A Probabilistic Approach to the Analysis of Propagation Delays in Large Cortical Axonal Trees

The most important means by which a neuron passes information to its postsynaptic partners is the propagation of action potentials from the soma to the synapses. Since the synapses are located at different distances from the soma, the action potential gets to them after different propagation delays. These propagation delays cause desynchronization in the computations performed by the cortical network. This paper presents an analysis of the propagation delays along a randomly branching axon. The model can describe propagation delays caused by both the randomness of the structure of the axon and by the changing electrical properties along its branches. The statistics of the branching axon are described by a subcritical branching process. It is shown that such a model does describe well some populations of axons. The propagation delays along the branching structure are random functions of the type of the branches. The main result is that the delays along the axon have an exponential or related distribution, which is wider than the Gaussian distribution with the same variance. Using some numerical estimates it follows that the coherence length in the cortex is in the order of magnitude of 1 mm. This result has implications concerning some current theories of brain function.