Extending Cable Theory to Heterogeneous Dendrites

Dendrites exhibit many types of electrical and morphological heterogeneities at the scale of a few micrometers. Models of neurons, even detailed models, rarely consider such heterogeneities. Small scale fluctuations in the membrane conductances and the diameter of dendrites are generally disregarded and spines merely incorporated into the dendritic shaft. Using the two scales method known as homogenization, we establish explicit expressions for the small scale fluctuations of the membrane voltage, and we derive the cable equation satisfied by the voltage, when these fluctuations are averaged out. We establish rigorously under which conditions an heterogeneous dendrite can be approximated by an homogeneous cable. We consider different distributions of synapses, orderly or random, on a passive dendrite, and we investigate when replacing excitatory and inhibitory synaptic conductances by their local averages leads to a small error in the voltage. This indicates in which regimes the approximations made in compartmental models are justified. We extend these results to active membranes endowed with voltagedependent conductances or NMDA receptors. Then, we show that a spiny dendrite behaves as a smooth dendrite when the conductance of the spine neck is large compared to the conductance of the synapses impinging on the spine head, but also in the opposite situation, when it is small compared to the synaptic conductance. In this last regime, the only effect of spines is to deliver an effective synaptic current to the dendrite. This suggests that pedunculated and stubby spines might play complementary roles in synaptic integration. Finally, we analyze how varicosities affect voltage diffusion in dendrites and discuss their impact on the spatiotemporal integration of synaptic input