I've recently discovered a family of quadrature rules that I don't recognize and may be novel. It is an interpolation based quadrature that uses local piecewise polynomial interpolation. It is distinct from splines and from Newton-Cotes type quadratures. It works on arbitrary nodes. For equally spaced nodes it appears to be the trapezoidal rule, but with corrections near the boundary, similar to Gregory's method though it appears to be distinct.

We apply the new wave response differential equation to the problem of apparent motion and find and asymptotic prediction for the threshold of entrainment. The predicted threshold to entrain to an apparently moving square stimulus is the same as the threshold to entrain to an actually moving stimulus with amplitude multiplied by the proportion of time that the stimulus is present.

Zack had a clever idea for the asymptotic expansion. This new expansion allows us to get a differential equation for the wave response which can successfully predict entrainment for weak stimuli. We walk through the derivation in the scalar case (no synaptic depression) for a moving Heaviside stimulus. For the pulse regime in the synaptic depression model we also consider a moving Heaviside, and a moving square wave.

We apply our asymptotic wave response approximation to a moving localized stimulus. We determine that the asymptotic formulation cannot differentiate between entrainment and non-entrainment and thus a new formulation is required.

We find an asymptotic approximation to the wave response function for the traveling front case.

We find the wave response function for the model incorporating synaptic depression. We test our derivation for a spatially homogeneous delta-pulse in the traveling pulse regime.

We have begun exploring stability of perturbations to the adaptive model. We are stuck showing inconsistency in the case where the eigenvalues have positive real component.

Zack found an error in the derivation for report 2021-06-10. We start over here, and derive the adjoint and begin computation of the nullspace.

We examine the difference-of-exponentials weight function $w(x,y) = M_1e^{-|x-y|/\sigma_1} - M_2e^{-|x-y|/\sigma_2}$. We derive the traveling pulse solution and the asymptotic approximation to the wave response function for this weight function in the case of the Heaviside firing-rate. The results do not match simulation, but we believe the simulation is not accurate enough to properly compare, and needs to be improved by using a semi-analytic version. Another error was corrected from the 2021-06-10 report.

We highlight the difficulties in finding a stationary bump solution to the adaptive model with a weight-kernel of the form $w(x,y) = M_1 e^{-|x-y|} - M_2 e^{-|x-y|/\sigma}$.

We attempt to identify difficulties in convergence of the numerical scheme. Discontinuity of the forcing term makes this a difficult task. In particular, that the forcing term is not Lipshitz continuous precludes applying most related theorems. We add detail to the previously suggested algorithm and now have a working cusp-detection implementation.

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.

Derived the wave-response approximation for a spatially localized delta pulse stimulus, and compared to simulations. Prepared for the July 8^{th} presentation to the research group.

We have begun using a semi-analytic implementation for simulations. The reduced computational complexity allows for efficient higher resolution simulations.

We numerically verify the wave response calculated in the previous report and motivate the need for a higher order method.

We have found an asymptotic approximation of the wave reponse function, though there is more work to be done determining the dimension of the null-space of the adjoint operator. The example that we have been simulating certainly has a one-dimensional null-space, though the particulars of that problem allow for a solution to be found. Comparison to simulations has proven difficult due to the inaccuracies in measuring the wave speed and front locations. We have outlined these difficulties here and proposed several possible solutions.

We finished deriving the traveling pulse solution. We encountered a bug in SymPy's integration routine involving the Heaviside function and variable assumptions, which cause delays. We submitted a bug report and proceeded integrating the forcing term by hand. The choice of parameters for our simulation makes determining the pulse width and wave speed an ill-conditioned process. We began simulations for spatially homogeneous temporally pulsatile stimuli.

We simulate the modified KB10 model (perhaps we will call it the $\beta$-null regime of the KB10 model) and find parameters that appear to admit traveling pulse solutions. We analyzed equations 3.30-3.31 from Kilpatrick & Bressloff 2010 in this case, but they appear inconsistent with observations. Perhaps the assumption that $\beta \ne 0$ was essential in their derivation. We may have to re-derive the traveling-pulse solution from scratch.

I examined two models incorporating negative feedback that alow 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 susspect a small bug in my implementation.