Effects of Several Classes of Voltage-Gated Ion Channel Conductances on Gamma and Theta Oscillations in a Hippocampal Microcircuit Model

Gamma and theta oscillations have been functionally associated with cognitive processes, such as learning and memory. Synaptic conductances play an important role in the generation of intrinsic network rhythmicity, but few studies have examined the effects of voltage-gated ion channels (VGICs) on these rhythms. In this report, we have used a pyramidal-interneuron-gamma (PING) network consisting of excitatory pyramidal cells and two types of inhibitory interneurons. We have constructed a conductance-based neural network incorporating a persistent sodium current (INaP), a delayed rectifier potassium current (IKDR), a inactivating potassium current (IA) and a hyperpolarization-activated current (IH). We have investigated the effects of several conductances on network theta and gamma frequency oscillations. Variation of all conductances of interest changed network rhythmicity. Theta power was altered by all conductances tested. Gamma rhythmogenesis was dependent on IA and IH. The IKDR currents in excitatory pyramidal cells as well as both types of inhibitory interneurons were essential for theta rhythmogenesis and altered gamma rhythm properties. Increasing INaP suppressed both gamma and theta rhythms. Addition of noise did not alter these patterns. Our findings suggest that VGICs strongly affect brain network rhythms. Further investigations in vivo will be of great interest, including potential effects on neural function and cognition.


NEURON MODELS
We use the following models for each type of the neurons in our study: • E-Cells: Based on the model of Olufsen et al. (2003) and extending it to incorporate the persistent sodium conductance (g N aP ) of the E-Cells as well, we use: where we have: f or x = m, h, or n and f or x = h or n (5) In the above equations, the letters C, V , t and τ , g, and I denote capacitance density, voltage, time, conductance density, and current density, respectively. The corresponding units for these quantities are µF cm , mV , ms, mS cm , and µA cm . For this part of the model, we use the following parameters: C = 1, g N a = 100, g K = 80, g L = 0.1, V N a = 50, V K = −100, V L = −67. Finally, for the α x and β x parameters we have: Finally, with the addition of persistent sodium conductance based on Li and Cleland (2013), we have: • I-Cells: Based on Wang and Buzsáki (1996), we follow the below equations for modeling the I-cells in this work: where we have: and where we have the rate functions as the following: The parameters utilized, are in the same units as for the E-cells and we have C = 1, g N a = 35, g K = 9, g L = 0.1, V N a = 55, V K = −90, V L = −65.
• O-Cells: We refer to the model of Tort et al. (2007) for defining the O-cells which is itself a reduction of the multi-compartmental model in Saraga et al. (2003). Therefore, we get: For x = m, n, h the functions x ∞ (V ) and τ x (V ) are the same as in Equations 4 and 5, and now the rate functions can be defined as follows: Eventually, for x = a, b, r, we present the x ∞ (V ) and τ x (V ) as below: The parameter values for the above equations are C = 1.3, g L = 0.05, g N a = 30, Next, we move on to modeling the synapses in this network.

SYNAPTIC MODEL
The utilized synaptic model in this work is adopted from Ermentrout and Kopell (1998). We represent the synaptic gating variable for each synapse associated with the presynaptic neuron with s where 0 ≤ s ≤ 1. This variable obeys: Here, τ R and τ D are the rise and decay time constants, respectively and ρ denotes a smoothed Heaviside function as the following: The synaptic input to every given neuron j from a neuron i in the model must be added to the right-hand side of the equation which governs the membrane potential V j . This term is added in the form of: where g ij denotes the maximal conductance associated with the synapse, s i represents the gating variable associated with neuron i, and V srev is the synaptic reversal potential. For AM P A receptor-mediated synapses, we have τ R = 0.1, τ D = 3, and V srev = 0; for GABA A receptor-mediated synapses, τ R = 0.2, τ D = 20, and V srev = -80, if the synapse originates from an O-cell.

PARAMETER SET
Here we bring the parameter values utilized in this work to model the hippocampal network which behaves as a PING network with nested gamma and theta rhythms. The model LFP exhibits mixed gamma and theta oscillations as observed experimentally in coronal slices of hippocampus Gloveli et al. (2005). Let us represent the number of E, I, and O-cells with N E , N I , and N O . We denote the maximal conductance of the synaptic connection between cell type X and cell type Y with G XY .
the employed g XY for each synapse type between different cell types are: g II = 0.1, g OI = 0.15, g IO = 0.5, g IE = 0.08, g OE = 0.15, g EI = 0.05, and g EO = 0.01 in mS cm 2 . All unspecified parameters are equal to zero.

ADDITIONAL RESULTS
In this section, we further explored the fluctuation of the network oscillations in response to variations in the drive currents I E and I I . We investigate the current value range between 0.2 and 1 µA cm 2 for both I E and I I . In this section, we only include the simultaneous effect of varying these currents concurrently with varying g A , g H , and g KDRe conductances as sample case studies. Due to the increased complexity with the addition of drive current variations, further analysis of these results is out of the scope of this study. Figures 1 and 2, represent the results comparing network oscillations in theta and gamma ranges while varying drive currents to E-cells and I-cells as well as g A , g H , and g KDRe . The theta range oscillations show higher vulnerability to drive current changes compared to gamma oscillations. This difference is more significant in the case of varying drive currents concurrently with g KDRe where the theta peak is nearly lost at many variations of I E and I I (Figures 1f and 2f). However, the power of the peak at gamma range is much less impacted by the changes in drive currents and this peak seems to be more robust in the network in all study cases of g A , g H , and g KDRe . These figures also confirm the expectations from a PING network discussed in the Introduction section where very weak I E or strong I I values abolish the PING rhythms and disturb the network activity. Due to the extra complexity by the addition of drive current variations, we leave further discussions about the clear impact of this variable in our model to future works. conductance values at each I I level and the red asterisks indicate both conductances and drive currents being at baseline. Graphs also display three adjusted drive currents for each conductance level for (a,b) g A , (c,d) g H , and (e,f) g KDRe . Fluctuations in peak frequency (left) and power (right) are given at each conductance and current pair for theta (top) and gamma (bottom) rhythms.