$\newcommand{vecx}{\vec{x}}$ $\newcommand{vecy}{\vec{y}}$

Group Update

Sage Shaw - May 30th, 2024

Kilpatrick Lab

Radial Basis Function Quadrature for Neural Field Equations

  • Neural Field Model
  • Radial Basis Function Quadrature Formulae
    (RBF-QF)
  • Experimental Results
  • Next Steps

Motivating Research

Spreading Depression


Zandt, Haken, van Putten, and Markus (2015)

Retinotopic Map


Zandt, Haken, van Putten, and Markus (2015)

Scintillating Scotoma

Reaction Diffusion Model

$$\begin{align*} u_t &= \underbrace{u - \frac{1}{3}u^3}_{\text{excitable}} - \underbrace{v}_{\text{recovery}} + \underbrace{D\nabla^2 u}_{\text{Diffusion}} \\ \frac{1}{\varepsilon} v_t &= u + \beta + \underbrace{K\int H(u) d \Omega}_{\substack{\text{neurovascular}\\\text{feedback}}} \end{align*}$$

Markus A. Dahlem (2013)

Reaction Diffusion on surfaces

$$\begin{align*} u_t &= 3u - u^3 - v + D \Delta_{\mathcal{M}}u \\ \frac{1}{\varepsilon} v_t &= u + \beta + K \int_{\mathcal{M}} H(u) \ d \mu_{\mathcal{M}} \end{align*}$$

  • Surface operators: $\Delta_{\mathcal{M}}, \int_{\mathcal{M}} \cdot d \mu_{\mathcal{M}}$
  • Affects speed and stability of waves

Kneer, Scholl, Dahlem (2014)

Coupled neural field and diffusion equation

$$\begin{align*} v_t &= -v + w \ast s_p(v, k) + g_v \\ k_t &= \delta k_{xx} + g_k(s, s_p, a, b) + I \end{align*}$$
  • Neural field model
  • Coupled potassium concentration
  • Models both ignition and propagation of CSD

Baspinar et al. (2023)

A Turing Reaction Diffusion System using RBFs

$$\begin{align*} u_t &= \delta_u \Delta_{\mathcal{M}} u + \alpha(1-\tau_1 v^2) + v(1-\tau_2 u)\\ v_t &= \delta_v \Delta_{\mathcal{M}} v + \beta(1-\frac{\alpha\tau_1}{\beta} uv) + u(\gamma-\tau_2 v)\\ \end{align*}$$

Neural Field Model

Neural Field Model

$\partial_t \color{blue}{u}(t, \vecx) = -\color{blue}{u} + \int_{\Omega} \color{green}{w}(\vecx, \vecy) \color{red}{f}[\color{blue}{u}(\vecy)] d \vecy$

Recreation of Coombes et al. (2012)

  • $\color{blue}{u}(t, \vecx)$ - Activity
  • $\color{green}{w}(\vecx, \vecy)$ - Connectivity kernel
  • $\color{red}{f}[\color{blue}{u}]$ - non-linear firing rate function

Projection Method (Avitabile 2023)

scalar: $\partial_t u(t, \vecx) = -u + \int_\Omega w(\vecx, \vecy) f[u(t, \vecy)] \ d\vecy$
Banach: $\dot{U}(t) = -U + W(f[U])$
projected: $\dot{U}_n(t) = -U_n + W_n(f[U_n])$
quadrature: $\dot{\tilde{U}}_n(t) = -\tilde{U}_n + Q(f[\tilde{U}_n])$

Error = projection error + quadrature error

Radial Basis Function Quadrature Formulae

RBF-QF Goal:

  • Given a set of points $\{\vecx_i\} \subset \Omega$
  • find weights $\{w_i\}$
  • such that $\int_\Omega f \approx \sum w_i \ f(\vecx_i)$

RBF-QF Algorithm

  • choose quadrature nodes
  • partition domain
  • choose stencils
  • integrate RBF interpolant
  • sum over stencil and elements

RBF Interpolation Properties

  • scattered nodes in any number of dimensions*
  • mesh-free*
  • arbitrary order of accuracy*

Experimental Results

Gaussian Test Functions

Gaussian Test Functions

Quadrature Convergence

Testing manufactured solution

Convergence

Next Steps

  • Adapt to surfaces.
  • Incorporate cortical spreading depression (CSD).
  • Study the effects of realistic cortical curvature on CSD wave generation and propagation.

Thank you!

Questions?