Employing the zeta-transform to optimize the calculation of the synaptic conductance of NMDA and other synaptic channels in network simulations

Calculation of the total conductance change induced by multiple synapses at a given membrane compartment remains one of the most time-consuming processes in biophysically realistic neural network simulations. Here we show that this calculation can be achieved in a highly efficient way even for multiply converging synapses with different delays by means of the zeta-transform. Using the example of an NMDA synapse, we show that every update of the total conductance is achieved by an iterative process requiring at most three recent multiplications, which together need only the history values from the two most recent iterations. A major advantage is that this small computational load is independent of the number of synapses simulated. A benchmark comparison to other techniques demonstrates superior performance of the zeta-transform. Nonvoltage-dependent synaptic channels can be treated similarly (Olshausen, 1990; Brettle & Niebur, 1994), and the technique can also be generalized to other synaptic channels.