Mathematical Modeling seminars

One of the fundamental functions of the brain is to flexibly plan and control movement production at different timescales to efficiently shape structured behaviors. I will present a model that clarifies how these complex computations could be performed in the mammalian brain, with an emphasis on the learning of an extendable library of autonomous motor motifs and the flexible stringing of these motifs in motor sequences. To build this model, we took advantage of the fact that the anatomy of the circuits involved is well known. Our results show how these architectural constraints lead to a principled understanding of how strategically positioned plastic connections located within motif-specific thalamocortical loops can interact with cortical dynamics that are shared across motifs to create an efficient form of modularity. This occurs because the cortical dynamics can be controlled by the activation of as few as one thalamic unit, which induces a low-rank perturbation of the cortical connectivity, and significantly expands the range of outputs that the network can produce. Finally, our results show that transitions between any motifs can be facilitated by a specific thalamic population that participates in preparing cortex for the execution of the next motif. Taken together, our model sheds light on the neural network mechanisms that can generate flexible sequencing of varied motor motifs.

Biological brains possess an exceptional ability to infer relevant behavioral responses to a wide range of stimuli from only a few examples. This capacity to generalize beyond the training set has been proven particularly challenging to realize in artificial systems. How neural processes enable this capacity to extrapolate to novel stimuli is a fundamental open question. A prominent but underexplored hypothesis suggests that generalization is facilitated by a low-dimensional organization of collective neural activity, yet evidence for the underlying neural mechanisms remains wanting. Combining network modeling, theory and neural data analysis, we tested this hypothesis in the framework of flexible timing tasks, which rely on the interplay between inputs and recurrent dynamics. We first trained recurrent neural networks on a set of timing tasks while minimizing the dimensionality of neural activity by imposing low-rank constraints on the connectivity, and compared the performance and generalization capabilities with networks trained without any constraint. We then examined the trained networks, characterized the dynamical mechanisms underlying the computations, and verified their predictions in neural recordings. Our key finding is that low-dimensional dynamics strongly increases the ability to extrapolate to inputs outside of the range used in training. Critically, this capacity to generalize relies on controlling the low-dimensional dynamics by a parametric contextual input. We found that this parametric control of extrapolation was based on a mechanism where tonic inputs modulate the dynamics along non-linear manifolds in activity space while preserving their geometry. Comparisons with neural recordings in the dorsomedial frontal cortex of macaque monkeys performing flexible timing tasks confirmed the geometric and dynamical signatures of this mechanism. Altogether, our results tie together a number of previous experimental findings and suggest that the low-dimensional organization of neural dynamics plays a central role in generalizable behaviors.

Neurons in the brain communicate through action potentials (spikes) that are transmitted through chemical synapses. Throughout the last decades, the question how networks of spiking neurons represent and process information has remained an important challenge. Some progress has resulted from a recent family of supervised learning rules (tempotrons) for models of spiking neurons. However, these studies have viewed synaptic transmission as static and characterized synaptic efficacies as scalar quantities that change only on slow time scales of learning across trials but remain fixed on the fast time scales of information processing within a trial. By contrast, signal transduction at chemical synapses in the brain results from complex molecular interactions between multiple biochemical processes whose dynamics result in substantial short-term plasticity of most connections. Here we study the computational capabilities of spiking neurons whose synapses are dynamic and plastic, such that each individual synapse can learn its own dynamics. We derive tempotron learning rules for current-based leaky-integrate-and-fire neurons with different types of dynamic synapses. Introducing ordinal synapses whose efficacies depend only on the order of input spikes, we establish an upper capacity bound for spiking neurons with dynamic synapses. We compare this bound to independent synapses, static synapses and to the well established phenomenological Tsodyks-Markram model. We show that synaptic dynamics in principle allow the storage capacity of spiking neurons to scale with the number of input spikes and that this increase in capacity can be traded for greater robustness to input noise, such as spike time jitter. Our work highlights the feasibility of a novel computational paradigm for spiking neural circuits with plastic synaptic dynamics: Rather than being determined by the fixed number of afferents, the dimensionality of a neuron's decision space can be scaled flexibly through the number of input spikes emitted by its input layer.

Self-organization is a universal concept spanning numerous disciplines including mathematics, physics and biology. Chromosomes are self-organizing polymers that fold into orderly, hierarchical and yet dynamic structures. In the past decade, advances in experimental biology have provided a means to reveal information about chromosome connectivity, allowing us to directly use this information from experiments to generate 3D models of individual genes, chromosomes and even genomes. In this talk I will present a novel data-driven modeling approach and discuss a number of possibilities that this method holds. I will discuss a detailed study of the time-evolution of X chromosome inactivation, highlighting both global and local properties of chromosomes that result in topology-driven dynamical arrest and present and characterize a novel type of motion we discovered in knots that may have applications to nanoscale materials and machines.

Active matter describes a class of systems that are maintained far from equilibrium by driving forces acting on the constituent particles. Here I will focus on confined active nematics, which exhibit especially rich flow behavior, ranging from structured patterns in space and time to disordered turbulent flows. To understand this behavior, I will take a deterministic dynamical systems approach, beginning with the hydrodynamic equations for the active nematic. This approach reveals that the infinite-dimensional phase space of all possible flow configurations is populated by Exact Coherent Structures (ECS), which are exact solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure; examples include unstable equilibria, periodic orbits, and traveling waves. The ECS are connected by dynamical pathways called invariant manifolds. The main hypothesis in this approach is that turbulence corresponds to a trajectory meandering in the phase space, transitioning between ECS by traveling on the invariant manifolds. Similar approaches have been successful in characterizing high Reynolds number turbulence of passive fluids. Here, I will present the first systematic study of active nematic ECS and their invariant manifolds and discuss their role in characterizing the phenomenon of active turbulence.

Conventionally, neurons are thought to be cellular units that process synaptic inputs into synaptic spikes. However, it is well known that neurons can also spike spontaneously and display a rich repertoire of firing properties with no apparent functional relevance e.g. in in vitro cortical slice preparations. In this talk, I will propose a hypothesis according to which intrinsic excitability in neurons may be a survival mechanism to minimize toxic byproducts of the cell’s energy metabolism. In neurons, this toxicity can arise when mitochondrial ATP production stalls due to limited ADP. Under these conditions, electrons deviate from the electron transport chain to produce reactive oxygen species, disrupting many cellular processes and challenging cell survival. To mitigate this, neurons may engage in ADP-producing metabolic spikes. I will explore the validity of this hypothesis using computational models that illustrate the implications of synaptic and metabolic spiking, especially in the context of substantia nigra pars compacta dopaminergic neurons and their degeneration in Parkinson's disease.

Neural circuits exhibit complex activity patterns, both spontaneously and in response to external stimuli. Information encoding and learning in neural circuits depend on the ability of time-varying stimuli to control spontaneous network activity. In particular, variability arising from the sensitivity to initial conditions of recurrent cortical circuits can limit the information conveyed about the sensory input. Spiking and firing rate network models can exhibit such sensitivity to initial conditions that are reflected in their dynamic entropy rate and attractor dimensionality computed from their full Lyapunov spectrum. I will show how chaos in both spiking and rate networks depends on biophysical properties of neurons and the statistics of time-varying stimuli. In spiking networks, increasing the input rate or coupling strength aids in controlling the driven target circuit, which is reflected in both a reduced trial-to-trial variability and a decreased dynamic entropy rate. With sufficiently strong input, a transition towards complete network state control occurs. Surprisingly, this transition does not coincide with the transition from chaos to stability but occurs at even larger values of external input strength. Controllability of spiking activity is facilitated when neurons in the target circuit have a sharp spike onset, thus a high speed by which neurons launch into the action potential. I will also discuss chaos and controllability in firing-rate networks in the balanced state. For these, external control of recurrent dynamics strongly depends on correlations in the input. This phenomenon was studied with a non-stationary dynamic mean-field theory that determines how the activity statistics and the largest Lyapunov exponent depend on frequency and amplitude of the input, recurrent coupling strength, and network size. This shows that uncorrelated inputs facilitate learning in balanced networks. The results highlight the potential of Lyapunov spectrum analysis as a diagnostic for machine learning applications of recurrent networks. They are also relevant in light of recent advances in optogenetics that allow for time-dependent stimulation of a select population of neurons.

Sensory systems reliably process incoming stimuli in spite of changes in context. Most recent models accredit this context invariance to an extraction of increasingly complex sensory features in hierarchical feedforward networks. Here, we study how context-invariant representations can be established by feedback rather than feedforward processing. We show that feedforward neural networks modulated by feedback can dynamically generate invariant sensory representations. The required feedback can be implemented as a slow and spatially diffuse gain modulation. The invariance is not present on the level of individual neurons, but emerges only on the population level. Mechanistically, the feedback modulation dynamically reorients the manifold of neural activity and thereby maintains an invariant neural subspace in spite of contextual variations. Our results highlight the importance of population-level analyses for understanding the role of feedback in flexible sensory processing.

Neural systems are remarkably robust against various perturbations, a phenomenon that still requires a clear explanation. Here, we graphically illustrate how neural networks can become robust. We study spiking networks that generate low-dimensional representations, and we show that the neurons’ subthreshold voltages are confined to a convex region in a lower-dimensional voltage subspace, which we call a ‘bounding box.’ Any changes in network parameters (such as number of neurons, dimensionality of inputs, firing thresholds, synaptic weights, or transmission delays) can all be understood as deformations of this bounding box. Using these insights, we show that functionality is preserved as long as perturbations do not destroy the integrity of the bounding box. We suggest that the principles underlying robustness in these networks—low-dimensional representations, heterogeneity of tuning, and precise negative feedback—may be key to understanding the robustness of neural systems at the circuit level.

To efficiently process sensory information, the brain relies on statistical regularities in the input. While generally improving the reliability of sensory estimates, this strategy also induces perceptual illusions that help reveal the underlying computational principles. Focusing on auditory and visual perception, in my talk I will describe how the brain exploits statistical regularities within and across the senses for the perception space, time and multisensory integration. In particular, I will show how results from a series of psychophysical experiments can be interpreted in the light of Bayesian Decision Theory, and I will demonstrate how such canonical computations can be implemented into simple and biologically plausible neural circuits. Finally, I will show how such principles of sensory information processing can be leveraged in virtual and augmented reality to overcome display limitations and expand human perception.

In behaving rodents, CA1 pyramidal neurons receive spatially-tuned depolarizing synaptic input while traversing a specific location within an environment called its place. Midbrain dopamine neurons participate in reinforcement learning, and bursts of action potentials riding a depolarizing wave of synaptic input signal rewards and reward expectation. Interestingly, slice electrophysiology in vitro shows that both types of cells exhibit a pronounced reduction in firing rate (adaptation) and even cessation of firing during sustained depolarization. We included a five state Markov model of NaV1.6 (for CA1) and NaV1.2 (for dopamine neurons) respectively, in computational models of these two types of neurons. Our simulations suggest that long-term inactivation of this channel is responsible for the adaptation in CA1 pyramidal neurons, in response to triangular depolarizing current ramps. We also show that the differential contribution of slow inactivation in two subpopulations of midbrain dopamine neurons can account for their different dynamic ranges, as assessed by their responses to similar depolarizing ramps. These results suggest long-term inactivation of the sodium channel is a general mechanism for adaptation.

Recent chronic imaging experiments in mice have revealed that the hippocampal code exhibits non-trivial turnover dynamics over long time scales. Specifically, the subset of cells which are active on any given session in a familiar environment changes over the course of days and weeks. While some cells transition into or out of the code after a few sessions, others are stable over the entire experiment. The mechanisms underlying this turnover are unknown. Here we show that the statistics of turnover are consistent with a model in which non-spatial inputs to CA1 pyramidal cells readily undergo plasticity, while spatially tuned inputs are largely stable over time. The heterogeneity in stability across the cell assembly, as well as the decrease in correlation of the population vector of activity over time, are both quantitatively fit by a simple model with Gaussian input statistics. In fact, such input statistics emerge naturally in a network of spiking neurons operating in the fluctuation-driven regime. This correspondence allows one to map the parameters of a large-scale spiking network model of CA1 onto the simple statistical model, and thereby fit the experimental data quantitatively. Importantly, we show that the observed drift is entirely consistent with random, ongoing synaptic turnover. This synaptic turnover is, in turn, consistent with Hebbian plasticity related to continuous learning in a fast memory system.

The richness of active matter's spatiotemporal patterns continues to capture our imagination. Shaping these emergent dynamics into pre-determined forms of our choosing is a grand challenge in the field. To complicate matters, multiple dynamical attractors can coexist in such systems, leading to initial condition-dependent dynamics. Consequently, non-trivial spatiotemporal inputs are generally needed to access these states. Optimal control theory provides a general framework for identifying such inputs and represents a promising computational tool for guiding experiments and interacting with various systems in soft active matter and biology. As an exemplar, I first consider an extensile active nematic fluid confined to a disk. In the absence of control, the system produces two topological defects that perpetually circulate. Optimal control identifies a time-varying active stress field that restructures the director field, flipping the system to its other attractor that rotates in the opposite direction. As a second, analogous case, I examine a small network of coupled Belousov-Zhabotinsky chemical oscillators that possesses two dominant attractors, two wave states of opposing chirality. Optimal control similarly achieves the task of attractor switching. I conclude with a few forward-looking remarks on how the same model-based control approach might come to bear on problems in biology.

A central goal in neuroscience is to understand how orchestrated computations in the brain arise from the properties of single neurons and networks of such neurons. Answering this question requires theoretical advances that shine light into the ‘black box’ of neuronal networks. In this talk, I will demonstrate theoretical approaches that help describe how cognitive and behavioral task implementations emerge from structure in neural populations and from biologically plausible learning rules. First, I will introduce an analytic theory that connects geometric structures that arise from neural responses (i.e., neural manifolds) to the neural population’s efficiency in implementing a task. In particular, this theory describes how easy or hard it is to discriminate between object categories based on the underlying neural manifolds’ structural properties. Next, I will describe how such methods can, in fact, open the ‘black box’ of neuronal networks, by showing how we can understand a) the role of network motifs in task implementation in neural networks and b) the role of neural noise in adversarial robustness in vision and audition. Finally, I will discuss my recent efforts to develop biologically plausible learning rules for neuronal networks, inspired by recent experimental findings in synaptic plasticity. By extending our mathematical toolkit for analyzing representations and learning rules underlying complex neuronal networks, I hope to contribute toward the long-term challenge of understanding the neuronal basis of behaviors.

Topographic maps are a prominent feature of brain organization, reflecting local and large-scale representation of the sensory surface. ​​Traditionally, such representations in early visual areas are conceived as retinotopic maps preserving ego-centric retinal spatial location while ensuring that other features of visual input are uniformly represented for every location in space. I will discuss our recent findings of a striking departure from this simple mapping in the secondary visual area (V2) of the tree shrew that is best described as a sinusoidal transformation of the visual field. This sinusoidal topography is ideal for achieving uniform coverage in an elongated area like V2 as predicted by mathematical models designed for wiring minimization, and provides a novel explanation for stripe-like patterns of intra-cortical connections and functional response properties in V2. Our findings suggest that cortical circuits flexibly implement solutions to sensory surface representation, with dramatic consequences for large-scale cortical organization. Furthermore our work challenges the framework of relatively independent encoding of location and features in the visual system, showing instead location-dependent feature sensitivity produced by specialized processing of different features in different spatial locations. In the second part of the talk, I will propose that location-dependent feature sensitivity is a fundamental organizing principle of the visual system that achieves efficient representation of positional regularities in visual input, and reflects the evolutionary selection of sensory and motor circuits to optimally represent behaviorally relevant information. The relevant papers can be found here: V2 retinotopy (Sedigh-Sarvestani et al. Neuron, 2021) Location-dependent feature sensitivity (Sedigh-Sarvestani et al. Under Review, 2022)

Optogenetics is a powerful tool that allows experimentalists to perturb neural circuits. What can we learn about a network from observing its response to perturbations? I will first describe the results of optogenetic activation of inhibitory neurons in mice cortex, and show that the results are consistent with inhibition stabilization. I will then move to experiments in which excitatory neurons are activated optogenetically, with or without visual inputs, in mice and monkeys. In some conditions, these experiments show a surprising result that the distribution of firing rates is not significantly changed by stimulation, even though firing rates of individual neurons are strongly modified. I will show in which conditions a network model of excitatory and inhibitory neurons can reproduce this feature.

Evidence-based education programmes, like many clinical interventions, are multi-faceted and can be expensive to implement. In this talk I will describe an alternative: distilling the common elements across many evidence-based programmes. Published programme manuals are selected through systematic review, then extensively coded and cross-referenced. Finally, the common elements that emerge are shared with practitioners as part of a ‘library’ of practices (rather than a holistic programme manual). Although the common elements methodology has been used in the prevention and intervention sciences, this project reflects the first attempt at applying this approach to early childhood education. I will describe the common elements methods and preliminary findings from our Nuffield-funded project, in collaboration with the Early Intervention Foundation. I will discuss the challenges and opportunities we have encountered, alongside our strategies for sharing evidence with practitioners in a digestible way.