Mathematical Modeling seminars

Homeostatic plasticity plays a key role in stabilizing neural network activity. But what is its role in neural information processing? We showed analytically how homeostasis changes collective dynamics and consequently information flow - depending on the input to the network. We then studied how input and homeostasis on a recurrent network of LIF neurons impacts information flow and task performance. We showed how we can tune the working point of the network, and found that, contrary to previous assumptions, there is not one optimal working point for a family of tasks, but each task may require its own working point.

Choosing among spatially distributed options is a central challenge for animals, from deciding among alternative potential food sources or refuges, to choosing with whom to associate. Here, using an integrated theoretical and experimental approach (employing immersive Virtual Reality), with both invertebrate and vertebrate models—the fruit fly, desert locust and zebrafish—we consider the recursive interplay between movement and collective vectorial integration in the brain during decision-making regarding options (potential ‘targets’) in space. We reveal that the brain repeatedly breaks multi-choice decisions into a series of abrupt (critical) binary decisions in space-time where organisms switch, spontaneously, from averaging vectorial information among, to suddenly excluding one of, the remaining options. This bifurcation process repeats until only one option—the one ultimately selected—remains. Close to each bifurcation the ‘susceptibility’ of the system exhibits a sharp increase, inevitably causing small differences among the remaining options to become amplified; a property that both comes ‘for free’ and is highly desirable for decision-making. This mechanism facilitates highly effective decision-making, and is shown to be robust both to the number of options available, and to context, such as whether options are static (e.g. refuges) or mobile (e.g. other animals). In addition, we find evidence that the same geometric principles of decision-making occur across scales of biological organisation, from neural dynamics to animal collectives, suggesting they are fundamental features of spatiotemporal computation.

Spike-driven adaptation involves intracellular mechanisms that are initiated by spiking and lead to the subsequent reduction of spiking rate. One of its consequences is the temporal patterning of spike trains, as it imparts serial correlations between interspike intervals in baseline activity. Surprisingly the hidden adaptation states that lead to these correlations themselves exhibit quasi-independence. This talk will first discuss recent findings about the role of such adaptation in suppressing noise and extending sensory detection to weak stimuli that leave the firing rate unchanged. Further, a matching of the post-synaptic responses to the pre-synaptic adaptation time scale enables a recovery of the quasi-independence property, and can explain observations of correlations between post-synaptic EPSPs and behavioural detection thresholds. We then consider the involvement of spike-driven adaptation in the representation of intervals between sensory events. We discuss the possible link of this time-stamping mechanism to the conversion of egocentric to allocentric coordinates. The heterogeneity of the population parameters enables the representation and Bayesian decoding of time sequences of events which may be put to good use in path integration and hilus neuron function in hippocampus.

The success of deep learning sparked interest in whether the brain learns by using similar techniques for assigning credit to each synaptic weight for its contribution to the network output. However, the majority of current attempts at biologically-plausible learning methods are either non-local in time, require highly specific connectivity motives, or have no clear link to any known mathematical optimization method. Here, we introduce Deep Feedback Control (DFC), a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of feedback connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing. By combining dynamical system theory with mathematical optimization theory, we provide a strong theoretical foundation for DFC that we corroborate with detailed results on toy experiments and standard computer-vision benchmarks.

Biological systems can self-organize in complex structures, able to evolve and adapt to widely varying environmental conditions. Despite the importance of fluid flow for transporting and organizing populations, few laboratory systems exist to systematically investigate the impact of advection on their spatial evolutionary dynamics. In this talk, I will discuss how we can address this problem by studying the morphology and genetic spatial structure of microbial colonies growing on the surface of a viscous substrate. When grown on a liquid, I will show that S. cerevisiae (baker’s yeast) can behave like “active matter” and collectively generate a fluid flow many times larger than the unperturbed colony expansion speed, which in turn produces mechanical stresses and fragmentation of the initial colony. Combining laboratory experiments with numerical modeling, I will demonstrate that the coupling between metabolic activity and hydrodynamic flows can produce positive feedbacks and drive preferential growth phenomena leading to the formation of microbial jets. Our work provides rich opportunities to explore the interplay between hydrodynamics, growth and competition within a versatile system.

Networks are widely used as mathematical models of complex systems across many scientific disciplines, and in particular within neuroscience. In this talk, we introduce two aspects of our collaborative research: (1) machine learning and networks, and (2) graph dimensionality. Machine learning and networks. Decades of work have produced a vast corpus of research characterising the topological, combinatorial, statistical and spectral properties of graphs. Each graph property can be thought of as a feature that captures important (and sometimes overlapping) characteristics of a network. We have developed hcga, a framework for highly comparative analysis of graph data sets that computes several thousands of graph features from any given network. Taking inspiration from hctsa, hcga offers a suite of statistical learning and data analysis tools for automated identification and selection of important and interpretable features underpinning the characterisation of graph data sets. We show that hcga outperforms other methodologies (including deep learning) on supervised classification tasks on benchmark data sets whilst retaining the interpretability of network features, which we exemplify on a dataset of neuronal morphologies images. Graph dimensionality. Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can be constrained by boundaries and distorted by inhomogeneities, or to intrinsically discrete systems such as networks. Deviating from approaches based on fractals, here, we present a new framework to define intrinsic notions of dimension on networks, the relative, local and global dimension. We showcase our method on various physical systems.

Chromosomes are extremely long, active polymers that are spatially organized across multiple scales to promote cellular functions, such as gene transcription and genetic inheritance. During each cell cycle, chromosomes are dramatically compacted as cells divide and dynamically reorganized into less compact, spatiotemporally patterned structures after cell division. These activities are facilitated by DNA/chromatin-binding protein motors called SMC complexes. Each of these motors can perform a unique activity known as “loop extrusion,” in which the motor binds the DNA/chromatin polymer, reels in the polymer fiber, and extrudes it as a loop. Using simulations and theory, I show how loop-extruding motors can collectively compact and spatially organize chromosomes in different scenarios. First, I show that loop-extruding complexes can generate sufficient compaction for cell division, provided that loop-extrusion satisfies stringent physical requirements. Second, while loop-extrusion alone does not uniquely spatially pattern the genome, interactions between SMC complexes and protein “boundary elements” can generate patterns that emerge in the genome after cell division. Intriguingly, these “boundary elements” are not necessarily stationary, which can generate a variety of patterns in the neighborhood of transcriptionally active genes. These predictions, along with supporting experiments, show how SMC complexes and other molecular machinery, such as RNA polymerase, can spatially organize the genome. More generally, this work demonstrates both the versatility of the loop extrusion mechanism for chromosome functional organization and how seemingly subtle microscopic effects can emerge in the spatiotemporal structure of nonequilibrium polymers.

Electro- and magneto-encephalography (EEG/MEG) are the leading methods to non-invasively record human neural dynamics with millisecond temporal resolution. However, it can be extremely difficult to infer the underlying cellular and circuit level origins of these macro-scale signals without simultaneous invasive recordings. This limits the translation of E/MEG into novel principles of information processing, or into new treatment modalities for neural pathologies. To address this need, we developed the Human Neocortical Neurosolver (HNN: https://hnn.brown/edu ), a new user-friendly neural modeling tool designed to help researchers and clinicians interpret human imaging data. A unique feature of HNN’s model is that it accounts for the biophysics generating the primary electric currents underlying such data, so simulation results are directly comparable to source localized data. HNN is being constructed with workflows of use to study some of the most commonly measured E/MEG signals including event related potentials, and low frequency brain rhythms. In this talk, I will give an overview of this new tool and describe an application to study the origin and meaning of 15-29Hz beta frequency oscillations, known to be important for sensory and motor function. Our data showed that in primary somatosensory cortex these oscillations emerge as transient high power ‘events’. Functionally relevant differences in averaged power reflected a difference in the number of high-power beta events per trial (“rate”), as opposed to changes in event amplitude or duration. These findings were consistent across detection and attention tasks in human MEG, and in local field potentials from mice performing a detection task. HNN modeling led to a new theory on the circuit origin of such beta events and suggested beta causally impacts perception through layer specific recruitment of cortical inhibition, with support from invasive recordings in animal models and high-resolution MEG in humans. In total, HNN provides an unpresented biophysically principled tool to link mechanism to meaning of human E/MEG signals.

Interfaces and membranes are ubiquitous in cellular systems across various scales. From lipid membranes to the interfaces of biomolecular condensates inside the cell, these borders not only protect and segregate the inner components from the outside world, but also are actively participating in mechanical regulation and biochemical reaction of the cell. Being part of a living system, these interfaces (membranes) are usually active and away from equilibrium. Yet, it's still not clear how activity can tweak their equilibrium dynamics. Here, I will introduce a model system to tackle this problem. We put together a passive fluid and an active nematics, and study the behavior of this liquid-liquid interface. Whereas thermal fluctuation of such an interface is too weak to be observed, active stress can easily force the interface to fluctuate, overhang, and even break up. In the presence of a wall, the active phase exhibits superfluid-like behavior: it can climb up walls -- a phenomenon we call activity-induced wetting. I will show how to formulate theories to capture these phenomena, highlighting the nontrivial effects of active stress. Our work not only demonstrates that activity can introduce interesting features to an interface, but also sheds light on controlling interfacial properties using activity.

Dictyostelium Discoideum (DD) are a fascinating single-cellular organism. When nutrients are plentiful, the DD cells act as autonomous individuals foraging their local vicinity. At the onset of starvation, a few (<0.1%) cells begin communicating with others by emitting a spike in the chemoattractant protein cyclic-AMP. Nearby cells sense the chemical gradient and respond by moving toward it and emitting a cyclic-AMP spike of their own. Cyclic-AMP activity increases over time, and eventually a spiral wave emerges, attracting hundreds of thousands of cells to an aggregation center. How DD cells go from autonomous individuals to a collective entity remains an open question for more than 60 years--a question whose answer would shed light on the emergence of multi-cellular life. Recently, trans-scale imaging has allowed the ability to sense the cyclic-AMP activity at both cell and colony levels. Using both the images as well as toy simulation models, this research aims to clarify whether the activity at the colony level is in fact initiated by a few cells, which may be deemed "leader" or "pacemaker" cells. In this talk, I will demonstrate the use of information-theoretic techniques to classify leaders and followers based on trajectory data, as well as to infer the domain of interaction of leader cells. We validate the techniques on toy models where leaders and followers are known, and then try to answer the question in real data--do leader cells drive collective behavior in DD colonies?

The CA1 pyramidal neurons are embedded in an intricate local circuitry that contains a variety of interneurons. The roles these interneurons play in the regulation of the excitatory synaptic plasticity remains largely understudied. Recent experiments showed that repeated cholinergic activation of 𝛼7 nACh receptors expressed in oriens-lacunosum-moleculare (OLM𝛼2) interneurons could induce LTP in SC-CA1 synapses. We used a biophysically realistic computational model to examine mechanistically how cholinergic activation of OLMa2 interneurons increases SC to CA1 transmission. Our results suggest that, when properly timed, activation of OLMa2 interneurons cancels the feedforward inhibition onto CA1 pyramidal cells by inhibiting fast-spiking interneurons that synapse on the same dendritic compartment as the SC, i.e., by disinhibiting the pyramidal cell dendritic compartment. Our work further describes the pairing of disinhibition with SC stimulation as a general mechanism for the induction of synaptic plasticity. We found that locally-reduced GABA release (disinhibition) paired with SC stimulation could lead to increased NMDAR activation and intracellular calcium concentration sufficient to upregulate AMPAR permeability and potentiate the excitatory synapse. Our work suggests that inhibitory synapses critically modulate excitatory neurotransmission and induction of plasticity at excitatory synapses. Our work also shows how cholinergic action on OLM interneurons, a mechanism whose disruption is associated with memory impairment, can down-regulate the GABAergic signaling into CA1 pyramidal cells and facilitate potentiation of the SC-CA1 synapse.

CA3-CA1 synapses in the hippocampus are the initial locus of episodic memory. The action of acetylcholine alters cellular excitability, modifies neuronal networks, and triggers secondary signaling that directly affects long-term plasticity (LTP) (the cellular underpinning of memory). It is therefore considered a critical regulator of learning and memory in the brain. Its action via M4 metabotropic receptors in the presynaptic terminal of the CA3 neurons and M1 metabotropic receptors in the postsynaptic spines of CA1 neurons produce rich dynamics across multiple timescales. We developed a model to describe the activation of postsynaptic M1 receptors that leads to IP3 production from membrane PIP2 molecules. The binding of IP3 to IP3 receptors in the endoplasmic reticulum (ER) ultimately causes calcium release. This calcium release from the ER activates potassium channels like the calcium-activated SK channels and alters different aspects of synaptic signaling. In an independent signaling cascade, M1 receptors also directly suppress SK channels and the voltage-activated KCNQ2/3 channels, enhancing post-synaptic excitability. In the CA3 presynaptic terminal, we model the reduction of the voltage sensitivity of voltage-gated calcium channels (VGCCs) and the resulting suppression of neurotransmitter release by the action of the M4 receptors. Our results show that the reduced initial release probability because of acetylcholine alters short-term plasticity (STP) dynamics. We characterize the dichotomy of suppressing neurotransmitter release from CA3 neurons and the enhanced excitability of the postsynaptic CA1 spine. Mechanisms underlying STP operate over a few seconds, while those responsible for LTP last for hours, and both forms of plasticity have been linked with very distinct functions in the brain. We show that the concurrent suppression of neurotransmitter release and increased sensitivity conserves neurotransmitter vesicles and enhances the reliability in plasticity. Our work establishes a relationship between STP and LTP coordinated by neuromodulation with acetylcholine.

To act effectively and flexibly in an imperfectly predictable environment with only incomplete and unreliable sensory information, animals must learn to form and compute with internal representations that reflect their necessarily uncertain beliefs about the state of the world. The optimal approach to handling uncertainty is rooted in Bayesian probability, and indeed humans and other animals often approach Bayes optimality with a degree of robustness and flexibility that continues to evade artificial systems. However, the question of how neural circuits organise to achieve this performance remains one of the fundamental mysteries of neuroscience. I will discuss a series of models built around the idea that distributional information is naturally encoded in a distributed fashion by neural population firing rates that converge on the mean values of non-linear functions of state. We will see that such representations emerge naturally in task-optimised systems, and also provide a simple and effective substrate for unsupervised learning. Finally, I will sketch ongoing work that links the emergence of such representations to the architecture of recurrent neural circuits. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2021-07-07. Recording duration: 00:52:28.

Neuromodulators control information processing in cortical microcircuits by regulating the cellular and synaptic physiology of neurons. Computational models and detailed simulations of neocortical microcircuitry offer a unifying framework to analyze the role of neuromodulators on network activity. In the present study, to get a deeper insight in the organization of the cortical neuropil for modeling purposes, we quantify the fiber length per cortical volume and the density of varicosities for catecholaminergic, serotonergic and cholinergic systems using immunocytochemical staining and stereological techniques. The data obtained are integrated into a biologically detailed digital reconstruction of the rodent neocortex (Markram et al, 2015) in order to model the influence of modulatory systems on the activity of the somatosensory cortex neocortical column. Simulations of ascending modulation of network activity in our model predict the effects of increasing levels of neuromodulators on diverse neuron types and synapses and reveal a spectrum of activity states. Low levels of neuromodulation drive microcircuit activity into slow oscillations and network synchrony, whereas high neuromodulator concentrations govern fast oscillations and network asynchrony. The models and simulations thus provide a unifying in silico framework to study the role of neuromodulators in reconfiguring network activity.

Waves and patterns in the brain are well known to subserve natural computation. Much attention in the theoretical neuroscience community has been devoted to analysing networks of relatively simple spiking neurons (IF type) or firing rate models (Wilson-Cowan type) and to great effect! Indeed, the understanding of how spatio-temporal patterns of neural activity may arise in the cortex has advanced significantly with the development and analysis of such models. To replicate this success for sub-cortical tissues requires an extension to include relevant ionic currents that can further shape firing response. Here I will advocate for two complementary approaches: i) that augments the approach for IF networks to include piecewise linear caricatures of gating dynamics for nonlinear ionic current models, ii) firing rate reductions for systems where the nonlinear ionic currents are slow. By way of illustration, I will show how to construct spatially periodic waves and patterns in i) a simple spiking tissue model of medial enthorinal cortex (with an I_h current), ii) a firing rate model of thalamus (with an I_T current). The biological commonality between these two models is that both express local 'rebound' currents that can usefully shape global tissue response. The mathematical commonality is the use of tools from non-smooth dynamical systems theory to make analytical progress in determining patterns and their stability. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2021-06-30. Recording duration: 00:40:23.

Individual bacteria transported in viscous flows, show complex interactions with flows and bounding surfaces resulting from their complex shape as well as their activity. Understanding these transport dynamics is crucial, as they impact soil contamination, transport in biological conducts or catheters, and constitute thus a serious health threat. Here we investigate the trajectories of individual E-coli bacteria in confined geometries under flow, using microfluidic model systems in bulk flows as well as close to surfaces using a novel Langrangian 3D tracking method. Combining experimental observations and modelling we elucidate the origin of upstream swimming, lateral drift or persistent transport along corners. [1] Junot et al, EPL, 126 (2019) 44003 [2] Mathijssen et al. 10:3 (2019) Nature Comm. [3] Figueroa-Morales et al., Soft Matter, 2015,11, 6284-6293 [4] Darnige et al. Review of Scientific Instruments 88, 055106 (2017) [5] Jing et al, Science Advances, 2020; 6 : eabb2012 [6] Figueroa-Morales et al, Sci. Adv. 2020; 6 : eaay0155, 2020, 10.1126/sciadv.aay0155

Threshold-linear networks (TLNs) are popular firing rate models of recurrent networks. They have been used to model associative memory, decision-making, and position coding in cortical and hippocampal networks. Unlike rate models with other choices of nonlinearity, TLNs are piecewise linear, making them more amenable to mathematical analysis. In this talk I will present ten theorems about TLNs from the past five years. Many of these theorems connect the fixed points of a network to the structure of an underlying connectivity graph. These results have enabled us to develop graph rules to predict both static and dynamic attractors from network motifs. The theorems will be complemented with examples that illustrate how the mathematical results can be used to analyze and design recurrent networks that support a rich variety of computations and dynamics. Examples include internally-generated sequences, neural integrators, and central pattern generator circuits. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2021-06-23. Recording duration: 00:53:51.

The typical goal of homeostatic mechanisms is to ensure a system operates at or in the vicinity of a stable set point, where a particular measure is relatively constant and stable. Neural firing rate homeostasis is unusual in that a set point of fixed firing rate is at odds with the goal of a neuron to convey information, or produce timed motor responses, which require temporal variations in firing rate. Therefore, for a neuron, a range of firing rates is required for optimal function, which could, for example, be set by a dual system that controls both mean and variance of firing rate. We explore, both via simulations and analysis, how two experimentally measured mechanisms for firing rate homeostasis can cooperate to improve information processing and avoid the pitfall of pulling in different directions when their set points do not appear to match.

There is little consensus on the level of spatial complexity at which dendrites operate. On the one hand, emergent evidence indicates that synapses cluster at micrometer spatial scales. On the other hand, most modelling and network studies ignore dendrites altogether. This dichotomy raises an urgent question: what is the smallest relevant spatial scale for understanding dendritic computation? We have developed a method to construct compartmental models at any level of spatial complexity. Through carefully chosen parameter fits, solvable in the least-squares sense, we obtain accurate reduced compartmental models. Thus, we are able to systematically construct passive as well as active dendrite models at varying degrees of spatial complexity. We evaluate which elements of the dendritic computational repertoire are captured by these models. We show that many canonical elements of the dendritic computational repertoire can be reproduced with few compartments. For instance, for a model to behave as a two-layer network, it is sufficient to fit a reduced model at the soma and at locations at the dendritic tips. In the basal dendrites of an L2/3 pyramidal model, we reproduce the backpropagation of somatic action potentials (APs) with a single dendritic compartment at the tip. Further, we obtain the well-known Ca-spike coincidence detection mechanism in L5 Pyramidal cells with as few as eleven compartments, the requirement being that their spacing along the apical trunk supports AP backpropagation. We also investigate whether afferent spatial connectivity motifs admit simplification by ablating targeted branches and grouping affected synapses onto the next proximal dendrite. We find that voltage in the remaining branches is reproduced if temporal conductance fluctuations stay below a limit that depends on the average difference in input resistance between the ablated branches and the next proximal dendrite. Consequently, when the average conductance load on distal synapses is constant, the dendritic tree can be simplified while appropriately decreasing synaptic weights. When the conductance level fluctuates strongly, for instance through a-priori unpredictable fluctuations in NMDA activation, a constant weight rescale factor cannot be found, and the dendrite cannot be simplified. We have created an open source Python toolbox (NEAT - https://neatdend.readthedocs.io/en/latest/) that automatises the simplification process. A NEST implementation of the reduced models, currently under construction, will enable the simulation of few-compartment models in large-scale networks, thus bridging the gap between cellular and network level neuroscience.