1. A novel approach for analyzing transients in passive structures called ''the method of moments'' is introduced. It provides, as a special case, an analytic method for calculating the time delay and speed of propagation of electrical signals in any passive dendritic tree without the need for numerical simulations. 2. 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, I, at point y[t(I)(y)] and the centroid of the transient voltage response, V, at point x [t(v)(x)]. The TD measured at the input point is nonzero and is called the local delay (LD). Propagation delay, PD(y, x), is then defined at TD(y, x) -LD(y) whereas the net dendritic delay, NDD(y, 0), of an input point, y, is defined as TD(y, 0) - LD(0), where 0 is the target point typically the soma. The signal velocity at a point x0 in the tree, theta(x0), is defined as /1/(dt(v)(x)/dx)/x=x0. 3. With the use of these definitions several properties od dendritic delay exists. First, the delay between any two points in a given tree is independent of the properties (shape and duration) of the transient current input. Second, the velocity of the signal at any given point (y) in a given direction from (y) does not depend on the morphology of the tree ''behind'' the signal, and of the input location. Third, TD(y, x) = TD(x, y), for any two points, x, y. 4. Two additional properties are useful for efficiently calculating delays in arbitrary passive trees. 1) The subtrees connected at the ends of any dendritic segment can each be functionally lumped into an equivalent isopotential R-C compartment. 2) The local delay at any given point (y) in a tree is the mean of the local delays of the separate structures (subtrees) connected at y, weighted by the relative input conductance of the corresponding subtrees. 5. Because the definitions for delays utilize difference between centroids, the local delay and the total delay can be interpreted as measures for the time window in which synaptic inputs affect the voltage response at a target/decision point. Large LD or TD is closely associated with a relatively wide time window, whereas small LD or TD imply that inputs have to be well synchronized to affect the decision point. the net dendritic delay may be interpreted as the cost (in terms of delay) of moving a synapse away from the target point. When this target point is the soma, the NDD is a rough measure for the contribution of the dendritic morphology to the overall delay introduced by the neuron. 6. The local delay (also TD) in an isopotential isolated soma is tau, the time constant of the membrane (R(m)C(m)), whereas the LD in an infinite cylinder is tau/2. In finite cylinders with both ends sealed, the TD from end to end is always larger than tau. When an isopotential soma with the same membrane properties is coupled to one end of the cylinder, the LD at any point is reduced, and the TD from any point to the soma is increased as compared with the corresponding point in the cylinder without a soma. As the soma size increases (rho(infinity) decreases), the LD at any given point decreases, and the TD from this point to the soma increases. 7. The velocity (theta) in an infinite cylinder is 2lambda/tau. In a semi-in-finite cylinder witha sealed end at its origin, theta is close to 2lambda.tau when the signal is electrically far from the boundary. As the signal approaches the origin, theta first decreases below this value then increases to infinity at the boundary. With a soma lumped at the origin, the velocity of the signal propagating toward the soma may first increase then decrease, or vice versa, or it may increase (or decrease) monotonically, depending on the size and membrane properties of the soma. Similar types of behavior are found in cylinders with a step change in their diameter. 8. In dendritic trees that are equivalent to a single cylinder, the TD from any input site to the soma is identical to the total delay in the equivalent cylinder for an input applied at the same electrotonic distance from the soma. The LD at any point in the full tree, however, is shorter than the LD in the corresponding input point in the cylinder. The LD at distal arbors steeply decreases and theta increases as a function of the order of branching. 9. In real dendritic trees with uniform R(m), the total delay between the synaptic input and the somatic voltage response is of the order of tau. In neuron models with a soma shunt (i.e., low somatic R(m)), this delay can be tree times the system time constant (tau0). In both models the local delay (which is a measure for the speed of electrical communication between adjacent synapses) at distal dendritic arbors is of the order of 0.1tau. Consequently, exact timing (synchronization) between inputs is critical for local dendritic computations (e.g., for triggering plastic processes) and is less important for the input-output (dendrites-to-axon0 function of the neuron. 10. Massive asynchronous background synaptic activity changes dynamically the dendritic delay as well as the temporal resolution of the tree. With increased background synaptic activity, the delays are reduced and the tree becomes more sensitive to the exact timing of its inputs. For example, without background synaptic activity, the net delay contributed by the dendrites (NDD) in a modeled layer 5 cortical pyramidal cell is 10-17 ms for distal apical arbors and approximately 1.5 ms for the basal dendrites (assuming tau = 20 ms). With background activity of 2 spikes/s in each of the 5,000 synapses that may contact this cell, the NDD is reduced by almost twofold (6-10 ms) for the apical arbors and by 15% (approximately 1.3 ms) for the basal arbors. 11. Excluding electrically distant dendritic locations, such as distal apical arbors of pyramidal cells, the NDD of a dendritic input is small compared with the local delay at the soma. The consequence is that placing the synapse at the dendrite rather than at the soma has only a minor effect on the time window for input integration at the soma. Furthermore, for proximal and intermediate inputs (e.g., on basal dendrites and proximal apical oblique dendrites of pyramidal cells) the time integral (but not the peak) of the resultant somatic voltage response is roughly the same as for a direct somatic input. We conclude that for the soma output, the location of the excitatory inputs at the tree is not very important. However, for decision points at the dendrites (e.g., where plastic processes may be triggered or where dendrodendritic synapses be activated), the localization and timing of inputs is very important. For these computations, electrically adjacent and well-synchronized inputs form the significant functional input. 12. the analytic treatment of the passive case presented here should serve as a reference case and a trigger for a study on the effect of the various dendritic nonlinearities, both synaptic and voltage dependent, on the problem of delay and synchronization in dendrites.
