The Information Theory of Developmental Pruning: Optimizing Global Network Architecture Using Local Synaptic Rules

Another paper from the Hennig lab is out, this one is from Carolin Scholl's master's thesis. Once again, we used an artificial neural network to get intuition about biology. The paper is on BioRiv, and you can also get the PDF here. 

Limit of an infinite chain of first-order exponential filters

First-order exponential filter

The simplest model how the voltage $x$ at a synapse responds to input $u$ is a first-order filter:

$$\tau \dot x = -x + u.$$

This corresponds to convolving signal $u(t)$ with exponential filter $\operatorname H(t) \exp(-t/\tau)$, where $\operatorname H(\cdot)$ is the Heaviside step function:

$$\begin{aligned}x(t) &= h(t) * u(t)\\h(t)&=\operatorname H(t) \exp(-t/\tau).\end{aligned}$$

The alpha function

A first-order filter has a discontinuous jump in response to an abrupt inputs (like spikes). A more realistic response is the "alpha function"  $t\cdot \exp(-t)$. The alpha function can be obtained by convolving two first decay functions (i.e. chaining together two first-order filters):