Control of traveling waves in adaptive neural fields

Math Bio Seminar

Sage Shaw - Aug 29^th, 2023

Outline

Neural field model
Traveling wave solutions
The wave response
Entrainment
Apparent Motion Entrainment

Neural Field Model

Biological Neural Networks

Image courtesy of Heather Cihak.

Retinotopic Map

Primary visual area (V1)
Sensory areas have spatially organized topologies

Dougherty et al. (2003)

Neural Field Models

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

Synaptic Depression

Rapid firing depletes pre-synaptic resources.

Model

\begin{align*} \tau_u \frac{\partial}{\partial t}\underbrace{u(x,t)}_{\text{Activity}} &= -u + \underbrace{\overbrace{w}^{\substack{\text{network}\\\text{connectivity}\\\text{kernel}}} \ast \big( q f[u] \big)}_{\substack{\text{network}\\\text{stimulation}}} \\ \tau_q \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*}

Model

$f[u] = H(u - \theta)$ - All or nothing firing-rate function.
$w(|x-y|) = \frac{1}{2}e^{-|x-y|}$ - network connectivity kernel
$\frac{1}{1+\beta} = \gamma \in (0, 1]$ - Relative timescale of synaptic depletion.

1D Neural Fields

Progressive fronts
Regressive fronts
Pulses

Model

Traveling wave solutions

Front solutions

$\xi = x - ct$
Active region: $(-\infty, 0)$
Restrict to $c > 0$

Linearizes equations
Decouples $q$ from $u$
$U(-\infty) = \gamma > \theta$

$$ \theta = \frac{\gamma + c\tau_q\gamma}{2(1+c\tau_q\gamma)(1+c\tau_u)} $$

Front Bifurcations

Pulse solutions

$\xi = x - ct$
Active region: $(-\Delta, 0)$

Linearizes equations
Decouples $q$ from $u$

Two consistency equations for $c$ and $\Delta$.

Pulse Speed and Width

Wave response

Correcting Position

Position encoded as a pulse
Must be corrected

Spatially homogeneous perturbation

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

Collect the $\mathcal{O}(\varepsilon)$ terms

$$\begin{align*} \underbrace{\begin{bmatrix}\tau_u & 0 \\ 0 & \tau_q\end{bmatrix}}_{T} \begin{bmatrix}\phi \\ \psi \end{bmatrix}_t + \mathcal{L}\begin{bmatrix}\phi \\ \psi \end{bmatrix} &= \begin{bmatrix} I_u + \tau_u U' \nu' \\ I_q + \tau_q Q' \nu ' \end{bmatrix} \end{align*}$$ $$ \mathcal{L}(\vec{v}) = \vec{v} - cT \vec{v} + \begin{bmatrix} -w Q f'(U) * \cdot & -w f(U) * \cdot \\ \beta Q f'(U) & \beta f(U) \end{bmatrix} \vec{v} $$

Bounded solutions exist if the inhomogeneity is orthogonal to $\mathcal{N}\{\mathcal{L^*}\}$. For $(v_1, v_2) \in \mathcal{N}\{\mathcal{L^*}\}$ $$\begin{align*} -c \tau_u v_1' &= v_1 - Qf'(U) \int w(y,\xi) v_1(y) \ dy + \beta Qf'(U) v_2 \\ -c \tau_q v_2' &= v_2 - f(U) \int w(y, \xi) v_1(y) \ dy + \beta f(U) v_2. \end{align*}$$

Wave response function

$$ \nu(t) = - \frac{\int_\mathbb{R} v_1 \int_0^t I_u(\xi, \tau) \ d\tau + v_2 \int_0^t I_q(\xi, \tau) \ d\tau \ d\xi}{\int_\mathbb{R} \tau_u U' v_1 + \tau_q Q' v_2 \ d\xi} $$

Spatially Homogeneous perturbation

$$ \varepsilon I_u = \varepsilon \delta(t - 1) $$

Spatially localized perturbation

Entrainment

Tracking a moving stimulus

old expasion

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

new expansion

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

old wave response

$$ \frac{\partial}{\partial t}\nu = \frac{ \langle v_1 ,I_u(\xi, \tau)\rangle + \langle v_2, I_q(\xi, \tau)\rangle }{-\tau_u \langle v_1, U' \rangle - \tau_q \langle v_2, Q' \rangle} $$

new wave response

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