Ayse Peker-Dobie

Large neuronal networks demonstrate complex dynamics across multiple scales, ranging from single-neuron excitability and spike-train variability to mesoscopic rhythms and whole-brain activity. Different types of differential equation models have been developed to comprehend these phenomena, connecting deterministic, stochastic, and mean-field descriptions. At the deterministic level, ordinary differential equation (ODE) models, including conductance-based neuron models, neural-mass systems, and whole-brain networks, summarize neural behavior through a reduced set of macroscopic variables. At the population level, mean-field partial differential equation (PDE) models such as Fokker-Planck, age-structured, kinetic, and neural field equations describe the evolution of probability or population densities over membrane-potentials, synaptic states, and other kinetic variables. These PDEs link single-neuron mechanisms to population-level activity and allow one to analyze bifurcations, oscillations and other collective patterns. Stochastic differential equation (SDE) models and their extensions that include jump-diffusion processes and stochastic PDEs (SPDEs) are widely used to describe random membrane fluctuations, irregular spike trains, synaptic plasticity and large-scale variability in neural activity. These stochastic models are also applied to neural data analysis, for example to quantify noise in electro-physiological recordings and to infer latent neural dynamics. Because variability and noise are central in neural systems, we devote more space to stochastic models but always relate them back to the surrounding ODE and PDE frameworks. This hierarchy of ODE, PDE, and SDE-SPDE models shows that the versatility of differential-equation-based approaches in neuroscience offers unified tools for multiscale modeling, neural signal processing, cognitive modeling, and the analysis of noisy neural systems. We also discuss some known numerical and computational approaches, especially for stochastic models and conclude by outlining open challenges, such as multiscale inference, control-oriented formulations and the integration of differential-equation models with modern machine-learning methods.

Using Braga’s control theory as a foundational approach, this paper investigates the control of Hopf and generalized Hopf bifurcations within a Susceptible–Infectious model featuring two distinct infectious stages, one being nonlethal and the other lethal. The model with control inputs is analyzed to understand the dynamics at one of its equilibrium points where such bifurcation phenomena occur. By applying Braga’s control theory, we explore how control inputs can influence the system’s behavior, specifically targeting the induction and modulation of limit cycles associated with Hopf bifurcations of codimension-1 and codimension-2. The research focuses on identifying how variations in control parameters affect these periodic oscillations and their characteristics. The results may guide public health interventions by elucidating how specific control functions and parameters can be optimized to stabilize disease dynamics and reduce undesirable fluctuations in epidemic models.

Goodwin's model is a cornerstone in the study of dynamical systems within macroeconomics, explaining the interaction between employment ratio and wage share in a closed economy. Analogous to predator-prey dynamics in mathematical economics, the Goodwin model, despite its simplicity, effectively captures the periodic behavior of state variables over specific time intervals. By relaxing the initial assumptions, the model can be adapted to account for more complex economic scenarios. In this article, we study a higher-dimensional extension of the Goodwin model that incorporates variable capacity utilization and capital coefficient alongside employment ratio and wage share. In particular instances, the wage share and employment rate equations decouple from the overall system. For these cases, by incorporating a delay effect in the Phillips curve, we demonstrate that while the equilibrium of the generalized system remains stable within certain parameter domains in the absence of delay, the introduction of delay can induce a Hopf bifurcation, leading to periodic oscillations. We analytically derive the critical delay parameter value that destabilizes the equilibrium point via a Hopf bifurcation.

• Performance Evaluation: Demonstrated that the proposed GNN model outperforms traditional baselines in forecasting accuracy on the developed dataset.This article presents a novel competitive information propagation model designed for multi-layer interconnected networks, addressing the complex dynamics of information spread in online social networks (OSNs). It highlights how both positive and negative information compete for attention and influence, with individual behaviors shaped by local and global information trends. The study mathematically analyzes the model's dynamics, calculates the basic reproduction number, and establishes stability conditions for different equilibria. It also formulates an optimal control strategy to manage the spread of negative information while minimizing costs.• Competitive Information Propagation Model: Developed a model that captures the coexistence of positive and negative information within multi-layer interconnected networks, reflecting real-world complexity.• Incorporation of Individual Herd Behavior: Introduced a compartmental model where individual tendencies to spread information are influenced by both local and global prevalence, enhancing understanding of adaptive behaviors in information dynamics.• Stability Analysis: Provided a rigorous mathematical foundation by calculating the basic reproduction number and using Lyapunov theory to discuss the stability of informationfree and endemic equilibria.• Optimal Control Framework: Formulated an optimization problem to effectively suppress negative information propagation, balancing control costs with propagation dynamics, and derived solutions to enhance practical decision-making in resource management.This study introduces the DNIRep model, a novel dynamic network representation framework designed to enhance the modeling of rumor propagation in social networks. Acknowledging the limitations of existing static representation methods, DNIRep effectively captures the temporal dynamics and the complex interplay between explicit and implicit relationships among users. The model incorporates a feedback mechanism to improve the performance of node representations and demonstrates how trust levels and interactive behaviors shape public opinion dynamics.• Dynamic Network Representation Model (DNIRep): Developed a new framework that integrates explicit and implicit relationships to more accurately represent information propagation in dynamic networks.• Feedback Mechanism: Introduced a novel feedback system that enhances the updating of node representations, improving overall network representation performance.• Alignment with Real-World Dynamics: Established a model that aligns more closely with real-world social networks, addressing the unique characteristics of rumor propagation and public opinion dynamics.• Experimental Validation: Extensive simulations were conducted demonstrating that higher trust levels stabilize group opinions and that certain user interactions, including novelty of topics and opinion leaders, significantly influence public sentiment evolution.This study explores the dynamics of epidemic spread on networks characterized by second-order properties, specifically focusing on assortative mixing-where nodes connect preferentially based on their degrees. The authors propose a stochastic algorithm to construct networks that reflect real human connections, enhancing the understanding of how these structures influence the spread of infectious diseases. The research examines the impact of these networks on epidemic trajectories and evaluates whether the spread of disease can be predicted using readily observable data.• Stochastic Network Construction: Developed a flexible algorithm for constructing networks based on individual preferences for connection, allowing for the creation of diverse assortative network profiles.• Marginal Degree Preference: Introduced a method to derive the marginal degree preference from a general preference function, accommodating multiple external factors influencing connections.• Epidemic Dynamics Simulation: Conducted simulations to analyze epidemic curves and overall epidemic sizes across various network types, supporting the hypothesis that the effective reproductive number can be predicted as a function of the susceptible population fraction over time.• Guidance for Future Research: Provided a framework for further theoretical investigations into the effects of network structures on epidemic spread, emphasizing the relevance of second-order network properties in epidemiological modeling.From a theoretical standpoint, the primary contribution of the article is the derivation of an infinite exponential, Dirichlet, series for the model variables, which are interrelated by logarithmic and exponential transformations to the R-variable. The finite truncation of the series results in a Prony approximation, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale.• A numerical Newton-Raphson approximation scheme for the R-variable is also derived and compared to the parametric solution.• The proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model's time constants, and an excitation timescale, dictated by the population size.• The DE solution is applied to estimate model parameters for a well-known epidemiological dataset-the boarding school flu outbreak.This special issue presents valuable insights into the application of compartment models beyond traditional epidemiology, focusing on their role in understanding social interactions and information dynamics. The featured articles explore various innovative approaches, including spatiotemporal graph neural networks and...

The investigation of infectious agents invading human and nonhuman populations represents a rich research domain within the framework of mathematical biology, captivating the interest of scientists across various disciplines. In this work, we examine the endemic equilibrium of feline coronavirus and feline infectious peritonitis by using a modified susceptible‐infected‐susceptible epidemiological model. We incorporate the concept of mutations from FCoV to FIP to enrich our analysis. We establish that the model, when subjected to reasonable parameter ranges, supports an endemic equilibrium wherein the FCoV group dominates. To demonstrate the stability of the equilibria under typical parameters and initial conditions, we employ the model SCF presented by Dobie in 2022 (Dobie, 2022). We ascertain that the equilibrium values reside within the interior domains of stability. Additionally, we displayed perturbed solutions to enhance our understanding. Remarkably, our findings align qualitatively with existing literature, which reports the prevalence of seropositivity to FCoV among stray cats (Tekelioglu et al. 2015, Oğuzoğlu et al. 2010, Pratelli 2008, Arshad et al. 2004).