Mathematical Modeling seminars

Colonies of ants and other eusocial insects are superorganisms, which perform sophisticated cognitive-like functions at the level of the group. In my talk I will review our efforts to establish the clonal raider ant Ooceraea biroi as a lab model system for the systematic study of the principles underlying collective information processing in ant colonies. I will use results from two separate projects to demonstrate the potential of this model system: In the first, we analyze the foraging behavior of the species, known as group raiding: a swift offensive response of a colony to the detection of a potential prey by a scout. By using automated behavioral tracking and detailed analysis we show that this behavior is closely related to the army ant mass raid, an iconic collective behavior in which hundreds of thousands of ants spontaneously leave the nest to go hunting, and that the evolutionary transition between the two can be explained by a change in colony size alone. In the second project, we study the emergence of a collective sensory response threshold in a colony. The sensory threshold is a fundamental computational primitive, observed across many biological systems. By carefully controlling the sensory environment and the social structure of the colonies we were able to show that it also appear in a collective context, and that it emerges out of a balance between excitatory and inhibitory interactions between ants. Furthermore, by using a mathematical model we predict that these two interactions can be mapped into known mechanisms of communication in ants. Finally, I will discuss the opportunities for understanding collective behavior that are opening up by the development of methods for neuroimaging and neurocontrol of our ants.

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.

TheVirtualBrain (TVB; thevirtualbrain.org) is a software platform for simulating whole-brain network dynamics. TVB models link biophysical parameters at the cellular level with systems-level functional neuroimaging signals. Data available from animal models can provide vital constraints for the linkage across spatial and temporal scales. I will describe the construction of a macaque cortical connectome as an initial step towards a comprehensive multi-scale macaque TVB model. I will also describe our process of validating the connectome and show an example simulation of macaque resting-state dynamics using TVB. This connectome opens the opportunity for the addition of other available data from the macaque, such as electrophysiological recordings and receptor distributions, to inform multi-scale models of brain dynamics. Future work will include extensions to neurological conditions and other nonhuman primate species.

Learning in real life is often sequential: A learner first learns task A, then task B. If the tasks are related, the learner may adapt the previously learned representation instead of generating a new one from scratch. Adaptation may ease learning task B but may also decrease the performance on task A. Such interference has been observed in experimental and machine learning studies. In the latter case, it is mediated by correlations between weight updates for the different tasks. In typical applications, like image classification with feed-forward networks, these correlated weight updates can be traced back to input correlations. For many neuroscience tasks, however, networks need to not only transform the input, but also generate substantial internal dynamics. Here we illuminate the role of internal dynamics for interference in recurrent neural networks (RNNs). We analyze RNNs trained sequentially on neuroscience tasks with gradient descent and observe forgetting even for orthogonal tasks. We find that the degree of interference changes systematically with tasks properties, especially with emphasis on input-driven over autonomously generated dynamics. To better understand our numerical observations, we thoroughly analyze a simple model of working memory: For task A, a network is presented with an input pattern and trained to generate a fixed point aligned with this pattern. For task B, the network has to memorize a second, orthogonal pattern. Adapting an existing representation corresponds to the rotation of the fixed point in phase space, as opposed to the emergence of a new one. We show that the two modes of learning – rotation vs. new formation – are directly linked to recurrent vs. input-driven dynamics. We make this notion precise in a further simplified, analytically tractable model, where learning is restricted to a 2x2 matrix. In our analysis of trained RNNs, we also make the surprising observation that, across different tasks, larger random initial connectivity reduces interference. Analyzing the fixed point task reveals the underlying mechanism: The random connectivity strongly accelerates the learning mode of new formation, and has less effect on rotation. The prior thus wins the race to zero loss, and interference is reduced. Altogether, our work offers a new perspective on sequential learning in recurrent networks, and the emphasis on internally generated dynamics allows us to take the history of individual learners into account.

Rapidly changing inputs such as visual scenes and auditory landscapes are transmitted over several synaptic interfaces and perceived with little loss of detail, but individual neurons are typically “noisy” and cortico-cortical connections are typically “weak”. To understand how information embodied in spike train is transmitted in a lossless manner, we focus on a single synaptic interface: between pyramidal cells and putative interneurons. Using arbitrary white noise patterns injected intra-cortically as photocurrents to freely-moving mice, we find that directly-activated cells exhibit precision of several milliseconds, but post-synaptic, indirectly-activated cells exhibit higher precision. Considering multiple identical messages, the reliability of directly-activated cells peaks at a timescale of dozens of milliseconds, whereas indirectly-activated cells exhibit an order-of-magnitude faster timescale. Using data-driven modelling, we find that error correction is consistent with non-linear amplification of coincident spikes.

The journey of development begins with sperm swimming through the female reproductive tract en-route to the egg. In order to successfully complete this journey sperm must beat a single flagellum, propelling themselves through a wide range of fluids, from liquified semen to viscous cervical mucus. It is well-known that the beating tail is driven by an array of 9 microtubule doublets surrounding a central pair, with interconnecting dynein motors generating shear forces and driving elastic wave propagation. Despite this knowledge, the exact mechanism by which coordination of these motors drives oscillating waves along the flagellum remains unknown; hypothesised mechanisms include curvature control, sliding control, and geometric clutch. In this talk we will discuss the mechanisms of flagellar bending, and present a simple model of active curvature that is able to produce many of the various sperm waveforms that are seen experimentally, including those in low and high viscosity fluids and after a cell has ‘hyperactivated’ (a chemical process thought to be key for fertilization). We will show comparisons between these simulated waveforms and sperm that have been experimentally tracked, and discuss methods for fitting simulated mechanistic parameters to these real cells.

Humans have a remarkable ability to figure out what happened and why. In this talk, I will shed light on this ability from multiple angles. I will present a computational framework for modeling causal explanations in terms of counterfactual simulations, and several lines of experiments testing this framework in the domain of intuitive physics. The model predicts people's causal judgments about a variety of physical scenes, including dynamic collision events, complex situations that involve multiple causes, omissions as causes, and causal responsibility for a system's stability. It also captures the cognitive processes underlying these judgments as revealed by spontaneous eye-movements. More recently, we have applied our computational framework to explain multisensory integration. I will show how people's inferences about what happened are well-accounted for by a model that integrates visual and auditory evidence through approximate physical simulations.

Animals possess the ability to effortlessly and precisely time their actions even though information received from the world is often ambiguous and is inadvertently transformed as it passes through the nervous system. With such uncertainty pervading through our nervous systems, we could expect that much of human and animal behavior relies on inference that incorporates an important additional source of information, prior knowledge of the environment. These concepts have long been studied under the framework of Bayesian inference with substantial corroboration over the last decade that human time perception is consistent with such models. We, however, know little about the neural mechanisms that enable Bayesian signatures to emerge in temporal perception. I will present our work on three facets of this problem, how Bayesian estimates are encoded in neural populations, how these estimates are used to generate time intervals, and how prior knowledge for these tasks is acquired and optimized by neural circuits. We trained monkeys to perform an interval reproduction task and found their behavior to be consistent with Bayesian inference. Using insights from electrophysiology and in silico models, we propose a mechanism by which cortical populations encode Bayesian estimates and utilize them to generate time intervals. Thereafter, I will present a circuit model for how temporal priors can be acquired by cerebellar machinery leading to estimates consistent with Bayesian theory. Based on electrophysiology and anatomy experiments in rodents, I will provide some support for this model. Overall, these findings attempt to bridge insights from normative frameworks of Bayesian inference with potential neural implementations for the acquisition, estimation, and production of timing behaviors.

Dendrites are thin processes that extend from the cell body of neurons, the main computing units of the brain. The role of dendrites in complex brain functions has been investigated for several decades, yet their direct involvement in key behaviors such as for example sensory perception has only recently been established. In my presentation I will discuss how computational modelling has helped us illuminate dendritic function. I will present the main findings of a number of projects in lab dealing with dendritic nonlinearities in excitatory and inhibitory and their consequences on neuronal tuning and memory formation, the role of dendrites in solving nonlinear problems in human neurons and recent efforts to advance machine learning algorithms by adopting dendritic features.

When listening to music, we typically lock onto and move to a beat (1-6 Hz). Behavioral studies on such synchronization (Repp 2005) abound, yet the neural mechanisms remain poorly understood. Some models hypothesize an array of self-sustaining entrainable neural oscillators that resonate when forced with rhythmic stimuli (Large et al. 2010). In contrast, our formulation focuses on event time estimation and plasticity: a neuronal beat generator that adapts its intrinsic frequency and phase to match the extermal rhythm. The model quickly learns new rhythms, within a few cycles as found in human behavior. When the stimulus is removed the beat generator continues to produce the learned rhythm in accordance with a synchronization continuation task.

Pain is a major clinical problem affecting 1 in 5 people in the world. There are unresolved questions that urgently require answers to treat pain effectively, a crucial one being how the feeling of pain arises from brain activity. Computational models of pain consider how the brain processes noxious information and allow mapping neural circuits and networks to cognition and behaviour. To date, they have generally have assumed two largely independent processes: perceptual and/or predictive inference, typically modelled as an approximate Bayesian process, and action control, typically modelled as a reinforcement learning process. However, inference and control are intertwined in complex ways, challenging the clarity of this distinction. I will discuss how they may comprise a parallel hierarchical architecture that combines pain inference, information-seeking, and adaptive value-based control. Finally, I will discuss whether and how these learning processes might contribute to chronic pain.

Threshold-linear networks (TLNs) display a wide variety of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. Over the past few years, we have developed a detailed mathematical theory relating stable and unstable fixed points of TLNs to graph-theoretic properties of the underlying network. In particular, we have discovered that a special type of unstable fixed points, corresponding to "core motifs," are predictive of dynamic attractors. Recently, we have used these ideas to classify dynamic attractors in a two-parameter family of inhibition-dominated TLNs spanning all 9608 directed graphs of size n=5. Remarkably, we find a striking modularity in the dynamic attractors, with identical or near-identical attractors arising in networks that are otherwise dynamically inequivalent. This suggests that, just as one can store multiple static patterns as stable fixed points in a Hopfield model, a variety of dynamic attractors can also be embedded in a TLN in a modular fashion.

The idea that behavior results from a dynamic interplay between automatic and controlled processing underlies much of decision science, but has also generated considerable controversy. In this talk, I will highlight behavioral and neural data showing how recently-developed computational models of decision making can be used to shed new light on whether, when, and how decisions result from distinct processes operating at different timescales. Across diverse domains ranging from altruism to risky choice biases and self-regulation, our work suggests that a model of prioritized attentional sampling and evidence accumulation may provide an alternative explanation for many phenomena previously interpreted as supporting dual process models of choice. However, I also show how some features of the model might be taken as support for specific aspects of dual-process models, providing a way to reconcile conflicting accounts and generating new predictions and insights along the way.

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.