Microcircuitry and oscillation features across model cortical regions

Abstract

We developed biophysically realistic 10,000 cell neuronal network models of three cortical regions to investigate microcircuitry features that support beta and gamma oscillations that are thought to subserve communication in the brain. The model single cells and networks were constrained using in vitro and in vivo cellular and synaptic neurophysiology data. The models suggested the key microcircuit parameters based on their impact on oscillation characteristics across all regions, as follows: short-term presynaptic plasticity, electrical and chemical coupling between individual inhibitory interneurons types, the mutual inhibition strengths between LTS and FSI cell types, and the rate of the afferent drive.

I. Introduction

Human brain has been optimized by evolution to have remarkable efficiency in performing a variety of tasks including sensory integration, computation, and communication. Interestingly, the brain follows design principles similar to those in human electronic and communication networks but with different ‘hardware’ [1, 2]. The question of how networked neural elements and circuits function and communicate, and ultimately give rise to brain function, is one of the most intriguing scientific challenges in reverse engineering [3, 4]. Oscillations are thought to play a key role in communication and the growing neural data is spurring investigations to begin exploring the details [5]. Cortex is thought to generate beta and gamma oscillations intrinsically in the local field potential (LFP) via reciprocal interactions between pyramidal cells (PNs) and interneurons, e.g., gamma rhythm via a pyramidal-interneuron-gamma (PING) mechanism. However, the linkage of microcircuitry and mechanisms to their genesis, variation across regions, or their role in communications and are unclear.

Computational network models of brain regions are beginning to provide important insights into the underlying microcircuits that might implement rhythms in local circuits. For instance, the gamma (40–80 Hz) rhythm is thought to arise in vivo via feedback interactions between pyramidal cells (PNs) and inhibitory fast spiking interneurons (FSI). Models also suggest conditions for the genesis of the rhythms, e.g., for the PING mechanism, excitatory drive to the interneurons has to largely come from pyramidal cells [6]. On the other hand, the beta oscillation (15–40 Hz) is thought to arise from the interaction between pyramidal and low threshold-spiking (LTS) cells [7, 8], and both the rhythms exhibit interesting spectro-laminar differences [9]. Such models can aid in the exploration of how microcircuits underlying rhythms could enable a network to form and manipulate cell assemblies [10]. Furthermore, larger models incorporating such insights have the potential to shed light on how the circuits might implement information flow across brain regions, and possibly support components of cognition [11]. Here we report on how microcircuitry in different model cortical regions support oscillations in the beta and gamma bands in three rodent brain regions: primary motor cortex (M1) layer 5 (L5) and layer 2/3 (M1-L2/3), and in L5 of its visual cortex (V1-L5). In vitro studies have reported the existence of beta and gamma rhythms in these regions (e.g., [12–14]), but the mechanistic underpinnings are unclear.

II. Biophysical models

Biophysical models of three cortical regions, V1-L5, M1-L5, and M1-L2/3 were developed using biological data, and approaches as in our previous models (e.g., [15, 16]). They were implemented using the NEURON simulator [17], with a fixed time step of 100 μs. Briefly, the dynamics for each compartment of a neuron (soma, dendrite) followed the Hodgkin-Huxley formulation in eqn. 1,

$$C_{m}d{V}_s/dt=-g_L\left(V_s-E_L\right)-g_c\left(V_s-V_d\right)-\sum I_{\mathit{cur},s}^{\mathit{int}}-\sum I_{\mathit{cur},s}^{\mathit{syn}}+I_{\mathit{inj}}$$ (1)

where $V_s/V_d$ are the somatic/dendritic membrane potential (mV), $I_{\mathit{cur},s}^{\mathit{int}}$ and $I_{\mathit{cur},s}^{\mathit{syn}}$ are the intrinsic and synaptic currents in the soma, $I_{\mathit{in}j}$ is the electrode current applied to the soma, $C_m$ is the membrane capacitance, $g_L$ is the conductance of the leak channel, and $g_c$ is the coupling conductance between the soma and the dendrite (similar term added for other dendrites connected to the soma). The intrinsic current $I_{\mathit{cur},s}^{\mathit{int}}$, was modeled as $I_{\mathit{cur},s}^{\mathit{int}}=g_{\mathit{cur}}{m}^p{h}^q\left(V_s-E_{\mathit{cur}}\right)$, where $g_{\mathit{cur}}$ is its maximal conductance, $m$ its activation variable (with exponent $p$), $h$ its inactivation variable (with exponent $q$), and $E_{\mathit{cur}}$ its reversal potential (a similar equation is used for the synaptic current $I_{\mathit{cur},s}^{\mathit{syn}}$ but without $m$ and $h$). The kinetic equation for each of the gating variables $x$ ($m$ or $h)$ takes the form but without $m$ and $h$).

The kinetic equation for each of the gating variables $x$ ($m$ or $h$) takes the form

$$\frac{dx}{dt}=\frac{x_{\mathrm{\infty }}\left(V,{\left[C{a}^{2+}\right]}_i\right)-x}{{\tau }_x\left(V,{\left[C{a}^{2+}\right]}_i\right)}$$ (2)

where $x_{\mathrm{\infty }}$ is the steady state gating voltage- and/or Ca²⁺-dependent gating variable and ${\tau }_x$ is the voltage- and/or Ca²⁺-dependent time constant. The equation for the dendrite follows the same equation with ‘$s$’ and ‘$d$’ switching positions in eqn. 1. Selection of the channel currents and their model parameters followed an approach we proposed recently that facilitates design while ensuring that the key neurocomputational properties of the cell are preserved [18].

Single cell models.

The single cells were of two overall classes, principal cells (PNs) of two types – corticopontine (CP) and corticostriatal (CS), and GABAergic interneurons (ITNs) of two types – fast-spiking (FSI) and low-threshold bursting (LTS) cells. A typical PN model had three compartments representing a soma (diameter, 24.75 μm; length, 25 μm), an apical dendrite (diameter, 3 μm and length, 270 μm) and another basal dendrite (diameter, 5 μm and length, 555 μm). These cells receive short-/long- term depressing synapses from local FSI/LTS interneurons, respectively. Here we use the M1-L5 region to illustrate the process of constraining the single-cell and network models; the same approach with different constraints applies to the V1 and L2/3 models. Model parameters for the M1-L5 PN were tuned to match passive properties and spiking behaviors in experimental reports [19, 20]. The FSI model had a soma compartment (diameter, 15 μm; length, 15 μm) and a dendrite compartment (diameter, 15 μm; length, 150 μm). LTS model had three compartments: a soma (diameter, 20 μm; length, 10 μm), an apical dendrite (diameter, 3 μm and length, 250 μm) and another basal dendrite (diameter, 3 μm and length, 250 μm). Model parameters for both FSI and LTS cells were tuned to match passive properties and current-clamp spiking behaviors from experimental results found in [20].

Network model and connectivity (Fig. 1).

Fig. 1.

Network model details [30]. A) Specific microcircuit focus of the study with 8 synaptic connection types. B) Placement of cells in 3-D space (only soma locations displayed for a smaller network of 1000). C. Extrinsic afferent types to the network was a train of 50Hz Poisson pulses of 1 sec duration (‘Long’), with a 500 ms gap.

All the 10,000-cell layer 5 network models had a PN/ITN ratio of 4:1, while the layer 2/3 model had a PN/ITN ratio of 87:13. For instance, the M1-L5 model had 8000 PN, 1200 FSI and 800 LTS cells (Fig 1B). The cell numbers for all regions are listed in Table 1. In all models connections were only chemical synapses except for FSI-FSI and LTS-LTS which had chemical and electrical synapses. Connection probabilities were based on biological data. [20–22].

Table 1.

Microcircuitry parameters and oscillation features of the models

Property	V1	M1-L5	M1-L2/3
Beta – Center Freq (Hz), Bandwidth (Hz), Relative Power (SNR)	CF: 28, BW:4 SNR: 0.30	CF: 19, BW: 4.00 SNR: 0.72	CF: 20, BW:6 SNR: 0.37
Gamma – Center Freq (Hz), Bandwidth (Hz), Relative Power (SNR)	CF: 61 BW: 21 SNR: 0.72	CF: 45, BW: 12 SNR: 0.61	CF:45,BW:12 SNR: 1.42
Numbers of cells: Total and individual, types	10,000:8500 PNs (CP/ CS 50:50), 850 FSI, 650 LTS	10,000: 8000 PN, 1200 FSI and 800 LTS	10,000: 8700 PNs (CP/ CS 50:50), 900 FSI, 400 LTS
Input resistance (MΩ) of cells: PN, FSI, LTS	CP:145, CS182 FSI,LTS: 144,243	CP:86, CS:215 FSI,LTS: 223,290	PN: 113 FSI,LTS: 223,290
Convergence: PN-LTS, LTS-PN, PN-FSI, FSI-PN	135, 26, 109, 42	135, 13, 180, 35	417, 23, 803, 43
Connection weights: LTS-FSI, FSI-LTS	LTS-FSI: 0.15 FSI-LTS: 4.5	LTS-FSI: 0.21 FSI-LTS: 2.85	LTS-FSI:0.24 FSI-LTS: 4.5
Gap junction weights: FSI-FSI, LTS-LTS	FSI-FSI: 0.00017 LTS-LTS: 0.00125	FSI-FSI:0.000066 LTS-LTS:0.00076	FSI-FSI: 0.000066 LTS-LTS: 0.00076

Synaptic currents and Synaptic Plasticity.

Excitatory transmission was mediated by AMPA/NMDA receptors, and inhibitory transmission by GABA_A receptors. Model AMPA and GABA_A synapses also exhibited short term pre-synaptic plasticity (STP) between FSI to PN (referred to as FSI2PN) and from PN-FSI, as in our previous models [23]. Specifically, PN-FSI and FSI-PN synapses had short-term depression, and PN-LTS, LTS-PN ones had short-term facilitation; none in afferent synapses. None of the synapses had long-term plasticity. The synaptic weights were set to follow a log-normal distribution for each connection.

Inputs.

A persistent background afferent drive generated from an independent Poisson process was delivered at 20 Hz to all neurons, PNs and ITNs. In the baseline case, the network received only the background afferent drive. The afferent synaptic weights were adjusted in setting the baseline firing rate of neurons in all three models [24–26]. Extrinsic afferent activity was simulated by delivering excitatory synaptic drive as a Poisson process at 50 Hz to 11.1% of PNs (representing ensembles; only the ‘Long’ case used here) for 1 sec followed by 500 ms gaps, repeated 10 times (Fig. 1), to simulate tonic drive similar to that seen in vivo [27].

Calculation of LFP and PSD.

We recorded transmembrane ionic currents from each compartment of the model cells using NEURON’s extracellular mechanism and the computed the extracellular potential and LFP [15]. Spectral parameterization (formerly Fooof) algorithm [28] was used to fit the aperiodic and periodic components of the PSD. The spectrogram was obtained using continuous wavelet transform with Morlet wavelet with 6.3 number of cycles [29]. The PSD plot (Fig. 2A) and the relative power in the bands were obtained by calculating the difference in area under the raw PSD and the aperiodic curve fits, within the specific band (see Fig. 2A), and then dividing it by the area under the aperiodic fit for the same band.

Figure 2.

Spectral analysis. A) (top right) Annotated output of Fooof PSD [23]. B) Fooof PSD plot for V1 region – in vivo (top middle) and model (top left). Note different y-axis scales. Both beta and gamma power profiles match in model and experiments. C) (bottom) Spectrograms from the models – V1, L5 and L2/3, respectively, showing gamma (transient) and beta (steady) power.

Model tuning.

The single cell models were tuned first to match experimental reports by adjusting the current channels conductances to match biological passive and firing properties (e.g., [30]); those parameters were fixed after integration into the network. Since all features of the network model except the strengths of the various connections, were constrained by biological data, the parameters of the lognormal distributions of the various connections were then tuned to reproduce baseline firing rates and oscillatory features known/expected with naturalistic extrinsic drive.

III. Results and Discussion

Large-scale biophysical network models were developed for M1-L5, M1-L2/3 and V1-L5 regions to investigate the linkage between microcircuitry and oscillations features across the cortex. The model structure in each improves on the homogeneous connectivity of our prior 1000-cell model [30] by including distance-dependent connectivity and scaling model size to 10,000 cells.

Local activity engages microcircuits differentially to engender beta and gamma rhythms.

The baseline models in each of the three cases had no beta or gamma rhythms. An increase in external drive caused local activity that engaged the PING mechanism. The log-normal parameters of the connections strengths were then successfully tuned for all cases to reproduce known overall rhythm features with the following center frequencies, respectively, for beta and gamma bands: for M1-L5 and for M1-L2/3 at 20 and 45 Hz; and for V1-L5 at 28 and 61 Hz [12–14, 26, 31]. The models were able to replicate the overall features of these rhythms as shown in Table 1 and Fig. 2 (V1 in vivo data from [31] matches model data that has a beta bump at 30 Hz).

We then explored the key microcircuitry features that affected the rhythms in each region using two approaches. Since the PING mechanism involving the PN-ITN-PN connection seems essential for rhythm generation, the first approach considered the effect of the other key microcircuit parameters on the power in the two rhythms by inactivating or substituting them individually (Table 2). The second approach used the standard sensitivity analysis for nonlinear systems by determining the change in oscillatory power with ±15% changes in parameter from nominal values. Both approaches are discussed separately below.

Table 2.

Effect of features on β/γ power

Region→Feature ↓	M1-L5	M1-L2/3	V1
PNs - all CS type	β 103%↑ γ 58%↓	β 835%↑ γ 21%↓	β 60%↓ γ 9%↓
PNs - all CP type	β 4%↓ γ 54%↓	–	β 40%↓ γ 17%↑
Gap Jns between ITNs x	β x γ 44%↓	β x γ 82%↓	β x γ 8% ↓
ChemSyn between ITNs x	β x γ x	β x γ 81%↓	β x γ x
STP x	β x γ 12%↓	β x γ 7% ↓	β x γ 62%↑
LTS-FSI and FSI-LTS x	β 182%↑ γ 8%↑	β 21%↓ γ 54% ↓	β 1% ↑ γ 40%↓

Impact of inactivation of microcircuit features on oscillatory power.

Replacing CP-type PNs with the CS-type resulted in a huge increase in beta power, e.g., +103% in L5, +835% in L2/3, and a decrease in gamma power (Table 2). Inactivation of either all gap junctions or all chemical synapses between LTS-LTS and FSI-FSI connections eliminated beta oscillations, and severely impaired power in the gamma rhythm in all three model regions. Removal of STP eliminated beta oscillations and decreased power in the gamma band. Elimination of the entire inhibition motif between LTS and FSI cells had mixed effects, requiring further study.

Cell numbers and convergence vs power in the bands.

LTS cells number difference between L2/3 vs L5 is 400 vs. 800, which may contribute to the 51% higher power for beta in L5 compared to L2/3 (0.72 vs 0.37). Although FSI cell numbers in L2/3 vs L5 is 900:1200, the corresponding convergence numbers for PN-FSI is an overwhelming 803:180, which may explain the 42% higher power for gamma in the L2/3 vs L5 (1.42 vs 0.61). The center frequencies are close because of similar decay time constants.

Sensitivity analysis reveals fine grained effect of the microcircuit parameters.

We explored the effect of the key parameters identified earlier at a finer grained level via a sensitivity analysis (±15%) (Table 3). The external drive rate had a large effect on beta but not gamma power. The convergence of PN-LTS and PN-FSI had a much larger effect than the LST-PN and FSI-PN connections. Both FSI-LTS and LTS-FSI connections bidirectionally controlled gamma/beta rhythms, respectively, e.g., increase/ decrease in FSI-LTS strength increased/decreased gamma power, with similar effect of LTS-FSI strength on beta power. In summary, although the microcircuitry implicated in the genesis of beta and gamma rhythms seems to be largely conserved across the three regions, the computational models helped provide deeper insights into the individual contributions of the circuit parameters to the oscillatory power (Tables 1–3), several of which are provided as predictions below.

Table 3.

Sensitivity analysis – effect of microcircuitry features on oscillation power

Change Rel. Power (SNR) → for ±15% change in feature (below)	V1				M1-L5				M1-L2/3
	Beta %		Gamma %		Beta %		Gamma %		Beta %		Gamma %
	−	+	−	+
External drive rate	−11	8	3	17	−14	90	15	−10	−30	139	−60	−50
PN-LTS convg.	−84	16	34	−33	9	150	0	−54	−43	104	23	−57
LTS-PN convg.	−96	188	10	8	1	−3	−20	19	−5	6	−2	5
PN-FSI convg.	206	−47	−44	55	124	−56	−37	15	138	−68	−75	46
FSI-PN convg.	50	−65	−2	14	86	33	−85	52	31	−15	−57	−9
lnhib: FSI-LTS	119	−26	−8	34	1	2	−13	26	97	−53	−61	5
lnhib: LTS-FSI	−33	13	12	1	1	1	29	−10	−20	95	−5	64

Model predictions and hypothesis.

The study provided six predictions. First, short-term plasticity is necessary for the generation of beta rhythm in all regions. Second, gap junction coupling, both LTS-LTS and FSI-FSI, was critical for generating beta oscillations, and its inactivation severely impaired power in the gamma rhythm in all three model regions, as seen in vivo [33]. Third, chemical synapses between the ITNs also had a profound effect, with their inactivation eliminating power in the beta band and decreasing it significantly in the gamma band. Fourth, as an example of an over-arching finding to illustrate the potential of such models - although it is known that M1-L2/3 exhibits much stronger gamma compared to the M1-L5 regions, the underlying mechanism is unclear. Our analyses shows that the M1-L2/3 model has 1.9x more PN-FSI compared to PN-LTS connections, while the same ratio is 1.3X in the M1-L5 model. This could explain the much stronger gamma in M1-L2/3 vs M1-L5 regions. Fifth, the mutual inhibition motif between LTS and FSI cells plays a key role in controlling the relative power between the two bands, and could explain the inverse relationship seen in Table 3 entries between beta and gamma power, also reported in vivo [32]. Sixth, among the various parameters, the key parameters affecting oscillatory power were found to be as follows: STP, PN-ITN (both FSI and LTS) connectivity, gap junction connectivity among ITNs (FSI-FSI and LTS-LTS), and the rate of the afferent drive. Finally, these findings lead to a hypothesis that the mutual inhibition microcircuit motif involving PN-PV-SOM cells (Fig. 1A) controls the power and peak frequency of the rhythm during the steady state portion of the 1-sec long pulse (300 ms-1000 ms in the spectrogram plots of Fig. 2A).

Experimental validation of the testable predictions and hypothesis, and further exploration of the role of oscillations in communication in neuronal networks [1, 2, 4], including how top-down control may implement ‘spatial computing’ in local circuits via beta and gamma bursts [5] are exciting directions for research in neural communication. Modeling and analysis tools, such as those presented here, have potential to contribute significantly to such research.