Resumo

Objectives:
Much experimental and computational work has been devoted to the description of morphologic and dynamic aspects of dendrites, specially after the discovery of dendritic active conductances. The conditions for generation and propagation of dendritic nonlinear excitations have been investigated by cable theory at the level of a dendritic branchlet. However, in a more realistic scenario of in vivo functioning, where synaptic inputs arrive with a reasonable degree of stochasticity, presumably generating several dendritic spikes which may propagate and interact, cable theory becomes analytically untreatable. We have recently studied a simplified model of such a scenario via computer simulations and shown that the excitable tree performs signal compression, coding over 50 dB of incoming stimulus rate in a single decade of output firing rate. Here we present a method to understand the model analytically.

Methods:
The model can be analytically dealt with owing to the simplifications in the description of its dynamics. Each branchlet is modelled as a simple excitable element which can be in one of three states: if the branchlet is active (s=1), in the next time step it becomes refractory (s=2), whose average period is controlled by the probability with which sites return to a quiescent state (s=0). Quiescent branchlets can become active due to activation by neighboring compartments (with some probability) or by synaptic inputs (modelled as independent Poisson processes with rate h). Formally, the model is a probabilistic cellular automaton which is described by a master equation. The probability of a branchlet being active in the next time step depends on the four-site joint probabilities, and these in turn depend on higher-order terms, leading to a hierarchy of equations. We apply standard techniques for truncating this hierarchy, namely the one- and two-site approximations (1S and 2S, respectively), as well as a novel excitable-wave mean-field approximation (EWMF).

Results:
Defining the apical activity F as the average rate of excitations produced at the basal site, we compare the analytical results from the approximation methods with those observed in the simulations for the response (transfer) function F(h) of the tree. We show that, on one hand, both 1S and 2S approximations incorrectly predict self-sustained activity (F different from zero) in the limit of vanishing stimulus intensity h. On the other hand, the EWMF approximation circumvents this problem, showing excellent agreement with simulation results.

Conclusions:
We present a method for calculating (approximately) the response of a model active dendritic tree under varying intensity of synaptic input. Specifically, the excitable-wave meand-field approximation can handle the interaction among dendritic spikes which are stochastically generated at different locations of the tree.

Financial support: CNPq, FACEPE, PRONEX, INCEMAQ, CAPES