Abstract
Gap junctions, also referred to as electrical synapses, are expressed along the entire central nervous system and are important in mediating various brain rhythms in both normal and pathological states. These connections can form between the dendritic trees of individual cells. Many dendrites express membrane channels that confer on them a form of sub-threshold resonant dynamics. To obtain insight into the modulatory role of gap junctions in tuning networks of resonant dendritic trees, we generalise the “sum-over-trips” formalism for calculating the response function of a single branching dendrite to a gap junctionally coupled network. Each cell in the network is modelled by a soma connected to an arbitrary structure of dendrites with resonant membrane. The network is treated as a single extended tree structure with dendro-dendritic gap junction coupling. We present the generalised “sum-over-trips” rules for constructing the network response function in terms of a set of coefficients defined at special branching, somatic and gap-junctional nodes. Applying this framework to a two-cell network, we construct compact closed form solutions for the network response function in the Laplace (frequency) domain and study how a preferred frequency in each soma depends on the location and strength of the gap junction.
Highlights
It has been known since the end of the nineteenth century and mainly from the work of Ramón y Cajal [1] that neuronal cells have a distinctive structure, which is different to that of any other cell type
We develop a more mathematical approach using the continuum cable description of a dendritic tree that can compactly represent the response of an entire dendro-dendritic gap junction coupled neural network to injected current using a response function
We introduce a method of ‘words’ to construct compact solutions for the Green’s function of this network and study how a preferred frequency in each soma depends on the location and strength of the gap junction
Summary
It has been known since the end of the nineteenth century and mainly from the work of Ramón y Cajal [1] that neuronal cells have a distinctive structure, which is different to that of any other cell type. Earlier theoretical studies demonstrate that neuronal gap junctions are able to synchronise network dynamics, they can contribute toward the generation of many other dynamic patterns including anti-phase, phase-locked and bistable rhythms [12] Such studies often ignore dendritic morphology and focus only on somato-somatic gap junctions. We develop a more mathematical approach using the continuum cable description of a dendritic tree (either passive or resonant) that can compactly represent the response of an entire dendro-dendritic gap junction coupled neural network to injected current using a response function This response function, often referred as a Green’s function, describes the voltage dynamics along a network structure in response to a delta-Dirac pulse applied at a given discrete location.
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