SIGNAL DELAY AND INPUT SYNCHRONIZATION IN PASSIVE DENDRITIC STRUCTURES

1. A novel approach for analyzing transients in passive structures cal led ''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 withou t the need for numerical simulations. 2. total dendritic delay (TD) be tween two points (y, x) is defined as the difference between the centr oid (the center of gravity) of the transient current input, I, at poin t 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 d efined 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 t arget point typically the soma. The signal velocity at a point x0 in t he tree, theta(x0), is defined as /1/(dt(v)(x)/dx)/x=x0. 3. With the u se 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 ''be hind'' 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 fu nctionally 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 lo cal delays of the separate structures (subtrees) connected at y, weigh ted by the relative input conductance of the corresponding subtrees. 5 . Because the definitions for delays utilize difference between centro ids, the local delay and the total delay can be interpreted as measure s for the time window in which synaptic inputs affect the voltage resp onse 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 poin t 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 infini te cylinder is tau/2. In finite cylinders with both ends sealed, the T D 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 cylind er, the LD at any point is reduced, and the TD from any point to the s oma is increased as compared with the corresponding point in the cylin der without a soma. As the soma size increases (rho(infinity) decrease s), 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 i s 2lambda/tau. In a semi-in-finite cylinder witha sealed end at its or igin, theta is close to 2lambda.tau when the signal is electrically fa r 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 propagat ing toward the soma may first increase then decrease, or vice versa, o r it may increase (or decrease) monotonically, depending on the size a nd membrane properties of the soma. Similar types of behavior are foun d in cylinders with a step change in their diameter. 8. In dendritic t rees that are equivalent to a single cylinder, the TD from any input s ite to the soma is identical to the total delay in the equivalent cyli nder for an input applied at the same electrotonic distance from the s oma. The LD at any point in the full tree, however, is shorter than th e LD in the corresponding input point in the cylinder. The LD at dista l arbors steeply decreases and theta increases as a function of the or der of branching. 9. In real dendritic trees with uniform R(m), the to tal 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 spee d of electrical communication between adjacent synapses) at distal den dritic arbors is of the order of 0.1tau. Consequently, exact timing (s ynchronization) between inputs is critical for local dendritic computa tions (e.g., for triggering plastic processes) and is less important f or the input-output (dendrites-to-axon0 function of the neuron. 10. Ma ssive asynchronous background synaptic activity changes dynamically th e dendritic delay as well as the temporal resolution of the tree. With increased background synaptic activity, the delays are reduced and th e tree becomes more sensitive to the exact timing of its inputs. For e xample, without background synaptic activity, the net delay contribute d 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 elec trically distant dendritic locations, such as distal apical arbors of pyramidal cells, the NDD of a dendritic input is small compared with t he local delay at the soma. The consequence is that placing the synaps e at the dendrite rather than at the soma has only a minor effect on t he time window for input integration at the soma. Furthermore, for pro ximal and intermediate inputs (e.g., on basal dendrites and proximal a pical oblique dendrites of pyramidal cells) the time integral (but not the peak) of the resultant somatic voltage response is roughly the sa me as for a direct somatic input. We conclude that for the soma output , the location of the excitatory inputs at the tree is not v