Approximating the effect of inputs and cortical geometry on wave propagation in neural fields Neural field models are systems of integrodifferential equations that model the dynamics of large-scale biological neural networks. We consider a neural field system incorporating a biologically realistic form of negative feedback called synaptic depression which gives rise to spatially localized traveling pulse (or bump) solutions capable of encoding the position of moving of objects in an animal's environment. Our asymptotic analysis characterizes the response of these marginally stable solutions to transient stimuli by quantifying the amount by which the stimuli shift the solution. We apply this analysis to a stimulus meant to model the apparent motion illusion - a phenomenon where a subject perceives a series of still objects presented in succession as a single object moving through space. Our analysis yields testable predictions of the perception of motion as a function of effective stimulus speed and stimulus amplitude. To assess the effects of cortical curvature on traveling wave solutions we develop a novel high-order neural field solver. Our solver is a method-of-lines collocation scheme using radial basis function interpolation and radial basis function quadrature formula. We establish theoretical convergence rates using the recently developed Projection Method framework for neural field solvers and present numerical evidence of convergence as well. Finally, we present a series of experiments on non-trivial cortical geometries, including a realistic cortical mesh, demonstrating the power and flexibility of this method. Neural field models are non-linear systems of integrodifferential equations intended to model large-scale neural activity. There is growing interest in identifying efficient and accurate schemes for simulating neural field models as they can capture activity dynamics that spread across wide swathes of tissues and that reflect highly complex neural architecture. Recently, a framework has been put forth for analyzing neural field solvers (Avitabile 2023) that separates the error due to the numerical representation of the solution (projection) and the error due to approximating the integral operator (quadrature). In this talk, we will discuss using Radial Basis Function (RBF) interpolation and quadrature methods to combine and simplify this error analysis and to create efficient, robust, and high-order-accurate neural field solvers. We will demonstrate their utility in solving neural fields over 2D manifolds and discuss their application to modeling cortical spreading depression.
June 12th, 2024 - ICNMS24: Radial Basis Function Methods for Neural Field Models
Neural field models are non-linear systems of integro-differential equations intended to model large-scale neural activity. There is growing interest in identifying efficient and accurate schemes for simulating neural field models as they can capture activity dynamics that spread across wide swathes of tissues and that reflect highly complex neural architecture. Recently, a framework has been put forth for analyzing neural field solvers (Avitable 2023) that separates the error due to the numerical representation of the solution (projection) and the error due to approximating the integral operator (quadrature). In this talk, we will discuss using Radial Basis Function (RBF) interpolation and quadrature methods to combine and simplify this error analysis and to create efficient, robust, and high-order-accurate neural field solvers. We will demonstrate their utility in solving neural fields over 2D manifolds and discuss their application to modeling cortical spreading depression.
Show recent results in neural field simulations on planar domains using radial basis function quadrature.
March 15th, 2024 - Poster: Radial Basis Function Techniques for Neural Field Models
My poster presentation for APPM Recruitment 2024.
October 26th, 2023 - Python in Scientific Computing: Why and How?
Python is one of the more popular scripting languages in scientific computing and data science, along with MATLAB, R, and Julia. An interesting difference is that Python was not designed with scientific computing in mind. Rather, it is a general purpose language designed to be simple and flexible. Strangely, I find that it is uncommon for scientists to take advantage of the features that make Python so successful, and instead to write procedural codes with an overreliance on the array data structure. In this talk I will focus on one feature in particular: generators (also called streams). Generators are built into Python syntax, and they are a natural way to represent mathematical sequences. I will explain what generators are, how they are baked into the Python syntax, how I use them, and why you should use them in your scientific Python codes.
We examine traveling wave solutions of a neural field model (integro-differential system) incorporating synaptic depression, allowing for biologically realistic traveling pulse solutions similar to those observed experimentally. We use these pulses as a biologically plausible model for visual motion processing, and develop an asymptotic wave response function characterizing how traveling pulse solutions respond to perturbative stimuli. For some spatially localized moving stimuli, traveling pulse solutions can accelerate to match the speed and location of the stimulus - a phenomenon known as entrainment. Generally speaking, traveling pulse solutions will entrain to spatially localized moving stimuli if the stimuli are sufficiently weak and sufficiently slow. We use our wave response function to identify this speed-magnitude threshold to first order. Finally, we apply our wave response function to a stimulus meant to evoke the apparent motion illusion - a phenomenon where a sequence of stationary stimuli are presented and motion is perceived.
August 29th, 2023 - Math-bio Seminar
We examine traveling wave solutions of a neural field model (integro-differential system) incorporating synaptic depression, allowing for biologically realistic traveling pulse solutions similar to those observed experimentally. We use these pulses as a biologically plausible model for visual motion processing, and develop an asymptotic wave response function characterizing how traveling pulse solutions respond to perturbative stimuli. For some spatially localized moving stimuli, traveling pulse solutions can accelerate to match the speed and location of the stimulus - a phenomenon known as entrainment. Generally speaking, traveling pulse solutions will entrain to spatially localized moving stimuli if the stimuli are sufficiently weak and sufficiently slow. We use our wave response function to identify this speed-magnitude threshold to first order. Finally, we apply our wave response function to a stimulus meant to evoke the apparent motion illusion - a phenomenon where a sequence of stationary stimuli are presented and motion is perceived.
August 16th, 2023 - Group Update
I show our asymptotic entrainment threshold agrees with simulations for the apparent motion stimulus.
I discuss the new formula for the wave response and the entrainment threshold.
May 16th, 2023 - SIADS 2023 Poster
My poster for the SIAD dynamical systems conference in 2023.
April 4th, 2023 - Comprehensive Exam Presentation
My presentation for my comprehenisve exam. Also, my thesis proposal.
March 10th, 2023 - APPM Recruitment Poster
My poster (and suplemental material) for the APPM 2023 recruitment poster session.
April 26th, 2022 - Group Update
I discuss my current work in incorporating adaptation into a neural field model. I recap my failed attempt at incorporating hyper-polarizing adaptation current, and discuss my current approach incorporating a synaptic depression variable.
March 10th, 2022 - CSCI 5314 Paper Presentation
I present a 15 minute summary of Ge and Liu's 2021 paper Foraging behaviours lead to spatiotemporal self-similar dynamics in grazing ecosystems.
December 6th, 2021 - APPM 5370 Presentation
The wave response function in an adaptive neural field model
A very trimmed down version of the Math-bio seminar talk.
November 8th, 2021 - APPM Math-bio Seminar
The wave response function in an adaptive neural field model
Neural field models are integro-differential equations describing the propagation of neural activity through the brain. The simplest of neural field models describes a single spatial dimension and exhibits traveling-front solutions where activity propagates into inactive regions then remains active indefinitely. More realistic models incorporate adaptation, which allows active regions to eventually decay back to base line inactivity, resulting in traveling pulse solutions. In this talk we will examine one such model that incorporates a hyperpolarizing adaptation current, explore some of its properties. In particular we will examine the wave response function: the shift in position of traveling pulses due to small perturbations.
August 26th, 2021 - APPM Graduate Student Seminar
Functional Programming in Python
Functional programming (FP) is a declarative programming paradigm, in which functions are said to be "first-class citizens", and function composition is used to create complex procedures while maintaining modularity and extensibility. Proponents of FP say that it reduces errors, simplifies debugging, and makes programming more mathematical. Python supports programming in the functional paradigm. In this inaugural presentation of the APPM Graduate Student Seminar we explore the potential of the functional paradigm in scientific computing with specific examples implemented in Python.
July 8th, 2021 - Research Group
Update the research group on my current work, including the $\mathcal{O}(\varepsilon)$ approximation to the wave response function for the adaptive model.