Neuronal networks in rodent primary visual cortex (V1) can generate oscillations in different frequency bands depending on the network state and the level of visual stimulation. High-frequency gamma rhythms, for example, dominate the network's spontaneous activity in adult mice but are attenuated upon visual stimulation, during which the network switches to the beta band instead. The spontaneous local field potential (LFP) of juvenile mouse V1, however, mainly contains beta rhythms and presenting a stimulus does not elicit drastic changes in network oscillations. We study, in a spiking neuron network model, the mechanism in adult mice allowing for flexible switches between multiple frequency bands and contrast this to the network structure in juvenile mice that lack this flexibility. The model comprises excitatory pyramidal cells (PCs) and two types of interneurons: the parvalbumin-expressing (PV) and the somatostatinexpressing (SOM) interneuron. In accordance with experimental findings, the pyramidal-PV and pyramidal-SOM cell subnetworks are associated with gamma and beta oscillations, respectively. In our model, they are both generated via a pyramidal-interneuron gamma (PING) mechanism, wherein the PCs drive the oscillations. Furthermore, we demonstrate that large but not small visual stimulation activates SOM cells, which shift the frequency of resting-state gamma oscillations produced by the pyramidal-PV cell subnetwork so that beta rhythms emerge. Finally, we show that this behavior is obtained for only a subset of PV and SOM interneuron projection strengths, indicating that their influence on the PCs should be balanced so that they can compete for oscillatory control of the PCs. In sum, we propose a mechanism by which visual beta rhythms can emerge from spontaneous gamma oscillations in a network model of the mouse V1; for this mechanism to reproduce V1 dynamics in adult mice, balance between the effective strengths of PV and SOM cells is required.

1  Introduction

Over the past decade, the primary visual cortex (V1) of mice has been intensively studied using transgenic techniques. Distinct subtypes of inhibitory interneurons that use gamma-aminobutyric acid (GABA) as a neurotransmitter were, for example, identified and linked to various spiking behaviors (Tremblay, Lee, & Rudy, 2016; van Versendaal & Levelt, 2016). These different subpopulations also have specific connectivity patterns with respect to one another; parvalbumin-expressing (PV) interneurons are mutually inhibited and receive inhibition from somatostatin-expressing (SOM) cells, while the latter subtype primarily receives inhibition from vasoactive intestinal peptide expressing (VIP) interneurons (Pfeffer, Xue, He, Huang, & Scanziani, 2013). These subtype-specific connection motifs also have an additional geometric component: PV interneurons in cortical layers 2 and 3 (L2/3) predominantly receive vertical inputs originating from cortical layer 4 (L4) compared to horizontal ones from other L2/3 cells, whereas the opposite is true for SOM cells as these receive more horizontally than vertically aligned inputs (Adesnik, Bruns, Taniguchi, Huang, & Scanziani, 2012).

This particular subtype-specific arrangement of the afferents and efferents within the network probably underlies the different synchronization regimes that have been observed in mouse V1 (Chen et al., 2015; Veit, Hakim, Jadi, Sejnowski, & Adesnik, 2017; Lensjø, Lepperød, Dick, Hafting, & Fyhn, 2017). One of these changes is the attenuation of gamma oscillations when visually stimulating the adult mouse; synchronization is instead achieved in the beta frequency band (see Figures 1C and 1D) (Chen et al., 2015; Veit et al., 2017). Here, we define the beta and gamma frequency range to be [15Hz;25Hz] and [40Hz;60Hz], respectively. The extent to which gamma and beta rhythms are decreased and enhanced, respectively, depends on the size of the stimulus, which led to the hypothesis that an inhibitory surround effect underlies this switch in synchrony (Adesnik et al., 2012; Veit et al., 2017). These gamma oscillations are not present in younger mice as they develop during the critical period (CP) (see Figure 1A), a period well known for its increased levels of ocular dominance plasticity (Chen et al., 2015; Hensch, 2005). Furthermore, before the CP, visual stimulation does not alter the network's synchronization frequency, even though beta power is still slightly enhanced (see Figure 1B) (Chen et al., 2015). What exactly initiates the development of gamma rhythms during this time window is unclear, but PV cells are likely involved: studies using optogenetic perturbations indicate that PV and SOM cells can be associated with the gamma and beta oscillations, respectively (Veit et al., 2017; Chen et al., 2017; Hakim, Shamardani, & Adesnik, 2018). PV cell progenitors transplanted into the V1 of adult mice differentiate into the PV (hence, not the SOM) subtype, integrate successfully in the network, and initiate a period with enhanced plasticity that resembles the CP (Davis et al., 2015; Howard & Baraban, 2016). These findings thus indicate that PV cells are involved in gamma oscillation generation and CP-associated plasticity.
Figure 1:

Visual stimulation elicits a change in the synchronization of the neural ensemble in the adult, but not juvenile, mouse primary visual cortex (V1). (A,B) Spectral density characterization of the spontaneous (A) and visually evoked (B) local field potential (LFP) in V1 of juvenile mice (<P20). (C,D) Spectral density characterization of the spontaneous (C) and visually evoked (D) LFP in adult mouse V1 (>P60). freq. = frequency, stim. = stimulus.

Figure 1:

Visual stimulation elicits a change in the synchronization of the neural ensemble in the adult, but not juvenile, mouse primary visual cortex (V1). (A,B) Spectral density characterization of the spontaneous (A) and visually evoked (B) local field potential (LFP) in V1 of juvenile mice (<P20). (C,D) Spectral density characterization of the spontaneous (C) and visually evoked (D) LFP in adult mouse V1 (>P60). freq. = frequency, stim. = stimulus.

Actually, the experimental literature is enriched with evidence emphasizing these associations. One other study, for instance, shows that gamma rhythms in V1 are transiently amplified right after the start of monocular deprivation (MD) in juvenile mice and in adult ones that have been depleted of the perineuronal nets (PNNs), which predominantly ensheath PV cells (Lensjø et al., 2017; Pizzorusso et al., 2002). Although the exact cause of this rise in gamma power is unknown, it has been shown that in the same time interval after the start of MD during the CP, the spike rates of pyramidal cells (PCs) were at first reduced but gradually rose to their original values through a decrease of the PV cell activity in the network (Kuhlman et al., 2013). With respect to the former of these two studies, however, it must be mentioned that there is a controversy as to whether the stronger gamma oscillations following PNN removal critically depend on MD. A recent study has demonstrated that merely removing the PNNs results in enhanced gamma power as well and that stronger thalamic inputs to the PV cells in L4 could account for this; this rise in gamma power is actually attenuated by depleting the PNNs and starting MD immediately thereafter (Faini et al., 2018).

So far, the phenomenon of surround inhibition and the shift from gamma to beta oscillations via visual stimulation has only been explained at the level of neural mass models (Veit et al., 2017; Litwin-Kumar, Rosenbaum, & Doiron, 2016). These models, however, do not reveal the spiking-dependent network mechanisms and also do not explain the emergence of gamma oscillations during the CP. Therefore, inspired by the observed induction of beta and subsequent attenuation of gamma oscillations following visual stimulation and the absence of such a switch in juvenile mice, we developed a spiking neuron network model of L2/3 from mouse V1 comprising pyramidal, PV, and SOM cells to determine the network configurations that enable the switch in synchronization from the gamma to the beta frequency band upon visual stimulation.

A separate consideration of the pyramidal-PV and pyramidal-SOM cell subnetworks in our model not only confirms the hypotheses that PV and SOM cells are involved in the generation of gamma and beta oscillations, respectively. In addition, it suggests that these rhythms are both generated via pyramidal-interneuron gamma (PING) mechanisms, in which the PCs drive the oscillations as their action potentials precede inhibitory spiking (Tiesinga & Sejnowski, 2009), and that the difference in frequency can mostly be attributed to the distinct connectivity profiles. Analogously, the results acquired through the variation of the size of the stimulated area agree with previous models that the respective enhancement and attenuation of beta and gamma rhythms through visual stimulation are a consequence of surround inhibition. In addition, they show that this switch is realized by the activation of SOM cells that consequently manipulate the gamma oscillations generated by the pyramidal-PV cell subnetwork so that the full network produces beta rhythms instead. Finally, by sampling network activity for various settings of the PV and SOM cell projection strength, we demonstrate that only a restricted range of values for these parameters gives rise to the attenuation of gamma and amplification of beta oscillations following visual stimulation; too strong PV and SOM cell projection strengths result in persistent gamma and beta oscillations, respectively. This indicates that these two interneuron subtypes must exert an approximately equal influence on the PCs in adult mice; if this balance is not present, the characteristic switch in the network synchronization frequency cannot be induced in one and the same network realization. Moreover, given that the spontaneous local field potential (LFP) of the pre-CP mouse V1 contains primarily beta oscillations and the post-CP one predominantly gamma, our results also suggest that PV cell projections are, on average, strengthened across the CP, which is consistent with the notion of PV cells being integrated in the local network of this cortical area during that time window.

2  Method

The results presented in this letter are obtained through means of a network model, which has 4500 neurons assumed to be located in L2/3 of mouse V1. These neurons are modeled using Izhikevich's model (Izhikevich, 2003) and receive input from three sources: background activity, visually induced currents, and recurrent inputs. With respect to the latter input source, the connectivity patterns depicted in Figure 2A are implemented. An elaboration on the model's equations is included in the remainder of this section and is followed by a brief description explaining how they are implemented and how the analysis is carried out.
Figure 2:

Schematic of the subtype-specific connectivity patterns in the network, cell positions, and graphical depiction of areas used for the analysis. (A) Schematic that shows the connectivity patterns embedded in the network. Long-distance connections are only shown for the center column. Arrow sizes are representative for adult V1 (SPC=SPV=SSOM=0.5). (B) Example of cell positioning. Green triangles, red dots, and blue squares represent pyramidal (PC), parvalbumin-expressing (PV), and somatostatin-expressing (SOM) neurons, respectively. The black and orange shaded areas additionally depict the center and reference regions. EC = excitatory cell from cortical layer 4, L2/3 = cortical layer 2 and 3, L4 = cortical layer 4.

Figure 2:

Schematic of the subtype-specific connectivity patterns in the network, cell positions, and graphical depiction of areas used for the analysis. (A) Schematic that shows the connectivity patterns embedded in the network. Long-distance connections are only shown for the center column. Arrow sizes are representative for adult V1 (SPC=SPV=SSOM=0.5). (B) Example of cell positioning. Green triangles, red dots, and blue squares represent pyramidal (PC), parvalbumin-expressing (PV), and somatostatin-expressing (SOM) neurons, respectively. The black and orange shaded areas additionally depict the center and reference regions. EC = excitatory cell from cortical layer 4, L2/3 = cortical layer 2 and 3, L4 = cortical layer 4.

2.1  Membrane Potential Dynamics

In our model, 4500 neurons are considered that are assumed to be located in L2/3 of mouse V1: NPC=3600 pyramidal, NPV=495 PV, and NSOM=405 SOM neurons, which roughly matches the reported relative abundances of these cell types in V1 (van Versendaal & Levelt, 2016; Rudy, Fishell, Lee, & Hjerling-Leffler, 2011). They are distributed across a 2×2 square patch by randomly assigning them positions in a Cartesian coordinate system (rjA) via
rjA=(xjA,yjA)=(ηx,j,ηy,j),
(2.1)
where the index j{1,2,,NA} denotes a cell within neuron population A{PC,PV,SOM} and ηx,j and ηy,j are independent random values drawn from a uniform distribution between -1 and 1. The square's size is defined to be dimensionless as we want to investigate how the extent of the activated area relative to the entire patch of cortex influences the oscillations generated by the model. An example of the resulting cell positions is shown in Figure 2B.
The dynamics of the membrane potentials are simulated using Izhikevich's model, which comprises two coupled differential equations (Izhikevich, 2003). The first of them specifies the membrane potential VjA(t),involts,
dVjA(t)dt=40,000·VjA(t)2+5,000·VjA(t)-ujA(t)+IjA(t)Cm,
(2.2)
and the other describes a slower gating variable (ujA(t),involts/second),
dujA(t)dt=aA·bA·VjA(t)-ujA(t).
(2.3)
In these two equations, Cm and IjA(t) represent the membrane capacitance and the total input current received by the neuron under consideration (see below), respectively. An action potential is detected when the membrane potential exceeds 30mV; such a threshold crossing imposes the following reset condition on the membrane potential and the gating variable:
ifVjA(t)0.030VthenVjA(t)cAujA(t)ujA(t)+dA
(2.4)

In the two preceding equations, aA, bA, cA, and dA are parameters chosen such that the modelled spiking behavior resembles its experimentally observed counterpart. The parameter values are given in Table 1. These values are based on those given in the literature for these cell types (Tremblay et al., 2016; Izhikevich, 2003), but have been adjusted in order to approximately match the oscillation frequencies emerging from the model to those observed in experimental studies (Chen et al., 2015; Veit et al., 2017; Chen et al., 2017). Specifically, the a-parameters corresponding to the PCs and SOM cells were doubled, and the b-parameter of the PV cells was raised from 200 to 250s-1. These parameter settings result in the neurons behaving as integrators rather than resonators. Additionally, the cell capacitance was set to Cm=100pF for all subtypes.

Table 1:

Values for the Subtype-Specific Parameters in the Model.

Neuron Typea (s-1)b (s-1)c (mV)d (V·s-1)RKbgA (Hz)g¯bgA (pS·s)
Pyr. 40±4 200 -65.0±6.5 8.0±0.8 1,500 
PV 100±10 250 -65.0±6.5 2.0±0.2 200 22 
SOM 40±4 250 -65.0±6.5 2.0±0.2 
Neuron Typea (s-1)b (s-1)c (mV)d (V·s-1)RKbgA (Hz)g¯bgA (pS·s)
Pyr. 40±4 200 -65.0±6.5 8.0±0.8 1,500 
PV 100±10 250 -65.0±6.5 2.0±0.2 200 22 
SOM 40±4 250 -65.0±6.5 2.0±0.2 

Notes: A ± sign designates a gaussian distributed random variable following the notation mean ± standard deviation. PV = parvalbumin expressing, Pyr. = pyramidal, SOM = somatostatin expressing.

We recognize that equations 2.2 to 2.4 are in a somewhat different form than they are in Izhikevich (2003). In the supplementary material, we explain how the forms shown here correspond to our choice of units.

2.2  Input Currents

The total input current received by each cell consists of three different: the background currents Ibg,jA(t), the visually induced input from L4 Ivis,jA(t), and the recurrent inputs Irec,jA(t):
IjA(t)=Ibg,jA(t)+Ivis,jA(t)+Irec,jA(t).
(2.5)

2.2.1  Background Input

The background represents inputs originating from an ensemble of KbgA neurons that are not explicitly modeled but whose spike trains are characterized by Poisson processes with a constant firing rate of RbgA. When the product of the latter two parameters is denoted by RKbgA, the resulting background current received by the neuron can be approximated via (Hansel & van Vreeswijk, 2012)
Ibg,jA(t)=gbg,jA(t)Ebg-VjA(t),wheregbg,jA(t)=g¯bgARKbgA+RKbgA·ξbg,jA(t).
(2.6)
In these equations, gbg,jA(t) denotes the conductance induced in the cell, and g¯bgA and Ebg are the overall scale of conductance induced by and the reversal potential of the background neuron ensemble, respectively. Finally, ξbg,jA(t) is a gaussian distributed random variable with zero mean and a temporal correlation of ξbg,jA(t)·ξbg,jA(t')=exp(-|t-t'|/τbg)/2τbg.

The values for the subtype-specific parameters are shown in Table 1. The remaining parameters, Ebg and τbg, were set to 0mV and 2ms, respectively. The values given to parameters Ebg, τbg, and g¯bgA(A{PC,PV,SOM}) are in line with the values assigned for PC projections (see section 2.2.3), though the values corresponding to the latter parameter have been scaled toward much lower values. As it is hard to designate a concrete source for background input, the background spike rate parameters (RKbgA,A{PC,PV,SOM}) are given values such that the network's excitability is raised to a dynamical range where recurrent inputs as well as visually induced currents are able to influence network dynamics. Finally, note that the value for RKbgPC is changed in some simulations to assess whether a particular circuit (pyramidal-PV cell or pyramidal-SOM cell subnetwork) produces rhythms via an interneuron gamma (ING) or a PING mechanism; if the excitability of the pyramidal cells is increased, an ING and PING mechanism give rise to similar and altered oscillation frequencies, respectively (Tiesinga & Sejnowski, 2009). In an ING mechanism, the oscillations are generated by the transient ceasing of inhibitory spiking, allowing PCs to produce action potentials (Tiesinga & Sejnowski, 2009). This also implies that in an ING mechanism, the inhibitory precede the excitatory spikes, whereas the reverse is true for PING mechanisms. Additionally, we change the background spike rates to the interneurons (RKbgA,A{PV,SOM}) in some simulations as well in order to characterize how altering these parameters impacts the oscillations generated by the individual circuits.

2.2.2  Visually Induced Input

The dorsal part of the lateral geniculate nucleus, which receives its input from the retina, projects to L4 of V1 (Seabrook, Burbridge, Crair, & Huberman, 2017). The signals are subsequently projected to L2/3 (Douglas & Martin, 2004; Olivas, Quintanar-Zilinskas, Nenadic, & Xu, 2012). Since the model represents L2/3 neurons, visually induced activity is assumed to be provided by the afferents originating from the same horizontal location but in L4. Following a previous study (Adesnik et al., 2012), it is decided that only pyramidal and PV cells (A{PC,PV}) receive visually induced currents that thus presumably originate from L4 (see Figure 2A). Not all the cells within these subgroups receive visual input: cells need to be within a square field with length 2·Dvis that is concentric with the square patch. Approximately half of the cells within this area are randomly selected to receive the visually induced currents. In sum, the following condition thus has to be met by a pyramidal or PV cell to receive visually induced input:
|xjA|Dvis|yjA|Dvis0.5ηvis,jA.
(2.7)
Here, is the logical AND operator, and ηvis,jA is a number randomly drawn from the standard uniform distribution (interval between 0 and 1). The visually induced current has the following form,
Ivis,jA(t)=Ivis,0·1+νvis,jA(t),
(2.8)
where Ivis,0 represents the magnitude of the current and νvis,jA(t) is gaussian white noise with a mean of 0 and a standard deviation of 1.

The variable Dvis is varied to assess its effect on the network's synchronisation. It approximately reflects the effects of stimulus size, without specifically taking into account the retinotopic mapping as well as other feature maps that are present in the mouse visual cortex (Basole, White, & Fitzpatrick, 2003). If not specified otherwise, Ivis,0 is set to 100 pA; similar values have been reported in an experimental study (Anderson, Carandini, & Ferster, 2000). Still, we investigate whether any observed effects critically depend on this particular value by varying it in one series of simulations.

2.2.3  Recurrent Inputs

The recurrent inputs include the currents that the explicitly modeled neurons in the network receive from one another. The probability of neuron j from subtype A to have neuron k from subtype B as an afferent is calculated via
PkjBA=G(xjA|xkB,σBA)·G(yjA|ykB,σBA)·PconBA,
(2.9)
where the factor PconBA has been included so that the relative number of connections approximately matches the subtype-specific connection probabilities listed in the experimental literature (Pfeffer et al., 2013; Jouhanneau, Kremkow, & Poulet, 2018; Jiang et al., 2015). In addition, the function G(x|m,s) represents the periodic gaussian probability distribution function,
G(x|m,s)=j=-j=12πsexp-(x-m-jL)22s2,
(2.10)
where L corresponds to the length of the square patch. In this context, one can interpret σBA as the width of the area around a cell from subpopulation B wherein it is possible for that neuron to connect to a neuron from subtype A. Note also that equations 2.9 and 2.10 imply that the probability distribution is not strictly circularly symmetric. Instead, the periodic gaussian increases the likelihood that a neuron close to one side of the square patch is connected to a cell that is near the opposite side so that the number of inputs to a neuron from a particular subtype is approximately equal throughout the square patch. The weights of the connections wkjBA are initialized via
wkjBA=SBwithprobabilityPkjBA0withprobability1-PkjBA,
(2.11)
where SB is the subtype-specific projection strength. This parameter thus scales all the outgoing projections of a neuron from subpopulation B. Given these weights, the recurrent input received by neuron j from subpopulation A{PC,PV,SOM} is determined via
Irec,jA(t)=Bg¯recBA·ErecB-VjA(t)kskB(t)·wkjBA.
(2.12)
In this expression, ErecB is the reversal potential of the synapse type associated with the cell type at the presynaptic side, and g¯recBA is the amplitude of the conductance induced in the postsynaptic cell by a presynaptic spike. In addition, skB(t) is the synaptic gating variable attached to presynaptic neuron k from population B, even though it represents changes at the postsynaptic side. It obeys the dynamics described by
dskB(t)dt=-skB(t)τrecBandskB(t)skB(t)+1iftheneuronspikes,
(2.13)
where τrecB is the synaptic decay time constant. It should be noted that equation 2.12 implies that this quantity is then also the decay time constant of the postsynaptic current.

The values for the subtype-specific parameters that have been introduced in this paragraph are given in Table 2 and are based on multiple papers that explored subtype-specific properties in mouse V1 (Pfeffer et al., 2013; Jouhanneau et al., 2018; Stern, Edwards, & Sakmann, 1992; Safari, Mirnajafi-Zadeh, Hioki, & Tsumoto, 2017). To be more explicit, the values corresponding to interneuron projections (PconPVA, PconSOMA, g¯recPVA, g¯recSOMA, A{PC,PV,SOM}) have been collected from Pfeffer et al. (2013) and those assigned to PC projection parameters (PconPCA,grecPCA,A{PC,PV,SOM}) have been derived from Jouhanneau et al. (2018). The synaptic time constants of the interneurons (τrecPV,τrecSOM) are inspired by Safari et al. (2017), that is, we use the same proportion between the two as proposed by the study, but scale them toward smaller values to make them more comparable to the synaptic time constant of the PCs (τrecPC); this latter value has been acquired from Stern et al. (1992) and is in accordance with an earlier modeling study to the mouse V1 (Martens, Houweling, & Tiesinga, 2017). In addition, σPCSOM=12 and σBPC=σBPV=16 where B{PC,PV,SOM}, which reflects the experimental finding that intralaminar projections to PC and PV cells are more spatially restricted than the ones to SOM cells. Furthermore, we set ErecPC=0 mV and ErecPV=ErecSOM=-80 mV. Finally, the overall synaptic strength scaling factor of the PCs is set to SPC=0.5, while those of the PV and SOM cells (SPV and SSOM) are varied to assess their effect on the network's synchronization. Figure 2A shows a schematic depiction of the network architecture that results from these parameter settings.

Table 2:

The Values for Subtype-Specific Parameters Associated with the Recurrent Inputs.

Presyn. cell (B)PconBPCPconBPVPconBSOMg¯recBPC (nS)g¯recBPV (nS)g¯recBSOM (nS)τrecB (ms)
Pyr. 0.07 0.44 0.44 0.2 0.8 0.2 
PV 1.00 1.00 0.00 2.4 2.4 0.0 
SOM 1.00 0.86 0.00 1.6 0.8 0.0 15 
Presyn. cell (B)PconBPCPconBPVPconBSOMg¯recBPC (nS)g¯recBPV (nS)g¯recBSOM (nS)τrecB (ms)
Pyr. 0.07 0.44 0.44 0.2 0.8 0.2 
PV 1.00 1.00 0.00 2.4 2.4 0.0 
SOM 1.00 0.86 0.00 1.6 0.8 0.0 15 

Note: Presyn. = presynaptic, PV = parvalbumin expressing, Pyr. = pyramidal, SOM = somatostatin expressing.

2.3  Parameter Variations

2.3.1  Assessing Stimulus Size Effects

For the first two parameter variations, which are carried out to investigate whether our model can reproduce the attenuation of gamma and the increase of beta power upon visual stimulation (network activation), the projection strengths of the interneurons are set to SPV=SSOM=0.5. We first vary the stimulus size parameter from 0.0 to 1.0 in steps of 0.1 (Dvis={0.0,0.1,,1.0}). For each parameter setting, 85s of network behavior are simulated. We also want to confirm that our model can dynamically reproduce the frequency switching. To do so, we change the stimulus size parameter during the simulation. Here, the data are acquired in epochs of 5s. The stimulus size parameter is set to Dvis=1.0, and we vary the magnitude of the visually induced current. It is set to 0pA in the first two and the last second, while during the third and fourth second of the epoch, it is set to various values to assess its influence on the network's synchronization; it can assume values between 20 and 180pA in steps of 40pA(Ivis,0={20pA,60pA,,180pA}). Seventeen epochs of 5s are acquired for each parameter setting.

2.3.2  Investigating the Role of Interneuron Projection Strengths

Next, we want to determine for which combinations of PV and SOM cell projection strengths the frequency switching behavior can be observed. In order to do so, we vary the PV cell projection strength from 0.3 to 0.9 in steps of 0.1 (SPV={0.3,0.4,,0.9}) and the SOM cell projection strength from 0.3 to 1.3 also in steps of 0.1 (SSOM={0.3,0.4,,1.3}). The network can be either in the resting (Dvis=0.0) or an activated (Dvis=1.0) state. For each parameter setting, 85s of network behavior are simulated. We also want to investigate in detail how increasing the PV cell projection strength influences the network's dynamical behavior. To do so, the network is simulated for a much longer period during which the PV strength is gradually increased from 0.3 to 0.7 in steps of 0.001 (SPV={0.300,0.301,,0.7}) while the SOM strength is fixed at SSOM=0.5. Again, the network can either be in the resting (Dvis=0.0) or the activated (Dvis=1.0) state. Five seconds of network behavior are simulated for each setting of the PV cell projection strength and the stimulus size parameter.

2.3.3  Determining the Mechanism of Oscillation Production

In order to comprehend how the switch in oscillation frequency upon visual stimulation is established, we first need to understand how either oscillation type is generated. We therefore consider the pyramidal-PV cell and pyramidal-SOM cell subnetworks individually. This is realized by setting SPV=0.5 and SSOM=0.0 to study the pyramidal-PV cell circuit and SPV=0.0 and SSOM=0.5 to obtain the dynamics associated with the pyramidal-SOM cell circuit. Additionally, the stimulated field parameter is set to Dvis=0.0. To determine which mechanism (interneuron gamma (ING) or pyramidal-interneuron gamma (PING)) underlies either type of oscillation, we alter the drive to the PCs by varying the background spike rate parameter corresponding to these cells (RKbgPC{1500Hz,2000Hz,3000Hz}) and examine the resulting power spectra and correlograms. If the oscillations are produced by an ING mechanism, this variation will not alter the oscillation frequency, whereas in a PING mechanism, the frequency would change (Tiesinga & Sejnowski, 2009). Additionally, an ING mechanism would suggest that excitation follows inhibition, which is the other way around for a PING mechanism. We also vary the drive to the interneurons (RKbgPV{100Hz,200Hz,300Hz}andRKbgSOM{1Hz,10Hz,50Hz}) to study how that influences the dynamics of the two subnetworks. For all parameter settings, 85s of network behavior are simulated.

Though this consideration reveals by what mechanism the oscillations are generated, it does not provide an intuition as to what dictates (the differences in) their frequencies. In other words, it does not resolve the question why the pyramidal-PV cell and pyramidal-SOM cell subnetworks produce rhythms that fall within the gamma and beta frequency band, respectively. To test if subtype-specific connectivity profiles or distinct internal cell parameters are responsible, we consider the whole network and switch the internal parameters of the PV and SOM cells. Given Table 1, this is realized by simply setting aPV=40±4s-1 and aSOM=100±10s-1. Subsequently, we vary the stimulus size parameter from 0 to 1 in steps of 0.1 (Dvis={0.0,0.1,,1.0}) and sample 85s of network dynamics for each field of visual stimulation.

2.3.4  Studying the Frequency Switching Mechanism

Finally, we seek to find out how the visually induced switch in oscillation frequency is realized in our model. For this examination, we first vary the natural frequency associated with the pyramidal-SOM cell subnetwork by altering the synaptic time constant of the SOM cell projections (τrecSOM{3.0ms, 7.5ms, 15.0ms, 30.0ms,75ms}). Note that changing the synaptic time constant may alter the SOM cells' influence within the network as well. To compensate for this effect, the SOM cell projection strength is varied from 0.3 to 1.3 in steps of 0.1 (SSOM{0.3,0.4,,1.3}), while keeping the one of the PV cell projections fixed at its standard value (SPV=0.5). For each parameter setting, 85s of resting-state (Dvis=0.0) and fully activated (Dvis=1.0) network dynamics are simulated. After the simulations are completed, one SOM cell projection strength is selected for each setting of the synaptic time constant. To determine an appropriate value for this projection strength, we first calculate the the mean beta (10–30 Hz) and gamma (40–60 Hz) power differences. Here, a positive difference means an increase in the power corresponding to the activated network with respect to the resting state. Subsequently, we multiply the mean beta power difference with the negative of the mean gamma power difference and select the SOM cell projection strength with the highest value for this product. Also, the beta frequency range mentioned here is broader than elsewhere in this letter since we need to account for any drastic shifts in beta oscillation frequency that can be induced by the synaptic time constant variation.

Finally, we determine which SOM cell projections are critical for the induction of beta and reduction of gamma oscillations upon visual simulation. To investigate this, we once again simulate 85s of resting-state (Dvis=0.0) and fully stimulated (Dvis=1.0) network dynamics, although now either the SOM-to-pyramidal cell projections or the SOM-to-PV cell projections are severed by setting wkjSOMPC=0 or wkjSOMPV=0, respectively. Also here, it must be recognized that severing these connections may affect the SOM cells' influence within the network; therefore, the SOM cell projection strength selection procedure described in the previous paragraph is performed for this investigation as well.

2.4  Implementation and Analysis

The model has been implemented using the Python (Python Software Foundation, https://www.python.org/) in combination with the C++ (Standard C++ Foundation, https://isocpp.org/) programming language. The integration follows Euler's method, where the integration time step size is set to 0.5ms (sampling rate of fs=2000Hz). The first 5s of every network simulation are removed prior to analysis so that the initial conditions do not influence the results. Each simulation is repeated for 12 different settings of the random seed, which controls the random variables in both the realization of network connectivity and the temporal dynamics, to estimate the variance in the results as a consequence of a particular choice of random variables. The mean potentials across the separate neuron subtypes and the spike times are stored for further analysis. We use two measures as LFP estimate V¯(t) to make sure that observed effect sizes do not critically depend on our choice of LFP model: the mean potential of the PCs and the peristimulus time histogram (PSTH) of the spikes produced by the PCs, which is calculated using a 1ms bin size (sampling rate of fs=1000Hz). Even though more refined LFP proxies have been reported (Mazzoni et al., 2015), we still choose the PSTHs and mean potentials as they robustly characterize the changes in the firing rate and the subthreshold dynamics induced by the different simulation conditions, respectively. Also, we only consider the PCs for the extraction of these quantities because superposition of their aligned dipoles does not lead to signal cancellation as opposed to the other cell types (Lindén, Pettersen, & Einevoll, 2010; Buzsáki, Anastassiou, & Koch, 2012). Both of these measures are used in two types of analysis: (time-resolved) spectral analysis and the calculation of spike-LFP pair-wise phase consistencies (PPCs). The PSTHs across the other cell types are also derived from the spike times and are used to calculate cross-correlation functions, so that these functions reflect the relative spike timings of the individual neuron subtypes with respect to one another. We now briefly describe each of the analysis methods we used.

2.4.1  Spectral Analysis

For spectral analysis, the spectral power is based on segments with a length of 5s, which are demeaned by subtracting the mean across time for each individual segment. Subsequently, each demeaned segment is subjected to the multitaper spectral density estimation method with five tapers that have a time half bandwidth product of 3 (Thomson, 1982). Finally, the mean is taken across the 16 spectra to obtain the average power spectrum for each setting of the parameters and the random seed.

2.4.2  Spike-LFP Pair-Wise Phase Consistencies (PPCs)

The calculation of the spike-LFP PPCs requires the use of the total length of the recordings; the data are not cut into smaller segments. For the analysis, two regions have to be defined with respect to the square patch. The spikes that are used to determine the PPCs are those corresponding to the neurons from the center area (see Figure 2B, black shaded area), while the LFP estimate (mean potentials or PSTH) is calculated on the basis of the PCs that are located in a reference region (see Figure 2B, orange shaded area). These distinct regions are defined to avoid spurious PPCs. The LFP estimate is demeaned and bandpass-filtered using two filters with distinct frequency bands: one filter only keeps the beta band (15–25 Hz) and another one only the gamma band (40–60 Hz) oscillations. The filtering is performed using an elliptic infinite impulse response filter with a passband ripple of 0.1 dB and a stopband attenuation of 60 dB via the functions iirdesign and filter included in the SciPy software package for Python (SciPy developers, https://www.scipy.org/about.html). The discrete Hilbert transform is taken of both bandpass-filtered signals (Marple, 1999). The intermediate result therefore comprises two bandpass-filtered, analytical signals (signals with both real and imaginary components), one for each frequency band. The phase of all complex values within these two signals are calculated so that estimates for the instantaneous phases (ϕβ and ϕγ) are retrieved. The spikes produced by neurons located in the center region are assigned their corresponding phases, that is, the phases that each of the oscillation types had when these action potentials appeared. Then the spike-LFP PPCs (ϒβ and ϒγ) are calculated for all neurons separately via Vinck, van Wingerden, Womelsdorf, Fries, and Pennartz (2010):
ϒfb=2M(M-1)j=1M-1k=j+1Mf(ϕjfb,ϕkfb).
(2.14)
In this expression, ϕjfb is the phase that is assigned to the jth spike from the considered neuron with respect to frequency band fb{β,γ} and f(ϕjfb,ϕkfb) represents the inner product of the two angles:
f(ϕjfb,ϕkfb)=cosϕjfbcosϕkfb+sinϕjfbsinϕkfb.
(2.15)
In addition, M is the total number of action potentials that were produced by that cell. Finally, the mean is taken across the PPCs from the individual neurons in the center region per subtype to yield the PPC of that group of cells for a given setting of the parameters and the random seed.

2.4.3  Cross-Correlations

Correlation analysis is also performed across the entire length of the recordings, though now the LFP estimates are constructed for each individual cell type by calculating the PSTHs on the basis of all the neurons of that particular type in the network. Each LFP estimate is again demeaned and, from the resulting signal, the cross-correlations between subpopulations RBA(τl) as functions of the lag between the postsynaptic and the presynaptic activity (τl) are determined via (Rabiner & Gold, 1975; Rabiner & Schafer, 1978)
RBA(τl)=tV¯A(t)·V¯B(t+τl).
(2.16)
These functions are normalized using the zero lag autocorrelations of each separate subpopulation yielding the normalized cross-correlation functions (R¯BA(τl)):
R¯BA(τl)=RBA(τl)RBB(0)RAA(0).
(2.17)

Correlation functions are in some cases subjected to spectral analysis by using the fast Fourier transform (FFT). First, the correlation function is transformed and, subsequently, the absolute values are taken of the complex valued outcome. Next, this two-sided spectrum is transformed to a one-sided one. Finally, the values are squared to get the power density spectrum. From such a spectrum, it can be determined to what extent the beta and gamma oscillations are present in the correlation functions.

2.4.4  Time Resolved Spectral Analysis

Spectrograms are constructed to confirm that the frequency changes following visual stimulation can be obtained dynamically and to determine the range of the PV projection strength that allows for this behavior while the SOM cell projection strength is fixed.

For the analysis used to confirm the dynamical nature of the frequency switching, a square window function with a size of 200ms is slid over the time series in steps of 100ms. As 16 usable 5s long segments are acquired for each setting of the random seed for this investigation (see section 2.3.1), 49 windows are obtained from each segment. The power density spectrum of each is calculated by demeaning the LFP estimates and subsequently subjecting them to the same procedure that is used to calculate the power density spectra of the correlation functions (see above). Finally, the average across these 16 segments is calculated to retrieve the average spectrogram for each setting of the random seed.

The simulations used for the assessment of the PV projection strengths allowing for the frequency switching behavior eventually yield 401 usable 5s long segments per setting of the random seed (see section 2.3.2). Each of these segments reflects the network behavior for a different setting of the PV projection strength. The segments are divided into five smaller 1s long segments. The power density spectra of these smaller segments are calculated by demeaning them and subsequently subjecting them to the same approach that is used to calculate the power density spectra of the correlation functions (see above) and, finally, their mean is taken. Using this procedure, a spectrogram consisting of 401 power density spectra is obtained for each setting of the random seed.

3  Results

Our spiking neuron network model of L2/3 from mouse V1 comprising 3600 PCs, 495 PV cells, and 405 SOM cells is introduced in section 2. All neurons are spread out across a 2×2 square patch of cortex, their membrane potential dynamics are simulated via Izhikevich's neuron model (Izhikevich, 2003), and each neuron within the network can receive input from three sources at most: the background, the visually induced activity, and the recurrent connections. The recurrent connections comprise the inputs the explicitly modeled neurons send to and receive from one another and are initialized according to the scheme depicted in Figure 2A. With regard to the visually induced activity, we defined a stimulus size parameter (Dvis), which can vary from 0 to 1 and determines the area within the simulated cortical patch in which neurons can receive visual input. This parameter thus approximately reflects the effects of stimulus size, without specifically taking into account any maps present in the mouse visual cortex (Seabrook et al., 2017). By varying the stimulus size parameter as well as the projection strengths of the interneurons, we explore the oscillatory properties of our model. For the specifics of our model and the parameter variations, we refer to section 2.

In this section, we discuss the results of this exploration. First, we show that our model reproduces the experimentally observed phenomenon of beta power enhancement and gamma power reduction following visual stimulation. Additional simulations confirm that our model is also able to simulate this frequency switching in a dynamical manner. By means of a grid search with respect to the PV and SOM cell projection strengths, we nonetheless demonstrate that the network only exhibits this frequency switching behavior for a small subset of all possible combinations of these two parameters. Then, by keeping the SOM cell projection strength fixed and increasing the one of the PV cells in small steps, we investigate the transitions with respect to the network behavior in more detail. Finally, we apply some modifications to the original model to study the mechanism that allows for flexible frequency switching.

3.1  Visual Stimulation Enhances Beta Rhythms via the Sculpting of the Temporal Structure of Inhibition

For the first series of simulations (see section 2.3.1), we set RKbgPC=1500Hz and SPV=SSOM=0.5. We examine how enlarging the stimulated field (i.e., increasing the value for the Dvis parameter), alters the synchrony in the network activity. The results indicate that the spike rates of the pyramidal and PV cells in the center region (see Figure 2B, black shaded area) initially increase but eventually decay when the extent of the stimulated field is increased (see Figures 3A and 3B). The firing rates of the SOM cells increase monotonously (see Figure 3C). These trends are in accordance to the empirically measured size-tuning curves (Adesnik et al., 2012). Subsequently, we analyze the power spectra obtained from the mean potentials of the PCs. Three peaks can be observed: a gamma peak at approximately 55Hz, a beta peak around 18Hz, and another one at roughly 36Hz, which is the harmonic of the beta peak as it appears at double that frequency (Figure 3D). Varying the stimulated field size reveals that the beta and gamma peak heights are increased and reduced as the area of activation increases, respectively (see Figure 3D). This effect is quantified by calculating the average power across the beta (15–25 Hz) and the gamma (40–60 Hz) frequency bands, which demonstrates that enlarging the visually activated area promotes beta and inhibits gamma oscillations (see Figures 3E and 3F). The power spectra obtained from the PSTHs of the PCs confirm that these findings do not critically depend on the use of the mean potentials of the PCs as the LFP estimate (see Figure S.1).
Figure 3:

Enlarging the model's stimulated field elicits a rise and fall of beta and gamma power, respectively. (A–C) Mean spike rates of the pyramidal (PC) (A), parvalbumin-expressing (PV) (B), and somatostatin-expressing (SOM) cells (C) in the center region as a function of the stimulus size relative to the entire simulated area (see Figure 2B). (D) Frequency spectra for three different relative stimulus sizes. (E,F) Mean power across the beta (15–25 Hz) (E) and the gamma (40–60 Hz) (F) frequency bands as a function of the relative stimulus size. Solid lines and bars depict the means, while error bars and shaded areas represent the ± mean standard deviation intervals across repeats of the simulation with different random seeds. Spectra were derived from the mean potentials of the PCs. a.u. = arbitrary unit, rel. = relative, stim. = stimulus.

Figure 3:

Enlarging the model's stimulated field elicits a rise and fall of beta and gamma power, respectively. (A–C) Mean spike rates of the pyramidal (PC) (A), parvalbumin-expressing (PV) (B), and somatostatin-expressing (SOM) cells (C) in the center region as a function of the stimulus size relative to the entire simulated area (see Figure 2B). (D) Frequency spectra for three different relative stimulus sizes. (E,F) Mean power across the beta (15–25 Hz) (E) and the gamma (40–60 Hz) (F) frequency bands as a function of the relative stimulus size. Solid lines and bars depict the means, while error bars and shaded areas represent the ± mean standard deviation intervals across repeats of the simulation with different random seeds. Spectra were derived from the mean potentials of the PCs. a.u. = arbitrary unit, rel. = relative, stim. = stimulus.

We wonder how this transition in network synchronization is accomplished and first study the spike time rastergrams to answer this question. When none of the neurons receive additional, visual input, no clear gamma-periodic volleys of action potentials originating from the pyramidal and PV cells can be observed (see Figure 4A). In contrast, a stimulated field that spans the whole area elicits strong synchronous, beta rhythmic spiking of SOM cells, which causes a suppression of pyramidal and PV cells (see Figure 4B). A larger stimulated field thus appears to enable SOM cells to produce their spikes in sync with the beta rhythm. To quantify this, the spike-LFP PPCs of the neurons in the center region are determined for each individual cell type by using the mean potential across PCs as the LFP estimate. The results indicate that all cell types gradually spike more and less coherent with the beta and the gamma oscillations, respectively, as the stimulated field increases (see Figures 4C–4E). The same effect is observed when the PPCs are calculated while using the PSTH of the PCs as the LFP estimate (see Figure S.2).
Figure 4:

Activation of somatostatin-expressing (SOM) cells transforms the gamma oscillations of the pyramidal and parvalbumin-expressing (PV) cells into beta rhythms. (A,B) Examples of spike time rastergrams for two stimulated field sizes: Dvis=0.0 (A) and Dvis=1.0 (B). Green, red, and blue dots represent action potentials from pyramidal, PV, and SOM cells, respectively. (C–E) Normalized spike-LFP pair-wise phase consistencies (PPCs) of the pyramidal (C), PV (D), and SOM (E) cells with respect to the beta (15–25 Hz) and gamma (40–60 Hz) frequency ranges as a function of the stimulus size. Blue and red colors correspond to the beta and gamma frequency band, respectively. PPCs were determined on the basis of the mean potentials of the pyramidal cells (PCs) and have been normalized to the maximum mean value. (F–H) Correlograms of the PC peristimulus time histogram with itself (F) and with those of the PV (G) and the SOM (H) cells for various stimulated field sizes. A common color legend is included on the right of panel F. Dots mark peaks that correspond to both beta and gamma rhythms, while downward arrows mark maxima that are only associated with gamma. (I,J) Mean beta (15–25 Hz) (I) and gamma (40–60 Hz) (J) power in the different types of correlograms as a function of the stimulated field size. Green, red, and blue colors correspond to the type of correlogram shown in panels F, G, and H, respectively (pyramidal with pyramidal, PV with pyramidal, and SOM with pyramidal, respectively). Solid lines and error bars depict the mean ± standard deviation across repeats of the simulation with different random seeds. a.u. = arbitrary unit, coef. = coefficient, corr. = correlation, norm. = normalised, rel. = relative, stim. = stimulus.

Figure 4:

Activation of somatostatin-expressing (SOM) cells transforms the gamma oscillations of the pyramidal and parvalbumin-expressing (PV) cells into beta rhythms. (A,B) Examples of spike time rastergrams for two stimulated field sizes: Dvis=0.0 (A) and Dvis=1.0 (B). Green, red, and blue dots represent action potentials from pyramidal, PV, and SOM cells, respectively. (C–E) Normalized spike-LFP pair-wise phase consistencies (PPCs) of the pyramidal (C), PV (D), and SOM (E) cells with respect to the beta (15–25 Hz) and gamma (40–60 Hz) frequency ranges as a function of the stimulus size. Blue and red colors correspond to the beta and gamma frequency band, respectively. PPCs were determined on the basis of the mean potentials of the pyramidal cells (PCs) and have been normalized to the maximum mean value. (F–H) Correlograms of the PC peristimulus time histogram with itself (F) and with those of the PV (G) and the SOM (H) cells for various stimulated field sizes. A common color legend is included on the right of panel F. Dots mark peaks that correspond to both beta and gamma rhythms, while downward arrows mark maxima that are only associated with gamma. (I,J) Mean beta (15–25 Hz) (I) and gamma (40–60 Hz) (J) power in the different types of correlograms as a function of the stimulated field size. Green, red, and blue colors correspond to the type of correlogram shown in panels F, G, and H, respectively (pyramidal with pyramidal, PV with pyramidal, and SOM with pyramidal, respectively). Solid lines and error bars depict the mean ± standard deviation across repeats of the simulation with different random seeds. a.u. = arbitrary unit, coef. = coefficient, corr. = correlation, norm. = normalised, rel. = relative, stim. = stimulus.

The inspection of the spike time rastergrams and their associated spike-LFP PPCs raises the question how the activities of the individual cell types are correlated to one another. Therefore, we determined the correlations between the PSTHs of the various cell types in the network. The autocorrelation function of the pyramidal cell activity reveals that a larger stimulated field attenuates the extrema in the correlogram that correspond to a gamma period, while the peaks appearing with a beta-like time interval are slightly amplified (see Figure 4F). The same is observed in the correlogram between the PV cell and PC activity (see Figure 4G). The correlation function between the SOM and pyramidal cell activity, on the contrary, seems to almost exclusively contain beta rhythms, especially if the stimulated field covers the whole area (see Figure 4H). To quantify the presence of both types of oscillations in these correlograms, their power spectra are determined using a FFT. Subsequently, the mean beta (15–25 Hz) and gamma (40–60 Hz) power are calculated from these spectra for each setting of the stimulated field. The outcome demonstrates that all three correlation functions become more dominated by the beta rhythm as the stimulated field grows (see Figure 4I). A complementary decrease in the gamma power is observed for the correlation functions of the PC activity with itself and with the PV cell activity (see Figure 4J, green and red lines), but not for the correlograms of the SOM with the pyramidal cell activity as they contain virtually no gamma oscillations for any size of the stimulated field (see Figure 4J, blue line). These results indicate that the SOM cells transform the gamma oscillations, which emerge through the interplay between pyramidal and PV cells, so that beta rhythms are amplified at the expense of gamma periodic network activity.

Finally, we perform additional checks to further assess the validity of our model. First, we confirm that our model can also reproduce the frequency-switching phenomenon dynamically. For this investigation, we set the activated area to cover the entire simulated cortical patch (Dvis=1.0) and alter the visually induced current (Ivis,0) within a 5s epoch: this parameter is set to 0pA in the first 2 s and the last second and to varying values in the intermediate 2 s. Inspection of the spectrograms reveals that beta and gamma power are indeed increased and decreased, respectively, when the visual current was higher than zero (see Figure S.3). Additionally, the results indicate that in our model, the extent to which the beta and gamma power are amplified and attenuated, respectively, not only depends on the stimulus size parameter, but also on the magnitude of the network activation current (Ivis,0) (see Figure S.3). In order to gain a complete understanding of the interplay between visual stimulus size and network activation current, we perform a grid search with respect to these two parameters, which reveals that the frequency-switching effect is inducible even when the latter parameter shrinks as the stimulus size increases (see Figure S.4). Finally, we test whether the results qualitatively depend on the proportion of pyramidal and PV cells that are located in the stimulated field and receive the visual activation current. To do so, we lower this proportion from 0.5 (see equation 2.7) to 0.3. The results show that besides an attenuation of the frequency-switching effect, no qualitative change is observed with regard to the spectral properties (see Figure S.5). One qualitative change, however, can be observed with regard to the SOM cells: their spike rate now increases linearly with the stimulus size (see Figure S.5C), which is different from the supralinear trend observed for the standard network (see Figure 3C).

3.2  To Amplify Beta and Attenuate Gamma Oscillations by Means of Visual Stimulation, the PV and SOM Cells Must Be Allowed to Compete Over Oscillatory Control of the PCs

We want to find out how generic the network motif is that allows beta oscillations to outcompete gamma upon visual stimulation. We therefore study the conditions under which the switch in oscillation frequency can be induced by varying the projection strengths of the PV (SPV) and the SOM (SSOM) cells, which scale the magnitude of all conductances induced by the considered interneuron subtype, either while none of the neurons receive input through visually induced activity (the resting-state network) or while the stimulated field size covers the entire square patch. At first, we look at the mean spike rates of the individual cell types. In addition to the pyramidal and PV cells being suppressed by stronger PV and SOM cell projections, their firing rates also exhibit a local minimum (see Figures 5A and 5B). The SOM cells, on the other hand, show a very different pattern: higher levels of SOM cell inhibition increase their firing rate, while stronger PV cell projections reduce it (see Figure 5C). The beta power follows the same pattern as the SOM cell firing rate (see Figures 5C and 5D), while the trend of the gamma power has more in common with those of the pyramidal and PV cells (see Figures 5A, 5B, and 5E). These spectral densities correspond to the resting-state (Dvis=0.0) network and are based on the mean potentials of the PCs. Also, the power that corresponds to a stimulated field that covers the entire square patch (Dvis=1.0) is determined using that same measure as the LFP estimate. Subsequently, the difference between the power associated with the visually activated and resting-state network is calculated. We observe that there is only a small subset of settings for the PV and SOM cell projection strengths that gives rise to higher beta and lower gamma power in the activated state (see Figures 5F and 5G). We determine which settings bring about each kind of behavior. The results of this assessment demonstrate that there is only a limited region with regard to the PV and SOM cell projection strengths that gives rise to enhanced beta and decreased gamma oscillations via activation of the network (see Figure 5H). Gamma and beta rhythms dominate the network's oscillatory activity when the strengths of the PV and SOM cell efferents, respectively, are too strong (see Figures 5D to 5H). Similar findings are obtained when the PSTHs of the PCs are used as the LFP estimate (see Figure S.6).
Figure 5:

Visual stimulation can induce an enhancement of beta and a reduction of gamma power for a restricted range of parvalbumin-expressing (PV) and somatostatin-expressing (SOM) interneuron projection strengths. (A–C) Mean firing rates of pyramidal (A), PV (B), and SOM (C) cells in the center region as a function of the projection strengths of the two interneuron types. Orange arrows point to local minima. (D,E) Beta (15–25 Hz) (D) and gamma (40–60 Hz) (E) power as a function of the PV and SOM cell projection strengths. The frequency spectra were obtained from the mean potentials of the PCs and have been normalized through a division by the mean resting-state power across the 5–95 Hz interval. For panels A to E, neurons did not receive additional visual input. (F–G) Difference in beta (F) and gamma (G) power between the situation wherein the stimulated field covered the entire square patch and the one without any additional visual input. A positive difference means that the power is larger for visual stimulation. Power differences were derived from the frequency spectra obtained from the mean potentials of the PCs and have been normalized through a division by the mean resting-state power across the 5–95 Hz interval. (H) Schematic depiction of the projection strength settings that resulted in the different types of network synchronization, that is, network synchronization behavior as a function of the PV and SOM cell projection strengths. Red and blue colors correspond to the network persistently oscillating in the gamma and beta frequency band, respectively, while yellow represents a system being able to switch between these two types of rhythms via activation of the network. In all panels, the black dot marks the standard network (SPV=SSOM=0.5). diff. = difference, norm. = normalised, proj. = projection.

Figure 5:

Visual stimulation can induce an enhancement of beta and a reduction of gamma power for a restricted range of parvalbumin-expressing (PV) and somatostatin-expressing (SOM) interneuron projection strengths. (A–C) Mean firing rates of pyramidal (A), PV (B), and SOM (C) cells in the center region as a function of the projection strengths of the two interneuron types. Orange arrows point to local minima. (D,E) Beta (15–25 Hz) (D) and gamma (40–60 Hz) (E) power as a function of the PV and SOM cell projection strengths. The frequency spectra were obtained from the mean potentials of the PCs and have been normalized through a division by the mean resting-state power across the 5–95 Hz interval. For panels A to E, neurons did not receive additional visual input. (F–G) Difference in beta (F) and gamma (G) power between the situation wherein the stimulated field covered the entire square patch and the one without any additional visual input. A positive difference means that the power is larger for visual stimulation. Power differences were derived from the frequency spectra obtained from the mean potentials of the PCs and have been normalized through a division by the mean resting-state power across the 5–95 Hz interval. (H) Schematic depiction of the projection strength settings that resulted in the different types of network synchronization, that is, network synchronization behavior as a function of the PV and SOM cell projection strengths. Red and blue colors correspond to the network persistently oscillating in the gamma and beta frequency band, respectively, while yellow represents a system being able to switch between these two types of rhythms via activation of the network. In all panels, the black dot marks the standard network (SPV=SSOM=0.5). diff. = difference, norm. = normalised, proj. = projection.

When considering Figure 5 in the context of the emergence of the spontaneous gamma oscillations during the CP (Chen et al., 2015), it must be recognized that the juvenile mouse V1 is represented by a network configuration similar to the blue region in Figure 5H: the LFP of the juvenile mouse V1 also mostly contains beta oscillations in both the resting and the activated state. The V1 of adult mice, on the other hand, has network dynamics resembling the yellow area in Figure 5H, because the mature, spontaneous LFP primarily contains gamma oscillations while beta oscillations are predominantly found in its visually evoked counterpart. Our results therefore imply that the influence of PV cells within the network increases across the critical period of mouse V1. Given this observation, we want to obtain a more detailed overview of how the PV cell projection strength influences the network dynamics when the SOM cell projection strength is fixed.

For this investigation, we set the SOM cell projection strength to SSOM=0.5 and vary the PV cell projection strength from 0.300 to 0.700 in steps of 0.001 (SPV{0.300,0.301,,0.700}). However, we change the latter parameter during one long simulation time series. More specifically, it is initially set to 0.300 and increased with the mentioned step size after every 5s of simulation until the upper limit of 0.700 is reached. We determine the power density spectrum using the mean potentials of the PCs for each of these epochs and plot them as a function of the PV cell projection strength. As expected, the results demonstrate that the power in the beta and gamma frequency band is reduced and enhanced, respectively, for increasing PV cell projection strengths, both when considering the resting-state and activated network (see Figures 6A and 6B). However, our findings also show that a distinct range of PV cell projection strengths exists wherein beta and gamma powers are substantially increased and decreased following network activation, respectively (see Figures 6A and 6B). To identify this range in more detail, we calculate the mean beta (15–25 Hz) and gamma (40–60 Hz) power for each setting of the PV cell projection strength and subsequently determine the difference between the powers corresponding to the activated and the resting-state network. The resulting plot clearly indicates that the beta power is never lower for the activated situation than for the resting state (see Figure 6C). The gamma power, on the other hand, could be lower, though only for a restricted range of PV cell projection strengths (see Figure 6C). Similar results were found when the PSTHs of the PCs were used as the LFP estimate (see Figure S.7).
Figure 6:

Varying parvalbumin-expressing (PV) cell projection strength yields spectrally different behavioral network modes. (A,B) Power density spectra of the resting-state (Dvis=0.0) (A) and the activated (Dvis=1.0) (B) network as a function of the PV cell projection strength for a fixed somatostatin-expressing (SOM) cell projection strength (SSOM=0.5). (C) Power differences between the activated and resting-state network activity with respect to the beta (15–25 Hz) (blue) and gamma (40–60 Hz) (red) frequency band as a function of the PV cell projection strength for the same fixed SOM cell projection strength. A positive and negative difference reflect a higher and lower power in the activated situation, respectively. Power has been normalized through a division by the mean resting-state power across the 5–95 Hz interval. The spectra based on the mean potentials of the PCs were used for the plots in this figure. diff. = difference, norm. = normalized.

Figure 6:

Varying parvalbumin-expressing (PV) cell projection strength yields spectrally different behavioral network modes. (A,B) Power density spectra of the resting-state (Dvis=0.0) (A) and the activated (Dvis=1.0) (B) network as a function of the PV cell projection strength for a fixed somatostatin-expressing (SOM) cell projection strength (SSOM=0.5). (C) Power differences between the activated and resting-state network activity with respect to the beta (15–25 Hz) (blue) and gamma (40–60 Hz) (red) frequency band as a function of the PV cell projection strength for the same fixed SOM cell projection strength. A positive and negative difference reflect a higher and lower power in the activated situation, respectively. Power has been normalized through a division by the mean resting-state power across the 5–95 Hz interval. The spectra based on the mean potentials of the PCs were used for the plots in this figure. diff. = difference, norm. = normalized.

3.3  The Model Produces Both the Beta and Gamma Oscillations via a PING

Now that we have shown the conditions under which our model is able to reproduce the experimentally observed reduction of gamma and induction of beta oscillations, we want to learn how it actually realizes this switch. However, for this investigation, we first need to understand how the oscillations are generated. We therefore consider the oscillations produced by the pyramidal-PV and pyramidal-SOM cell subnetwork separately. Whether they are produced by an ING or PING mechanism is first examined by varying the drive to the PCs in the form of the corresponding background activity (RKbgPC) (see section 2.3.3). Increasing this quantity leads to an increase of the oscillation frequency for both circuits irrespective of the LFP estimate used (see Figures 7A, 7D, S.8A, and S.8C). This indicates that the circuits embedded in the model use a PING mechanism to generate the rhythms; if an ING mechanism was involved, the peak frequency would not have been affected as strongly and hence would have been observed at approximately the same frequencies (Tiesinga & Sejnowski, 2009). The examination of the cross-correlograms of the PSTHs of the PCs and the interneurons confirms this notion: PCs consistently produce spikes before the interneurons associated with the considered subnetwork (see Figures 7B and 7E), which evidences a PING mechanism underlying the oscillations too (Tiesinga & Sejnowski, 2009). We also test how the background activity to the interneurons alters the oscillations. This examination reveals that the oscillations remain mostly unaffected by a change in the drive to the interneurons when the separate subnetworks are considered, though with the exception that the power of the rhythms associated with the pyramidal-PV cell subnetwork diminishes with increasing drive to the PV cells (see Figures 7C, 7E, S.8B, and S.8D).
Figure 7:

Both individual excitatory-inhibitory cell circuits produce oscillations via a pyramidal-interneuron gamma mechanism (PING), and differences between the oscillation frequencies of both circuits can mostly be attributed to the distinct connectivity profiles. (A,B) Spectral density (A) and pyramidal-interneuron correlogram (B) corresponding to the isolated circuit comprising pyramidal cells (PCs) and parvalbumin-expressing (PV) cells. The drive to the PCs was altered by varying RKbgPC. (C) Spectral density corresponding to the isolated pyramidal-PV cell circuit while varying the drive to the PV cells (RKbgPV). (D,E) Spectral density (D) and pyramidal-interneuron correlogram (E) corresponding to the isolated circuit comprising pyramidal cells (PCs) and somatostatin-expressing (SOM) cells. The drive to the PCs was altered by varying RKbgPC. (F) Spectral density corresponding to the isolated pyramidal-SOM cell circuit while varying the drive to the SOM cells (RKbgSOM). Resting-state (Dvis=0.0) network dynamics were considered for panels A to F. (G) Spectral density for various stimulated field sizes (Dvis) for a network where the PV cells were assigned the internal parameters of the SOM cells and vice versa. Solid lines and shaded areas depict the mean ± standard deviation across repeats of the simulation with different random seeds. The spectral density estimates were derived from the mean potentials across the PCs. a.u. = arbitrary unit, corr. = correlation, coef. = coefficient, norm. = normalised.

Figure 7:

Both individual excitatory-inhibitory cell circuits produce oscillations via a pyramidal-interneuron gamma mechanism (PING), and differences between the oscillation frequencies of both circuits can mostly be attributed to the distinct connectivity profiles. (A,B) Spectral density (A) and pyramidal-interneuron correlogram (B) corresponding to the isolated circuit comprising pyramidal cells (PCs) and parvalbumin-expressing (PV) cells. The drive to the PCs was altered by varying RKbgPC. (C) Spectral density corresponding to the isolated pyramidal-PV cell circuit while varying the drive to the PV cells (RKbgPV). (D,E) Spectral density (D) and pyramidal-interneuron correlogram (E) corresponding to the isolated circuit comprising pyramidal cells (PCs) and somatostatin-expressing (SOM) cells. The drive to the PCs was altered by varying RKbgPC. (F) Spectral density corresponding to the isolated pyramidal-SOM cell circuit while varying the drive to the SOM cells (RKbgSOM). Resting-state (Dvis=0.0) network dynamics were considered for panels A to F. (G) Spectral density for various stimulated field sizes (Dvis) for a network where the PV cells were assigned the internal parameters of the SOM cells and vice versa. Solid lines and shaded areas depict the mean ± standard deviation across repeats of the simulation with different random seeds. The spectral density estimates were derived from the mean potentials across the PCs. a.u. = arbitrary unit, corr. = correlation, coef. = coefficient, norm. = normalised.

Next, we answer the question how the distinct oscillation frequencies result from the separate subnetworks. As the modeled PV and SOM cells only differ in their values for the a-parameter included in Izhikevich's model of the spiking neuron, we hypothesized that connectivity profiles rather than internal properties are responsible. To test this, the a-parameter values are switched between the interneurons, that is, the PV cells are given the a-parameter values corresponding to the SOM cells and vice versa (see section 2.3.3). The network dynamics should still contain gamma oscillations in the resting state, and network activation should still result in beta rhythm induction if connectivity profiles are indeed decisive for the frequencies of the oscillations. Our findings show that for the greater part, this is indeed the case: even when this difference in network configuration is carried out, the resting state primarily contains gamma rhythms, and even beta power amplification is observed (see Figure 7G), though the power differences following visual stimulation are notably smaller than for the standard network (see Figures 3D and 7G). These findings demonstrate that in our model, the pyramidal-PV cell and pyramidal-SOM cell subnetworks produce oscillations in the gamma and beta band, respectively, because of their distinct connectivity profiles. Nonetheless, as we show in the next section, relatively small frequency changes may be induced by varying the parameters associated with the individual cell types and these can nevertheless have large effects on the frequency-switching behavior.

3.4  Interactions of Beta with Gamma Oscillations Enhance the Effect of Increased Beta and Decreased Gamma Power Following Visual Stimulation

With the investigation of the individual oscillatory mechanisms completed, we continue to find out how our model establishes the switch in oscillation frequency. We first manipulate the oscillation frequency associated with the pyramidal-SOM cell circuit to find out whether interactions between beta and gamma rhythms exist. This is done by altering the synaptic time constant associated with the SOM cells (τrecSOM) and subsequently simulating the resting-state (Dvis=0.0) and activated (Dvis=1.0) network. The results show that both the frequency and the power of the visually induced beta oscillations are influenced by this parameter, while the gamma oscillations are relatively mildly affected in terms of both frequency and power (see Figures 8A and 8C). We quantify this influence and reveal that the frequencies of the beta oscillations corresponding to both the resting-state and activated network linearly decrease with the logarithm of the considered synaptic time constant (see Figure 8D). Their power difference exhibits a maximum value (see Figure 8E). In contrast to the beta oscillations, the variation of the synaptic time constant of the SOM cell projections leaves the gamma frequency relatively untouched (see Figures 8F). Still, enlarged gamma power differences are found at positions that give rise to enlarged beta power differences as well (see Figures 8E and 8G). Moreover, at those positions, the activated gamma frequencies are close to the harmonic frequencies of the beta rhythms: the beta and gamma frequencies are approximately 18Hz and 54Hz, respectively, for τrecSOM=15.0ms and 14Hz and 46Hz, respectively, for τrecSOM=30ms (see Figures 8D and 8F). These results are obtained using the mean potentials across PCs as LFP estimate, but similar observations result from the PSTHs of the PCs (see Figure S.9). The findings presented in this paragraph indicate that the beta can interact with the gamma oscillations and that this can considerably enhance the effect of increased beta and reduced gamma power upon visual stimulation.
Figure 8:

Beta can interact with gamma oscillations to enhance the effect induced by visual stimulation. (A–C) Spectral density for the resting state (Dvis=0.0) and the activated (Dvis=1.0) network dynamics for various settings of the synaptic time constant of the SOM cell projections: τrecSOM=75.0ms (A), τrecSOM=15.0ms (B) and τrecSOM=3.0ms (C). (D,E) Main frequency (D) and mean power difference (E) in the beta frequency range (10–30 Hz) as a function of the synaptic time constant of the SOM cell projections. (F,G) Main frequency (F) and mean power difference (G) in the gamma frequency range (40–60 Hz) as a function of the synaptic time constant of the SOM cell projections. For all the plots included in this figure, the mean potentials across the PCs were used for the spectral density estimation. In panels A, B, C, D, and F, red and blue colors represent the resting-state and activated network, respectively. Powers have been normalized by the mean resting-state power across the 5–95 Hz interval. Solid lines and error bars depict the mean ± standard deviation across repeats of the simulation with different random seeds. diff. = difference, freq. = frequency, norm. = normalized.

Figure 8:

Beta can interact with gamma oscillations to enhance the effect induced by visual stimulation. (A–C) Spectral density for the resting state (Dvis=0.0) and the activated (Dvis=1.0) network dynamics for various settings of the synaptic time constant of the SOM cell projections: τrecSOM=75.0ms (A), τrecSOM=15.0ms (B) and τrecSOM=3.0ms (C). (D,E) Main frequency (D) and mean power difference (E) in the beta frequency range (10–30 Hz) as a function of the synaptic time constant of the SOM cell projections. (F,G) Main frequency (F) and mean power difference (G) in the gamma frequency range (40–60 Hz) as a function of the synaptic time constant of the SOM cell projections. For all the plots included in this figure, the mean potentials across the PCs were used for the spectral density estimation. In panels A, B, C, D, and F, red and blue colors represent the resting-state and activated network, respectively. Powers have been normalized by the mean resting-state power across the 5–95 Hz interval. Solid lines and error bars depict the mean ± standard deviation across repeats of the simulation with different random seeds. diff. = difference, freq. = frequency, norm. = normalized.

Finally, we want to find out whether both types of SOM cell projections within the network are required for our model to reproduce proper V1 dynamics. For this investigation, first the SOM-to-pyramidal-cell connections are eliminated from the model and the resting-state (Dvis=0.0) and activated (Dvis=1.0) network dynamics are simulated. Spectral analysis of the simulation results indicates that this deletion eliminates the network's ability to enhance beta oscillations upon visual stimulation (see Figures S.10A and S.10C to S.10F), thus revealing the critical role of the SOM to pyramidal cell projections. In contrast, eliminating the SOM-to-PV cell connections from the original network did not drastically alter the dynamics (see Figures S.10B and S.10C, S.10E, and S.10F).

4  Discussion

In sum, our model of L2/3 from mouse V1 is able to reproduce the empirically observed switch from gamma to beta dominated synchronization following visual stimulation (see Figures 3D, 3F, and S.3). We have shown that this change in the network's oscillatory behavior can only occur in one and the same network for a restricted range of PV and SOM cell projections; if PV and SOM cell projections become too powerful, the network produces primarily gamma and beta oscillations, respectively, irrespective of the amount of network activation (see Figures 5 and 6). Afterward, we have demonstrated that a PING mechanism underlies the production of both types of oscillations and that the distinct frequencies with which they are generated are primarily a consequence of differences in connectivity rather than internal neuron parameters (see Figure 7). Finally, we have acquired results that enable us to formulate a mechanism that explains how this switch is realized in our model of mouse V1 (see Figures 4, 8, and S.10).

All these results, however, still require an interpretation in the context of the model that has been introduced in this letter and the available experimental literature. In the following, we propose a possible mechanism for flexible frequency switching in the mouse V1 and discuss the relevance of this study and any future prospects that will arise from it.

4.1  The Relation between the Stimulated Field Size and the Induced Rise in Beta and Fall in Gamma Power

Our model reproduces the experimentally observed enhancement of beta and the reduction of gamma power in the LFP of mouse V1 following the presentation of a visual stimulus to the animal (see Figures 3D to 3F) (Chen et al., 2015; Veit et al., 2017). Additionally, the simulations also qualitatively reproduce the subtype-specific size tuning curves described by the literature (see Figure 3A and 3C) (Adesnik et al., 2012). However, one must be careful in interpreting these findings. In the model, the stimulated field size is a relative measure that cannot be related directly to a physical stimulus size. The results presented in this letter therefore demonstrate that when the magnitude of the visually induced current is fixed (see Figure S.3), the activated area of V1 determines to what extent beta oscillations are induced in the visually evoked LFP. The model thus imitates electrophysiological properties of mouse V1 through its biologically plausible connectivity patterns, which have been derived from the experimental literature (see section 2.2.3).

Nevertheless, very recent experimental literature suggests that the projection ranges corresponding to the neuron subtypes considered in this study have values close to 100 μm (Billeh et al., 2020) and that optogenetic stimulation of cortical patches leads to increasing SOM cell spike rates even when the radius of the stimulated area grows beyond 300 μm (Adesnik et al., 2012). These findings imply that the proportion between these empirical projection ranges (approximately 3) roughly matches the one used in this study (see section 2.2.3). In addition, studies have found the cortical surface area of mouse V1 to be approximately 4mm2 (Garrett, Nauhaus, Marshel, & Callaway, 2014), which is larger than the area represented by the model here relative to the projection ranges. This is consistent with our initial goal, since our model was not designed to be a direct representation of mouse V1 but instead was set up to investigate the synchronization phenomena that it exhibits. The spike rates of the SOM cells, for example, also do not exactly match the experimental literature: SOM cell spike rates typically increase linearly as the stimulus grows larger until they saturate (Adesnik et al., 2012; Keller, Roth, & Scanziani, 2020) and are even reported to be surround suppressed (Dipoppa et al., 2018). By lowering the proportion of pyramidal and PV cells receiving visual stimulation, we replicated the linear trend (see Figure S.5C). It is therefore useful to conduct further studies of larger networks, incorporating the parameter sets published by the Allen Institute, to study the robustness of the mechanism proposed here and how it integrates with other feature maps in visual cortex.

4.2  A Possible Mechanism for the Dynamical Transition between Beta and Gamma Oscillations Following Visual Stimulation

We first confirmed that our model can reproduce the experimentally observed phenomenon of beta rhythm induction and gamma power reduction upon visual stimulation in mouse V1. Afterward, we studied extensively the conditions that allow the model to do so. Here, we summarize our findings regarding this investigation and propose the mechanism by which our model establishes the switch in oscillation frequency.

First, we claim that competition between PV and SOM cells over the oscillatory control of the PCs constitutes the key principle of this mechanism. This competition becomes evident when one considers the indispensable and disposable nature of, respectively, the SOM-to-pyramidal and SOM-to-PV cell connections (see Figure S.10) together with the observation that the PV and SOM cell inhibitions should have comparable magnitudes (see Figure 5). Especially the result of SOM-to-PV cell connections being dispensable is vital to this inference: if they were necessary, the switch could also have been caused by SOM cells controlling the dynamics of the pyramidal-PV cell subnetwork. Nevertheless, since the synchronisation switch following visual stimulation does not critically depend on them, this possibility vanishes.

Second, we conclude that the collateral excitation of SOM cells through network activation triggers the production of beta oscillations by the network: the isolated pyramidal-SOM cell subnetwork generates rhythms in a beta band frequency (see Figures 7D to 7F) and network activation leads to synchronized, beta periodic firing of SOM cells (see Figures 4A and 4B). We furthermore observe that beta and gamma power can only be considerably enhanced and reduced, respectively, if the latter type of oscillations has a frequency that is (almost) a multiple of the one of the former (see Figure 8), a property evidenced by the experimental literature as well (Chen et al., 2015; Veit et al., 2017). Taken together, these two findings indicate that in our model, the increase of beta and decrease of gamma power following visual stimulation is established as follows.

In the resting state, gamma oscillations are generated via a PING mechanism (see Figures 7A and 7B): PCs fire, are followed by PV cells, and are subsequently silenced for a short period of time (Tiesinga & Sejnowski, 2009). However, as the field of visual stimulation increases, the PV cells become unable to effectively suppress the PCs; the SOM cells become activated (see Figures 3A and 3C). Consequently, the pyramidal-SOM cell subnetwork starts generating beta rhythms alongside the gamma oscillations via a PING mechanism as well (see Figures 3D, 7D and 7E, and 8A to 8C). Now consider the gamma being a multiple of, for example, three times, the beta frequency. Because of this property, PCs firing more synchronized with either the beta or gamma rhythm are collectively excited every third cycle of the gamma oscillations. This benefits the beta oscillations as more PC spiking leads to more SOM cell activity and thus to larger beta power. It also inhibits the gamma oscillations, because more SOM cell spiking induces a longer-lasting recuperation period in affected PCs, which are then unable to activate PV cells. In contrast, when the gamma is not a multiple of the beta frequency, the effect is substantially reduced since the collective excitation of PCs synchronized to both the beta and gamma rhythm is not established in the first place. Note that this mechanism also leaves a role for the PV cells in the induction of the beta oscillations at the expense of gamma oscillations: together with the PCs, they lay the framework with which the SOM cells have to interact.

4.3  Network Configuration Alterations May Explain the Emergence of Gamma Oscillations During the Critical Period

One of the motivations for this study was to find the parameter settings of a spiking neuron network model that are responsible for V1 dynamics in juvenile and adult mice. Specifically, we aimed to find the connectivity changes explaining the establishment of spontaneous, high-frequency gamma oscillations, which are disrupted at the benefit of the beta rhythms following visual stimulation, in mouse V1 during the CP (Chen et al., 2015). Since stronger PV cell projections were found to increase the gamma power (see Figure 5E), our results indicate that an overall strengthening of PV cell projections across this time window may very well be the reason for the emergence of these rhythms. At the same time, the outcomes of our simulations and analyses also demonstrate that the PV and SOM cell associated influences on the PCs should be balanced at the end of the CP. It is exactly after this period of enhanced plasticity that gamma power should be suppressed and beta power augmented during visual stimulation of the mouse (Chen et al., 2015), and our model only exhibits such behaviors for a restricted range of PV and SOM cell projection strengths (see Figure 5H). Therefore, this study shows that during the CP, PV cell inhibitory contributions become stronger until the network reaches that balanced state.

Plasticity mechanisms are one method to reinforce these projections, and additional experimental evidence supports the notion that PV cell-related plasticity underlies the enhancement of gamma powers during the CP. For instance, the opening of the CP has been linked to the maturation of a subset of the GABAergic interneuron population (Hensch, 2005; Hensch et al., 1998), and there is evidence that that maturing subset comprises the PV cells. When stem cells derived from the medial ganglionic eminence, the embryonic brain region that produces PV and SOM cells during development (Kriegstein & Noctor, 2004), are transplanted into the V1 long after the CP, they differentiate to a large extent toward this interneuron subtype and functionally integrate themselves into the host network (Davis et al., 2015; Howard & Baraban, 2016). A consequence of this transplantation and subsequent integration is the putative induction of a time window with enhanced plasticity that resembles the CP (Davis et al., 2015). Likewise, the closure of the CP is marked by molecular and cellular advancements too. The appearance of molecular “brakes on plasticity,” like myelin sheaths, that have Nogo-A as an associated protein, and the PNNs, which were mentioned in section 1, namely coincides with the end of the CP (Pizzorusso et al., 2002; McGee, Yang, Fischer, Daw, & Strittmatter, 2005; Carulli et al., 2010). Especially the latter type of consolidators, the PNNs, has recently gained much interest in multiple studies (Lensjø et al., 2017; Faini et al., 2018; Thompson et al., 2018). It has been shown that these nets primarily enwrap PV cells and that their removal reactivates ocular dominance plasticity in the V1 of mice (Pizzorusso et al., 2002). More recent studies have demonstrated that PNN removal also increases gamma power right after, but not for longer periods of, MD (Lensjø et al., 2017), that it disrupts the retrieval of remote fear memory (Thompson et al., 2018), and that in L4 it leads to increased thalamic PV cell recruitment (Faini et al., 2018).

Nevertheless, other explanations for the increase of the gamma oscillations during the CP are possible. In other brain areas, it is, for example, known that the decay time constant of the IPSC of PV cells declines during development (Jiao, Zhang, Yanagawa, & Sun, 2006; Doischer et al., 2008). Since one of these studies investigated the barrel cortex of mice, which shares many developmental aspects with the V1 (Fox & Wong, 2005), this potential mechanism should be assessed. However, a quick, mathematical evaluation of the effect that such a development would elicit reveals that it only further weakens the influence that PV cells have on the PCs. A more promising study has found that the SOM cells lose cholinergic responsiveness during the CP, which would lower their excitability (Yaeger, Ringach, & Trachtenberg, 2019). As a consequence, these cells would have weaker control of the PCs, and, complementarily, the influence of the PV cells on the excitatory cells would increase. This developmental loss may therefore contribute to the emergence of gamma oscillations during the CP.

In sum, the CP thus seems to be marked by high amounts of PV cell-related plasticity- and our results provide a new insight as to how this plasticity may change the network dynamics.

4.4  The Precise Function of the Oscillation Frequency Switch Following Visual Stimulation Remains Unknown and Its Elucidation Requires More Study

The function of the switch in main oscillation frequency is not fully understood, but some ideas have been presented. For instance, it has been argued that beta oscillations in primates are related to the maintenance of the current cognitive state (Engel & Fries, 2010). More interestingly, it has been shown in rodents that because of their horizontally aligned afferents (Adesnik et al., 2012), SOM cells promote synchronization across cortical space (Veit et al., 2017; Hakim et al., 2018). This property of the SOM cells and the fact that in our model SOM cells are activated when PV cells are unable to effectively suppress PC activity together imply that strong visual stimulation triggers the generation of beta oscillations so that more distant cortical areas also become increasingly synchronized with V1. The latter, in its own right, would then putatively improve the information transfer between cortical areas. Though improved information transmission is typically observed for gamma oscillations (Buehlmann & Deco, 2010), the beta oscillations considered in this letter may facilitate this as well, for their peak frequencies are intermediate between the experimentally determined beta and gamma frequency bands.

The experimental finding (Chen et al., 2015; Veit et al., 2017; Chen et al., 2017) that frequency switching is facilitated in mature cortices by an altered balance between PV and SOM neurons begs the question whether this poses computational advantages. First, the mechanism allows for each of the two types of neurons to facilitate the switch, each of which could be under control of specific neuromodulatory projections (Yaeger et al., 2019; Disney & Aoki, 2008) or top-down projections (Jiang et al., 2015; Gonchar & Burkhalter, 2003; Jiang, Wang, Lee, Stornetta, & Zhu, 2013) and could thus serve a particular computational role that needs to be examined further. There have been proposals that slower rhythms (i.e., alpha/theta/beta versus gamma) are more appropriate for long-range synchronization (Kopell, Ermentrout, Whittington, & Traub, 2000; Stein, Chiang, & König, 2000), hence that a switch to beta could improve long-range communication. Furthermore, in nonhuman primates, evidence has been reported that the direction of communication determines the frequency band, with feedforward mediated by fast rhythms and feedback by slower frequencies (van Kerkoerle et al., 2014; Bastos et al., 2015); a switch such as the one reported here could therefore enhance the efficiency of, for instance, feedback projections. Other studies have suggested that because oscillation frequency is dependent on stimulus conditions (Gieselmann & Thiele, 2008; Ray & Maunsell, 2015) these oscillations cannot serve computational roles such as binding and communication through coherence (Fries, 2005; Tiesinga & Sejnowski, 2010; Akam & Kullmann, 2014) that would need constant frequencies (Ray & Maunsell, 2010; Roberts et al., 2013; ter Wal & Tiesinga, 2017). From this perspective, frequency switches would not be beneficial. Taken together, the model framework presented here suggests new avenues for investigations of frequency-dependent communication between cortical networks to address some of the issues we have posed here.

4.5  How Our Model Relates to Other Computational Studies to Mouse V1

In section 1, some neural mass models of mouse V1 were mentioned. These models successfully reproduced the phenomenon of surround inhibition and the increased beta and attenuated gamma power on visual stimulation (Veit et al., 2017; Litwin-Kumar, Rosenbaum, & Doiron, 2016). Though firing rate models may be used to study neural oscillations, it must be acknowledged that spike timing is a determining factor in the generation of LFP signals. Moreover, it has been demonstrated that firing rate and synchrony can be modulated independently, which makes neural mass models less fit to study oscillations (Tiesinga & Buia, 2008).

By using a spiking neuron model, we have obtained new insights into beta and gamma rhythms and the roles that PV and SOM cells play in them; specifically, our results indicate that the relative PV and SOM cell inhibitions should satisfy a particular constraint at the end of the CP for a proper functioning of mouse V1. More generally, we have shown that the main synchronization frequency of oscillations generated via a PING mechanism can be altered via network activation. It has already been demonstrated that such an effect cannot be observed when the network hosts a combination of ING and PING mechanisms: in that situation, the high frequency of the two then dominates the synchronization (Viriyopase, Memmesheimer, & Gielen, 2016). Whether ING mechanisms can facilitate peak frequency shifts on network activation is unclear. Here, it must also be mentioned that the extent to which PCs are involved in the generation of oscillations in the neocortex should be limited in terms of the number of spikes per cycle; as a consequence, it is believed that neural rhythms in the visual system are produced neither purely by a PING, nor purely by an ING mechanism (Whittington, Traub, Kopell, Ermentrout, & Buhl, 2000). Note that this does not rule out the applicability of our study to mouse V1; it merely gives it a more nuanced perspective.

To our knowledge, this is the first study of oscillations in V1 during the critical period that involves the explicit modeling of three distinct neuron types. Spiking neuron models that were inspired by this cortical region investigated other properties. One of these, for example, proposed a possible mechanism as to how orientation selectivity can be established in cortices that lack an organized map with regard to this feature (Hansel & van Vreeswijk, 2012). This model merely had two neuron classes: excitatory and inhibitory neurons. Another example investigated how stimulus detection performance can be enhanced in noisy spiking neural networks; this model consisted of the same neuron types as have been included in this study (Martens et al., 2017). Still, we are not the first to investigate the coexistence of oscillations through a spiking neuron network model comprising three distinct cell types: one model that was based on the hippocampus already demonstrated that the coexistence of theta and gamma oscillations requires a balance in the effective strengths of the different inhibitory neurons in the network (Gloveli et al., 2005). Our work shows that the same principle is applicable to the beta and gamma rhythms in mouse V1 and also demonstrates that network activation can alter the synchronization of the neural ensemble too.

There are multiple types of interneurons; the ones classified as parvalbumin positive (PV), somatostatin positive (SOM), and vasoactive intestinal peptide positive (VIP) have received the most attention (Tremblay et al., 2016) (note that there are alternative labels in use for each of these types). Optogenetic approaches to transgenic animals in which specific cells are either labeled by GFP or express Cre have elucidated the functional role of each type and identified structural motifs in different cortical layers (for a perspective, see Womelsdorf, Valiante, Sahin, Miller, & Tiesinga, 2014; Kim, Adhikari, & Deisseroth, 2017; or Adesnik & Naka, 2018). These motifs need to be developmentally established, and this may happen both within critical periods as well as outside. The vagueness of this description derives from the fact that the development of these motifs has not been studied extensively. Here we interpret our simulation results in terms of motifs and the ocular dominance critical period experiments that have been reported in the literature.

Even though there are many types of interneurons, we focus on two groups: the PV and the SOM cells. The literature on CP plasticity identifies PV neurons as prime actors (Hensch, 2005), and the electrophysiological literature identifies SOM cells as prime actors in visually induced beta oscillations and horizontal projections mediating surround inhibition (Veit et al., 2017; Chen et al., 2017). This means we omit from the model VIP interneurons, which do, however, play an important role in the effects of locomotion on visual responses (Dipoppa et al., 2018) and have a specific neuromodulator sensitivity (Batista-Brito, Zagha, Ratliff, & Vinck, 2018). Additionally, defects in their function influence other types of cortical plasticity relevant for cognitive function (Batista-Brito et al., 2018), and a recent study has also demonstrated the large influence of the SOM-VIP circuit on PC activity, wherein weak inputs to VIP neurons can lead to large changes in the somato-dendritic inhibition of PCs (Hertäg & Sprekeler, 2019). We defer a computational investigation of their role in the context of modulating oscillations to a future study. Here, the model in Wang, Tegnér, Constantinidis, and Goldman-Rakic (2004) may serve as inspiration.

5  Conclusion

In this letter, the differential roles of PV and SOM cells in the generation of oscillations have been investigated. From our results, three main conclusions can be drawn. First, the emergence of gamma oscillations during the CP (Chen et al., 2015) is most likely caused by an overall increase in the influence that PV cells have on the PCs in the network. Given the available knowledge of the CP, plasticity presumably underlies this development, which would concretely imply a general strengthening of PV cell projections across this time window. Second, this increase in influence has a limit: persistent gamma oscillations emerge if PV cells become relatively too powerful. This would prevent visually stimulating the animal from inducing the SOM cell-associated beta rhythms in the V1, which, as the available literature demonstrates, should actually be possible (Chen et al., 2015; Veit et al., 2017). Hence, the inhibitory contributions of PV and SOM cells must be balanced at the end of the CP in order for spontaneous gamma and visually evoked beta oscillations to coexist in V1. Finally, we have presented evidence for a mechanism by which these visually evoked beta oscillations are realized. The results of this study indicate that SOM cells transform the dynamic circuit motif laid out by pyramidal and PV cells for the production of gamma oscillations so that it then produces beta oscillations instead. In addition, it has been argued that this implies that beta rhythms emerge when the PV cells are unable to effectively suppress the PCs before they collaterally activate the SOM cells.

In conclusion, our study links many experimental studies together into one comprehensive model that has biologically plausible connectivity patterns. It also provides new insight into how specific members of neural ensembles in the brain can be mobilized to produce different types of oscillations. Furthermore, it demonstrates that experimental observations in electrophysiological studies may be explained by mechanisms that are sensitive to a precise parameter setting and presumably require careful fine-tuning of the network configuration in order to emerge and be maintained, which may occur during maturation of neural circuits.

Acknowledgments

We thank C. Bollen and M. J. ter Wal for their comments and suggestions on the manuscript. This study fell under the project Light after Dark; Restoring Visual Perception in inherited Retinal Dystrophies (NWO 058-14-002).

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