Figure 1.
Diagrammatic representation of the PAC model (A) and the choice of sigmoidal response function (B).
A: grey arrows represent excitatory connections (+), black circles represent inhibitory connections (−). All weights in the model are positive, excitatory or inhibitory connections appear as +/− signs in the model equations (Equation (1)). External input can be received by either the E or the I population. Mid-grey arrows represent the leak in activity levels as a result of passive membrane properties. B: the exact shape of the sigmoid chosen (Equation (2)). The mean threshold is at x = 1. β = 4.
Figure 2.
Analysis of the system's nullclines.
A: phase plane of the system for θE = 0.7, θI = 0. E-nullcline (Equation (5)) (light grey), I-nullcline (Equation (6)) (black), trajectory beginning at E = 0,I = 0 (mid grey), vector field (black arrows). Intersection of nullclines corresponding to equilibrium at E = 0.46, I = 0.42. The eigenvalues of the system at this point are λ1 = 1.16+2.64i, λ2 = 1.16−2.64i, hence this equilibrium is unstable and trajectories converge to a limit cycle. B: effect on the E-nullcline of increasing input to the E population (θE) from 0.2 to 1. C: effect on the I-nullcline of increasing input to the I population (θI) from 0.2 to 1.
Figure 3.
Behaviour of the model for a range of constant inputs, received by the E population only.
Pictures A–D: output activity of the two populations E (grey) and I (black) when receiving constant input to E population of (A) 0.2, (B) 0.4, (C) 0.7 and (D) 1.18. E: maximum and minimum values of the E population's output activity are plotted (grey line) in order to display the region where these values differ and oscillations appear. Arrows indicate examples A–D. F: bifurcation diagram generated by continuation of the model's steady state equilibrium (θE = θI = 0, initial conditions: E = I = 0, equilibrium reached: E = 0.0181, I = 0.0207). Labelled points: (1) Hopf bifurcation, θE = 0.399974, (2) Hopf bifurcation, θE = 1.199932.
Figure 4.
Behaviour of the model for a range of oscillatory inputs, received by the E population only.
A–E: top panel shows the theta frequency oscillatory input to the E population; bottom panel shows the output of the E (grey) and I (black) populations. F: maximum and minimum values of the E population's output activity are plotted (grey line) in order to display the region where these values differ and oscillations appear. Brackets indicate the extent of the theta frequency input's amplitude in each of the examples A–E.
Figure 5.
Behaviour of the model for a range of constant inputs, received by the I population (θE = 1.3).
Pictures A–D: output activity of the two populations E (grey) and I (black) when receiving constant input to I population of (A) 0.05, (B) 0.12, (C) 0.4 and (D) 0.6. E: maximum and minimum values of the E population's output activity are plotted in order to display the region where oscillations appear. Arrows indicate examples A–D. F: bifurcation diagram generated by continuation of the model's steady state equilibrium (θE = 1.3, θI = 0, initial conditions: E = I = 0, equilibrium reached: E = 0.8873, I = 0.9568). Labelled points: (1) Hopf bifurcation, θI = 0.105812, (2) Hopf bifurcation, θI = 0.523650.
Figure 6.
Behaviour of the model for a range of oscillatory inputs, received by the I population only (θE = 1.3).
A–E: top panel shows the theta frequency oscillatory input to the E (grey) and I (black) populations; bottom panel shows the output of the E (grey) and I (black) populations. F: maximum and minimum values of the E population's output activity are plotted in order to display the region where oscillations appear. Brackets indicate the extent of the theta frequency input's amplitude in each of the examples A–E.
Figure 7.
Range of behaviour of the model when weight and input parameters are varied.
A: constant input values θE & θI were increased in incremental steps of size 0.01; if the resulting model activity was oscillatory the values are marked in red; if instead the E and I populations converged to constant values then the θE & θI values are marked in blue. B: same experiment as in A but for all simulations wEI = 2.5 (all other parameters take default values). The red region demonstrating oscillatory activity is smaller in comparison to that shown in A. C: example simulation showing theta-gamma PAC when all model parameters are set to default values. Modulation Index (MI) calculated on E's activity = 3.0. D: example simulations showing theta-gamma PAC when wEI = 2.5 (all other parameters take default values); in comparison to D the gamma activity is lower amplitude. MI calculated on E's activity = 2.333. E: simulation in which both θE & θI are oscillatory; θE = 4 Hz, θI = 2 Hz (amplitude and mean of the two input oscillations also differs, refer to top half of plot). Gamma activity appears locked to alternate theta cycles. F: simulation in which both θE & θI are oscillatory; both input oscillations have a frequency of 4 Hz but which both θE lags θI by 30° (amplitude and mean again differ slightly, refer to top half of plot). Whilst E demonstrates peak-locked PAC, I demonstrated gamma activity locked to the ascending phase of theta.