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Approximating the effect of inputs and cortical geometry on wave propagation in neural fields

Sage Shaw - University of Colorado Boulder, USA

May 9th, 2025

Mathematical Models of Neural Encodings

Outline

  • Neural Field Models
  • One Dimension
    • Stimulus response
    • Entrainment
  • Two Dimension
    • Numerical Analysis: the Projection Method
    • RBF-QF method-of-lines solver for neural fields
    • Curvature showcase
  • Future Work

Neural Field Models

Biological Neural Networks

Image courtesy of Heather Cihak.

Neural Field Models

  • Organize neural populations on a line
  • Connectivity is determined by distance
  • Extend to a continuum limit

Model

\begin{align*} \tau_u \frac{\partial}{\partial t}\underbrace{u(t, x)}_{\text{Activity}} &= -u + \underbrace{ \int_\Omega \color{green}{w}(x, y) \quad \color{red}{f}[u(t, y)] \ d y}_{\substack{\text{network}\\\text{stimulation}}} \\ \end{align*}

Synaptic Depression

Rapid firing depletes pre-synaptic resources.

Model with Synaptic Depression

\begin{align*} \tau_u \frac{\partial}{\partial t} u(t, x) &= -u + \int_\Omega w(x, y) \quad \color{blue}{q}(y) \quad f[u(t, y)] \ d y \\ \underbrace{\tau_q}_{\text{slow}} \frac{\partial}{\partial t}\underbrace{q(x,t)}_{\substack{\text{Synaptic}\\\text{Efficacy}}} &= 1 - q - \underbrace{\beta}_{\substack{\text{rate of}\\\text{depletion}}} q \underbrace{f(u)}_{\substack{\text{firing-rate}\\\text{function}}} \end{align*}

1D Neural Fields

Pulse solutions if

$\frac{1}{1+ \beta} < \theta$

Stimulus Response

Correcting Position

  • Position encoded as a pulse
  • Must be corrected

Localized Pulse Stimulus

Asymptotic Approximation

add stimulus terms $$ \begin{align*} \tau_u u_t &= -u + w * (q f[u]) + \varepsilon I_u(x, t) \label{eqn:forced_u} \\ \tau_q q_t &= 1 - q - \beta q f[u] + \varepsilon I_q(x, t) \label{eqn:forced_q} \end{align*} $$

substitute with the expansion $$ \begin{align*} u(\xi, t) &= U\big( \xi - \varepsilon \nu(t) \big) + \varepsilon \phi\big(\xi - \varepsilon \nu(t), t\big) + \mathcal{O}(\varepsilon^2) \\ q(\xi, t) &= Q\big( \xi - \varepsilon \nu(t) \big) + \varepsilon \psi\big(\xi - \varepsilon \nu(t), t\big) + \mathcal{O}(\varepsilon^2) \end{align*} $$

Stimulus Response Function

$$ \frac{\partial}{\partial t}\nu = \frac{ \langle v_1 ,I_u(\xi + \color{blue}{\varepsilon \nu}, t)\rangle + \langle v_2, I_q(\xi +\color{blue}{\varepsilon \nu}, t)\rangle }{\underbrace{-\tau_u \langle v_1, U' \rangle - \tau_q \langle v_2, Q' \rangle}_{K}} $$

Stimulus Response

Entrainment

Tracking a moving stimulus

Tracking a moving stimulus

Entraining to a moving stimulus

Asymptotic threshold

$$ \Delta_c \lt \varepsilon \frac{c\tau_u}{K}$$

Asymptotic Entrainment Threshold

 

 

$ \Delta_c < \varepsilon \frac{c\tau_u}{K} \color{green}{\frac{T_\text{on}}{T_\text{on} + T_\text{off}}} $

Numerical Analysis

The Projection method

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)

  • True Solution
  • Collocation
  • Projection
  • Error
    • Projection
    • Quadrature
    • Time Integration
  • $\text{Error} \sim \mathcal{O}(n^{-\text{order}/\text{dim}})$

RBF-QF method-of-lines neural field solver

Radial Basis Function Quadrature Formulae (RBF-QF)

  • 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

The brain is not flat!

Extending to Manifolds

Reeger and Fornberg (2016)

Curvature Showcase

Curvature affects the trajectory

Future Work

Spreading Depression


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

Retinotopic Map


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

Scintillating Scotoma

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

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

shawsa.github.io

Slides:

Auxiliary Slides

Retinotopic Map

Dougherty et al. (2003)

Reaction Diffusion on surfaces

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

  • Surface operators: $\Delta_{\mathcal{M}}, \int_{\mathcal{M}} \cdot d \mu_{\mathcal{M}}$
  • Curvature 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

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

Radial Basis Funtion Interpolation

  • Function to approximate
  • Sample at scatterd nodes
  • $\phi_j(\vecx) = \Phi(||\vecx - \vecx_j||)$
  • $f(\vecx) \approx s(\vecx) = \sum_j c_j \phi_j(\vecx)$
  • $\text{Error} \to 0$ as $n \to \infty$

RBF Interpolation Properties

  • scattered nodes in any number of dimensions*
  • mesh-free*
  • arbitrary order of accuracy*
  • can be used to find
    • finite difference formulae
    • quadrature formulae

Radial Basis Function Quadrature Formulae

Experimental Results

Gaussian Test Functions

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.

Our Contributions

  • Neural Field Solver
    • High Order Accurate
    • Flexible Geometry
  • Simplifed Error Analysis
  • Adapt to Surfaces
  • Couple with CSD

Auxiliary Slides

shawsa.github.io

sage.shaw@colorado.edu

The Kilpatrick Lab

Prof. Zack Kilpatrick

Dr. Tahra Eissa

Sage Shaw

Noah Parks

Will Magrogan

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)