Supplementary

Proposition 1. Membrane potentials without spike reset are computed as a convolution Ṽ We proceed our proof in two steps. In step 1, we unroll the discretized LIF difference equation (without reset) in time and in step 2, we show how this is equivalent to the proposed convolution. This implies equivalence between Equations 1 and 2 for t = k + 1 assuming equivalence between Equations 1 and 2 holds true for t = k. By the principle of induction, equivalence is established given that both the base case and inductive step hold true. This is identical to Equation 2 and (by step 1) identical to Equation 1.