$ \newcommand{\RR}{\mathbb{R}} \newcommand{\NN}{\mathbb{N}} \newcommand{\OO}{\mathcal{O}} \newcommand{\mathcow}{\OO} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\CC}{\mathbb{C}} \newcommand{\KK}{\mathbb{K}} \newcommand{\PP}{\mathcal{P}} \newcommand{\TT}{\mathcal{T}} \newcommand{\BB}{\mathcal{B}} \newcommand{\LL}{\mathcal{L}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\veca}{\vec{a}} \newcommand{\vecb}{\vec{b}} \newcommand{\vecd}{\vec{d}} \newcommand{\vece}{\vec{e}} \newcommand{\vecf}{\vec{f}} \newcommand{\vecn}{\vec{n}} \newcommand{\vecp}{\vec{p}} \newcommand{\vecr}{\vec{r}} \newcommand{\vecu}{\vec{u}} \newcommand{\vecv}{\vec{v}} \newcommand{\vecx}{\vec{x}} \newcommand{\vecy}{\vec{y}} \newcommand{\vecz}{\vec{z}} \renewcommand{\vec}[1]{\mathbf{#1}} $ Shaw Research Notes

July 15th, 2021

We derive periodic solutions to the adaptive model. This derivation has yet to be verified numerically. It appears that the period is a free parameter. The limitations remain to be determined.

To-Do List Snapshot

Periodic Soluions

In characteristic coordinates $\xi = x - ct$ our adaptive model becomes \begin{align*} -c \mu U_\xi &= - U + w \ast f(U-A) \\ -c \alpha A_\xi &= -A + \gamma f(U-A). \end{align*} We will use the exponential weight function $w(x,y) = \tfrac{1}{2} e^{-|x-y|}$ and a Heaviside firing-rate function with threshold $\theta$.

We will assume that the solution is periodic with period $\lambda$, and that on the periodic domain, there is a single connected active region of width $\Delta$. Without loss of generality, we will say that the active region is $(-\Delta, 0)$ in the interval $(-\Delta, \lambda-\Delta)$. The active regions can then be described by the set $\bigcup\limits_{i=-\infty}^\infty (i \lambda - \Delta, i \lambda)$.

For $\xi \in (-\lambda, 0)$, this gives us the coupled system of ODEs \begin{align*} -c\mu U_\xi &= -U + \sum_{i=-\infty}^{-1} \int_{i\lambda - \Delta}^{i \lambda} \frac{1}{2} e^{-|\xi - y|} \ dy + \sum_{i=1}^{\infty} \int_{i\lambda - \Delta}^{i \lambda} \frac{1}{2} e^{-|\xi - y|} \ dy + \int_{-\Delta}^0 \tfrac{1}{2} e^{-|\xi - y|} H(U(y) - A(y) - \theta) \ dy \\ -c \alpha A_\xi &= -A + \gamma H(U-A-\theta). \end{align*}

Simplifying the first equation we have \begin{align*} -c\mu U_\xi &= -U + \sum_{i=-\infty}^{-1} \int_{i\lambda - \Delta}^{i \lambda} \frac{1}{2} e^{-\xi}e^{y} \ dy + \sum_{i=1}^{\infty} \int_{i\lambda - \Delta}^{i \lambda} \frac{1}{2} e^{\xi} e^{-y} \ dy + \begin{cases} \int_{-\Delta}^\xi \frac{1}{2} e^{-\xi}e^{y} \ dy + \int_{\xi}^0 \frac{1}{2} e^{\xi}e^{-y} \ dy, & \xi < 0 \\ \int_{- \Delta}^{0} \frac{1}{2} e^{-\xi}e^{y} \ dy, & 0 \le \xi \end{cases}\\ &= -U + \frac{1}{2} e^{-\xi}\sum_{i=-\infty}^{-1} \left(e^{i \lambda} - e^{i\lambda - \Delta} \right) + \frac{1}{2} e^{\xi} \sum_{i=1}^{\infty} \left(e^{\Delta - i\lambda} - e^{-i \lambda} \right) + \begin{cases} \frac{1}{2} e^{\xi} - \frac{1}{2}e^{-\Delta}e^{-\xi}, & \xi < 0 \\ \frac{1}{2} \left(1 - e^{-\Delta} \right) e^{-\xi}, & 0\le \xi \end{cases}\\ &= -U + \frac{1}{2} e^{-\xi} (1 - e^{-\Delta}) \sum_{i=1}^{\infty} e^{-i \lambda} + \frac{1}{2} e^{\xi} (e^{\Delta} - 1) \sum_{i=1}^{\infty} e^{-i \lambda} + \begin{cases} \frac{1}{2} e^{\xi} - \frac{1}{2}e^{-\Delta}e^{-\xi}, & \xi < 0 \\ \frac{1}{2} \left(1 - e^{-\Delta} \right) e^{-\xi}, & 0\le \xi \end{cases}\\ &= -U + \frac{1}{2} e^{-\xi} (1 - e^{-\Delta}) \frac{e^{-\lambda}}{1 - e^{-\lambda}} + \frac{1}{2} e^{\xi} (e^{\Delta} - 1) \frac{e^{-\lambda}}{1 - e^{-\lambda}} + \begin{cases} \frac{1}{2} e^{\xi} - \frac{1}{2}e^{-\Delta}e^{-\xi}, & \xi < 0 \\ \frac{1}{2} \left(1 - e^{-\Delta} \right) e^{-\xi}, & 0\le \xi \end{cases}\\ &= -U + \begin{cases} \left( \frac{1}{2} (e^{\Delta} - 1) \frac{e^{-\lambda}}{1 - e^{-\lambda}} + \frac{1}{2} \right)e^{\xi} + \left( \frac{1}{2} (1 - e^{-\Delta}) \frac{e^{-\lambda}}{1 - e^{-\lambda}} - \frac{1}{2}e^{-\Delta} \right)e^{-\xi}, & \xi < 0 \\ \left( \frac{1}{2} (e^{\Delta} - 1) \frac{e^{-\lambda}}{1 - e^{-\lambda}} \right)e^{\xi} + \left( \frac{1}{2} (1 - e^{-\Delta}) \frac{e^{-\lambda}}{1 - e^{-\lambda}} + \tfrac{1}{2} (1-e^{-\Delta}) \right)e^{-\xi}, & 0 \le \xi. \end{cases}\\ &= -U + \begin{cases} \left( \frac{1}{2} (e^{\Delta} - 1) \frac{e^{-\lambda}}{1 - e^{-\lambda}} + \frac{1}{2} \right)e^{\xi} + \left( \frac{1}{2} (1 - e^{-\Delta}) \frac{e^{-\lambda}}{1 - e^{-\lambda}} - \frac{1}{2}e^{-\Delta} \right)e^{-\xi}, & \xi < 0 \\ \left( \frac{1}{2} (e^{\Delta} - 1) \frac{e^{-\lambda}}{1 - e^{-\lambda}} \right)e^{\xi} + \left( \frac{1}{2} (1 - e^{-\Delta}) \frac{1}{1 - e^{-\lambda}} \right)e^{-\xi}, & 0 \le \xi. \end{cases}\\ \end{align*}

Solving the system, we obtain the solution in terms of 7 unkown constants $$\begin{align*} U{\left(\xi \right)} &= \begin{cases} C_{1} e^{\frac{\xi}{\mu c}} + \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{\xi}}{- \mu c + 1} + \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{- \xi}}{\mu c + 1} & \text{for}\: \xi < 0 \\C_{2} e^{\frac{\xi}{\mu c}} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \xi}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{\xi}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)} & \text{otherwise} \end{cases}\\ A{\left(\xi \right)} &= \begin{cases} C_{3} e^{\frac{\xi}{\alpha c}} + \gamma & \text{for}\: \xi < 0 \\C_{4} e^{\frac{\xi}{\alpha c}} & \text{otherwise} \end{cases}. \end{align*}$$ Enforcing continuity of $U$ at $\xi=0$ and over the boundary $\xi = -\Delta, \lambda-\Delta$ we obtain the following equations for $C_1$ and $C_2$ $$ \left[\begin{matrix}1 & -1\\e^{- \frac{\Delta}{\mu c}} & - e^{\frac{- \Delta + \lambda}{\mu c}}\end{matrix}\right]\left[\begin{matrix}C_{1}\\C_{2}\end{matrix}\right]=\left[\begin{matrix}- \frac{\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}}{- \mu c + 1} + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}\\- \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{- \Delta}}{- \mu c + 1} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta - \lambda}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{- \Delta + \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}\end{matrix}\right] $$ which gives $$\begin{align*} C_{1} &= - \frac{\left(- \frac{\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}}{- \mu c + 1} + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}\right) e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}}}{- e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + e^{- \frac{\Delta}{\mu c}}} + \frac{- \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{- \Delta}}{- \mu c + 1} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta - \lambda}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{- \Delta + \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + e^{- \frac{\Delta}{\mu c}}}\\ C_{2} &= \frac{- \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{- \Delta}}{- \mu c + 1} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta - \lambda}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{- \Delta + \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + e^{- \frac{\Delta}{\mu c}}} - \frac{- \frac{\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}}{- \mu c + 1} + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{\frac{\Delta}{\mu c}} e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + 1}. \end{align*}$$ Enforcing continuity of $A$ at $\xi=0$ and over the boundary $\xi = -\Delta, \lambda-\Delta$ we obtain the following equations for $C_3$ and $C_4$ $$ \left[\begin{matrix}1 & -1\\e^{- \frac{\Delta}{\alpha c}} & - e^{\frac{- \Delta + \lambda}{\alpha c}}\end{matrix}\right]\left[\begin{matrix}C_{3}\\C_{4}\end{matrix}\right]=\left[\begin{matrix}- \gamma\\- \gamma\end{matrix}\right] $$ which gives $$\begin{align*} C_{3} &= \frac{\gamma e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}}}{- e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + e^{- \frac{\Delta}{\alpha c}}} - \frac{\gamma}{- e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + e^{- \frac{\Delta}{\alpha c}}}\\ C_{4} &= - \frac{\gamma}{- e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + e^{- \frac{\Delta}{\alpha c}}} + \frac{\gamma}{- e^{\frac{\Delta}{\alpha c}} e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + 1}. \end{align*}$$ We now have the solution in terms of $\lambda, \Delta$ and $c$. Enforcing threshold crossing of $U-A$ at $\xi = 0, -\Delta$ gives $$\begin{align*} 0 &= \frac{\gamma}{- e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + e^{- \frac{\Delta}{\alpha c}}} - \frac{\gamma}{- e^{\frac{\Delta}{\alpha c}} e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + 1} - \theta + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} + \frac{- \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{- \Delta}}{- \mu c + 1} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta - \lambda}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{- \Delta + \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + e^{- \frac{\Delta}{\mu c}}} - \frac{- \frac{\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}}{- \mu c + 1} + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{\frac{\Delta}{\mu c}} e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}\\ 0 &= - \theta + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \left(- \frac{\gamma}{- e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + e^{- \frac{\Delta}{\alpha c}}} + \frac{\gamma}{- e^{\frac{\Delta}{\alpha c}} e^{- \frac{\Delta}{\alpha c} + \frac{\lambda}{\alpha c}} + 1}\right) e^{- \frac{\Delta}{\alpha c}} + \left(\frac{- \frac{\left(\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}\right) e^{- \Delta}}{- \mu c + 1} + \frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{\Delta - \lambda}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\left(\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}\right) e^{\Delta}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda} e^{- \Delta + \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + e^{- \frac{\Delta}{\mu c}}} - \frac{- \frac{\frac{1}{2} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}}}{- \mu c + 1} + \frac{\frac{1}{2} - \frac{e^{- \Delta}}{2}}{\left(1 - e^{- \lambda}\right) \left(\mu c + 1\right)} - \frac{\frac{\left(\frac{1}{2} - \frac{e^{- \Delta}}{2}\right) e^{- \lambda}}{1 - e^{- \lambda}} - \frac{e^{- \Delta}}{2}}{\mu c + 1} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}}{- e^{\frac{\Delta}{\mu c}} e^{- \frac{\Delta}{\mu c} + \frac{\lambda}{\mu c}} + 1}\right) e^{- \frac{\Delta}{\mu c}} + \frac{\left(\frac{e^{\Delta}}{2} - \frac{1}{2}\right) e^{- \Delta} e^{- \lambda}}{\left(1 - e^{- \lambda}\right) \left(- \mu c + 1\right)}. \end{align*}$$