# Is summation linear in a passive membrane?

I am not too sure what it means for summation to be linear? I am running a simulation and as I decrease time for the second EPSP the amplitude decreases. Does summation being linear mean that there is a direct negative linear relationship between delay of onset and amplitude? I also noticed there is no change when I double or triple my conductance value.

I am just not too sure what the relationship is telling me.

## 1 Answer

For a membrane to satisfy the conditions for linear summation implies that when two EPSPs of amplitude $A$ and $B$ are summed at the integration zone, the resulting output signal will be of amplitude $A+B$. Since the signal dissipates due to time and length constants, increasing the delay between the two inputs means that the later input will be dissipated so $A+B$ will be smaller than it would have if the delay was shorter. Under this scenario, inputs $A$ and $B$ are still summed linearly. However, if the delay is shorter than the time constants or the inputs are delivered closer in space than the length constant, then you may get non-linear summation since the output signal at the integration zone will be larger than $A+B$.

Have a look here for more information (especially Figure 7):
Silver RA. Neuronal arithmetic. Nature reviews Neuroscience. 2010;11(7):474-489. doi:10.1038/nrn2864.