Organized by: Laurent PUJO-MENJOUET and Suzanne SINDI (Claude Bernard Lyon 1 University (Lyon, FRANCE))

1. Théo LOUREAUX University of California, Merced
   "Modeling the Prion Aggregation Process During Polymerization Experiments Using Delay Differential Equations"
Prion proteins are notorious for their ability to induce neurodegenerative diseases by forming long fibrillar aggregates that accumulate in the brain. While the aggregation of these proteins and their fragmentation by oligomeric species is central to disease progression, the underlying mechanisms remain poorly understood. To better interpret experimental data, mathematical models have been developed to translate the key chemical reactions governing this process. In this talk, I present a novel modeling approach based on delay differential equations (DDEs), designed to capture the time-dependent features of prion polymerization dynamics. I will demonstrate how this framework aligns with experimental observations from polymerization assays in which prion monomers are thermally induced to aggregate. The model not only fits the data well but also suggests an alternative perspective on the interplay between aggregation and fragmentation, offering a new theoretical lens on prion dynamics.
2. Ashish Raj University of California San Francisco
   "Biophysical modeling of pathology progression in dementia and its implementation using physics-informed neural networks"
This presentation will focus on developing mathematical and computational models that use the brain’s structural connectivity to predict the development of brain diseases, including Alzheimer’s, Parkinson’s, Huntington’s, ALS and other neurodegenerative diseases. I will first describe our original proposal that Alzheimer and other dementias are underpinned by misfolded pathologies that spread on the brain structural connectome. This process can be mathematically captured by the so-called 'Network Diffusion Model'. Several examples from AD, ALS, Huntington's, Parkinson's and other dementias will be demonstrated. I will then present new extensions of this model in many meaningful ways, incorporating protein aggregation, clearance, active axonal transport, and mediation by external genes, cells and neuroinflammation. Deep neural network implementations of these complex and computationally prohibitive models will be motivated, and preliminary work on physics-informed neural networks will be presented. I will also briefly describe recent work in modeling brain electrophysiology using similar graph spectral models. All above models centrally involve the brain’s complex network Laplacian eigen-spectrum and “graph harmonics.” Through this work, we have found significant differences in the model’s parameters that relate healthy brains to Alzheimer’s disease, sleep, epilepsy and infant brain maturation. The related papers will be briefly highlighted.
3. Human Rezaei INRAE, Jouy-en-Josas FRANCE
   "Intrinsic Dynamics and Deterministic Diversification Drive a New Model of Prion Replication and Dissemination"
Through experimental approaches combining nanoscale infrared spectroscopy and dynamic atomic force microscopy, we investigated the intrinsic dynamics of PrPSc assemblies outside the context of templated replication. These studies revealed that PrPSc assemblies exhibit an inherent capacity for structural diversification and material exchange, leading us to establish a new replication model. This model incorporates deterministic structural diversification and proposes that prion assemblies can undergo catalytic conformational transitions independently of replication events, challenging the notion that replication alone governs strain specificity. Building on these findings, we developed a stochastic reaction-diffusion framework that integrates nonlinear replication dynamics and tissue responses. Using the Gillespie algorithm, we modeled neuroinvasion as a complex and emergent process shaped by strain-specific PrPSc behavior and anatomical connectivity. This integrative approach offers new insights into how structural diversity is maintained within populations of prions and how it contributes to strain-dependent tropism and pathogenesis. By shifting the focus from purely replication-driven models to those considering intrinsic structural dynamics, this work proposes a revised conceptual framework for prion propagation, with broader implications for other protein misfolding diseases.
4. Laurent Pujo-Menjouet University Claude Bernard Lyon 1 - Camille Jordan Institute
   "Modeling the formation of perinuclear crowns made of agglutinated ATM proteins observed in fibroblasts from patients affected by Alzheimer’s disease"
Alzheimer’s disease is a progressive neurodegenerative disorder marked by the irreversible loss of brain cells. In response to oxidative stress, ATM proteins typically migrate to the nucleus to detect and repair double-strand DNA breaks. However, recent studies suggest that APOE proteins may accumulate at the nuclear envelope, blocking the entry of ATM proteins and resulting in the formation of a characteristic perinuclear crown. To better understand this phenomenon and evaluate potential therapeutic interventions, we propose two modeling approaches: a compartmental model and a reaction-diffusion system that capture the physical interactions between ATM and APOE proteins. Both models incorporate key biological processes, including protein transport, monomer aggregation, and the dissociation of dimers and complexes. We explore the effects of irradiation and antioxidant treatments on the disintegration of the perinuclear crown. Our simulations suggest that the combined use of these two strategies is the most effective in delaying crown reformation, highlighting a promising therapeutic avenue for Alzheimer’s disease.

Organized by: Richard Bertram (Florida State University), Yangyang Wang, Brandeis University

1. Na Yu Toronto Metropolitan University
   "Exploring the Roles of Noise and Coupling Strength in the Emergent Dynamics of Clustered Neural Motifs"
Understanding how neural networks process information involves exploring how their structure and external influences shape collective dynamics. Research has shown that neuronal networks are not always randomly connected; instead they are organized into highly clustered motifs that act as fundamental building blocks. These motifs play a critical role in shaping collective dynamics such as synchrony and coherence. In this study, we modeled a neural network composed of six representative types of clustered motifs and examined how key factors, including intrinsic noise, inter-motif connectivity, network size, and coupling strength, affect the emergence of synchrony and coherence at both the motif and network levels. Our results reveal that synchrony is optimized when noise intensity and inter-motif connectivity are at intermediate levels. We also find that network performance is enhanced when the ratio of intra- to inter-motif coupling strength is within a specific range.  These results reveal how the interplay between structure and noise shapes coherent neural dynamics.
2. Amin Akhshi McGill University
   "From Chaos to Neural Code: Exploring the Role of Gamma-Frequency Burst Oscillations in Sensory Pyramidal Cells"
Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish fire action potentials in the form of bursts (i.e., repeated clusters of spikes) through a mechanism known as ghostbursting. This mechanism enables ELL pyramidal cells to fire gamma-frequency burst oscillations (20–80 Hz) in response to constant applied current, characterized by progressively increasing spike frequencies within bursts. To efficiently investigate the emergence of these complex burst dynamics, we developed a phenomenological model based on the Hindmarsh-Rose (HR) formalism by incorporating dual adaptation variables and an extracellular noise term representing random excitatory and inhibitory inputs. Using parameter optimization against in vivo recordings, we demonstrated the model reliably reproduced key spiking features of ELL pyramidal cells, including chaotic burst firing patterns comparable to experimental observations. Subsequently, bifurcation analysis confirmed that chaotic burst firing is an intrinsic property of the model, consistent with the behavior of ELL pyramidal cells, while parameter sensitivity analysis revealed that external stochastic input modulates spiking activity, collectively shaping the firing patterns observed in vivo. Finally, to investigate the implications of our model for population coding, we performed network simulations in response to electrocommunication stimuli and found that burst firing significantly enhances synchronization among model neurons receiving common input. In this talk, I will provide an overview of these results.
3. Adam Stinchcombe University of Toronto
   "Modelling Insights into Two Behaviour Rhythm Phenomena"
Ultradian behavioural rhythms are highly-flexible oscillations in goal-directed behaviour with periods shorter than a day. They remain mysterious in both their biochemical mechanisms and their functional significance, but are generally believed to be a reflection of neural dynamics. We propose that D2 autoreceptor-dependent dopamine self-regulation in the midbrain-striatal synapses gives rise to ultradian rhythmicity. We express this hypothesis in an ordinary differential equation based mathematical model in a dual-negative feedback-loop structure. Numerical integration and bifurcation analysis shows that the oscillations have a flexible and parameter-sensitive period in agreement with experimental observation. The model also demonstrates the masking-entraining effects of circadian (approximately 24 hour) regulation on ultradian rhythms and the rapid-resetting effect of transient excitation. This reveals the crucial role of circadian-ultradian interaction in consolidating behavioural activity and coordinating the motivation to engage in recurring, albeit not highly predictable events, such as social interactions.
4. Richard Bertram Florida State University
   "Dynamic Homeostasis in Relaxation and Bursting Oscillations"
Homeostasis is typically thought of as the invariance of an equilibrium across a range of input values. However, many biological systems oscillate, and dynamic homeostasis refers to the invariance of some feature of the oscillation across a range of input values. In this presentation, we demonstrate that in fast/slow systems, in which the variables can be partitioned into those that change rapidly and those that change slowly, invariance can be produced in the mean value of a slow variable responsible for driving the oscillations. We use fast/slow analysis to explain this form of dynamic homeostasis and consider the effects of noise. The biological relevance lies in the fact that the slow variable driving the relaxation or bursting oscillation is often a quantity, such as the intracellular calcium or ATP concentration, that plays many roles in a cell and for which homeostasis is advantageous to cell behavior or health.

1. Alexander Ginsberg The University of Utah, Department of Mathematics
   "A predictive propensity measure to enter REM sleep"
During sleep periods, most mammals alternate multiple times between rapid-eye-movement (REM) sleep and non-REM (NREM) sleep. A common theory proposes that these transitions are governed by an ``hourglass-like'' homeostatic need to enter REM sleep that accumulates during the inter-REM interval and partially discharges during REM sleep. However, markers or mechanisms for REM homeostatic pressure remain undetermined. Recently, an analysis of sleep in mice demonstrated that the cumulative distribution function (CDF) of the amount of NREM sleep between REM bouts correlates with REM bout duration, suggesting that time in NREM sleep influences REM sleep need. Here, we build on those results and construct a predictive measure for the propensity to enter REM sleep as a function of time in NREM sleep since the previous REM episode. The REM propensity measure is precisely defined as the probability to enter REM sleep before the accumulation of an additional pre-specified amount of NREM sleep. Analyzing spontaneous sleep in mice, we find that, as NREM sleep accumulates between REM bouts, the REM propensity exhibits a peak value and then decays to zero with further NREM accumulation. We show that the REM propensity at REM onset predicts features of the subsequent REM bout under certain conditions. Specifically, during the light phase and for REM propensities occurring before the peak in propensity, the REM propensity at REM onset is correlated with REM bout duration, and with the probability of the occurrence of a short REM cycle called a sequential REM cycle. Further, we also find that proportionally more REM sleep occurs during sequential REM cycles, supporting a correlation between high values of our REM propensity measure and high REM sleep need. These results support the theory that a homeostatic need to enter REM sleep accrues during NREM sleep, but only for a limited range of NREM sleep accumulation. Time permitting, we will discuss current research directions.

1. Brandon Imstepf University of California, Merced
   "Accelerating Solutions of Nonlinear PDEs Using Machine Learning: A Case Study with the Network Transport Model"
Alzheimer’s Disease (AD) is a progressive neurodegenerative disorder affecting approximately 10% of Americans over age 65, leading to memory loss, cognitive decline, and impaired daily function. Disease progression correlates with the spread of tau and amyloid-beta proteins, which aggregate into neurofibrillary tangles. While macroscopic whole-brain network models predict large-scale protein deposition patterns, they lack the specificity to capture individual disease progression. Conversely, microscale neuron-neuron models offer highly detailed biochemical aggregation and transport simulations but are computationally prohibitive for whole-brain parameter inference. In this work, we explore using machine learning to accelerate whole-brain simulations by approximating explicit solutions to the microscopic Two-Neuron Transport Model (TNTM), a partial differential equation describing tau flux along a neuron, incorporating biochemical aggregation, fragmentation, and transport. We simulate a single-edge model across physiological ranges of biochemical parameters and boundary conditions, then compare regression methods with varying levels of interpretability, from neural networks (low) to symbolic regression via PySR (high). Neural networks achieve the lowest error but lack biological insight. Linear and polynomial regression compute rapidly but yield high errors with limited interpretability. Symbolic regression achieves a balance between accuracy and transparency. This work demonstrates the potential of machine learning for computationally scalable AD modeling, opening avenues for patient-specific parameterization using AD data repositories.
2. Youngmin Park University of Florida
   "Phase Reduction of Heterogeneous Coupled Oscillators"
We introduce a method to identify phase equations for heterogeneous oscillators beyond the weak coupling regime. This strategy is an extension of the theory from [Y. Park and D. Wilson, SIAM J. Appl. Dyn. Syst., 20 (2021), pp. 1464--1484] and yields coupling functions for N general limit-cycle oscillators with arbitrary types of coupling, with similar benefits as the classic theory of weakly coupled oscillators. These coupling functions enable the study of oscillator networks in terms of phase-locked states, whose stability can be determined using straightforward linear stability arguments. We demonstrate the utility of this approach by reducing and analyzing conductance-based thalamic neuron model. The reduction correctly predicts the emergence of new phase-locked states as a function of coupling strength and heterogeneity. We conclude with a brief remark on recent extensions to n:m phase-locking and N-body interactions.