Figures and data
Tradeoff between two inhibitory motifs in the E – PV – SOM cortical circuit.
A.Sketch of the full E – PV – SOM network model. A positive or negative modulatory input is applied to the SOM neurons. B. Transfer function (top) and population gain bX (bottom) for neuron population X = {E, P, S} (see Eq. (4)). C. Top: Relation between modulation of input to the SOM population 
Gain and stability in E – PV – SOM circuits.
A. Left: Sketch of a disinhibitory network with stimulus input onto E and PV populations and positive SOM modulation. Right: Numerical E (red), PV (blue) and SOM (green) rate dynamics of the case with positive SOM modulation at 0.05s (solid line), and the case without modulation (dashed line). Stimulus presentation at 0.35s. Symbols indicate calculated values based on Eq. (1) and Eq. (2). B.Measures to quantify the effect of SOM modulation: (i) Effect of modulation on E 
Network gain and stability in the E – PV network.
A. Network sketch (i), firing rate grid (rE, rP) in the form of a heatmap for normalized network gain gE (ii) and normalized stability λmax (iii), and the eigenvalues for changing PV rates rP (iv) for a network without connections between the E – PV network and SOM. Every value in the heatmap is a fixed point of the population rate dynamics. The color denotes normalized network gain (Eq. (2)) or normalized stability (Fig. 2Biii). Lines of constant network gain and stability are shown in gray (from dark to light gray in steps of 0.2). The black line marks where the rate dynamics become unstable. The black dashed line separates ISN from non-ISN regime. Blue line in iii indicates the parameters for which the eigenvalues are shown in iv.
Modulation of SOM neurons with feedforward SOM connectivity.
A. Network sketch (i), firing rate grid (rE, rP) in the form of a heatmap for normalized network gain and stability (ii), and modulation measures Δ Gain (Δg) and Δ Stability (Δλ) (iii), for a network with SOM → E connection (inhibitory pathway). The arrows indicate in which direction a fixed point of the rate dynamics is changed by a positive SOM modulation. All arrow lengths are set to the same value. The modulation measure quantifies the change in stability and gain from an initial condition in the (rE, rP) grid for a positive (black dots) and negative (gray dots) SOM modulation. Q1-Q4 indicates the percent of data points in the respective quadrant (only |Δg| > 0.1 and |Δλ| > 0.01 are considered). B. Same as B for a network with SOM → PV connection (disinhibitory pathway). The purple dot in Ciii is the case of Fig. 2A. C. Same as A for a network with SOM → E and SOM → PV connections. SOM rate r2 = 2 Hz in all panels.
Modulation of SOM neurons with E to SOM feedback.
Heatmaps and modulation measures as defined in Fig. 3 and Fig. 4 for a network with an inhibitory pathway and E → SOM feedback. Left to right: Network sketch (i), normalized network gain (gE) and stability (λmax) (ii), and modulation measures Δ Gain (Δg) and Δ Stability (Δλ) (iii). Top to bottom: increase of the SOM firing rate from rS = 1 Hz (A), to rS = 2 Hz (B), rS = 3 Hz (C). The arrows indicate in which direction a fixed point of the rate dynamics is changed by a positive SOM modulation.
Effect of synaptic weight strength on network gain and stability.
A. Effect of synaptic weight change on network gain (gE) and stability (λmax) in a network biased to inhibitory SOM influence (wES > wPS). We change the strength of one weight at a time, either wEP or wPE (i), wES or wSE (ii), wPS or wSP (iii), or wPP or wSS (iv). Colorbar indicates the weight strength, red corresponds to weights onto E, blue onto PV, and green onto SOM. B. Same as A but in a network biased to disinhibitory SOM influence (wES < wPS). The networks are in the non-ISN regime (wEE is weak) and all the rates are fixed rE = 3, rP = 5, rS = 0.5.
Tuning curve changes induced by SOM modulation depend on network connectivity.
A. Top: Tuning curves of E (red), PV (blue) and SOM (green) populations in a network with connections SOM → E and SOM → PV and a feedback connection E → SOM (wES, wPS, wSE ≠ 0). Solid lines represent the tuning curve before modulation and dashed lines after a negative SOM modulation. Bottom: Linear regression of unmodulated versus modulated rates (black dots: unmodulated versus modulated rate pairs, gray solid line: fit, gray dashed line: unity line). B. Multiplicative/divisive component versus additive/subtractive component for different network connectivities. Add/sub component is normalized to the maximum rate response. Diamond case shown in panel A.
Weight parameters
Modulation of SOM neurons with PV to SOM feedback.
Same as Fig. 5 for a network with a disinhibitory pathway and PV → SOM feedback (wSP). Left to right: Network sketch (i), Network gain (gE) and stability (λmax) (ii), and modulation measures Δ Gain (Δg) and Δ Stability (Δλ) (iii). Top to bottom: increase of the SOM firing rate from rS = 1 Hz (A), to rS = 2 Hz (B), rS = 3 Hz (C). The purple dot corresponds to the case in Fig. 2C.
Effect of synaptic weight strength on network gain and stability (ISN regime).
A. Effect of synaptic weight change on network gain (gE) and stability (λmax) in a network biased to inhibitory SOM influence (wES > wPS). We change the strength of one weight at a time, either wEP or wPE (i), wES or wSE (ii), wPS or wSP (iii), or wPP or wSS (iv). Colorbar indicates the weight strength, red corresponds to weights onto E, blue onto PV, and green onto SOM. B. Same as A but in a network biased to disinhibitory SOM influence (wES < wPS). The networks are in the ISN regime (wEE is strong) and all the rates are fixed rE = 3, rP = 5, rS = 0.5.
