Mathematical Modeling seminars

A myriad of pathological changes associated with epilepsy, including the loss of specific cell types, improper expression of individual ion channels, and synaptic sprouting, can be recast as decreases in cell and circuit heterogeneity. In recent experimental work, we demonstrated that biophysical diversity is a key characteristic of human cortical pyramidal cells, and past theoretical work has shown that neuronal heterogeneity improves a neural circuit’s ability to encode information. Viewed alongside the fact that seizure is an information-poor brain state, these findings motivate the hypothesis that epileptogenesis can be recontextualized as a process where reduction in cellular heterogeneity renders neural circuits less resilient to seizure onset. By comparing whole-cell patch clamp recordings from layer 5 (L5) human cortical pyramidal neurons from epileptogenic and non-epileptogenic tissue, we present the first direct experimental evidence that a significant reduction in neural heterogeneity accompanies epilepsy. We directly implement experimentally-obtained heterogeneity levels in cortical excitatory-inhibitory (E-I) stochastic spiking network models. Low heterogeneity networks display unique dynamics typified by a sudden transition into a hyper-active and synchronous state paralleling ictogenesis. Mean-field analysis reveals a distinct mathematical structure in these networks distinguished by multi-stability. Furthermore, the mathematically characterized linearizing effect of heterogeneity on input-output response functions explains the counter-intuitive experimentally observed reduction in single-cell excitability in epileptogenic neurons. This joint experimental, computational, and mathematical study showcases that decreased neuronal heterogeneity exists in epileptogenic human cortical tissue, that this difference yields dynamical changes in neural networks paralleling ictogenesis, and that there is a fundamental explanation for these dynamics based in mathematically characterized effects of heterogeneity. These interdisciplinary results provide convincing evidence that biophysical diversity imbues neural circuits with resilience to seizure and a new lens through which to view epilepsy, the most common serious neurological disorder in the world, that could reveal new targets for clinical treatment.

Large-scale recordings of neural activity are providing new opportunities to study network-level dynamics with unprecedented detail. However, the sheer volume of data and its dynamical complexity are major barriers to uncovering and interpreting these dynamics. I will present machine learning frameworks that enable inference of dynamics from neuronal population spiking activity on single trials and millisecond timescales, from diverse brain areas, and without regard to behavior. I will then demonstrate extensions that allow recovery of dynamics from two-photon calcium imaging data with surprising precision. Finally, I will discuss our efforts to facilitate comparisons within our field by curating datasets and standardizing model evaluation, including a currently active modeling challenge, the 2021 Neural Latents Benchmark [neurallatents.github.io].

This talk will focus on the generative modelling of resting state time series or endogenous neuronal activity. I will survey developments in modelling distributed neuronal fluctuations – spectral dynamic causal modelling (DCM) for functional MRI – and how this modelling rests upon functional connectivity. The dynamics of brain connectivity has recently attracted a lot of attention among brain mappers. I will also show a novel method to identify dynamic effective connectivity using spectral DCM. Further, I will summarise the development of the next generation of DCMs towards large-scale, whole-brain schemes which are computationally inexpensive, to the other extreme of the development using more sophisticated and biophysically detailed generative models based on the canonical microcircuits.

Ever since Bliss and Lømo discovered the phenomenon of long-term potentiation (LTP) in rabbit dentate gyrus in the 1960s, Hebb’s rule—neurons that fire together wire together—gained popularity to explain learning and memory. Accumulating evidence, however, suggests that neural activity is homeostatically regulated. Homeostatic mechanisms are mostly interpreted to stabilize network dynamics. However, recent theoretical work has shown that linking the activity of a neuron to its connectivity within the network provides a robust alternative implementation of Hebb’s rule, although entirely based on negative feedback. In this setting, both natural and artificial stimulation of neurons can robustly trigger network rewiring. We used computational models of plastic networks to simulate the complex temporal dynamics of network rewiring in response to external stimuli. In parallel, we performed optogenetic stimulation experiments in the mouse anterior cingulate cortex (ACC) and subsequently analyzed the temporal profile of morphological changes in the stimulated tissue. Our results suggest that the new theoretical framework combining neural activity homeostasis and structural plasticity provides a consistent explanation of our experimental observations.

Selection is the hallmark of attention: processing improves for attended items but is relatively impaired for unattended items. It is well known that visual spatial attention changes sensory signals and perception in this selective fashion. In the work I will present, we asked whether and how attentional selection happens across time. First, our experiments revealed that voluntary temporal attention (attention to specific points in time) is selective, resulting in perceptual tradeoffs across time. Second, we measured small eye movements called microsaccades and found that directing voluntary temporal attention increases the stability of the eyes in anticipation of an attended stimulus. Third, we developed a computational model of dynamic attention, which proposes specific mechanisms underlying temporal attention and its selectivity. Lastly, I will mention how we are testing predictions of the model with MEG. Altogether, this research shows how precisely timed voluntary attention helps manage inherent limits in visual processing across short time intervals, advancing our understanding of attention as a dynamic process.

Brain microstructure plays a key role in driving the transport of drug molecules directly administered to the brain tissue as in Convection-Enhanced Delivery procedures. This study reports the first systematic attempt to characterize the cytoarchitecture of commissural, long association and projection fiber, namely: the corpus callosum, the fornix and the corona radiata. Ovine samples from three different subjects have been imaged using scanning electron microscope combined with focused ion beam milling. Particular focus has been given to the axons. For each tract, a 3D reconstruction of relatively large volumes (including a significant number of axons) has been performed. Namely, outer axonal ellipticity, outer axonal cross-sectional area and its relative perimeter have been measured. This study [1] provides useful insight into the fibrous organization of the tissue that can be described as composite material presenting elliptical tortuous tubular fibers, leading to a workflow to enable accurate simulations of drug delivery which include well-resolved microstructural features. As a demonstration of the use of these imaging and reconstruction techniques, our research analyses the hydraulic permeability of two white matter (WM) areas (corpus callosum and fornix) whose three-dimensional microstructure was reconstructed starting from the acquisition of the electron microscopy images. Considering that the white matter structure is mainly composed of elongated and parallel axons we computed the permeability along the parallel and perpendicular directions using computational fluid dynamics [2]. The results show a statistically significant difference between parallel and perpendicular permeability, with a ratio about 2 in both the white matter structures analysed, thus demonstrating their anisotropic behaviour. This is in line with the experimental results obtained using perfusion of brain matter [3]. Moreover, we find a significant difference between permeability in corpus callosum and fornix, which suggests that also the white matter heterogeneity should be considered when modelling drug transport in the brain. Our findings, that demonstrate and quantify the anisotropic and heterogeneous character of the white matter, represent a fundamental contribution not only for drug delivery modelling but also for shedding light on the interstitial transport mechanisms in the extracellular space. These and many other discoveries will be discussed during the talk." "1. https://www.researchsquare.com/article/rs-686577/v1, 2. https://www.pnas.org/content/118/36/e2105328118, 3. https://ieeexplore.ieee.org/abstract/document/9198110

Every thought is associated with a unique pattern of brain activity. Thus, in principle, it should be possible to use these activity patterns as "brain fingerprints" for different thoughts and to read out what a person is thinking based on their brain activity alone. Indeed, using machine learning considerable progress has been made in such "brainreading" in recent years. It is now possible to decode which image a person is viewing, which film sequence they are watching, which emotional state they are in or which intentions they hold in mind. This talk will provide an overview of the current state of the art in brain reading. It will also highlight the main challenges and limitations of this research field. For example, mathematical models are needed to cope with the high dimensionality of potential mental states. Furthermore, the ethical concerns raised by (often premature) commercial applications of brain reading will also be discussed.

When we remember a stimulus we rarely maintain a full fidelity representation of the observed item. Our working memory instead maintains a mixture of the observed feature values and categorical/gist information. I will discuss evidence from computational models supporting a mix of categorical and fine-detail information in working memory. Having established the need for two memory formats in working memory, I will discuss whether categorical and fine-detailed information for a stimulus are represented separately or as a single unified representation. Computational models of these two potential cognitive structures make differing predictions about the pattern of responses in visual working memory recall tests. The present study required participants to remember the orientation of stimuli for later reproduction. The pattern of responses are used to test the competing representational structures and to quantify the relative amount of fine-detailed and categorical information maintained. The effects of set size, encoding time, serial order, and response order on memory precision, categorical information, and guessing rates are also explored. (This is a 60 min talk).

Macromolecular protein complexes perform essential biological functions across life forms. A fundamental, though yet unsolved question in biology is how the function of such complexes is regulated by intracellular or extracellular signals. For instance, we have little understanding of how forces affect multi-protein machines whose function is often mechanical in nature. We address this question by studying the bacterial flagellar motor, a large complex that powers swimming motility in many bacteria. This rotary motor autonomously adapts to changes in mechanical load by adding or removing force-generating ‘stator’ units that power rotation. In the bacterium Escherichia coli, up to 11 units drive the motor at high load while all the units are released at low load. We manipulate motor load using electrorotation, a technique in which a rapidly rotating electric field applies an external torque on the motor. This allows us to change motor load at will and measure the resulting stator dynamics at single-unit resolution. We found that the force generated by the stator units controls their unbinding, forming a feedback loop that leads to autoregulation of the assembly. We complemented our experiments with theoretical models that provide insight into the underlying molecular interactions. Torque-dependent remodeling takes place within seconds, making it a highly responsive control mechanism, one that is mediated by the mechano-chemical tuning of protein interactions.

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.

There is an avalanche of new data on activity in neural networks and the biological brain, revealing the collective dynamics of vast numbers of neurons. In principle, these collective dynamics can be of almost arbitrarily high dimension, with many independent degrees of freedom — and this may reflect powerful capacities for general computing or information. In practice, neural datasets reveal a range of outcomes, including collective dynamics of much lower dimension — and this may reflect other desiderata for neural codes. For what networks does each case occur? We begin by exploring bottom-up mechanistic ideas that link tractable statistical properties of network connectivity with the dimension of the activity that they produce. We then cover “top-down” ideas that describe how features of connectivity and dynamics that impact dimension arise as networks learn to perform fundamental computational tasks.

Neuronal dendritic trees display a wide range of nonlinear input integrations due to their voltage-dependent active calcium channels. We reveal that in vivo-like fluctuating input enhances nonlinearity substantially in a single dendritic compartment and shifts the input-output relation to exhibiting nonmonotonous or bistable dynamics. In particular, with the slow activation of calcium dynamics, we analyze noise-induced bistability and its timescales. We show bistability induces long-timescale fluctuation that can account for observed dendritic plateau potentials in vivo conditions. In a multicompartmental model neuron with realistic synaptic input, we show that noise-induced bistability persists in a wide range of parameters. Using Fredholm's theory to calculate the spiking rate of multivariable neurons, we discuss how dendritic bistability shifts the spiking dynamics of single neurons and its implications for network phenomena in the processing of in vivo–like fluctuating input.

Interest in psychedelic compounds is growing due to their remarkable potential for understanding altered neural states and their breakthrough status to treat various psychiatric disorders. However, there are major knowledge gaps regarding how psychedelics affect the brain. The Computational Neuroscience Laboratory at the Turner Institute for Brain and Mental Health, Monash University, uses multimodal neuroimaging to test hypotheses of the brain’s functional reorganisation under psychedelics, informed by the accounts of hierarchical predictive processing, using dynamic causal modelling (DCM). DCM is a generative modelling technique which allows to infer the directed connectivity among brain regions using functional brain imaging measurements. In this webinar, Associate Professor Adeel Razi and PhD candidate Devon Stoliker will showcase a series of previous and new findings of how changes to synaptic mechanisms, under the control of serotonin receptors, across the brain hierarchy influence sensory and associative brain connectivity. Understanding these neural mechanisms of subjective and therapeutic effects of psychedelics is critical for rational development of novel treatments and for the design and success of future clinical trials. Associate Professor Adeel Razi is a NHMRC Investigator Fellow and CIFAR Azrieli Global Scholar at the Turner Institute of Brain and Mental Health, Monash University. He performs cross-disciplinary research combining engineering, physics, and machine-learning. Devon Stoliker is a PhD candidate at the Turner Institute for Brain and Mental Health, Monash University. His interest in consciousness and psychiatry has led him to investigate the neural mechanisms of classic psychedelic effects in the brain.

Spike-driven adaptation involves intracellular mechanisms that are initiated by neural firing and lead to the subsequent reduction of spiking rate followed by a recovery back to baseline. We report on long (>0.5 second) recovery times from adaptation in a thalamic-like structure in weakly electric fish. This adaptation process is shown via modeling and experiment to encode in a spatially invariant manner the time intervals between event encounters, e.g. with landmarks as the animal learns the location of food. These cells also come in two varieties, ones that care only about the time since the last encounter, and others that care about the history of encounters. We discuss how the two populations can share in the task of representing sequences of events, supporting path integration and converting from ego-to-allocentric representations. 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. Finally we discuss how all the cells of this gateway to the pallium exhibit mixed selectivity of social features of their environment. The data and computational modeling further reveal that, in contrast to a long-held belief, these gymnotiform fish are endowed with a corollary discharge, albeit only for social signalling.

In decision-making under risk (DMR) participants' choices are based on probability values systematically different from those that are objectively correct. Similar systematic distortions are found in tasks involving relative frequency judgments (JRF). These distortions limit performance in a wide variety of tasks and an evident question is, why do we systematically fail in our use of probability and relative frequency information? We propose a Bounded Log-Odds Model (BLO) of probability and relative frequency distortion based on three assumptions: (1) log-odds: probability and relative frequency are mapped to an internal log-odds scale, (2) boundedness: the range of representations of probability and relative frequency are bounded and the bounds change dynamically with task, and (3) variance compensation: the mapping compensates in part for uncertainty in probability and relative frequency values. We compared human performance in both DMR and JRF tasks to the predictions of the BLO model as well as eleven alternative models each missing one or more of the underlying BLO assumptions (factorial model comparison). The BLO model and its assumptions proved to be superior to any of the alternatives. In a separate analysis, we found that BLO accounts for individual participants’ data better than any previous model in the DMR literature. We also found that, subject to the boundedness limitation, participants’ choice of distortion approximately maximized the mutual information between objective task-relevant values and internal values, a form of bounded rationality.

The intensity and features of sensory stimuli are encoded in the activity of neurons in the cortex. In the visual and piriform cortices, the stimulus intensity re-scales the activity of the population without changing its selectivity for the stimulus features. The cortical representation of the stimulus is therefore intensity-invariant. This emergence of network invariant representations appears robust to local changes in synaptic strength induced by synaptic plasticity, even though: i) synaptic plasticity can potentiate or depress connections between neurons in a feature-dependent manner, and ii) in networks with balanced excitation and inhibition, synaptic plasticity determines the non-linear network behavior. In this study, we investigate the consistency of invariant representations with a variety of synaptic states in balanced networks. By using mean-field models and spiking network simulations, we show how the synaptic state controls the emergence of intensity-invariant or intensity-dependent selectivity by inducing changes in the network response to intensity. In particular, we demonstrate how facilitating synaptic states can sharpen the network selectivity while depressing states broaden it. We also show how power-law-type synapses permit the emergence of invariant network selectivity and how this plasticity can be generated by a mix of different plasticity rules. Our results explain how the physiology of individual synapses is linked to the emergence of invariant representations of sensory stimuli at the network level.

Macroscopic brain organisation emerges early in life, even prenatally, and continues to develop through adolescence and into early adulthood. The emergence and continual refinement of large-scale brain networks, connecting neuronal populations across anatomical distance, allows for increasing functional integration and specialisation. This process is thought crucial for the emergence of complex cognitive processes. But how and why is this process so diverse? We used structural neuroimaging collected from a large diverse cohort, to explore how different features of macroscopic brain organisation are associated with diverse cognitive trajectories. We used diffusion-weighted imaging (DWI) to construct whole-brain white-matter connectomes. A simulated attack on each child's connectome revealed that some brain networks were strongly organized around highly connected 'hubs'. The more children's brains were critically dependent on hubs, the better their cognitive skills. Conversely, having poorly integrated hubs was a very strong risk factor for cognitive and learning difficulties across the sample. We subsequently developed a computational framework, using generative network modelling (GNM), to model the emergence of this kind of connectome organisation. Relatively subtle changes within the wiring rules of this computational framework give rise to differential developmental trajectories, because of small biases in the preferential wiring properties of different nodes within the network. Finally, we were able to use this GNM to implicate the molecular and cellular processes that govern these different growth patterns.

It is generally agreed upon that qualities of conscious experience instantiate structural properties, usually called relations. They furnish a representation of qualities (or qualia, in fact) in terms of a mathematical space Q (rather than a set), which is crucial to both modelling and measuring of conscious experience." "What is usually disregarded is that “only such structural properties generalize across individuals” (Austen Clark), but that qualities themselves as differentiated by stimulus specifications, behavior or reports do not. We show that this implies that only the part of Q which is invariant with respect to the automorphism group has a well-defined referent, while individual elements do not. This poses a prima facie limitation of any theory or experiment that aims to address individual qualities. We show how mathematical theories of consciousness can overcome this limitation via symmetry groups and group actions, making accessible to science what is properly called private language.

To prevail in a dynamic and noisy environment, the brain must create reliable and meaningful representations from sensory inputs that are often ambiguous or corrupt. Since only information that permeates the cortical hierarchy can influence sensory perception and decision-making, it is critical that noisy external stimuli are encoded and propagated through different processing stages with minimal signal degradation. Here we hypothesize that stimulus-specific pathways akin to cortical topographic maps may provide the structural scaffold for such signal routing. We investigate whether the feature-specific pathways within such maps, characterized by the preservation of the relative organization of cells between distinct populations, can guide and route stimulus information throughout the system while retaining representational fidelity. We demonstrate that, in a large modular circuit of spiking neurons comprising multiple sub-networks, topographic projections are not only necessary for accurate propagation of stimulus representations, but can also help the system reduce sensory and intrinsic noise. Moreover, by regulating the effective connectivity and local E/I balance, modular topographic precision enables the system to gradually improve its internal representations and increase signal-to-noise ratio as the input signal passes through the network. Such a denoising function arises beyond a critical transition point in the sharpness of the feed-forward projections, and is characterized by the emergence of inhibition-dominated regimes where population responses along stimulated maps are amplified and others are weakened. Our results indicate that this is a generalizable and robust structural effect, largely independent of the underlying model specificities. Using mean-field approximations, we gain deeper insight into the mechanisms responsible for the qualitative changes in the system’s behavior and show that these depend only on the modular topographic connectivity and stimulus intensity. The general dynamical principle revealed by the theoretical predictions suggest that such a denoising property may be a universal, system-agnostic feature of topographic maps, and may lead to a wide range of behaviorally relevant regimes observed under various experimental conditions: maintaining stable representations of multiple stimuli across cortical circuits; amplifying certain features while suppressing others (winner-take-all circuits); and endow circuits with metastable dynamics (winnerless competition), assumed to be fundamental in a variety of tasks.