This study investigates neural population dynamics through mathematical modeling, building on foundational work by Winfree (1967) and Kuramoto (1984) on oscillator synchronisation. The research focuses on developing neural mass models that describe large-scale brain activity using population firing rates.

Key Approaches:

• Neural mass models provide macroscopic representations of brain networks with mathematical tractability

• Phase oscillator networks offer detailed analysis of heterogeneous conduction delays affecting neural synchronisation

• Combined framework integrates both approaches to bridge macroscopic descriptions with phase-based models

Research Focus: The work examines how heterogeneous delays influence synchronisation patterns, particularly relevant for pathological conditions like multiple sclerosis where altered axonal conduction affects brain dynamics. Using coupled oscillator theory extended to infinite-dimensional delayed differential equations, the study explores synchronisation mechanisms in complex neural networks.

Applications: This unified framework provides insights into large-scale brain dynamics, with potential applications in computational neuroscience projects and understanding neurological disorders through event-related synchronisation/desynchronisation phenomena observed in brain imaging studies.

The research advances next-generation neural population models by incorporating biologically realistic network dynamics and distributed delays.

YAPKO (Publication, Research, and Project Coordination Board of Istanbul Ticaret University)
22-2018/34

 This project explores the analytical structure of nonlinear fractional partial differential equations (FPDEs) using symmetry-based methods. Building on the classical theory of Lie groups developed by Sophus Lie and its later generalisations by Bluman and Ovsyannikov, the research focuses on extending these approaches to fractional-order models relevant in modern mathematical physics.

Key Approaches:

• Lie group analysis is applied to nonlinear FPDEs to derive invariant solutions, symmetries, and conservation laws.

• Caputo fractional derivatives are used to handle initial and boundary value problems in a physically meaningful way.

• Symbolic computation tools, such as Mathematica, are employed for calculating infinitesimal generators, verifying invariance conditions, and constructing exact solutions.

Research Focus:

The work systematically investigates how symmetry methods can be adapted to fractional differential models that describe diffusion, wave propagation, and memory effects in complex media. Special emphasis is placed on the generalised Burgers, Korteweg–de Vries (KdV), Harry-Dym, and K(m,n) equations with time-fractional dynamics. The methodology addresses the challenges posed by nonlocality in time, which is inherent to fractional operators.

Applications:

This framework enhances the theoretical toolkit for analysing fractional models arising in physics, engineering, and applied sciences. The ability to derive exact invariant solutions provides insight into nonlinear wave phenomena, anomalous diffusion, and viscoelastic systems. Additionally, the results support further development of symmetry-based techniques for boundary value problems in fractional calculus.

Scientific Impact:

The project led to several scientific outputs, including a publication in Turkish Journal of Mathematics and submissions to Physica A and International Journal of Computer Mathematics. These contributions provide a foundation for ongoing research on the integration of group-theoretic methods with fractional differential modeling.

We applied and implemented these methods using the Mathematica software for symbolic manipulation and verification.

As a result of our collaborative efforts, several research articles were produced, including submissions to Physica A and International Journal of Computer Mathematics, and one published paper in the Turkish Journal of Mathematics.

This work has provided a solid theoretical foundation for further research on symmetry methods applied to fractional differential equations.