Top–down modulation in canonical cortical circuits with short-term plasticity

Significance

In cortical circuits, interacting cell types create intriguing nonlinear behaviors. Understanding how these nonlinear phenomena arise has been a puzzle. Through a combination of mathematical analyses and computer simulations, we uncover how experimentally identified short-term plasticity mechanisms can give rise to a nonlinear phenomenon called response reversal. In this case, top–down modulation during animal locomotion affects the response of interneurons expressing somatostatin (SST) differently based on visual stimulation conditions. Our study not only sheds light on the origins of response reversal but also reveals links between this phenomenon and key features of how the underlying cortical network operates. Specifically, we find that a key factor is the inhibitory influence of SST interneurons on the stability of network dynamics.

Abstract

Cortical dynamics and computations are strongly influenced by diverse GABAergic interneurons, including those expressing parvalbumin (PV), somatostatin (SST), and vasoactive intestinal peptide (VIP). Together with excitatory (E) neurons, they form a canonical microcircuit and exhibit counterintuitive nonlinear phenomena. One instance of such phenomena is response reversal, whereby SST neurons show opposite responses to top–down modulation via VIP depending on the presence of bottom–up sensory input, indicating that the network may function in different regimes under different stimulation conditions. Combining analytical and computational approaches, we demonstrate that model networks with multiple interneuron subtypes and experimentally identified short-term plasticity mechanisms can implement response reversal. Surprisingly, despite not directly affecting SST and VIP activity, PV-to-E short-term depression has a decisive impact on SST response reversal. We show how response reversal relates to inhibition stabilization and the paradoxical effect in the presence of several short-term plasticity mechanisms demonstrating that response reversal coincides with a change in the indispensability of SST for network stabilization. In summary, our work suggests a role of short-term plasticity mechanisms in generating nonlinear phenomena in networks with multiple interneuron subtypes and makes several experimentally testable predictions.

Inhibitory neurons in the cortex are highly diverse in anatomy, electrophysiology, and function (1–4). In the mouse cortex, three major classes of interneurons expressing parvalbumin (PV), somatostatin (SST), and vasoactive intestinal peptide (VIP) make up more than 80% of GABAergic interneurons (4). Together with excitatory (E) neurons, they form a canonical microcircuit relevant for various cortical computations, including locomotion-induced gain modulation (5), selective attention (6), context-dependent modulation (7, 8), predictive processing (9, 10), novelty detection (11, 12), flexible routing of information flow (13, 14), regulating global coherence (15, 16), and gating of synaptic plasticity (17). Interactions between different cell types in the canonical microcircuit can give rise to counterintuitive nonlinear phenomena. More specifically, in darkness, locomotion-induced top–down modulation via VIP decreases the activity of SST neurons in layer 2/3 of the mouse primary visual cortex (5); Fig. 1]. In contrast, when animals receive visual stimuli, locomotion leads to an increase in SST activity (18, 19); Fig. 1]. This phenomenon in which the same locomotion-induced top–down modulation via VIP affects SST response oppositely depending on the visual stimulation condition is known as response reversal (20).

Fig. 1.

Schematic diagrams illustrating that under different stimulus conditions, locomotion-induced modulatory input via VIP affects SST response oppositely. Top: In darkness, locomotion-induced top–down modulation increases VIP activity but decreases SST activity (5). Bottom: In the presence of visual stimulation, locomotion-induced top–down modulation increases both VIP and SST activity (18, 19).

Previous computational work has shown that networks with nonlinear neuronal input–output functions can generate response reversal (20). However, cortical neurons exhibit highly irregular spiking (21) and heterogeneous firing rates (22, 23) that are hallmarks of tightly balanced networks in which population-averaged responses are linear in the input (24). This raises the possibility that other factors, such as dynamically changing synapses, may contribute to nonlinear population responses like response reversal. On a perceptually and behaviorally relevant timescale from milliseconds to seconds, synapses are subject to short-term plasticity (STP) (25, 26). Different types of synapses can experience different degrees of short-term depression (STD) or short-term facilitation (STF) (25). In particular, inhibitory synapses exhibit more pronounced short-term dynamics than excitatory synapses, and synapses from different interneuron subtypes can undergo different short-term plastic changes (27). However, little is known about how these experimentally identified STP mechanisms shape network dynamics and computations in recurrent neural circuits of multiple interneuron subtypes.

Response reversal of SST induced by the same top–down modulation may suggest that the network operates in different regimes under different stimulation conditions. Increasing evidence suggests that cortical networks operate in an inhibition-stabilized regime, in which feedback inhibition generated by the network is imperative to stabilize excitatory activity (28, 29). In networks with one excitatory and one inhibitory population and fixed connectivity, an identifying characteristic of inhibition stabilization is that increasing (decreasing) excitatory input to the inhibitory population decreases (increases) inhibitory firing, known as the paradoxical effect (28, 30, 31). Yet, it is unclear whether response reversal can be linked to inhibition stabilization and whether there exists a relationship between response reversal and the paradoxical effect. In addition, how STP shapes inhibition stabilization in networks with multiple interneuron subtypes, particularly how specific interneuron subtypes contribute to network stabilization (which we refer to as interneuron-specific stabilization), is unknown.

Here, we use analytical calculations and numerical simulations to demonstrate that inhibitory short-term plasticity (iSTP) enables response reversal without requiring neuronal nonlinearities. We find that despite not directly affecting SST and VIP activity, PV-to-E STD has a crucial influence on response reversal. We further reveal the relationship between response reversal, the paradoxical effect, and the interneuron-specific stabilization property of the network. Interestingly, when the SST response to top–down modulation switches from suppression to enhancement, the network undergoes an interneuron-specific change in stabilization, and SST is required for network stabilization. In summary, our model suggests that iSTP enables the network to perform nonlinear computations and makes several experimentally testable predictions.

Results

To study how response reversal emerges in canonical cortical circuits, we used rate-based population models consisting of one excitatory (E) and three different inhibitory (PV, SST, VIP) populations with network connectivity constrained by previous experimental studies [Fig. 2A; (1)]. This type of model allows for a trade-off between sufficient biological detail and mathematical analysis and has previously been used with great success to study cortical computations (20, 32–35). Consistent with experimental work (29), network connectivity was chosen so that the network operates in an inhibition-stabilized regime defined as the regime where feedback inhibition generated by the network is needed to stabilize recurrent excitation (28). As proposed by influential modeling work on cortical dynamics (24, 36), the network’s population-averaged responses can be approximated by a rectified linear function of the input (Fig. 2A, Inset). To account for activity-dependent changes in network connectivity on a perceptually and behaviorally relevant timescale, we modeled STP based on recent experimental work from the Allen Institute (27). We incorporated the four most pronounced STP mechanisms: PV-to-E STD, PV-to-PV STD, PV-to-VIP STD, and SST-to-VIP STF (Fig. 2A and SI Appendix, Fig. S1). Since all the prominent synapses undergoing STP are inhibitory, we refer to the plasticity mechanisms as iSTP. The dynamics of the network with iSTP can be described as follows:

$$\begin{array}{c}\hfill {\tau }_E\frac{d{r}_E}{\mathit{dt}}=-r_E+{\left[J_{\mathit{EE}}{r}_E-x_{\mathit{EP}}{J}_{\mathit{EP}}{r}_P-J_{\mathit{ES}}{r}_S+g_E+\alpha \right]}_{+},\end{array}$$ [1]

$$\begin{array}{c}\hfill {\tau }_P\frac{d{r}_P}{\mathit{dt}}=-r_P+{\left[J_{\mathit{PE}}{r}_E-x_{\mathit{PP}}{J}_{\mathit{PP}}{r}_P-J_{\mathit{PS}}{r}_S+g_P+\alpha \right]}_{+},\end{array}$$ [2]

$$\begin{array}{c}\hfill {\tau }_S\frac{d{r}_S}{\mathit{dt}}=-r_S+{\left[J_{\mathit{SE}}{r}_E-J_{\mathit{SV}}{r}_V+g_S\right]}_{+},\end{array}$$ [3]

$$\begin{array}{c}\hfill {\tau }_V\frac{d{r}_V}{\mathit{dt}}=-r_V+{\left[J_{\mathit{VE}}{r}_E-x_{\mathit{VP}}{J}_{\mathit{VP}}{r}_P-u_{\mathit{VS}}{J}_{\mathit{VS}}{r}_S+g_V+c\right]}_{+},\end{array}$$ [4]

for the rates $r_i$ of excitatory, PV, SST, and VIP populations with $i\in \{E,P,S,V\}$ and ${[·]}_{+}$ denotes linear rectification. ${\tau }_i$ represents the corresponding time constant of the rate dynamics, $J_{\mathit{ij}}$ denotes the synaptic strength from population $j$ to population $i$, and $g_i$ is the individual background input. Importantly, we distinguish between bottom–up input to E and PV to represent different stimulation conditions, denoted as $\alpha$, and top–down input to VIP mimicking locomotion-induced top–down modulation, denoted as $c$ (Fig. 2A).

Fig. 2.

iSTP enables response reversal of SST induced by top–down modulation via VIP. (A) Schematic of network model with one excitatory (E) population and three distinct inhibitory populations, including PV, SST, and VIP. STD and STF connections are indicated by the dashed lines. Each population receives a background input $g$. E and PV receive bottom–up input $\alpha$ depending on sensory stimulation, and VIP receives top–down input $c$ during locomotion. Top Right: Rectified linear input–output function; Bottom Right: Cartoons showing how inhibitory connection strength changes with presynaptic stimulation under STD and STF. (B) Network responses to top–down modulation without any bottom–up input ($\alpha =0$), corresponding to darkness without sensory stimulation. Top–down modulation via VIP is applied during the interval from 2 to 4 s (gray bar). Different colors denote the activity of different populations. The dashed line represents the initial activity level of SST. (C) Same as B but at $\alpha =15$ corresponding to sensory stimulation. (D) Change in SST response induced by top–down modulation to VIP as a function of bottom–up input $\alpha$ in networks with iSTP. (E) Same as B but for networks without iSTP. (F) Same as C but for networks without iSTP. (G) Same as D but for networks without iSTP. (H) Input to the E population at $\alpha =0$. Different colors indicate different sources: input from the E population, input from the I populations, and the sum of the inputs from the E and I populations. (I) Same as E but at $\alpha =15$. (J) Change in different sources of recurrent inputs to the E population measured between baseline and at steady state during top–down modulation as a function of bottom–up input $\alpha$.

STP mechanisms are implemented based on the Tsodyks–Markram model (37):

$$\begin{array}{c}\hfill \frac{d{x}_{\mathit{EP}}}{\mathit{dt}}=\frac{1-x_{\mathit{EP}}}{{\tau }_x}-U_d{x}_{\mathit{EP}}{r}_P,\end{array}$$ [5]

$$\begin{array}{c}\hfill \frac{d{x}_{\mathit{PP}}}{\mathit{dt}}=\frac{1-x_{\mathit{PP}}}{{\tau }_x}-U_d{x}_{\mathit{PP}}{r}_P,\end{array}$$ [6]

$$\begin{array}{c}\hfill \frac{d{x}_{\mathit{VP}}}{\mathit{dt}}=\frac{1-x_{\mathit{VP}}}{{\tau }_x}-U_d{x}_{\mathit{VP}}{r}_P,\end{array}$$ [7]

$$\begin{array}{c}\hfill \frac{d{u}_{\mathit{VS}}}{\mathit{dt}}=\frac{1-u_{\mathit{VS}}}{{\tau }_u}+U_f(U_{\text{max}}-u_{\mathit{VS}})r_S,\end{array}$$ [8]

where $x_{\mathit{ij}}$ is a STD variable limited to the interval (0,1] for the synaptic connection from population $j$ to population $i$. Biophysically, the STD variable $x$ represents the fraction of vesicles available for release. ${\tau }_x$ is the time constant of STD, and $U_d$ is the depression factor controlling the degree of depression induced by the presynaptic activity. Similarly, $u_{\mathit{ij}}$ is a STF variable constrained to the interval [1, $U_{\mathit{max}}$) for the synaptic connection from population $j$ to population $i$. Unlike STD, the STF variable $u$ biophysically represents the ability to release neurotransmitters. ${\tau }_u$ is the time constant of STF, $U_f$ is the facilitation factor controlling the degree of facilitation induced by the presynaptic activity, and $U_{\mathit{max}}$ is the maximal facilitation value.

iSTP Enables Response Reversal of SST.

To represent different stimulus conditions (e.g., darkness vs. visual stimulation), we varied the bottom–up input $\alpha$ to E and PV. Increasing $\alpha$ leads to a supralinear increase in the baseline activity in all populations (SI Appendix, Fig. S2). We modeled the effect of locomotion-induced top–down modulation on network activity by increasing the input to VIP by a positive value $c$. In our network model with iSTP, for a low $\alpha$, corresponding to low baseline activity in the absence of bottom–up input, top–down modulation via additional excitatory input to VIP decreases SST activity (Fig. 2B). In contrast, for a high $\alpha$ corresponding to high baseline activity, the same top–down modulation leads to an increase in SST activity and, thus, response reversal (Fig. 2C). Our modeling results suggest that under different stimulus conditions regulated by bottom–up inputs, identical top–down modulation reversely affects the change of SST activity (Fig. 2D), consistent with previous experiments (5, 18, 19).

To highlight the role of iSTP in generating response reversal of SST activity, we further simulated the same network while disabling iSTP (Fig. 2 E and F). In contrast to networks with iSTP, the change of SST activity is largely unaffected for different values of $\alpha$ when iSTP is disabled (Fig. 2G). Interestingly, in our model, for a high $\alpha$ during the stimulation period, despite the increased activity of all inhibitory populations, the steady state of excitatory activity also increases (Fig. 2C). This observation appears to differ from what would be predicted by a classical disinhibition mechanism in which reducing inhibition increases excitatory activity. We confirm this by plotting the amount of recurrent excitation, recurrent inhibition, and the sum of recurrent excitation and inhibition that the excitatory population receives during the simulation (Fig. 2 H and I). Surprisingly, even for a low $\alpha$, despite decreased SST activity, top–down modulation via VIP increases the total inhibition to the excitatory population at the steady state (Fig. 2H). Enhanced inhibition to the excitatory population at the steady state during top–down modulation is also observed for a high $\alpha$ (Fig. 2I). We systematically investigated the change in the input to the excitatory population due to top–down modulation at different levels of bottom–up input $\alpha$. We found that top–down modulation always increases the amount of inhibition to the excitatory population irrespective of whether it increases or decreases the activity of the SST population as $\alpha$ changes (Fig. 2J). These results suggest that rather than the decrease in the total inhibition, the increase in the recurrent excitation contributes to the elevated excitatory activity (Fig. 2H–J). Importantly, a joint increase in the excitation and inhibition of the excitatory population is a distinctive feature in inhibition-stabilized networks (31, 33, 38). In contrast, noninhibition-stabilized networks do not exhibit an increase in the total inhibitory inputs to the excitatory population (SI Appendix, Fig. S3).

Taken together, our numerical simulation results reveal that experimentally identified forms of iSTP enable response reversal, a nonlinear computation observed in cortical circuits.

Theoretical Analysis.

To better understand how iSTP enables our model network to perform response reversal of SST activity, we sought to mathematically analyze how top–down modulation affects SST activity. Locomotion-induced top–down modulation can be considered a form of perturbation to the VIP population. Investigating how top–down modulation inversely affects SST activity under different stimulus conditions can therefore be mathematically formulated as how perturbations to VIP affect SST activity under varying levels of $\alpha$. To this end, we extended previous studies on static networks (20, 39) and developed a general theoretical framework for networks with iSTP. More specifically, we derived how the steady state response of any population changes with a perturbation of the external input to a given population while including iSTP (Materials and Methods). Using this approach, we formulated the change of SST activity induced by top–down modulation via the change of the input to VIP, ${\mathbf{R}}_{\mathit{SV}}$, as follows:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SV}}=D{\mathbf{K}}_{\mathit{SV}}\delta g_V.\end{array}$$ [9]

Here, $D$ is a positive quantity for any stable network (Materials and Methods and SI Appendix, Fig. S4A), $\delta g_V$ represents the perturbation of the VIP population’s input which is a positive number $c$ in our network setting, and ${\mathbf{K}}_{\mathit{SV}}$ is the response factor which is given by

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\mathit{SV}}& =(x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P-1)(J_{\mathit{EE}}-1)J_{\mathit{PP}}{J}_{\mathit{SV}}\hfill & \hfill \\ \hfill & -(x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P-1)J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}}\hfill & \hfill \\ \hfill & +(J_{\mathit{EE}}-1)(J_{\mathit{PP}}{J}_{\mathit{SV}}+J_{\mathit{SV}})-J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}},\hfill \end{array}$$ [10]

with $x_{\mathit{ij}}^{\ast }$ the STP variable from population $j$ to population $i$ at steady state before perturbation, and $x_{\mathit{ij}}^{{\ast }^{\prime }}$ the derivative of the STP variable with respect to the activity of population $j$, evaluated at the steady state (SI Appendix, SI Text).

A negative (positive) ${\mathbf{R}}_{\mathit{SV}}$ denotes a decrease (increase) in SST activity caused by top–down modulation. To have response reversal, ${\mathbf{R}}_{\mathit{SV}}$ must switch its sign for different values of $\alpha$, corresponding to different stimulus conditions. More specifically, when animals perceive no visual stimulus in darkness, i.e., when $\alpha$ is low, top–down modulation via VIP decreases SST activity. Therefore, ${\mathbf{R}}_{\mathit{SV}}$ is expected to be negative for a low $\alpha$. In contrast, when animals receive a visual stimulus, namely when $\alpha$ is high, top–down modulation via VIP increases SST activity. Thus, ${\mathbf{R}}_{\mathit{SV}}$ is expected to be positive for a high $\alpha$.

In agreement with our simulation results (Fig. 2 B and C), we observed that ${\mathbf{R}}_{\mathit{SV}}$ changes its sign when calculated at different values of $\alpha$ (Fig. 3). As our theoretical framework is based on the linearization of the network dynamics around the steady state and higher order terms are ignored (Materials and Methods), the computed ${\mathbf{R}}_{\mathit{SV}}$ agrees well with the numerical simulation results for small perturbations (Fig. 3A) and diverges for large perturbations (Fig. 3B). Yet, it qualitatively captures the key aspect of modeling behaviors: the sign switch of the change in SST activity induced by top–down modulation with different values of $\alpha$. Note that while here we are interested in how neural activity changes in response to a given perturbation, several other studies have investigated the contributions of higher-order motifs to the perturbation-induced change of neural activity in excitatory and inhibitory networks (40, 41).

Fig. 3.

Comparison between analytical predictions and numerical simulations on the change in SST activity and identification of PV-to-E STD as the crucial STP mechanism for the generation of response reversal. (A) Analytical prediction of the change in SST population response induced by the perturbation to VIP (${\mathbf{R}}_{\mathit{SV}}$) matches closely with numerical simulation for a small perturbation. (B) Same as A but with a large perturbation. (C) Analytical contributions of the STP-dependent term ${\mathbf{R}}_{\mathit{SV}}^{\text{STP}}$ and the STP-independent term ${\mathbf{R}}_{\mathit{SV}}^{\text{nonSTP}}$ to the change in SST activity as a function of bottom–up input $\alpha$ for a small perturbation. (D) Same as C but with a large perturbation. (E) Analytical contributions of the PV-to-E STD-dependent term ${\mathbf{R}}_{\mathit{SV}}^{\text{PED}}$, the PV-to-PV STD-dependent term ${\mathbf{R}}_{\mathit{SV}}^{\text{PPD}}$, and the overall STD-dependent term ${\mathbf{R}}_{\mathit{SV}}^{\text{STP}}$ to the change in SST activity as a function of bottom–up input $\alpha$ for a small perturbation. (F) Change in SST response induced by top–down modulation to VIP as a function of bottom–up input $\alpha$ with a large perturbation for different network configurations marked with different colors. Here, for small perturbations $\delta g_V=0.1$ and for large perturbations $\delta g_V=3$.

As shown in Eq. 9, since $D$ and $\delta g_V$ are positive, ${\mathbf{K}}_{\mathit{SV}}$ is the only term that can change the sign of ${\mathbf{R}}_{\mathit{SV}}$. To further investigate the influence of the iSTP mechanisms on response reversal, we rewrote ${\mathbf{K}}_{\mathit{SV}}$ as a sum of a STP-dependent and a STP-independent term:

$$\begin{array}{c}\hfill {\mathbf{K}}_{\mathit{SV}}={\mathbf{K}}_{\mathit{SV}}^{\text{STP}}+{\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}},\end{array}$$ [11]

where

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\mathit{SV}}^{\text{STP}}& =(x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P-1)(J_{\mathit{EE}}-1)J_{\mathit{PP}}{J}_{\mathit{SV}}\hfill & \hfill \\ \hfill & -(x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P-1)J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}},\hfill \end{array}$$ [12]

and

$$\begin{array}{c}\hfill {\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}}=(J_{\mathit{EE}}-1)(J_{\mathit{PP}}{J}_{\mathit{SV}}+J_{\mathit{SV}})-J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}}.\end{array}$$ [13]

Analogously to ${\mathbf{K}}_{\mathit{SV}}$, we have

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SV}}={\mathbf{R}}_{\mathit{SV}}^{\text{STP}}+{\mathbf{R}}_{\mathit{SV}}^{\text{nonSTP}}=D{\mathbf{K}}_{\mathit{SV}}^{\text{STP}}\delta g_V+D{\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}}\delta g_V.\end{array}$$ [14]

As the STP-independent term ${\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}}$ is governed by the static network weights, it is constant over the entire range of change in bottom–up input, i.e., ${\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}}$ does not change with $\alpha$. Note that because of Eq. 9 and since $D$ is always positive but subject to change in magnitude (SI Appendix, Fig. S4A), ${\mathbf{R}}_{\mathit{SV}}^{\text{nonSTP}}$ changes in magnitude as well (Fig. 3B). To match recent experimental findings that the network is inhibition stabilized when animals receive no stimulus in darkness (29), we set $J_{\mathit{EE}}$ to be larger than 1. In this case, the STP-independent term ${\mathbf{K}}_{\mathit{SV}}^{\text{nonSTP}}$ is always negative, which implies that ${\mathbf{R}}_{\mathit{SV}}^{\text{nonSTP}}$ is also always negative. Thus, the STP-dependent term ${\mathbf{K}}_{\mathit{SV}}^{\text{STP}}$, and as a result, ${\mathbf{R}}_{\mathit{SV}}^{\text{STP}}$ too, is the only part that can influence the sign of ${\mathbf{R}}_{\mathit{SV}}$ and enable the network to perform response reversal of SST activity (Fig. 3 C and D).

In conclusion, our theoretical framework enables us to analyze how perturbations affect the activity of individual populations and hence reveals how iSTP enables response reversal of SST activity.

PV-to-E STD Plays a Key Role in the Generation of Response Rseversal.

Next, we sought to dissect the role of individual iSTP mechanisms in response reversal. To this end, we separated the STP-dependent term ${\mathbf{K}}_{\mathit{SV}}^{\text{STP}}$ (Eq. 12) into a PV-to-PV STD-dependent part ${\mathbf{K}}_{\mathit{SV}}^{\text{PPD}}$ and a PV-to-E STD-dependent part ${\mathbf{K}}_{\mathit{SV}}^{\text{PED}}$ as follows:

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\mathit{SV}}^{\text{STP}}& =\underset{{\mathbf{K}}_{\mathbf{SV}}^{\mathbf{\text{PPD}}}}{\underbrace{(x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P-1)(J_{\mathit{EE}}-1)J_{\mathit{PP}}{J}_{\mathit{SV}}}}\hfill & \hfill \\ \hfill & \underset{{\mathbf{K}}_{\mathbf{SV}}^{\mathbf{\text{PED}}}}{\underbrace{-(x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P-1)J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}}}}.\hfill \end{array}$$ [15]

Since both $x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P-1$ and $x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P-1$ are always negative (Materials and Methods and SI Appendix, Fig. S4B), the PV-to-PV STD-dependent part ${\mathbf{K}}_{\mathit{SV}}^{\text{PPD}}$ is always negative and the PV-to-E STD-dependent part ${\mathbf{K}}_{\mathit{SV}}^{\text{PED}}$ is always positive. Importantly, ${\mathbf{K}}_{\mathit{SV}}^{\text{PPD}}$ decreases with increasing bottom–up input $\alpha$, whereas ${\mathbf{K}}_{\mathit{SV}}^{\text{PED}}$ increases with increasing bottom–up input $\alpha$ (SI Appendix, SI Text and Fig. S4C).

Similarly, we can write:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SV}}^{\text{STP}}={\mathbf{R}}_{\mathit{SV}}^{\text{PPD}}+{\mathbf{R}}_{\mathit{SV}}^{\text{PED}}=D{\mathbf{K}}_{\mathit{SV}}^{\text{PPD}}\delta g_V+D{\mathbf{K}}_{\mathit{SV}}^{\text{PED}}\delta g_V.\end{array}$$ [16]

${\mathbf{R}}_{\mathit{SV}}^{\text{PPD}}$ and ${\mathbf{R}}_{\mathit{SV}}^{\text{PED}}$ show similar changes as ${\mathbf{K}}_{\mathit{SV}}^{\text{PPD}}$ and ${\mathbf{K}}_{\mathit{SV}}^{\text{PED}}$, respectively (Fig. 3E). Therefore, when bottom–up input $\alpha$ increases from a low to a high level (e.g., switching from darkness to visual stimulation condition), to display response reversal ${\mathbf{R}}_{\mathit{SV}}^{\text{STP}}$ must overcome in magnitude the negative ${\mathbf{R}}_{\mathit{SV}}^{\text{nonSTP}}$, resulting in an overall switch of ${\mathbf{R}}_{\mathit{SV}}$ from negative to positive. The increasing PV-to-E STD-dependent term rather than the decreasing PV-to-PV STD-dependent term is imperative for this switch (Fig. 3E). As no terms directly associated with PV-to-VIP STD and SST-to-VIP STF appear in Eq. 15, our analysis shows that these two mechanisms are unimportant for the generation of response reversal.

We performed the same simulations as with the intact network while inactivating specific iSTP mechanisms to confirm our analysis. The inactivation of particular mechanisms was implemented by freezing the respective plasticity variables at their baseline values when bottom–up input is high, ensuring that the steady-state activities of all populations are positive at the baseline and during the top–down modulation period. We then varied the bottom–up inputs from high to low and found that SST response reversal still occurs despite inactivating PV-to-PV STD, PV-to-VIP STD, or SST-to-VIP STF. Such networks show similar patterns to networks with intact iSTP (Fig. 3F). In contrast, when PV-to-E STD is inactivated, the change of SST activity manifests an opposite trend from that in networks with intact iSTP (Fig. 3F). Furthermore, we found that PV-to-E STD is crucial for generating the effective supralinear input–output relation observed in the baseline state for varying bottom–up input $\alpha$ (SI Appendix, Fig. S2). Inactivating PV-to-E STD completely diminished the supralinearity of the effective input–output relations in contrast to inactivating other iSTP mechanisms (SI Appendix, Fig. S5A). In addition, as bottom–up input increases, the resulting inhibitory current from PV to E is suppressed by PV-to-E STD (SI Appendix, Fig. S5B). This suppression is greater for stronger bottom–up inputs leading to a sublinear increase in PV current, which is important for the generation of the effective supralinear input–output relation (SI Appendix, Fig. S5B).

Taken together, our analysis and numerical simulations reveal that PV-to-E STD is the determining mechanism for generating response reversal. In contrast, the effects of PV-to-PV STD, PV-to-VIP STD, and SST-to-VIP STF on response reversal are negligible.

Relationship between Response Reversal and the Paradoxical Effect.

Locomotion-induced top–down modulation excites VIP and effectively inhibits SST due to the mutually inhibitory connections between VIP and SST. However, when animals receive visual stimuli at a high baseline activity state (high $\alpha$), additional VIP inhibition induced by top–down modulation increases the activity of SST. This phenomenon is reminiscent of the paradoxical effect (28, 42). We thus sought to identify the relationship between response reversal and the paradoxical effect. To this end, we derived the change of SST activity induced by a change in the input to SST itself, ${\mathbf{R}}_{\mathit{SS}}$, as follows:

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SS}}=D{\mathbf{K}}_{\mathit{SS}}\delta g_S,\end{array}$$ [17]

where

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\mathit{SS}}& =-[(x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P-1)(J_{\mathit{EE}}-1)J_{\mathit{PP}}\hfill & \hfill \\ \hfill & -(x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P-1)J_{\mathit{PE}}{J}_{\mathit{EP}}\hfill & \hfill \\ \hfill & +(J_{\mathit{EE}}-1)(J_{\mathit{PP}}+1)-J_{\mathit{PE}}{J}_{\mathit{EP}}]\hfill \\ \hfill & =-{\mathbf{K}}_{\mathit{SV}}/J_{\mathit{SV}},\hfill \end{array}$$ [18]

and $\delta g_S$ represents the change of input to SST. Furthermore,

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SS}}=-\frac{\delta g_S}{J_{\mathit{SV}}\delta g_V}{\mathbf{R}}_{\mathit{SV}}.\end{array}$$ [19]

When $\delta g_S$ is positive, to obtain a paradoxical response of SST (i.e., to have a negative ${\mathbf{R}}_{\mathit{SS}}$), ${\mathbf{K}}_{\mathit{SS}}$ has to be negative. As ${\mathbf{K}}_{\mathit{SS}}$ is equal to $-{\mathbf{K}}_{\mathit{SV}}/J_{\mathit{SV}}$, for low $\alpha$ corresponding to the darkness condition (${\mathbf{K}}_{\mathit{SV}}$ and ${\mathbf{R}}_{\mathit{SV}}$ are negative), ${\mathbf{K}}_{\mathit{SS}}$ and ${\mathbf{R}}_{\mathit{SS}}$ are positive, hence, no paradoxical response is observed (Fig. 4 A and B, Left). In contrast, for high $\alpha$ corresponding to the visual stimulation condition (${\mathbf{K}}_{\mathit{SV}}$ and ${\mathbf{R}}_{\mathit{SV}}$ are positive), ${\mathbf{K}}_{\mathit{SS}}$ and ${\mathbf{R}}_{\mathit{SS}}$ are negative. Therefore, SST exhibits a paradoxical response (Fig. 4 A and B, Right).

Fig. 4.

Relationship between response reversal and paradoxical response of SST. (A) Analytical predictions of the change in SST response induced by an excitatory perturbation ($\delta g_S$) to SST, ${\mathbf{R}}_{\mathit{SS}}$, and change in SST response induced by an excitatory perturbation ($\delta g_V$) to VIP, ${\mathbf{R}}_{\mathit{SV}}$, as a function of bottom–up input $\alpha$. Here, $\delta g_V=\delta g_S=3$. (B) Left: Normalized activity when injecting additional excitatory current into SST at a low baseline state corresponding to $\alpha =0$ marked with triangular in A. SST does not show a paradoxical response. Right: Same as left but at a high baseline state corresponding to $\alpha =15$ marked with a dot in A. SST shows a paradoxical response.

We have mathematically proven a correspondence between response reversal and the paradoxical response of SST. More specifically, the SST population will not show a paradoxical response when top–down modulation via VIP decreases SST activity but will respond paradoxically when top–down modulation via VIP increases SST activity.

Relationship between Response Reversal, the Paradoxical Effect, and Interneuron-Specific Stabilization.

The paradoxical effect is a defining characteristic of inhibition stabilization in networks with fixed connectivity (28). We, therefore, sought to investigate the relationship between response reversal and inhibition stabilization. Identifying the relationship may shed light on how response reversal relates to other cortical functions as inhibition-stabilized networks can perform a variety of computations (43). As the network in our study consists of three different inhibitory populations, the network can, in principle, be stabilized by any type of interneuron. Beyond identifying inhibition stabilization, we particularly aimed to ascertain the specific interneuron subtype that stabilizes the model networks in different stimulation conditions.

To this end, we computed the leading eigenvalue of the Jacobian of individual subnetworks with the corresponding firing rates and STP dynamics while excluding specific interneuron subtypes. Such eigenvalues can be used to determine the stability of the subnetwork. A negative leading eigenvalue implies that the fixed point of the network dynamics is stable and a transient perturbation to the system does not result in a deviation from the original fixed point. In contrast, a positive leading eigenvalue means that the fixed point is unstable, and a transient perturbation causes a deviation from the original fixed point. We found that the leading eigenvalue of the Jacobian of the E subnetwork in the model (defined as the network without any interneurons) is positive for all values of $\alpha$, suggesting that the E subnetwork is unstable and the network is inhibition-stabilized for all stimulation conditions (SI Appendix, Fig. S6). Furthermore, we found that the E subnetwork being unstable (i.e., $J_{\mathit{EE}}>1$) at the high bottom–up input is a necessary condition to observe response reversal (SI Appendix, SI Text). VIP does not stabilize the network, as the leading eigenvalue of the Jacobian of the E-VIP subnetwork (the network without PV and SST interneurons) is always positive (SI Appendix, Fig. S6). By computing the leading eigenvalue of the Jacobian of the E-PV-VIP subnetwork (the network without SST interneurons), we found that the eigenvalue switches from negative to positive when the response of SST to top–down modulation is reversed (Fig. 5A), indicating that SST is required for network stabilization when top–down modulation via VIP increases SST activity. Furthermore, the leading negative eigenvalue of the Jacobian of the E-PV-VIP subnetwork for low $\alpha$ suggests that in the regime in which top–down modulation via VIP decreases SST activity, the network does not require SST for stabilization and can be stabilized by PV. To determine whether PV could be the only interneuron subtype stabilizing the network in that regime, we calculated the leading eigenvalue of the Jacobian of the E-SST-VIP subnetwork (the network without PV interneurons). We found that this eigenvalue is always negative in the current model (Fig. 5A), suggesting that SST can serve the stabilization role in that regime as well as PV.

Fig. 5.

The network undergoes a change in the indispensability of SST for network stabilization with increasing bottom–up input. (A) Leading eigenvalues of the E-PV-VIP subnetwork and the E-SST-VIP subnetwork as a function of bottom–up input $\alpha$. The response reversal boundary extracted from analytical calculations (${\mathbf{R}}_{\mathit{SV}}=0$) is indicated by the vertical dashed line. (B) Left: Normalized activity when injecting an additional transient excitatory current into E while freezing PV for networks at a low baseline state corresponding to $\alpha =0$ marked with a triangle in A. The small transient excitatory input is introduced at the time marked with arrows. The periods in which PV is frozen are marked with the gray bar. Right: Same as Left but for networks at a high baseline state corresponding to $\alpha =15$ marked with a dot in A. (C) Similar to B but with frozen SST. (D) Same as A but for the second-largest eigenvalue. (E) Parity of the number of unstable modes in the E-PV-VIP subnetwork as a function of bottom–up input $\alpha$. Numbers indicate the amount of unstable modes.

We confirmed these results by injecting a transient excitatory perturbation into the excitatory population while clamping the activity of either PV or SST. We found that when clamping PV activity, the fixed point in the given network is stable to perturbations over the entire range of $\alpha$ and reaches the same fixed point after the transient perturbation (Fig. 5B). In contrast, when clamping SST activity, while the fixed point at low $\alpha$ is stable to perturbations, a transient perturbation at high $\alpha$ leads to unstable dynamics (Fig. 5C).

Consistent with the change in the requirement of SST for network stabilization, we observed a transition in the prevalence of inhibition received by the excitatory population from PV to SST with increasing $\alpha$ (SI Appendix, Fig. S7). More specifically, at the low baseline state, the excitatory population receives more inhibition from PV than SST (SI Appendix, Fig. S7A). Top–down modulation via VIP leads to increases in the overall inhibition and the inhibition from PV at the steady state but a decrease in the inhibition from SST (SI Appendix, Fig. S7A). In contrast, at the high baseline state, the excitatory population receives more inhibition from SST than PV (SI Appendix, Fig. S7B). Top–down modulation increases the overall inhibition at the steady state as well as the inhibition from both PV and SST (SI Appendix, Fig. S7B). This increase in total inhibition at the steady state observed during top–down modulation is a unique characteristic of inhibition-stabilized networks in contrast to noninhibition-stabilization networks [SI Appendix, Fig. S3; (31, 33, 38)]. In inhibition-stabilized networks with iSTP, top–down modulation induces a transient disinhibition enabling the growth of recurrent excitation and increasing excitation and inhibition to the excitatory population at the steady state.

To systematically investigate how response reversal and paradoxical effects of SST relate to interneuron-specific stabilization, we conducted mathematical analyses and found that ${\mathbf{K}}_{\mathit{SV}}$ and ${\mathbf{K}}_{\mathit{SS}}$ are linked to the determinant of the Jacobian of the E-PV-VIP network, $det({\mathbf{M}}_{\text{E-PV-VIP}})$ (SI Appendix, SI Text). In the network we considered here, because of the STP mechanisms, the Jacobian of the E-PV-VIP subnetwork is a 6-by-6 matrix. When ${\mathbf{K}}_{\mathit{SV}}$ is positive (i.e., top–down modulation increases SST activity), ${\mathbf{K}}_{\mathit{SS}}$ is negative (i.e., the network exhibits paradoxical effects of SST), and $det({\mathbf{M}}_{\text{E-PV-VIP}})$ is negative. Note that in high-dimensional systems, the determinant of the Jacobian matrix alone is not sufficient to determine network stability. For a six-dimensional system, a negative $det({\mathbf{M}}_{\text{E-PV-VIP}})$ implies an odd number of positive eigenvalues corresponding to unstable eigenvectors/modes and thus the necessity for SST stabilization. However, the network can also require SST for stabilization in the presence of a positive $det({\mathbf{M}}_{\text{E-PV-VIP}})$, for instance, when the Jacobian of the E-PV-VIP subnetwork has an even number of unstable modes. To confirm the change in the number of unstable modes, we examined the second-largest eigenvalue of the E-PV-VIP subnetwork and found that the second-largest eigenvalue is always negative in the given network (Fig. 5D). As a result, the number of unstable modes changes from even to odd (Fig. 5E) when the SST response reverses from suppression to enhancement, and the network exhibits the paradoxical effect in the response of SST. Note that we did not find a direct mathematical relationship between PV stabilization and response reversal of SST (SI Appendix, SI Text). In other words, response reversal does not imply a change in PV stabilization. Consequently, PV stabilization and how it changes with bottom–up inputs, as presented in our study, are contingent on specific parameters.

Taken together, these results suggest that with increasing bottom–up input, representing a change in stimulation condition, the impact of top–down modulation on SST activity transitions from suppression to enhancement, the network exhibits a paradoxical response of SST, requires SST for stabilization, and the E-PV-VIP subnetwork has an odd number of unstable modes (Figs. 4A, 5A, and 6).

Fig. 6.

The relationship between response reversal, paradoxical effect, and inhibition stabilization. At low bottom–up input, top–down modulation decreases SST activity, and the network does not exhibit a paradoxical response of SST such that SST may not be required for stabilization, or SST may be required for stabilization, but the E-PV-VIP subnetwork has an even number of unstable modes. As demonstrated in Fig. 5, in this regime, the network is inhibition stabilized and stabilized by either PV or SST. With increasing bottom–up input, the response of SST induced by top–down modulation is reversed from suppression to enhancement, the network exhibits a paradoxical response of SST and requires SST for stabilization with an odd number of unstable modes in the E-PV-VIP subnetwork. As demonstrated in Fig. 5, in this regime, the network is inhibition stabilized and stabilized by SST but not PV. Note that while the relationship between response reversal, paradoxical effects, and inhibition stabilization marked in blue boxes does not depend on the choice of parameters, the possible interneuron-specific stabilization regimes shaded in gray are contingent on specific parameters (SI Appendix, SI Text).

Modeling Results Are Robust to Variations in STP Mechanisms, Inputs, and Network Connectivity.

To demonstrate that our results are valid for a variety of perturbations, we performed different sensitivity analyses on STP mechanisms, inputs, and network connectivity.

We first investigated whether additional STP mechanisms affect our results. In this study, we used a rate-based population model, ignoring the large number of connections between individual neurons on a microscopic level. Given the dominant number of excitatory neurons in the cortex, we might have underestimated the effective depression of the E-to-E connection and facilitation of the E-to-SST connection compared to real circuits (27). We therefore sought to examine their influence on our results by analyzing how they might affect the analytical expression of ${\mathbf{K}}_{\mathit{SV}}$ and network simulations. We found that the response reversal of SST from suppression to enhancement with increasing bottom–up input, as reported experimentally, is preserved in the presence of E-to-E STD (Fig. 7A). However, the change in SST activity evolves nonmonotonically with increasing bottom–up input, starting to decrease and eventually being reversed from enhancement to suppression at high $\alpha$ (Fig. 7A). Due to E-to-E STD, the effective excitatory-to-excitatory coupling decreases, resulting in a stable E subnetwork, and the network eventually becomes a noninhibition-stabilized network (non-ISN) as demonstrated by the leading eigenvalues of the E subnetwork and E-VIP subnetwork switching from positive to negative with increasing bottom–up input (Fig. 7B). Interestingly, the response reversal of SST from enhancement to suppression does not occur at the same time as the network transitions from ISN to non-ISN. We proved that being an ISN is a necessary but not a sufficient condition to generate enhanced SST activity induced by top–down modulation, and non-ISNs cannot generate enhanced SST activity induced by top–down modulation (SI Appendix, SI Text). Consistent with our previous results, in the presence of E-to-E STD, the paradoxical response of SST is also linked to the change in SST activity induced by top–down modulation (Fig. 7C). More specifically, the network exhibits (no) paradoxical response of SST when top–down modulation increases (decreases) SST activity (Fig. 7C).

Fig. 7.

Modeling results are robust in the presence of E-to-E STD. (A) Change in SST activity as a function of bottom–up input $\alpha$ for networks also including E-to-E STD, showing numerical results and analytical predictions. The response reversal boundaries extracted from analytical calculations (${\mathbf{R}}_{\mathit{SV}}=0$) are indicated by the vertical dashed lines. Here, $\delta g_V=0.1$. (B) Leading eigenvalue of the E subnetwork and E-VIP subnetwork as a function of bottom–up input $\alpha$. The leading eigenvalue eventually turns negative, indicating that the network becomes noninhibition stabilized. The dot represents the $\alpha$ level at which the leading eigenvalues are zero. (C) Relationship between response reversal and the paradoxical response of SST. Analytical predictions of the change in SST response induced by an excitatory perturbation ($\delta g_S$) to SST, ${\mathbf{R}}_{\mathit{SS}}$, and change in SST response induced by an excitatory perturbation ($\delta g_V$) to VIP, ${\mathbf{R}}_{\mathit{SV}}$, as a function of bottom–up input $\alpha$. Here, $\delta g_V=\delta g_S=0.1$. (D) Similar to B but for the E-PV-VIP subnetwork and the E-SST-VIP subnetwork. (E) Similar to D but for the second-largest eigenvalues. (F) Parity of the number of unstable modes in the E-PV-VIP subnetwork as a function of bottom–up input $\alpha$. Numbers in the brackets indicate the amount of unstable modes.

Different from networks without E-to-E STD, by examining the leading eigenvalue of the E-PV-VIP and E-SST-VIP subnetwork (Fig. 7D), we observed a repertoire of interneuron-specific stabilization regimes and some regime transitions (Fig. 8). For instance, we observed a transition from being stabilized by PV but not SST (as reflected by a leading positive eigenvalue of the E-SST-VIP subnetwork and a leading negative eigenvalue of the E-PV-VIP subnetwork) to being stabilized by both PV and SST (as reflected by leading positive eigenvalues of the E-PV-VIP and E-SST-VIP subnetwork). We also observed a transition from being stabilized by SST but not PV (as reflected by a leading positive eigenvalue of the E-PV-VIP subnetwork and a leading negative eigenvalue of the E-SST-VIP subnetwork) to being stabilized by either PV or SST (as reflected by leading negative eigenvalues of the E-PV-VIP and E-SST-VIP subnetwork) (Figs. 7D and 8). We further confirmed these distinct regimes by injecting a transient excitatory perturbation into the excitatory population while clamping the activity of PV, or SST, or both PV and SST (SI Appendix, Fig. S8). Despite novel regimes observed in the presence of E-to-E STD, the link between response reversal, paradoxical effects of SST, and the parity of unstable modes in the E-PV-VIP subnetwork remains unchanged (Fig. 7C–F). When top–down modulation decreases SST activity, the network exhibits no paradoxical response of SST, and the E-PV-VIP subnetwork has an even number of unstable modes. When top–down modulation increases SST activity, the network exhibits a paradoxical response of SST, and the E-PV-VIP subnetwork has an odd number of unstable modes implying that SST is required for network stabilization (Fig. 8).

Fig. 8.

The relationship between response reversal, inhibition stabilization, and paradoxical response in networks also including E-to-E STD. At the low bottom–up input, top–down modulation decreases SST activity, and the network does not exhibit a paradoxical response of SST; thus, SST may not be required for stabilization, or SST may be required for stabilization, but the E-PV-VIP subnetwork has an even number of unstable modes. As demonstrated in Fig. 7, in this regime, the network is inhibition stabilized and stabilized by PV but not SST. With increasing bottom–up input, the response of SST induced by top–down modulation is reversed from suppression to enhancement. The network further exhibits the paradoxical effect in the response of SST and requires SST for stabilization with an odd number of unstable modes in the E-PV-VIP subnetwork. As demonstrated in Fig. 7, in this regime, the network is inhibition stabilized and stabilized by both PV of SST and then transitions into being stabilized by SST but not PV. Further increasing bottom–up input, the response of SST induced by top–down modulation is reversed from enhancement to suppression, and the network does not exhibit a paradoxical response of SST: thus, SST may not be required for stabilization, or SST may be required for stabilization, but the E-PV-VIP subnetwork has an even number of unstable modes. As demonstrated in Fig. 7, in this regime, the network is inhibition stabilized and stabilized by either PV or SST and finally transitions into a non-ISN. Note that while the relationship between response reversal, paradoxical effects, and inhibition stabilization marked in blue boxes does not depend on the choice of parameters, the possible interneuron-specific stabilization regimes shaded in gray are contingent on specific parameters.

In the presence of E-to-SST STF, the analytical expression of ${\mathbf{K}}_{\text{SV}}$ (Eq. 10), dictating the sign of the change of SST induced by top–down modulation, remains unchanged. It does not contain any E-to-SST dependent terms, suggesting that the emergence of response reversal is unaffected by E-to-SST STF (SI Appendix, Fig. S9A). Consistent with the analysis, our simulation results show that adding E-to-SST STF does not alter the dynamics and the generation of response reversal (SI Appendix, Fig. S9 B and C). Given the omnipresence of STP mechanisms in the mouse visual cortex among various populations and the centrality of SST to response reversal, we further incorporated STP mechanisms in all connections considered in our model. These simulations show that response reversal can still be observed (SI Appendix, Fig. S10). In addition, we examined how different inputs and network connectivity affect our results and found that response reversal is preserved in networks with varying inputs and connectivity strengths (SI Appendix, Figs. S11 and S12).

In conclusion, through multiple sensitivity analyses, we demonstrated the robustness of our findings to variations in STP mechanisms, inputs, and network connectivity.

Discussion

In this paper, we investigated how experimentally measured iSTP mechanisms enable model networks with one excitatory and three types of interneuron populations to perform a nonlinear computation known as response reversal. Using analytical calculations and numerical simulations, we identified that PV-to-E STD is the iSTP mechanism critical for generating response reversal. We further clarified the relationship between response reversal, the paradoxical response of SST, and the interneuron-specific stabilization property of the network, making important links between well-known operating regimes of cortical network dynamics.

We made several assumptions that enabled us to analytically understand response reversal. First, we studied responses in the presence of bottom–up and top–down inputs relative to a baseline state, assuming that the network activity has reached a fixed point, and we did not consider scenarios like multistability (44, 45) or oscillations (16). While multistability and oscillations have been observed in the brain (46, 47), the single stable fixed point assumed here is considered to be a realistic approximation of the awake sensory cortex (48).

Concerning the modeled STP mechanisms, our analysis primarily focused on PV-to-E STD, PV-to-PV STD, PV-to-VIP STD, and SST-to-VIP STF. Additional simulations of networks also including E-to-E STD, or E-to-SST STF, or STP mechanisms in all existing synapses demonstrated the robust occurrence of response reversal in SST. In addition to the incorporated STP mechanisms, substantial PV-to-SST STD has also been reported (27). However, experimental studies demonstrated negligible inhibition from PV to SST (1), and hence, we did not consider PV-to-SST STD.

Furthermore, our work models the neural input–output function as a rectified linear function, a characteristic feature of tightly balanced networks (24, 36). Without iSTP, our model network behaves like a linear network when all populations have positive activity. In addition to iSTP proposed in our study, several other factors can induce nonlinearities in the population response and, therefore, could contribute to the studied response reversal. Recent studies have suggested that cortical networks may operate in a loosely balanced regime, resulting in a supralinear input–output function (49–52). Response reversal can also be generated by such a nonlinear input–output function (20).

Finally, we modeled neurons of the same type as a homogeneous population governed by the same dynamics. In contrast, even within the same cell type, biological neurons have highly heterogeneous time constants and firing thresholds (53, 54). Such heterogeneity can theoretically also give rise to nonlinear population responses (55, 56). Moreover, biological neurons possess complex morphologies (3, 57) and manifest nonlinear dendritic integrations (58–61). This suggests that the complete set of underlying mechanisms behind response reversal can be even richer and remains to be examined experimentally.

Our study makes several predictions. First, during top–down modulation, along with decreased SST activity, we also observed that the inhibition of the excitatory population increased at the steady state. Top–down modulation via VIP induces transient disinhibition, facilitating the growth of recurrent excitation and resulting in increased excitatory activity. This increased recurrent excitation is balanced by the concurrent increase in inhibition, which is a characteristic of inhibition-stabilized networks. This prediction can be tested experimentally by measuring excitatory and inhibitory currents to the excitatory neurons during top–down modulation.

Second, due to iSTP, locomotion-induced top–down modulation via VIP can reversely regulate SST response under different stimulus conditions. Although PV-to-E STD does not directly affect SST and VIP activity, surprisingly, our analysis suggests that PV-to-E STD is the determining mechanism underlying the generation of response reversal. Theoretical studies have demonstrated that inhibitory-to-inhibitory connections have the dominant impact on cortical dynamics, memory capacity, and working memory maintenance (62, 63). Here, our work suggests that the dynamics of inhibitory to excitatory synapses can be more important than those of inhibitory to inhibitory synapses to generate certain nonlinear phenomena.

Third, our theory reveals a correspondence between response reversal and a paradoxical response of SST in the presence of iSTP. More specifically, when the bottom–up stimulation condition switches from darkness to visual input, the impact of locomotion-induced top–down modulation via VIP on SST activity changes from suppression to enhancement. Once SST activity induced by top–down modulation gets elevated, the network exhibits a paradoxical response of SST. This correspondence can therefore be tested directly in future optogenetic experiments to see whether injecting excitatory (inhibitory) currents into the SST population indeed decreases (increases) its activity.

Fourth, our analysis shows that response reversal is tightly linked to the indispensability of SST for network stabilization. In darkness, when top–down modulation decreases SST activity, SST may not be required for network stabilization (i.e., solely PV can stabilize the network). In contrast, in the presence of visual stimuli, top–down modulation increases SST activity, the network stabilization requires SST, and the E-PV-VIP subnetwork has an odd number of unstable modes. It is worth noting that when the network requires SST for stabilization, the network can require only SST but not PV (Fig. 5 and SI Appendix, Fig. S8) or both PV and SST for stabilization (SI Appendix, Fig. S8). The observation that the network requires both PV and SST for stabilization is interesting and will require further investigation. Consistent with recent studies (29), the network is inhibition-stabilized in all bottom–up stimulation conditions, even in darkness without visual stimuli. However, contrary to recent studies suggesting that PV is positioned to stabilize network activity (64), our work suggests that the specific inhibitory cell type stabilizing the network can change dynamically depending on the stimulation condition. As SST primarily targets dendrites of excitatory neurons (4, 65), stabilization through SST can be mechanistically realized via establishing a spatially precise E/I balance within individual dendritic segments. Recent experimental observations support the existence of such localized E/I balance at the dendritic segment level (66), and SST is ideal for establishing the dendritic E/I balance and thus can provide an important source of stabilization. Furthermore, our results suggest a shift in inhibition source from PV to SST, typically accompanied by the occurrence of response reversal. Exploring the computational implications of this interneuron-specific inhibition shift and response reversal raises intriguing questions. Since PV neurons preferentially target perisomatic regions of excitatory neurons, whereas SST neurons target distal dendritic regions of excitatory neurons, the switch of dominant inhibition from soma to dendrite might prioritize inputs to perisomatic regions over inputs to distal dendritic regions and thus could be important for gating information (67). Furthermore, inhibition also plays an important role in controlling plasticity (68). The redistribution of inhibition sources might imply different abilities of distinct interneurons to control plasticity in different regimes.

Last, for the network connectivity and the set of STP mechanisms considered here, our results show that when SST is required for network stabilization and the E-PV-VIP subnetwork has an odd number of unstable modes, the network exhibits a paradoxical response of SST. Several studies have investigated the relationship between inhibition stabilization and the paradoxical effect in networks with multiple interneuron subtypes (33–35, 39), in networks with STP while ignoring different cell types (29, 69), as well as in networks with multiple interneuron subtypes and STP while ignoring cell type specificity (38). How paradoxical effects of a given cell type relate to the number of unstable modes in subnetworks excluding that cell type has been studied in networks without STP in a recent theoretical study (31). Here, we revealed the relationship between interneuron-specific stabilization and the paradoxical effect in networks with multiple interneuron subtypes in the presence of a set of STP mechanisms.

Taken together, our work sheds light on how experimentally identified iSTP mechanisms can generate response reversal, reveals the roles of individual iSTP mechanisms in response reversal, and uncovers the relationship between response reversal, the paradoxical effect, and interneuron-specific stabilization properties.

Materials and Methods

Response Matrix.

To investigate how the input to one particular population affects the response of any given population in the presence of v, we developed a general theoretical framework using linear perturbation theory. Using the separation of time scales for the rate dynamics and the STP dynamics, we can write the system of equations introduced before (Eqs. 1–4) in matrix form while replacing the STP variables with their steady-state values, as follows:

$$\begin{array}{c}\hfill \mathbf{T}\frac{d}{\mathit{dt}}\mathbf{r}=-\mathbf{r}+\mathbf{f}(\mathbf{P} ᴏ Jr+g),\end{array}$$ [20]

where $\mathbf{T}$ is a diagonal matrix of time constants of the firing rate dynamics, $\mathbf{r}$ a vector of firing rates of different populations, $\mathbf{f}(x)$ a vector of the rectified linear input–output function of the respective populations, $\mathbf{P}$ a matrix of the STP variables, $\mathbf{J}$ the connectivity matrix, and $\mathbf{g}$ a vector of inputs to different populations. $ᴏ$ denotes the element-wise product. The steady states of the STP variables are obtained by setting the Eqs. 5–8 to 0. Note that since the steady states of the STP variables are determined by the presynaptic activity, $x_{\mathit{EP}}^{\ast }$, $x_{\mathit{PP}}^{\ast }$, and $x_{\mathit{VP}}^{\ast }$ are the same. If STP is not present on the synapses from $j$ to $i$, the corresponding element ${\mathbf{P}}_{\mathit{ij}}$ is 1 (for further details, see SI Appendix, SI Text).

By linearizing about the fixed point and ignoring higher-order terms, we obtain the following equation:

$$\begin{array}{c}\hfill \mathbf{T}\frac{d}{\mathit{dt}}\delta \mathbf{r}=-\delta \mathbf{r}+\mathbf{F}(P ᴏ J)\delta \mathbf{r}+\mathbf{F}({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r}))\delta \mathbf{r}+\mathbf{F}\delta \mathbf{g}.\end{array}$$ [21]

Here, $\delta \mathbf{r}$ is a vector containing the deviations of firing rates from their fixed point values. $\mathbf{F}$ is a diagonal matrix containing the derivatives of the input–output functions evaluated at the fixed point. ${\mathbf{P}}^{\prime }$ is a matrix containing the derivative of the STP variables with respect to the corresponding presynaptic firing rate, evaluated at the fixed point. $\text{diag}(r)$ is a diagonal matrix containing the firing rates of different populations. And $\delta \mathbf{g}$ is a vector containing the changes/perturbations of external inputs to different populations.

The fixed point solution of Eq. 21 quantifies the change in population rates $\delta \mathbf{r}$ to an input perturbation $\delta \mathbf{g}$:

$$\begin{array}{ccc}\hfill \delta \mathbf{r}& =(\mathbf{1}-\mathbf{F}\left(P ᴏ J\right)-\mathbf{F}\left({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r})\right){)}^{-1}\mathbf{F}\delta \mathbf{g}\hfill & \hfill \\ \hfill & =\frac{1}{det(\mathbf{L})}\text{adj}(\mathbf{L})\mathbf{F}\delta \mathbf{g},\hfill \end{array}$$ [22]

with

$$\begin{array}{c}\hfill \mathbf{L}=\mathbf{1}-\mathbf{F}(P ᴏ J)-\mathbf{F}\left({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r})\right),\end{array}$$ [23]

where $\mathbf{1}$ denotes the identity matrix, and “det” and “adj” represent the matrix’s determinant and adjugate, respectively.

By replacing $\delta \mathbf{g}$ with a diagonal matrix $\delta \mathbf{G}$ whose diagonal elements are $\delta \mathbf{g}$, we can obtain a response matrix $\mathbf{R}$ as follows:

$$\begin{array}{c}\hfill \mathbf{R}=\frac{1}{det\mathbf{L}}\text{adj}(\mathbf{L})\mathbf{F}\delta \mathbf{G}.\end{array}$$ [24]

Importantly, the element ${\mathbf{R}}_{\mathit{ij}}$ provides the change in the steady-state rate response of population $i$ caused by an input perturbation $\delta {\mathbf{G}}_{\mathit{jj}}$ to population $j$.

We further define a scalar $D$ and a response factor matrix $\mathbf{K}$ as follows:

$$\begin{array}{c}\hfill D=\frac{1}{det(\mathbf{L})}=\frac{1}{det(\mathbf{1}-\mathbf{F}(P ᴏ J)-\mathbf{F}({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r})))}\end{array}$$ [25]

and

$$\begin{array}{c}\hfill \mathbf{K}=\text{adj}(\mathbf{L})=\text{adj}(\mathbf{1}-\mathbf{F}(P ᴏ J)-\mathbf{F}({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r})))\mathbf{F}.\end{array}$$ [26]

Then the response matrix $\mathbf{R}$ can be expressed as

$$\begin{array}{c}\hfill \mathbf{R}=D\mathbf{K}\delta \mathbf{G}.\end{array}$$ [27]

Note that if the network is stable, all eigenvalues of the Jacobian ${\mathbf{T}}^{-1}(-\mathbf{1}+\mathbf{F}(P ᴏ J)+\mathbf{F}({\mathbf{P}}^{\prime } ᴏ J\text{diag}(\mathbf{r})))$ have negative real parts. Therefore, all eigenvalues of $\mathbf{L}$ have positive real parts, and $D$ is always positive (SI Appendix, Fig. S4A).

To investigate how top–down modulation via VIP affects SST response in networks with iSTP, we can apply the theoretical framework introduced above and write the change of SST activity as a function of the change of the input to VIP. Since the derivatives of the rectified linear input–output functions are 1 in regimes where all cell populations have positive firing rates, we have

$$\begin{array}{c}\hfill {\mathbf{R}}_{\mathit{SV}}=D{\mathbf{K}}_{\mathit{SV}}\delta g_V\end{array}$$ [28]

with

$$\begin{array}{ccc}\hfill {\mathbf{K}}_{\mathit{SV}}& =(x_{\mathit{PP}}^{\ast }+x_{\mathit{PP}}^{{\ast }^{\prime }}{r}_P)(J_{\mathit{EE}}-1)J_{\mathit{PP}}{J}_{\mathit{SV}}\hfill & \hfill \\ \hfill & -(x_{\mathit{EP}}^{\ast }+x_{\mathit{EP}}^{{\ast }^{\prime }}{r}_P)J_{\mathit{SV}}{J}_{\mathit{PE}}{J}_{\mathit{EP}}+(J_{\mathit{EE}}-1)J_{\mathit{SV}}.\hfill \end{array}$$ [29]

Simulations.

Simulations were performed in Python. All differential equations were implemented by Euler integration with a time step of 0.1 ms. The simulation duration was 9 s for each experiment. Top–down modulation was applied in the interval of 5 to 7 s. Networks were initialized using the parameters in SI Appendix, Tables. STP variables were initially set to 1 and reached their steady-state values within the first second. Figures depict 6 s of network activity following 3 s of relaxation after initialization. Bottom–up input $\alpha$ was modeled in the interval $[0,20]$ with a step size of $0.5$ unless stated otherwise. All simulation parameters are listed in SI Appendix, Tables.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Model simulations data have been deposited in GitHub (https://github.com/comp-neural-circuits/top-down-modulation-with-iSTP)(70). All other data are included in the manuscript and/or SI Appendix.