Signal Delay in Passive Dendritic Trees

A novel approach for analyzing transients in passive structures is introduced. It provides, as a special case, an analytical method for calculating the signal delay in any passive dendritic tree. Total dendritic delay (TD) between two points (y,x) is defined as the difference between the centroid (the center of gravity) of the transient current input at point y and the centroid of the voltage response at point x. The TD for x = y is the local delay (LD) and the propagation delay, PD(y,x), is TD (y,x) — LD (y). With these definitions, the delay between any two points in a given tree is independent of the properties of the transient current input. Also, TD(y,x) = TD(x,y) for any two points, x,y. The local delay (also TD) in an isopotential isolated soma is τ, the time constant of the membrane whereas the LD in an infinite cylinder is τ/2. In finite cylinders coupled to a soma the TD from end-to-soma is always larger than τ. In dendritic trees equivalent to a single cylinder (Rall [1]), the TD from a given input site (X = x/λ) at an individual branch to the soma is identical to the total delay in the equivalent cylinder for current injected at the same X. The LD(X), however, is shorter in the full tree for any X ≠ 0. In real dendritic trees the total delay between the synaptic input and the voltage response at the soma is on the order of τ. However, electrical communication between adjacent synapses in distal arbors can be more than 10-times faster. Consequently, exact timing between inputs is critical for local dendritic computations and less important for the input-output function of the neuron. These results have important functional significance for both the input-output characteristics of neurons and for processes underlying learning and memory.