Network structure
For the construction of the model we interpret the connections of the network as a bipartite directional graph where the two sets of nodes are the TC and nRT cells. We used a 1-dimensional network: both sets of cells are on a one unit long line with equal distances between them, and the two lines of the two sets are parallel to each other. For the sake of reproducibility, we used a deterministic random number generator based on the Mersenne Twister algorithm to choose target cells. Our examined networks consist of 100 TC cells and 20 nRT cells. Each TC cell targets 8 different nRT cells from the closest 20, based on normal distribution. The nRT cells target 20 different TCs from the closest 50.
Network structure
The network generation algorithm first creates the set of cells based on the given parameters than iterates through every one of them. At each iteration step, the algorithm choses the n closest cells from the opposite set than selects m ≤ n cells that will get an input from the currently iterated one. The m cells are selected via a combination that is based on normal distribution in a way that cells closer to the iterated one have a higher chance to be selected.
We did not allow reciprocal connections in our networks. If a TC cell gives an input to an nRT cell, then it cannot receive input from the same cell, and vice-versa. The algorithm creates the network without any consideration of the reciprocal rule, and it trims these pairs after the completion of the network. If a TC and nRT pair is connected both ways, one of the directions is removed based on a coin flip.
The strength of a given synaptic connection in the network is given by the weight associated with it. Connection weights are normally distributed with a mean as the default synaptic strength (0.0028 umho for AMPA and 0.0088 umho for GABAa). This approach makes it theoretically possible to create topologically analogous networks that have completely different behaviors because of the different synapses.
| Number of TC cells | 100 |
| Number of nRT cells | 20 |
| Number of closest TC to nRT connections to choose from | 20 |
| Number of closest TC to nRT connections to actually choose | 8 |
| Number of closest nRT to TC connections to choose from | 50 |
| Number of closest nRT to TC connections to actually choose | 20 |
| AllowAutoCorrelation | False |
| Circular | False |
For the purpose of this study we adapted the network model of thalamocortical (TC) and thalamic reticular (nRT) cells, constructed by Destexhe et al (1996) [1] in the NEURON modeling environment [2].
Single compartment TC and NRT cell membrane potential was governed by the following equations:
$$C_m \dot{V}_{TC}=-g_L (V_{TC}-E_L)-I_{KL}-I_T-I_h-I_{Na}-I_K-I_ {{ GABA}_A}$$ $$C_m \dot{V}_{nRT}=-g_L (V_{nRT}-E_L)-I_{T2}-I_{CAN}-I_{AHP}-I_{Na}-I_K-I_{{GABA}_A}$$
To support our findings, we created a thalamic network model consisting of single-compartment thalamocortical and thalamic reticular cells based on the work of Alain Destexhe [1]. The original Destexhe model was capable of producing sustained oscillations through the interaction of the TC and nRT cells. nRT cell parameters were taken from an earlier model of the same author. TC cells elicited AMPA-mediated EPSPs in the nRT cells while nRT cells inhibited the TC cells through a mixture of GABAa and GABAb receptors. Thanks to the characteristics of the T current – found in both cell types – a sufficiently strong IPSP could elicit a rebound burst from the TC cell. Following an EPSP the T current in the nRT cells results in an oscillatory bursting behavior that stops after a few bursts. The TC rebounds refresh the nRT burst train thus keeping the oscillation of the network alive.
TC cells activated nRT cell via AMPA receptors while nRT cells inhibited TC cell via GABA-A receptors. We disregarded GABA-B receptors since their blockade hardly altered spindle oscillations experimentally [3], [4]. Synaptic currents were based on the models constructed by Destexhe et al [1] supplemented with temperature dependency. Both currents can be described by the following equations:
$$I_{syn}=\overline{g}_{syn}r(V-E_{syn})$$ $$\dfrac{dr}{dt}=\alpha{[C]}(1-r)-\beta{r}$$Where \(syn\) can be substituted for AMPA or GABA-A and \(I_{syn}\) is the synaptic current, \(g_{syn}\) is the maximal conductance and \(E_{syn}\) is the reversal potential respectively. The fraction of open channels is denoted by \(r\) and \([C]\) is the concentration of neurotransmitters in the synaptic cleft, while \(\alpha\) and \(\beta\) are the forward and backward binding rates of neurotransmitters for the given receptor type. We also define the following variables:
$$\tau_{r}=\dfrac{1}{\alpha{C}_{max}+\beta}$$ $$r_{\infty}=\dfrac{\alpha{C}_{max }}{\alpha{C}_{max}+\beta}$$ $$\varphi_h=Q_{10}^{\dfrac{T-T_{exp}}{10\,^{\circ}\mathrm{C}}}$$Where \(\tau_{r}\) is the time constant of channel binding, \(r_{\infty}\) is the steady-state fraction of open channels and \(\varphi_h\) is the temperature dependent multiplier that was added to the original model. \(C_{max}\) is the maximum concentration of transmitters, \(T\) is the actual temperature and \(T_{exp}\) is the experimental temperature the binding rates got measured on. The temperature coefficients (\(Q_{10}\)) of AMPA and GABA-A receptors were estimated based on the work of Postlethwaite et al. (2007) [5] and Otis and Mody (1992) [6], respectively.
An analytical expression exists for the fraction of open receptors based on the phase of transmitter release, which is approximated with a simple pulse. For further details on the deduction of the following equations refer to the original model. [1]
$$r_{ON}=r_{\infty}+(r-r_{\infty})exp(-\dfrac{\varphi_h{\Delta{t}}}{\tau_{r}})$$ $$r_{OFF}={r}{\;}{exp(-\varphi_h{\beta}{\Delta{t}})}$$Conductances were adjusted to reproduce spindle oscillation with the characteristics described by Bartho et al (2014). [7] Model cells matched the firing patterns of both TC and nRT cells, and spindles generated by the model reproduced waxing and waning behavior, but were slightly longer than the measured counterparts.