the conditions for its stability. Increasing the synaptic gain, the unique fixed point destabilizes, and the network exhibits chaotic activity for zero or negative levels of symmetry (i.e., random or antisymmetric). Instead, for positive levels of symmetry, there is multi-stability among a large number of marginally stable fixed points. In this regime, ergodicity is broken and the network exhibits non-exponential relaxational dynamics. We discuss the potential relevance of such a “glassy” phase to explain some features of cortical activity.

Understanding the properties of active matter is a challenge which is currently driving a rapid growth in soft- and bio-physics. Some of the most important examples of active matter are at the microscale, and include active colloids and suspensions of microorganisms, both as a simple active fluid (single species) and as mixed suspensions of active and passive elements. In this last class of systems, recent experimental and theoretical work has started to provide a window into new phenomena including activity-induced depletion interactions, phase separation, and the possibility to extract net work from active suspensions. Here I will present our work on a paradigmatic example of mixed active-passive system, where the activity is provided by swimming microalgae. Macro- and micro-scopic experiments reveal that microorganism-colloid interactions are dominated by rare close encounters leading to large displacements through direct entrainment. Simulations and theoretical modelling show that the ensuing particle dynamics can be understood in terms of a simple jump-diffusion process, combining standard diffusion with Poisson-distributed jumps. Entrainment length can be understood within the framework of Taylor dispersion as a competition between advection by the no-slip surface of the cell body and microparticle diffusion. Building on these results, we then ask how external control of the dynamics of the active component (e.g. induced microswimmer anisotropy/inhomogeneity) can be used to alter the transport of passive cargo. As a first step in this direction, we study the behaviour of mixed active-passive systems in confinement. The resulting spatial inhomogeneity in swimmers’ distribution and orientation has a dramatic effect on the spatial distribution of passive particles, with the colloids accumulating either towards the boundaries or towards the bulk of the sample depending on the size of the container. We show that this can be used to induce the system to de-mix spontaneously.