Neural Field Theory and Brain Network Modelling for Human Brain Mapping

Understanding the complex dynamics of cortico-hippocampal interactions is essential for exploring cognitive processes and their disruptions in neurological disorders. This section of the educational course provides a practical introduction to neural field modeling, focusing on the cortico-hippocampal interactions. More specifically, we focus on how to derive and apply eigenmodes—a mathematical framework that captures the intrinsic oscillatory patterns of any geometric surface.

To achieve this, we go through a step-by-step implementation of the following:

1 - Preparing Anatomical Surfaces
We begin by learning techniques to import and preprocess 2-D anatomical meshes of cortical and hippocampal surfaces. Preprocessing would involve ensuring a smooth medial wall boundary for the cortical mesh and a smooth outer boundary for the hippocampal mesh.

2 - Computing Eigenmodes for Spatial Activity Patterns
Eigenmodes will be computed by solving the Helmholtz equation for the cortical and hippocampal meshes. Hands-on exercises will be provided, covering eigenmode computation, visualization, and decomposition of neural activity.

3 - Integrating Eigenmodes into the Neural Field Theory Framework

After the first two steps, we will integrate eigenmodes into the Neural Field Theory (NFT) framework by decomposing cortical and the hippocampal NFT equations into eigenmode domain. We will briefly detail the mathematical steps to transform NFT equations into eigenmode-specific dynamics, including the derivation of temporal evolution equations for each eigenmode. With these equations, we will simulate the spatio-temporal dynamics of eigenmode evolution on these surfaces. Some of these dynamical behaviours involve looking at mode-mixing, and pathological states (e.g., epileptogenic activity) using hands-on examples. The specific effects of a proximity preserving coupling between cortical and hippocampal surfaces on the dynamics across these two surfaces would be subsequently discussed.

In summary, this section of the course will bridge theoretical concepts with practical, hands-on techniques, equipping attendees with tools to model and interpret cortico-hippocampal dynamics.
 

Neural field theories (NFT) model the waves of activity that sweep across the cortical surface. This talk will trace the formal derivation of "classic NFT" from the physiological and anatomical properties of the brain, including synaptic filtering of inputs, somatic summation, nonlinear gain, signal propagation and time-delayed feedback in loops and circuits. Under broad assumptions, NFT can be written as a damped wave equation, whose behaviour predicts the abrupt transitions observed in sleep-wake cycles and epilepsy; the spatial and temporal spectra of healthy cortical and hippocampal activity; and the impact of electrical and pharmacological neuromodulation. The damped wave equation comprises temporal and spatial components: The former captures the temporal power spectrum of neural activity, as seen in EEG. The spatial component can be expressed as a sum of weighted "geometric eigenmodes" that provide a parsimonious description of task- and resting state EEG. This talk will walk through these principles and derivations, providing the neuroimaging community with new tools and concepts to study functional neuroimaging data.
 

This presentation offers a didactic account of a neural field theory (NFT) approach, developed progressively and systematically for more than 20 years, to studying whole-cortex activity patterns driven by the rhythmogenic circuitry of the corticothalamic (CT) system. It has been known since Berger that the ~10Hz alpha rhythm is the most prominent type of brain activity visible in human electroencephalography (EEG) and related recording modalities. Concentrated primarily over visual cortex, alpha sometimes displays a split peak; suppressed by visual inputs; and is often accompanied by a beta rhythm occurring at its harmonic. Later, the ~10 Hz mu and tau rhythms were found, respectively concentrated over motor and auditory cortex, and suppressed by corresponding sensory input. Early theories argued that separate groups of neurons fire at ~10 Hz at the relevant locations, but these lacked explanatory power. More recently, the alpha rhythm was argued to be a natural mode of activity in the cortex (Nunez) or corticothalamic system (Robinson), and analyzed using NFT. Our most recent work on this shows that just 4 corticothalamic activity eigenmodes are sufficient to explain the key features of alpha, mu, and tau rhythms - namely their frequency, structure, and topography. CT loops account for the basic 10 Hz frequency and alpha-beta correlations, with splitting arising from breaking of degeneracy due to cortical folding. We have observed split-beta and predict split-mu, split-tau, and harmonic tau rhythms to occur, plus split harmonic mu and tau. One of the key recent advances in this NFT variant is the idea that spatial peaks (e.g. the posterior concentration of alpha power) in the EEG appear due to constructive interference of modes in the relevant sensory region, which is which is suppressed (blocked) when CT gains are reduced by bottom-up sensory input and attention.  

This session provides an introduction to a general framework for macro-scale neural field modelling, whereby cortical patches or brain areas are represented as translationally invariant neural fields, and are interconnected by long-range white matter connections of the kind reconstructed by tractography-based anatomical connectomes. Topics covered include:
Neural field model fundamentals, including (a high-level summary of) the Jirsa-Haken approach to formulating neural field PDEs
The progression from simple two-patch neural field systems to connectome-based networks with neural fields embedded at their nodes.
Connections between these models and homogeneous neural fields defined on the folded cortical surface
Practical considerations, including the choice of patch sizes, spatial resolutions, regularization techniques, Laplace operators, and connectome density mappings.
Current advancements in the field, including applications to high-resolution neural field modeling for brain stimulation and epilepsy
This session will bridge theoretical concepts with their computational and clinical implications, equipping participants with foundational knowledge and insights into the future directions of neural field modeling.
 

Deep brain stimulation (DBS), which modulates dysfunction in the brain network by applying chronic high-frequency electrical stimulation to a specific location in the brain, is being explored as a groundbreaking therapy for drug-resistant neurological and neuropsychiatric diseases. DBS has been applied to several brain diseases such as Parkinson's disease, epilepsy, obsessive compulsive disorder, and major depressive disorder, and has demonstrated positive effects in improving symptoms. However, despite many efforts to provide personalized treatment through advanced neuroimaging techniques and accumulated clinical expertise, the therapeutic effects still vary from patient to patient. A potential reason for these inconsistent findings may be the individual variation in the brain structure and functional network organization. Because the stimulation response depends not only on the external conditions including its location, type, and parameters, but also on the dynamic state of the brain network being stimulated, a systematic approach to investigate individualized impacts of stimulation is required. Therefore, we propose a virtual brain modeling approach that enables personalized exploration of DBS. In particular, we examine the feasibility of this modeling approach in the context of treatment-resistant depression (TRD).
Virtual brain models were constructed from patient-specific structural data including brain anatomy and connectome, and then equipped with computational neural mass models for each brain region, thereby reproducing the functional dynamics of the brain. Given the recent neuroimaging findings reporting that the engagement of specific fiber tracts at the stimulation site is associated with the efficacy of DBS, we extend the existing virtual brain modeling framework to incorporate the geometry of fiber tracts and investigate stimulus-induced network effects. In particular, we combine two factors: high resolution and explicit fiber tract modeling. A high-resolution brain model at a mm-scale is built by placing neural mass models at the vertices of the brain surface mesh, regardless of brain parcellation (region). Through explicit modeling of fiber tracts considering their locations and geometries, neural mass models located at both terminals of each fiber tract are coupled taking into account connection strengths and transmission delays. This approach also allows the stimulation of segments of the fiber tract, including signal propagation along its entire length in both directions.
Numerical simulations were run examining regional activations and EEG-projected activity patterns, evoked by stimulation of specific fiber tracts by DBS. Tracts were distinguished by their elicitation of distinct spatiotemporal activity patterns in the network. Our results indicate that, while there are some challenges in validating against empirical data (e.g., individual parameter tuning and variable accuracy across stimulation sites), the high-resolution connectome-based neural field modeling with TVB effectively captures the functional network effects of DBS across different stimulation locations.
Code for replicating these results is available online, and a specially tailored, streamlined version will also be provided and presented to workshop attendees for exploration during the hands-on tutorial accompanying this talk at the end of the workshop’s afternoon session.  

The loose collection of neurobiological insights and mathematical framings nowadays referred to as Neural Field Theory (NFT) saw their first concrete expression in print in the late 1940s and early 1950s. Interestingly, these ‘field theories of neural nets’ were much closer in spirit to the cognitive science-oriented ideas of McCulloch & Pitts, connectionist psychology, and what eventually became modern deep learning, than to those of their more biochemistry and physiology-focused contemporaries such as Hodgkin & Huxley or Goldman & Cole. In subsequent decades, some scientists would look to ground NFT formulations on a more concrete neurobiological footing, whereas others would focus more on their properties as pattern-forming dynamical systems and computational units. It was not until some time in the early 1970s that NFT had its Annus Mariabis - with Amari, Wilson-Cowan, and Nunez respectively introducing key ideas into the mix around inhibition, meso-scale columnar lateral connections, columnar physiology, and macroscopic brain geometry and connectivity. The history of NFT’s development since then, particularly in the area or human neurophysiology and neuroimaging, might reasonably described as a series of footnotes to Nunez. 50 years on, although NFT has arguably still not yet quite ‘hit the big time’ amongst neuroscientists and brain mappers at large, its influence on many major recent developments in the field are undeniable. As steady progress is made on techniques for imaging brain structure, connectivity, and activity dynamics, the contributions of NFT to discovery and understanding in human brain mapping research looks set to increase at a steady clip.
In this introductory lecture, I offer a largely non-technical survey of the key conceptual pillars of neural field theory (NFT), following the chronology of ideas from the 1940s through to 2025 and beyond. The aim is to lay foundations and set the scene for subsequent talks in the workshop, where the core mathematics, neurobiology, and concrete cutting-edge applications of NFT are discussed at length. 

Neural Field Theory (NFT) represents a powerful mathematical framework for understanding brain dynamics, yet its practical implementation often poses challenges for researchers. In this educational course, I will lead two hands-on sessions that bridge theoretical concepts with practical applications using interactive Python notebooks in Google Colab. Building on the success of our previous educational courses at OHBM 2023 (https://griffithslab.github.io/OHBM-whole-brain-modelling-course/) and OHBM 2024 (https://griffithslab.github.io/OHBM2024-educational-course/), these sessions will provide participants with hands-on experience in implementing and exploring NFT concepts.

The first hands-on session will focus on fundamental NFT implementations, following the morning's theoretical foundations. Participants will learn to construct basic neural field models, integrate them with connectome data, and visualize the resulting dynamics. The second session will build upon this foundation to explore more advanced applications, including the modeling of brain rhythms and cortico-hippocampal interactions. Both sessions will utilize pre-prepared Jupyter notebooks, allowing participants to modify parameters and immediately observe their effects on brain dynamics.

These interactive sessions are designed to make NFT accessible to researchers from diverse backgrounds, from those new to computational modeling to experienced modelers looking to incorporate NFT into their work.
Following OHBM's commitment to open science and reproducibility, all materials will be freely available through Google Colab. Our previous workshops' materials remain openly accessible, demonstrating our ongoing commitment to the open-science initiative:

OHBM 2023 (Montreal):
https://drive.google.com/drive/folders/1Dnk5HyudcVPXVNT3l7E1FwW1FVIf01Lg

OHBM 2024 (Seoul):
https://drive.google.com/drive/folders/1uR-NMDuaQtJs0wJAwG8NKWQ9x5xbeysQ

Through these hands-on sessions, participants will gain practical experience with NFT implementations that they can directly apply to their own research questions, while contributing to a growing community of open-source computational neuroscience.