Mathematical Modeling seminars

My research focuses, broadly speaking, on how emotions impact decision making. Specifically, I am interested in affective biases, a phenomenon known to be important in depression. Using a rodent decision-making task, combined with computational modelling I have investigated how different antidepressant and pro-depressant manipulations that are known to alter mood in humans alter judgement bias, and provided insight into the decision processes that underlie these behaviours. I will also highlight how the combination of behaviour and modelling can provide a truly translation approach, enabling comparison and interpretation of the same cognitive processes between animal and human research.

Classical models of perceptual decision-making assume that animals use a single, consistent strategy to integrate sensory evidence and form decisions during an experiment. In this talk, I aim to convince you that this common view is incorrect. I will show results from applying a latent variable framework, the “GLM-HMM”, to hundreds of thousands of trials of mouse choice data. Our analysis reveals that mice don’t lapse. Instead, mice switch back and forth between engaged and disengaged behavior within a single session, and each mode of behavior lasts tens to hundreds of trials.

How can neural networks exhibit persistent activity on time scales much larger than allowed by cellular properties? We address this question in the context of larval zebrafish, a model vertebrate that is accessible to brain-scale neuronal recording and high-throughput behavioral studies. We study in particular the dynamics of a bilaterally distributed circuit, the so-called ARTR, including hundreds neurons. ARTR exhibits slow antiphasic alternations between its left and right subpopulations, which can be modulated by the water temperature, and drive the coordinated orientation of swim bouts, thus organizing the fish spatial exploration. To elucidate the mechanism leading to the slow self-oscillation, we train a network graphical model (Ising) on neural recordings. Sampling the inferred model allows us to generate synthetic oscillatory activity, whose features correctly capture the observed dynamics. A mean-field analysis of the inferred model reveals the existence several phases; activated crossing of the barriers in between those phases controls the long time scales present in the network oscillations. We show in particular how the barrier heights and the nature of the phases vary with the water temperature.

A key challenge in modern soft matter is to identify the principles that govern the organisation and functionality in non-equilibrium systems. Current research efforts largely focus on non-equilibrium processes that occur either at the single-molecule scale (e.g. protein and DNA conformations under driving forces), or at the scale of whole tissues, organisms, and active colloidal and microscopic objects. However, the range of the scales in-between — from molecules to large-scaled molecular assemblies that consume energy and perform work — remains under-explored. This is, nevertheless, the scale that is crucial for the function of a living cell, where molecular self-assembly driven far from equilibrium produces mechanical work needed for cell reshaping, transport, motility, division, and healing. Today I will discuss physical modelling of active elastic filaments, called ESCRT-III filaments, that dynamically assemble and disassemble on cell membranes. This dynamic assembly changes the filaments’ shape and mechanical properties and leads to the remodelling and cutting of cells. I will present a range of experimental comparisons of our simulation results: from ESCRT-III-driven trafficking in eukaryotes to division of evolutionary simple archaeal cells.

The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function. However, quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of rectified-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. We then use this analytical characterization to rigorously analyze the solution space geometry and derive certainty conditions guaranteeing a non-zero synapse between neurons.

The vIRt nucleus in the medulla, composed of mainly inhibitory neurons, is necessary for whisking rhythm generation. It innervates motoneurons in the facial nucleus (FN) that project to intrinsic vibrissa muscles. The nearby pre-Bötzinger complex (pBötC), which generates inhalation, sends inhibitory inputs to the vIRt nucleus which contribute to the synchronization of vIRt neurons. Lower-amplitude periodic whisking, however, can occur after decay of the pBötC signal. To explain how vIRt network generates these “intervening” whisks by bursting in synchrony, and how pBötC input induces strong whisks, we construct and analyze a conductance-based (CB) model of the vIRt circuit composed of hypothetical two groups, vIRtr and vIRtp, of bursting inhibitory neurons with spike-frequency adaptation currents and constant external inputs. The CB model is reduced to a rate model to enable analytical treatment. We find, analytically and computationally, that without pBötC input, periodic bursting states occur within a certain ranges of network connectivities. Whisk amplitudes increase with the level constant external input to the vIRT. With pBötC inhibition intact, the amplitude of the first whisk in a breathing cycle is larger than the intervening whisks for large pBötC input and small inhibitory coupling between the vIRT sub-populations. The pBötC input advances the next whisk and shortens its amplitude if it arrives at the beginning of the whisking cycle generated by the vIRT, and delays the next whisks if it arrives at the end of that cycle. Our theory provides a mechanism for whisking generation and reveals how whisking frequency and amplitude are controlled.

General anesthesia is a drug-induced, reversible condition comprised of five behavioral states: unconsciousness, amnesia (loss of memory), antinociception (loss of pain sensation), akinesia (immobility), and hemodynamic stability with control of the stress response. Our work shows that a primary mechanism through which anesthetics create these altered states of arousal is by initiating and maintaining highly structured oscillations. These oscillations impair communication among brain regions. We illustrate this effect by presenting findings from our human studies of general anesthesia using high-density EEG recordings and intracranial recordings. These studies have allowed us to give a detailed characterization of the neurophysiology of loss and recovery of consciousness due to propofol. We show how these dynamics change systematically with different anesthetic classes and with age. As a consequence, we have developed a principled, neuroscience-based paradigm for using the EEG to monitor the brain states of patients receiving general anesthesia. We demonstrate that the state of general anesthesia can be rapidly reversed by activating specific brain circuits. Finally, we demonstrate that the state of general anesthesia can be controlled using closed loop feedback control systems. The success of our research has depended critically on tight coupling of experiments, signal processing research and mathematical modeling.

A wealth of experimental studies show that the trial-to-trial variability of neuronal activity is quenched during stimulus evoked responses. This fact has helped ground a popular view that the variability of spiking activity can be decomposed into two components. The first is due to irregular spike timing conditioned on the firing rate of a neuron (i.e. a Poisson process), and the second is the trial-to-trial variability of the firing rate itself. Quenching of the variability of the overall response is assumed to be a reflection of a suppression of firing rate variability. Network models have explained this phenomenon through a variety of circuit mechanisms. However, in all cases, from the vantage of a neuron embedded within the network, quenching of its response variability is inherited from its synaptic input. We analyze in vivo whole cell recordings from principal cells in layer (L) 2/3 of mouse visual cortex. While the variability of the membrane potential is quenched upon stimulation, the variability of excitatory and inhibitory currents afferent to the neuron are amplified. This discord complicates the simple inheritance assumption that underpins network models of neuronal variability. We propose and validate an alternative (yet not mutually exclusive) mechanism for the quenching of neuronal variability. We show how an increase in synaptic conductance in the evoked state shunts the transfer of current to the membrane potential, formally decoupling changes in their trial-to-trial variability. The ubiquity of conductance based neuronal transfer combined with the simplicity of our model, provides an appealing framework. In particular, it shows how the dependence of cellular properties upon neuronal state is a critical, yet often ignored, factor. Further, our mechanism does not require a decomposition of variability into spiking and firing rate components, thereby challenging a long held view of neuronal activity.

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.

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.

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.

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.

Framing effects in individual decision-making have puzzled economists for decades because they are hard, if at all, to explain with rational choice theories. Why should mere changes in the description of a choice problem affect decision-making? Here, we examine the hypothesis that changes in framing cause changes in the allocation of attention to the different options – measured via eye-tracking – and give rise to changes in decision-making. We document that the framing of a sure alternative as a gain – as opposed to a loss – in a risk-taking task increases the attentional advantage of the sure option and induces a higher choice frequency of that option – a finding that is predicted by the attentional drift-diffusion model (aDDM). The model also correctly predicts other key findings such as that the increased attentional advantage of the sure option in the gain frame should also lead quicker decisions in this frame. In addition, the data reveal that increasing risk aversion at higher stake sizes may also be driven by attentional processes because the sure option receives significantly more attention – regardless of frame – at higher stakes. We also corroborate the causal impact of framing-induced changes of attention on choice with an additional experiment that manipulates attention exogenously. Finally, to study the precise mechanisms underlying the framing effect we structurally estimate an aDDM that allows for frame and option-dependent parameters. The estimation results indicate that – in addition to the direct effects of framing-induced changes in attention on choice – the gain frame also causes (i) an increase in the attentional discount of the gamble and (ii) an increased concavity of utility. Our findings suggest that the traditional explanation of framing effects in risky choice in terms of a more concave value function in the gain domain is seriously incomplete and that attentional mechanisms as hypothesized in the aDDM play a key role.