Two tufted neurons

Now consider a more realistic neuronal network consisting of two identical tufted or mitral cells. Each neuron has one primary dendritic branch, which branches at its end and forms a tuft. The two cells are coupled in their tufts by dendro-dendritic gap junctions (see Fig. 4.13A). Although it is possible to apply the method of local point matching to find the Green’s functions for this tufted network, it is more convenient to perform first the model reduction with equivalent cylinders (discussed in§2.4.3), and the reduced network topology is shown in Fig. 4.13B. Notably, after model reduction, a tufted neuron becomes a “Y”-shaped neuron studied in§4.3.

To be specific, if the input and output are not located in the tufts, the tufted branches for each neuron can be merged into two equivalent cylinders, with an equivalent gap junction located on one of them, or explicitly,

z∗_{T ,GJ} =NGJzT, (4.34a) R∗_GJ =RGJ/NGJ, (4.34b) wherezT, zT,GJ∗ are the characteristic admittances of the individual tufted branches, and RGJ, R∗_GJ are the gap-junctional resistances. The superscript ∗ in this section denotes the equivalent reduced model. The Green’s functions obtained under the conditions (4.34) are identical for the two models, that is,

G(x0, yk) =G∗(x0, y2). (4.35) If the input is in the tuft but the output is not, it is easy to check that the constraints (4.34) would give the same Jy. However, Green’s functions are dependent on the

Cell 2 Cell 1

A

B

l₀

l

l₀

l

J

J

J

J

J

J

J

J

Figure 4.13: (A): A schematic of a simplified network of two neurons coupled by gap junctions in their tufts. Each neuron hasN dendritic branches attached to its soma (x= 0). One of theN branches is the primary dendrite, which branches at the end (x=l0) and forms a tuft. The other branches attached to the somata and the secondary branches in the tufts are semi-infinite. Each tuft has NT branches, and NGJ of them form dendro-dendritic gap junctions (one on each). All the gap junc- tions are identical in strengthgGJ and located at x=lGJ+l0. (B): An equivalent reduced model to (A). Everything outside the tufts are unchanged from (A). Either tuft consists of only two branches, one of which forms a gap junction to the other tuft. The gap junction still locates at lGJ, whereas its strength g∗GJ = NGJgGJ. The arrows denote the terms in Eqs. (A.8) according to the local point matching method.

Gap-junctional conductance (μS) 0 0.1 0.2 200 400 300 0.06 0 Gap-junctional location ( μ m)

Figure 4.14: Coupling ratio as a function of gap-junctional location lGJ ≥ l0 and conductancegGJ =R_GJ−1. The two neurons are passive and identical. Their geome- tries are shown in Fig. 4.13, with the primary dendritic branch of length l0 = 200

µm, and their passive electrical parameters are the same as in Fig. 4.11. zj(y) at the input location due to Eq. (2.83), which leads to

G(x0, yk) = 1 NT −NGJ

G∗(x0, y1), (4.36) if the input atyk is applied to the branch without a gap junction, and

G(x0, yk) = 1 NGJ

G∗(x0, y2), (4.37) if the input at yk is applied to the branch with a gap junction. Here the reduced model is constructed so that the stimuli in the full and reduced models are placed at the same location, i.e. y1 = yk and y2 = yk. The point x0 < l0 is located on the primary dendritic branch of either of the neurons. If the output is in the tufts but the input is not, the input-output reciprocity (2.67) can be employed. If both input and output are in the tufts, the symmetry among the branches in the tufts is broken, and thus the model reduction fails.

After obtaining the Green’s functions by the method of local point matching on the reduced model, those on the full model can be easily found by Eqs. (4.35) - (4.37) (explicit expressions and deductions omitted here, details to be found inAppendix A.2). To investigate the effect of gap junctions in the tufts, the coupling ratio (CR) between the two neurons are shown in Fig. 4.14, where CR is defined in [99] as

CR = maxtL

−1_{G

2(0,0;ω)}(t) maxtL−1{G1(0,0;ω)}(t)

Although increasinggGJ strengthens the coupling (as expected),lGJ actually plays an even more important role. Comparing to the results in Fig. 4.12, the domination oflGJ over the system occurs for relatively largegGJ. Nonetheless, all the observa- tions in this section suggest that gap-junctional location has stronger impact than gap-junctional strength on voltage spreading between the coupled neurons.

4.5

Summary

In this chapter, I have computed the numerical results on neuronal models with different dendritic geometries. Firstly, a single neuron with a single dendritic branch is investigated in§4.2. By defining the resonant frequency to quantify the resonant behaviours of the neuronal systems, it is found that different geometry of a single branch (tapered or cylindrical) has little impact on signal modulation. However, a tapered dendrite is better at current transfer from its distal (thinner) end to the proximal (thicker) end. This effect is found to be a local property of tapered dendrites in§4.3, by investigating a single neuron with a “Y”-shaped dendritic tree and varying the geometries of the primary and the secondary dendritic branches.

Secondly, a simplified neuronal network of two gap junctional coupled neurons is considerd in§4.4. Mimicking a realistic experimental protocal, injecting currents and recording responses at the two somata, the relationship between the gap junc- tional properties and the output measurements can be computed. This procedure can be used to assist the estimation of gap junctional parameters. The numerical results suggest that the location of the gap junction has a noticeable effect on the signal transmission between the coupled neurons, whereas little impact is seen on the resonant behaviours. The strong modulation on voltage attenuation by gap junctional location, i.e. the distance between the gap junction and the soma, is also observed in a more realistic model consisting of tufted neurons.