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.
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).
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.
2.1 Membrane Potential Dynamics
In the two preceding equations, , , , and 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 -parameters corresponding to the PCs and SOM cells were doubled, and the -parameter of the PV cells was raised from 200 to . These parameter settings result in the neurons behaving as integrators rather than resonators. Additionally, the cell capacitance was set to for all subtypes.
|Neuron Type .||(s) .||(s) .||(mV) .||(Vs) .||(Hz) .||(pSs) .|
|Neuron Type .||(s) .||(s) .||(mV) .||(Vs) .||(Hz) .||(pSs) .|
Notes: A sign designates a gaussian distributed random variable following the notation mean standard deviation. PV parvalbumin expressing, Pyr. pyramidal, SOM somatostatin expressing.
2.2 Input Currents
2.2.1 Background Input
The values for the subtype-specific parameters are shown in Table 1. The remaining parameters, and , were set to and , respectively. The values given to parameters , , and 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 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 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 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 variable 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, 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 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 , , , , have been collected from Pfeffer et al. (2013) and those assigned to PC projection parameters ) have been derived from Jouhanneau et al. (2018). The synaptic time constants of the interneurons 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 ; 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, and where , 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 mV and mV. Finally, the overall synaptic strength scaling factor of the PCs is set to , while those of the PV and SOM cells and 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.
|Presyn. cell () .||.||.||.||(nS) .||(nS) .||(nS) .||(ms) .|
|Presyn. cell () .||.||.||.||(nS) .||(nS) .||(nS) .||(ms) .|
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 . We first vary the stimulus size parameter from 0.0 to 1.0 in steps of 0.1 . For each parameter setting, 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 . The stimulus size parameter is set to , and we vary the magnitude of the visually induced current. It is set to 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 in steps of . Seventeen epochs of 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 and the SOM cell projection strength from 0.3 to 1.3 also in steps of 0.1 . The network can be either in the resting or an activated state. For each parameter setting, 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 while the SOM strength is fixed at . Again, the network can either be in the resting or the activated 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 and to study the pyramidal-PV cell circuit and and to obtain the dynamics associated with the pyramidal-SOM cell circuit. Additionally, the stimulated field parameter is set to . 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 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 to study how that influences the dynamics of the two subnetworks. For all parameter settings, 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 and . Subsequently, we vary the stimulus size parameter from 0 to 1 in steps of 0.1 and sample 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 , , , . 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 , while keeping the one of the PV cell projections fixed at its standard value . For each parameter setting, of resting-state and fully activated 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 of resting-state and fully stimulated network dynamics, although now either the SOM-to-pyramidal cell projections or the SOM-to-PV cell projections are severed by setting or , 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 (sampling rate of ). The first 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 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 bin size (sampling rate of ). 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 , 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)
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 is slid over the time series in steps of . As 16 usable 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 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 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.
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 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 (), 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
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 and alter the visually induced current within a epoch: this parameter is set to 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 (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
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.
3.3 The Model Produces Both the Beta and Gamma Oscillations via a PING
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 -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 -parameter values are switched between the interneurons, that is, the PV cells are given the -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
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 and activated 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).
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 m (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 m (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 (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.
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.
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).