May 20th, 2021

I examined two models incorporating negative feedback that allow for traveling pulses: the Pinto-Ermentrout model (reference: Kilpatrick et al. 2008) and the Kilpatrick-Bressloff model (reference: Kilpatrick & Bressloff 2010). I have simulated each numerically and found parameters and initial conditions that induce traveling pulses. My pulse-profiles do not seem to quite match the profiles in Kilpatrick & Bressloff 2010 Figures 2 and 4. I suspect a small bug in my implementation.

To-Do List Snapshot

Pinto-Ermentrout Model

The Pinto-Ermentrout model is given in Kilpatrick et al. 2008 by $$\begin{align*} \tau_m u_t &= -u + \int_{-\infty}^{\infty} w(x,x^\prime) f( u(x^\prime,t) ) dx^\prime - \beta v \\ \frac{1}{\alpha} v_t &= -v + u \end{align*}$$ where $u$ is a measure of synaptic activity, $v$ is a local negative-feedback mechanism, $w$ describes the connectivity between neurons, $f$ is the firing-rate function, and $\tau_m, \beta,$ and $\alpha$ are constants.

Below, we have chosen $w(x,x^\prime) = \frac{1}{2}\exp(|x-x^\prime|)$, $f(u) = H(u - 0.2)$ where $H$ is the Heaviside function. The constants are $\alpha = 0.04$, $\beta = 2$, and $\tau_m = 1$. The initial conditions for both $u$ and $v$ are cosine bells, centered at $x=0$ and $x=-\pi$ respectively. The profile of $u$ seems to match the dashed line in Figure 6 of Kilpatrick et al. 2008.

The Kilpatrick-Bressloff Model

Kilpatrick & Bressloff 2010 use a model incorporating a "hyper-polarizing adaptation current" (similar to the negative-feedback variable $v$ from the Pinto-Ermentrout model above) and also a synaptic scaling factor that represents available synaptic resources. The model is given by $$\begin{align*} \mu u_t &= -u + \int_{-\infty}^\infty w(x,x^\prime) q(x^\prime,t) f( u(x^\prime,t) - a(x^\prime,t)) \ dx^\prime \\ q_t &= \frac{1 - q}{\alpha} - \beta q f(u - a) \\ \epsilon a_t &= -a + \gamma f(u - a) \end{align*}$$ where $u$ is the synaptic activity, $q$ is the available synaptic resources, and $a$ is the adaptation current (negative feed-back).

Again we choose $w(x,x^\prime) = \frac{1}{2}\exp(|x-x^\prime|)$, $f(u) = H(u - 0.2)$, and the constants $\alpha = 20, \beta = 0.2, \epsilon = 5, \gamma = 0.05,$ and $\mu = 1$. For the initial conditions, we take $u$ to be a cosine bell with width 20, $a=0$ and $q=1$. The simulation below does exhibit traveling pulses, but they do not quite match the profile of Figure 2 in Kilpatrick & Bressloff 2010. In the limit as $x \to -\infty$ $u \to 0.2$ instead of $0.15$. I'm not sure why.

Changing the parameters to $\alpha = 20, \beta = 0.4, \epsilon = 5, \gamma = 0.1,$ and $\mu = 1$; we also exhibit a traveling pulse, close to but Figure 4 in Kilpatrick & Bressloff 2010. This profile again is close, but is it does not appear to dip below $u = 0$ near the trailing edge. I'm also not sure why.

Questions

Meeting Notes

For simplicity of analysis, I will remove the synaptic depression $q$ from the Kilpatrick and Bressloff 2010 (KB10) model. This may necessitate increasing $\gamma$ in order to allow for traveling pulse solutions. We discussed three possible topics going forward: wave-train analysis, wave-responses to stimuli, and colliding pulses.

Wave-Train Analysis

Looking for non-trivial periodic solutions to KB10 could be interesting. In particular, what is the largest frequency that allows for non-trivial solutions. I think I read the phrase "wave-train" in one of the papers. Perhaps start with a literature review to see what has been done and with which models.

Wave-Response Function

Similar to Kilpatrick and Ermentrout 2012, examine the effects of stimuli on the KB10 model.

Colliding Pulses

These seems to be the least interesting idea, but should be easy enough to simulate. Since stable pulses are unique up to direction and translation invariance, the only way they can collide is if they are traveling in opposite directions. We expect that they will simply annihilate, but it should be easy enough to simulate.