TY - JOUR
T1 - Signal delay and input synchronization in passive dendritic structures
AU - Agmon-Snir, H.
AU - Segev, I.
PY - 1993
Y1 - 1993
N2 - 1. A novel approach for analyzing transients in passive structures called 'the method of moments' is introduced. It provides, as a special case, an analytical 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 as 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, θ(x0), is defined as |1/(dt(V)(x)/dx|(x=λ0). 3. With the use of these definitions, several properties of dendritic delay exist. 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 τ, the time constant of the membrane (R(m)C(m)), whereas the LD in an infinite cylinder is τ/2. In finite cylinders with both ends sealed, the TD from end to end is always larger than τ. 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 (ρ(∞) decreases), the LD at any given point decreases, and the TD from this point to the soma increases. 7. The velocity (θ) in an infinite cylinder is 2λ/τ. In a semi-infinite cylinder with a sealed end at its origin, θ is close to 2λ/τ when the signal is electrically far from the boundary. As the signal approaches the origin, θ 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.
AB - 1. A novel approach for analyzing transients in passive structures called 'the method of moments' is introduced. It provides, as a special case, an analytical 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 as 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, θ(x0), is defined as |1/(dt(V)(x)/dx|(x=λ0). 3. With the use of these definitions, several properties of dendritic delay exist. 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 τ, the time constant of the membrane (R(m)C(m)), whereas the LD in an infinite cylinder is τ/2. In finite cylinders with both ends sealed, the TD from end to end is always larger than τ. 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 (ρ(∞) decreases), the LD at any given point decreases, and the TD from this point to the soma increases. 7. The velocity (θ) in an infinite cylinder is 2λ/τ. In a semi-infinite cylinder with a sealed end at its origin, θ is close to 2λ/τ when the signal is electrically far from the boundary. As the signal approaches the origin, θ 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.
UR - http://www.scopus.com/inward/record.url?scp=0027521708&partnerID=8YFLogxK
U2 - 10.1152/jn.1993.70.5.2066
DO - 10.1152/jn.1993.70.5.2066
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C2 - 8294970
AN - SCOPUS:0027521708
SN - 0022-3077
VL - 70
SP - 2066
EP - 2085
JO - Journal of Neurophysiology
JF - Journal of Neurophysiology
IS - 5
ER -