Using the Green's function to simplify and understand dendrites

Neurons are endowed with dendrites: tree-like structures that collect and transform inputs. These arborizations are believed to substantially enhance the computational repertoire of neurons. While it has long been known that dendrites are not iso-potential units, only in the last few decades it was shown experimentally that dendritic branches can transform local inputs in a non-linear fashion. This finding led to the subunit hypothesis, which states that within the dendritic tree, inputs arriving in one branch are transformed non-linearly and independently from what happens in other branches. Recent progress in experimental recording techniques shows that this localized dendritic integration contributes to shaping behavior. While it is generally accepted that the dendritic tree induces multiple subunits, many questions remain unanswered. For instance, it is not known how much separation there needs to be between different branches to be able to function as subunits. Consequently, there is no information on how many subunits can coexist along a dendritic arborization. It is also not known what the input-output relation of these subunits would be, or whether these subunits can be modified by input patterns. As a consequence, assessing the effects of dendrites on the workings of networks of neurons remains mere guesswork. During this work, we choose a theory-driven approach to advance our knowledge about dendrites. Theory can help us understand dendrites by deriving accurate, but conceptually simple models of dendrites that still capture their main computational effects. These models can then be analyzed and fully understood, which in turn teaches us how actual dendrites function computationally. Such simple models typically require less computer operations to simulate than highly detailed dendrite models. Hence, they may also increase the speed of network simulations that incorporate dendrites. The Green's function forms the basis for our theory driven approach. We first explored whether it could be used to reduce the cost of simulating dendrite models. One mathematically interesting finding in this regard is that, because this function is defined on a tree graph, the number of equations can be reduced drastically. Nevertheless, we were forced to conclude that reducing dendrites in this way does not yield new information about the subunit hypothesis. We then focused our attention on another way of decomposing the Green's function. We found that the dendrite model obtained in this way reveals much information on the dendritic subunits. In particular, we found that the occurrence of subunits is well predicted by the ratio of input over transfer impedance in dendrites. This allowed us to estimate the number of subunits that can coexist on dendritic trees. We also found that this ratio can be modified by other inputs, in particular shunting conductances, so that the number of subunits on a dendritic tree can be modified dynamically. We finally were abl