THE CONCEPT OF DECISION POINTS - DEFINITIONS AND METHODS

The method of moments

In the METHOD OF MOMENTS, dendritic transients are characterized by properties that are defined using the moments of the transient signal. The

moment,

, of a transient function f(t) is

The moment is the area (time integral) of f(t), and serves as a measure for the strength of the signal; the ratio between the and the moment is the "center of gravity", or centroid, of f(t) which is a measure for the characteristic time of the transient signal; The moment can be used to give a measure for the width of the signal, and using higher moments, parameters of the shape of the signal can be introduced.

The METHOD OF MOMENTS yields general theorems, formulae and algorithms concerning the analysis of these moments-based properties. These results provide a very general analysis of attenuation, delays and signal broadening in models of arbitrary passive dendritic trees with transient current inputs. The analysis using the method of moments is also efficient in computing time, compared with analysis based on compartmental modeling and numerical methods for evaluating the voltage response in arbitrary passive dendritic structures.

The main moments-based properties of a transient signal, f(t), are demonstrated in the next figure

where is the strength of the signal, is the characteristic time of the signal and is the width of the signal.

In the figure below, the current I(t) is injected at point y. The resultant voltage response at point y is denoted V(y,t). The input resistance, (y), is defined as the ratio between the strengths of V(y,t) and I(t). The input delay, (y) (also named the local delay, LD(y)), is defined as the difference between the characteristic times of V(y,t) and I(t).

Considering the point x (when the current is injected at point y) we define the transfer resistance, (y,x), as the ratio between the strengths of V(x,t) and I(t). The transfer delay, (y,x) (also named The total delay, TD(y,x)), is defined as the difference between the characteristic times of V(x,t) and I(t).

Similarly, we define the attenuation factor, (y,x), as the ratio between the strengths of V(y,t) and V(x,t). The propagation delay, PD(y,x), is defined as the difference between the characteristic times of V(x,t) and V(y,t).

The , , defined here are equal to the classical , , defined with respect to the steady-state case.

The method of moments is described in:

Agmon-Snir, H. and Segev, I. 1993. Signal delay and input synchronization in passive dendritic structures. J. Neurophysiol. 70: 2066-2085.

Agmon-Snir, H. 1994. A novel approach to the analysis of dendritic transients. Ph.D. Thesis, Hebrew University of Jerusalem.

The concept of decision points in the dendritic tree

1. The soma as a decision point

For the input-output relationship of the neuron, we typically emphasize the role of the soma as a site where synaptic currents are transformed into a pattern of action potentials. Because of the convergence of the many synaptic inputs at the soma, the soma cannot be regarded as a simple relay that delivers all the arriving information to the next "station". Many combinations of synaptic inputs can result in the same output. In other words, only part of the spatio-temporal information that arrives at the soma is expressed in the output pattern of action potentials at the axon. Hence, in the soma, a decision is made - what part of the information should be delivered to the axon. For example, in a simple isopotential integrate-and-fire model of the neuron, the only information delivered by the neuron is when the sum of the inputs reaches a threshold.

2. Dendritic sites as local decision points

The soma, however, is not the only decision point in the cell. Many types of nonlinear interactions between inputs at the dendritic tree (e.g., local "veto" effect of synaptic inhibition, dendritic "hot spots", nonlinear summation of excitatory inputs at a partly isolated dendritic arbors, activity-dependent plastic changes, dendro-dendritic interactions) may be implemented at local decision points. These processes are triggered by a potential change at a decision point that results from integration of several inputs in the proximity of this point.

The dendritic decision points are triggered by a subset of the inputs that impinge on the dendrites, and in many cases, the voltage response at the decision point affects other decision points. Decision points cannot be regarded as faithful relay points because their output does not include all the information contents of their inputs. We may say that such decision points "decide" what part of the information is to be transmitted to other decision points (specifically, to the soma).

3. Analyzing spatio-temporal integration using the concept of decision points

Characterization of computation in dendrites is difficult because much of the biophysical information is still missing. The computational function of the neuron seems to be dependent on many complicated factors (morphology, channel density and distribution as well as input architecture). Thus, carefully simplified and reduced models should be constructed to understand the underlying principles that govern this computation. The idea that the processing in the dendrite is done by "decisions" at "decision points" is an example of such a simplification. It does not convey the entire biophysical properties of realistic neurons, but it allows one to state some rules for dendritic computation.

In the present work, the dendritic tree and the soma were assumed to be passive, and the inputs were modeled by current injections (rather than by conductance changes). To this passive model we added the concept of decision points. Hence, although the voltage response in the model is still calculated using the elegant methods of passive cable theory, the interpretation of the results are made from the point of view of the decision points, which may be regarded as nonlinear. We demonstrate here some examples for this "hybrid" point of view.

The use of the method of moments in the analysis of decision points

There are two good reasons for analyzing decision points using the method of moments:

1. The computational interpretation of the moments-based definitions

The strength of the PSP at a decision point, defined as the time-integral of the voltage response, may serve as a measure for the contribution of this PSP to the total voltage response that may yield a "decision", typically when reaching a threshold. Hence, the

(y) and the

(y,x) are measures for comparing the effect of a given input injected at y on the voltage response at various decision points.

The TD(y,x) (total delay) is a measure for the width of the time window in which the input at point y will affect the voltage response at the decision point x. The LD(y) is, thus, a measure for the width of the time window for local interactions. Short time window implies that good synchronization is required for effective integration of inputs at the decision point.

2. The generality of the results of the method of moments

The method of moments yields important theorems for the moments-based properties. The most important one is the Shape Invariance Theorem:

The transfer resistance, the attenuation factor, the total delay and the propagation delay between any two given points (y,x) in a passive structure are independent of the shape of the injected transient current. As a special case, this theorem holds for the input resistance and the local delay at any point.

Another important theorem is the Equivalence Theorem:

When analyzing input/transfer resistance, attenuation and delays in a passive structure, one can compute input resistance, attenuation factor and delays in any cylindrical segment, replacing the structures (subtrees) at its boundaries by equivalent structures (e.g., isopotential compartments) that have the same input resistance and input delays as the corresponding original structures (subtrees).

These theorems, together with few others, make the analysis of decision points using the method of moments very general. The results are independent of the input shape and analyzing simple structures may yield general insights for complicated structures.

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