deletions | additions       diff --git a/section_Thresholding_RateSAIL_label_thresh__.tex b/section_Thresholding_RateSAIL_label_thresh__.tex  index 59b38aa..8846371 100644  --- a/section_Thresholding_RateSAIL_label_thresh__.tex  +++ b/section_Thresholding_RateSAIL_label_thresh__.tex 
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y_i &= 1 \text{ if } u_i > \theta_i \text{ else } 0  \end{split}  \end{equation}  which are leakiness, linear filtering, and inhibition. $y_j$ is the instantaneous spiking variable for neuron $i$. This circuit may not always descend the objective function, as we can show. Given the analog optimization problem in \ref{desc}, there are a number of different ways to instantiate a LIF circuit to approximate inference. The negative gradient can be interpreted as the driving force in the circuit  \begin{equation}  \frac{du_i}{dt} = -\frac{\partial E(a|X; \Phi, W, \theta)}{\partial u_i} = -\frac{\partial E(a|X; \Phi, W, \theta)}{\partial a_i} \frac{\partial a_i}{\partial u_i}  \end{equation}