Mathematical Modeling seminars

Galaxy images and spectra map large-scale cosmic structure and constrain fundamental physics. While galaxy-clustering bias models are well developed, forthcoming datasets can also trace the intergalactic medium and recover complementary cosmic-web information. Jamie Sullivan discusses models for both neutral and ionized intergalactic material. Neutral-hydrogen work includes the environments of Lyman-alpha emitters at redshifts 2.5–5, relevant to future DESI extensions and Spec-S5, together with the measured Lyman-alpha forest and an observationally practical compression of its bispectrum. Ionized-gas research studies momentum through an effective-field-theory kinetic Sunyaev–Zel’dovich bispectrum, and density through fast-radio-burst clustering and cross-correlations. Together, these approaches aim to map intergalactic modes alongside galaxy tracers and improve precision cosmological constraints.

This seminar discusses a proof-of-concept gravitational-waveform model described in arXiv:2511.10522. It augments post-Newtonian equations using physics-informed neural-network methods and universal differential equations. The talk also covers ongoing development of a precessing surrogate model designed to evaluate in less than one millisecond while covering a broader portion of the LVK frequency band.

The neurobiological mechanisms of arousal and anesthesia remain poorly understood. Recent evidence highlights the key role of interactions between the cerebral cortex and the diffusely projecting matrix thalamic nuclei. Here, we interrogate these processes in a whole-brain corticothalamic neural mass model endowed with targeted and diffusely projecting thalamocortical nuclei inferred from empirical data. This model captures key features seen in propofol anesthesia, including diminished network integration, lowered state diversity, impaired susceptibility to perturbation, and decreased corticocortical coherence. Collectively, these signatures reflect a suppression of information transfer across the cerebral cortex. We recover these signatures of conscious arousal by selectively stimulating the matrix thalamus, recapitulating empirical results in macaque, as well as wake-like information processing states that reflect the thalamic modulation of largescale cortical attractor dynamics. Our results highlight the role of matrix thalamocortical projections in shaping many features of complex cortical dynamics to facilitate the unique communication states supporting conscious awareness.

Measuring the degree of consciousness an organism possesses has remained a longstanding challenge in Neuroscience. In part, this is due to the difficulty of finding the appropriate mathematical tools for describing such a subjective phenomenon. Current methods relate the level of consciousness to the complexity of neural activity, i.e., using the information contained in a stream of recorded signals they can tell whether the subject might be awake, asleep, or anaesthetised. Usually, the signals stemming from a complex system are correlated in time; the behaviour of the future depends on the patterns in the neural activity of the past. However these past-future relationships remain either hidden to, or not taken into account in the current measures of consciousness. These past-future correlations are likely to contain more information and thus can reveal a richer understanding about the behaviour of complex systems like a brain. Our work employs the "epsilon-machines” framework to account for the time correlations in neural recordings. In a nutshell, epsilon-machines reveal how much of the past neural activity is needed in order to accurately predict how the activity in the future will behave, and this is summarised in a single number called "statistical complexity". If a lot of past neural activity is required to predict the future behaviour, then can we say that the brain was more “awake" at the time of recording? Furthermore, if we read the recordings in reverse, does the difference between forward and reverse-time statistical complexity allow us to quantify the level of time asymmetry in the brain? Neuroscience predicts that there should be a degree of time asymmetry in the brain. However, this has never been measured. To test this, we used neural recordings measured from the brains of fruit flies and inferred the epsilon-machines. We found that the nature of the past and future correlations of neural activity in the brain, drastically changes depending on whether the fly was awake or anaesthetised. Not only does our study find that wakeful and anaesthetised fly brains are distinguished by how statistically complex they are, but that the amount of correlations in wakeful fly brains was much more sensitive to whether the neural recordings were read forward vs. backwards in time, compared to anaesthetised brains. In other words, wakeful fly brains were more complex, and time asymmetric than anaesthetised ones.

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.

A major neuroscientific challenge is to identify the neural mechanisms that support consciousness. Though experimental studies have accumulated evidence about the location of the neural substrate of consciousness, we still lack a full understanding of why certain brain areas, but not others, can support consciousness. In this talk, I will give an overview of our approach, taking advantage of the theoretical framework provided by Integrated Information Theory (IIT). First, I will introduce results showing that a maximum of integrated information within the human brain matches our best evidence concerning the location of the NSC, supporting the IIT’s prediction. Furthermore, I will discuss the possibility that the NSC can change its location and even split into two depending on the task demand. Finally, based on some graph-theoretical analyses, I will argue that the ability of different brain regions to contribute or not to consciousness depends on specific properties of their anatomical connectivity, which determines their ability to support high integrated information.

The COINTOB (conditional intention and integration to bound) model provides a heuristic framework of processes in Libet-style experiments. The model is based on three assumptions. First, brain activation preceding conscious intentions in Libet-style experiments does not reflect an unconscious decision but rather the unfolding of a decision process. Second, the time of conscious decision (W) reflects the moment in time when the decision boundary is crossed. This interpretation of W is consistent with our apparent intuition that we decide in the moment we experience the conscious intention to act. Third, the decision process is configured by conscious intentions that participants form at the beginning of the experiment based on the experimental instruction. Brass and Mele discuss the model, conceptual background for it, and the model’s bearing on free will.

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.